Flight Night: Israël Campain Day 1 TASMO

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Flight Path 2019 Episode 3

Hello and welcome to “Flight Path.” I’m Chris Jones the chief marketing officer at McCarran International Airport. After Terminal 3 opened in 2012 plans to upgrade Terminal 1 soon followed. Let’s take a look at some of the latest renovations. Since its opening in 1963 McCarran’s Terminal 1 has grown and changed with the Las Vegas community. The facility’s recent evolution has included a new modern look and state-of-the-art technology. To keep up with the ever-increasing passenger volume crews have been hard at work replacing carpet with sparkling terrazzo flooring, installing new wall coverings and charging stations and renovating restrooms. When we built T3 of course the airlines had to approve that project and one of the conditions was that we were going to go ahead and improve Terminal 1 afterwards. So that’s what we’ve been doing ever since T3 opened. We started with ticketing and we’re improving not only the looks of it but also the operational part of it. And we redid just about everything. Of course we started with the terrazzo floors, put a little bit of Las Vegas sparkle in them, but we really opened it up with natural light, a lot of window walls, a lot more efficient, compact check-in units and brightened up sky cap as well. Digital signage, upgraded flight information displays, self-service bag-tagging kiosks and new gate podiums all contribute to a seamless airport experience for travelers and employees alike. The floor surfaces are easier to clean, easier on the eyes and easier for passengers to roll suitcases on Significant improvements to the A/B security checkpoint will offer a more spacious area for travelers as well as interlocking tile flooring designed to withstand heavy traffic. Expansion of the checkpoint will allow for a dedicated TSA PreCheck lane and installation of automated screening lanes, or ASLs The A/B checkpoint is the last one that we’re finally getting to modernize with all new equipment from TSA. The lanes will become automatic screening lanes, that helps the passenger flow more than anything. As our traffic’s been growing we’re going to be able to have dedicated lanes for PreCheck, for first class, for special needs, done in a proper manner that helps the efficiency of the flow Also on the horizon in Terminal 1 are exciting new restaurants, including Shake Shack and Jamba Juice in the A/B gates and Jimmy John’s in the public shopping area. New retail will bring Kalifano, which sells crystals gemstones and art, as well as a third XpresSpa location. What we’re most excited about with the renovations is that we’re upgrading all of our food, beverage and retail opportunities. You know, that’s part of the experience. The customer wants a smooth, seamless experience and to have some of the national brands that they’re used to. With an eye toward future growth, the Department of Aviation will continue to plan and execute projects to efficiently operate McCarran’s facilities and offer visitors a phenomenal first and last look at Las Vegas Beverages are not allowed through airport security checkpoints, but thanks to conveniently located water bottle filling stations, passengers McCarran can travel with a reusable bottle and stay hydrated on-the-go. Located throughout McCarran’s terminals, these popular stations dispense cold water over a small sink allowing passengers to fill a reusable bottle before or after a flight The stations have multiplied in numbers as travelers have called for a more economical and environmentally friendly alternative to single-use water bottles for purchase. The Department of Aviation started installing water bottle fill stations in 2015. Since then we’ve installed a total of 50 stations throughout our airport facilities. It also includes our general aviation facilities, as well as our consolidated Car Rental Center. The airport embraces its role in social responsibility and strives to be sustainable. By offering an alternative to single-use water bottles, McCarran helps cut down on plastic waste in the environment. Water bottle filling stations are a definite benefit to our passengers. They provide a fast and efficient way for them to fill up an empty bottle of water that they take through security before they get on a flight. Using water bottle fill stations reduces environmental impact through the airport operations. Placement of these stations around the airport means customers have easy access to inexpensive clean water. The water comes from Southern Nevada’s municipal water supply. It’s the same supply that reaches our homes and it also meets or surpasses all federal Safe Drinking Water Act standards. To cater to the tastes of international passengers, McCarran offers another unique amenity for its thirsty travelers in Terminal 3. In addition to the water bottle fill stations we installed at Terminal 1, we also install two hot water stations at T3. They are equipped with a special handle so our passengers can fill the water safely. The airport installed the stations to encourage passengers to rehydrate by refilling their bottles and reducing waste. That saves passengers money and is good for the environment

At a busy airport like McCarran, keeping vehicles moving on the arrivals and departures curbs is a big job. Here’s a look at the team of transportation specialists who manage the traffic flow As one of the busiest airports in the nation, McCarran has thousands of travelers arriving and departing each day Hi! Number 17. You’re welcome! To assist passengers and drivers McCarran has a team of more than 60 transportation specialists who oversee the airport curbs and roadways. From directing traffic to overseeing taxi lines, these hard-working employees keep traffic moving safely and efficiently When they’re on the departure curb our employees are making sure that the cars that are waiting at the curb are not waiting for too long. We’re making sure that the traffic flow is going as smooth as possible for all the other passengers that are being dropped off at the curb Left-hand Lane right over there where that green open sign is. Have a good week. The team of transportation specialists give directions and assist with the inter-terminal shuttle service which, transports customers between Terminals 1 and 3. They also manage the airport’s busy taxi operation. We make it very, very swift and effortless for the passengers to come out of the terminal and use the taxi service. Hello, two people? Number 7 While the line of passengers waiting for taxis at times looks daunting, McCarran’s ground transportation team keeps it moving by adjusting the queue as needed and directing cabs and travelers to the appropriate loading spaces Transportation specialists also monitor the Passenger Pickup and Departure curbs. They respond to vehicle accidents and medical emergencies, and oversee the towing of unattended vehicles. These employees must have great communication skills and the ability to listen to travelers’ needs. I like dealing with people so at McCarran it’s more we’re gonna see it 95 percent of the folks so I the customer service aspect. I love that it’s a different pace. To give good customer service you have to realize you might hear the same question a thousand times, but it’s from a different person each time. So it’s like really hearing the same question only once. It’s up on the second floor so why don’t you spin around, go up to valet, it’ll take you right out there. I’m a people person. Bringing a customer service to the airport, it’s a good job. McCarran’s friendly transportation specialists and supervisors play a vital role in airport operations. As vehicles and passengers come and go, they’re also key to maintaining safety and security at McCarran. Airport lounges offer passengers comforts beyond what’s generally available in the terminal. Here’s a look at a brand new high-end lounge located in the D Concourse. The club at LAS, a 7,000 square foot shared use lounge, is an ideal place to relax, unwind or get some work done before a flight. The spacious lounge, located near Gate D33, is open to McCarran customers through the purchase of a forty dollar day pass A second Club at LAS location in Terminal 3 is also available. If you have a Priority Pass membership, then we welcome you complimentary. We also contract with several of our airline partners as well, with many airline relationships in place Or you can simply purchase a day pass and once you get inside everything’s complimentary. The space is divided into distinct zones for visitors to enjoy Featuring economic tasks chairs, a desk-height counter and an abundance of electrical outlets, the productivity zone serves as a dedicated space for business travelers. A relax zone offers comfortable seating where guests can enjoy sweeping views of the airport and Las Vegas. A resting zone features recliners and ottomans with convenient reading lamps and electrical outlets Yeah we’re comfortable, that’s the best part. Since we have a long layover we’re very happy to be here. I’ve always been very well-received at the front door and then the variety is always really good They have good food and snacks and they’ve got pretty much everything. And of course a good cold beer, so it’s a good spot. In the replenish zone guests can enjoy a selection of food and beverages from a hot and cold buffet, and a variety of all-day snack items. A separate bar with dedicated seating allows guests to enjoy complimentary cocktails and premium spirits. A hydration station offers fruit- and herb-infused waters. A refresh zone provides private restrooms and shower suites, offering spa-quality soaps and lotions to help travelers feel at home One of the unique features of this lounge is we actually have a couple of shower suites that we’ve modeled on the spa-like quality. So if you’ve been out and it’s hot and you just want to refresh before your flight we’re delighted to provide that amenity for our customers. The atmosphere at the Club at LAS is enhanced with images by local photographer Jeff Greene who has lived in Las Vegas for more than 30 years. The photos in the lounge capture a variety of subject matter ranging from Las Vegas landmarks and iconic locations to the natural beauty surrounding the city. Providing a world-class customer experience is central to the mission at McCarran. The addition of this new club greatly enhances the level of services and amenities available to Las Vegas travelers. As part of our ongoing efforts to reach out to the community, two groups of students interested in aviation careers recently had the unique opportunity to take a behind-the-scenes tour of the airport McCarran was proud to host these young men and women from the Civil Air Patrol and the Aviation Career Education Academy. So this whole area is run by

east ramp control. This group of future aviation professionals from Nevada Civil Air Patrol 802nd Squadron are on an airfield tour of McCarran International Airport. This is an outstanding opportunity for our cadets to see what opportunities they have in the future, once they get out of our cadet program. They have opportunities in the Air Force, they have opportunities in the civil airline industry and other aerospace careers The enthusiastic group saw firsthand how a busy ramp operates and how planes are taxied to their gates. It’s all made possible with careful coordination and planning between the Federal Aviation Administration and the Department of Aviation. The agencies work together to make one of the nation’s busiest airports operate smoothly The cadets toured the airfield’s fire station where they got to see specialized firefighting equipment in action during a routine training exercise. The Civil Air Patrol cadet program focuses on leadership, while promoting education in aerospace and aviation with engaging hands-on curriculum and activities. For these cadets the tour helped inspire them to pursue their goals in the military, aerospace and civil aviation careers I found it fun because we actually got to be inside the airport. I’ve always seen the airport from outside the fence or from an airplane but actually seeing all the private jets and hangars like these up close was definitely a new experience for me. I want to be commercial airline pilot for Southwest and this helps me because I kind of get to see it more. It’s better than it just being on a video screen. This is Flight 1920 coming in and we’re taking it back to Chicago as flight 1396. During another recent tour for young people interested in aviation, a stop by Southwest Airlines operation center provided insight into how mccarran’s largest carrier manages its daily flights. The group, from the Aviation Career Education, or ACE Academy, were taking part in a summer program that exposes middle and high school students to careers in the aerospace industry through hands-on experiences Activities features science, technology and engineering, as well as field trips such as this one to McCarran. During one stop on the tour students learned how flight attendants are trained in all aspects of aircraft safety. The demonstration included climbing into a floatation raft to see how the devices work in the event of a water evacuation The behind-the-scenes tour is amazing. I get to see all these different things that you would never even know go into how the planes work, how like they’re getting to different gates, the safety aspect behind it, all the different things are very fascinating to learn about Students also saw the airline’s maintenance base where the airline stores thousands of parts. Students learned how the airport uses technology to keep track of luggage, and toured portions of the airfield with an Airport Operations Coordinator The goal of the summer program is to expose students to the wide variety of career opportunities in the aviation industry When I was a kid there weren’t a lot of opportunities available to learn about aviation so once I became a pilot I decided that I wanted to go ahead try and give something back and expose the next generation of youth to careers in aviation. The tours offered to the Las Vegas ACE Academy and Nevada’s Civil Air Patrol enable the Department of Aviation to give back to the community by working with airport partners and local young people to inspire the next generation of aviation professionals Take one world-class destination, add millions of monthly travelers, and you get the nation’s ninth-busiest airport Here are some other fun facts about McCarran, by-the-numbers. Aircraft movements and the desert sun take their toll on McCarran’s runway and taxiway markings, but the airport’s airfield maintenance team is on the job using more than 7,000 gallons of paint per year to keep the airfield up to date That much paint would fill a large, above-ground swimming pool! Bar patrons at McCarran enjoy raising a glass before boarding a flight. In 2018 they consumed nearly 3 million fluid ounces of draft beer and more than 25,000 Bloody Marys Keeping such a large airport at a comfortable temperature is a big job In fact, the heating, ventilation and air conditioning system at McCarran consists of more than 25 miles of pipe running throughout roughly 4.7 million square feet of climate-controlled space The airport’s HVAC central plants have the capacity to cool or heat approximately four thousand six hundred and eighty-eight residential homes McCarran uses radio frequency identification, or RFID, technology in the tags on checked bags. The technology ensures bags get from the check-in counter to the correct aircraft. Since implementing the system in 2005, nearly 210 million bags have been tagged That’s enough tags to circle the earth almost three times! If there’s trouble in the airport, McCarran’s K9 team is there to sniff it out! The officers and their four-legged partners responded to two-thousand-twenty-six reports of unattended items in 2018 Last year they also screened 3,692,000 pounds of mail and cleared 673,920 pounds of

domestic cargo. For more fun facts about McCarran, check us out on Twitter and Facebook and tune in to future episodes of “Flight Path!” Two of Clark County’s general aviation facilities recently received high honors — inclusion in the Air Elite network of airports. Here’s a look at celebrations marking this milestone and what these airports mean to the Las Vegas community. Here’s to Clark County Airports! With a toast to the future, Henderson Executive and North Las Vegas Airports rolled out the red carpet celebrating their entrance into an exclusive club. Air Elite is a global system of airports dedicated to delivering diamond-level customer service. By joining this pre-eminent group of fixed base operators, or FBOs, Clark County airports and Henderson and North Las Vegas gain global recognition among the best of the best. So it really is a network of mostly independent FBOs from around the world that like to collaborate and network with each other They want to exemplify a service level that’s recognized by the aircraft operators around the world. So when the pilots and the flight departments see Air Elite they know that there’s a facility there with all the amenities, that there’s a customer service level there that they need to take care of their passengers. With more than 70 FBOs around the world, Air Elite airports offer top-notch service. Each location is dedicated to providing an exceptional experience. Delivering consistently on the air elite diamond service promise across the network is accomplished through a customer service culture with training to create engaged members. Every Air Elite member completes this training to ensure the customer experience is memorable. Well, it comes down to training We engage with our employees, so we give them intense training provided by the Ritz Carlton. And with that training they’re able to service the customer, and with the upgraded facilities and amenities we have here we are becoming a world-class FBO which will attract customers from around the world. The airports have access to clientele from Air Elite, customers who fly exclusively to facilities within the network. Those customers expect and receive first-class treatment each time they fly. Henderson Executive Airport has been part of the Clark County Department of Aviation since 1996. Its convenient location offers easy access to freeways, hotels, convention spaces and the world-famous Las Vegas Strip. It also serves as a reliever to busy McCarran International Airport. The entire aviation industry is so vital to Southern Nevada. They worked so hard to achieve this status, it’s not something that just happened overnight, it’s something they worked hard to achieve, it’s the quality and the service that we provide to those coming and going through this Airport that helped earn that status After a 2 million dollar renovation, North Las Vegas Airport was primed for Air Elite status. It’s the second busiest airport in Nevada and is the gateway to downtown Las Vegas. It offers the services and amenities most requested by pilots and their passengers, and is a vital part of the community serving as host for support services such as a Civil Air Patrol squadron, air ambulance flights, police search and rescue and the Bureau of Land Management We appreciate all of our community partners. To the sponsors that have helped make this possible, thank you all so very much. On behalf of the Board of County Commissioners, we are extremely grateful for the partnership as well as the sponsorship. Air Elite locations have strict service expectations and are audited to assure that the service dedication remains at the highest level This means that at any time airport customers are in Henderson or North Las Vegas, they can expect only the best McCarran offers a unique opportunity for business owners who want to showcase their products and services to a diverse selection of travelers. Here’s look at one of our many outreach programs that help educate entrepreneurs who want to do business at the airport. Business opportunities at the airport are abundant. From concessions and advertising to construction and engineering, working with McCarran is an ideal way to partner with local government. When you start to know all of the resource partners and everyone who’s available to you, you can really start getting a wealth of information to help you grow. Outreach events like this one are hosted by the Department of Aviation to give entrepreneurs information on how to do business at the airport. I’ve come here to the workshop because I need to get information on all the different resources that they offer for people who are starting a new business at the airport level. I do know that selling retail or import in a brick-and-mortar is much different than the airport concessions, so that’s why I want to get here and get all the resources that you have. How many of you have a written business plan? In addition to giving small business owners important information about partnering with the airport community resource partners offered their assistance to small and startup businesses. From the IRS to the Nevada Department of Business and Industries, the seminar covered a wide range of business management tips for those who attended. The event outlined useful information to prospective entrepreneurs. And for those who already have a business at the airport it was a chance to network. For me it’s finding the right talent, making sure to increase my network a little bit so I’m able to find people. After working

here for a while, getting to know that the people that work here that are in the different departments, different organizations that’s been helpful as well. During this outreach seminar airport staff provided information regarding how McCarran’s request for proposals, or RFP, process works and how proposals are evaluated. One thing about the DBE certification, it creates a level playing field for all DBEs and ACDBEs so they can fairly compete The Department of Aviation has established a Disadvantaged Business Enterprise program and an airport concession Disadvantaged Business Enterprise program to ensure that small, disadvantaged, minority- and women-owned business enterprises can compete fairly for federally funded transportation- related projects. The airport has a dedicated business section on its website packed with helpful information visit McCarran-dot-com-slash-business to learn about concession, construction and purchasing opportunities. In addition, the website also lists upcoming business outreach events, public notices and meetings that current and future business owners will find informative Las Vegas is a must-visit destination known throughout the world. With the start of new nonstop flights from two additional airlines this summer, travelers from even more cities can experience what the US Southwest has to offer McCarran recently celebrated the arrival of two new airline partners in true Las Vegas fashion. First up was KLM’s nonstop service from Amsterdam, the airline’s first scheduled flights to Las Vegas With an estimated annual economic impact of 65 million dollars, the flights operate on Sundays, Tuesdays and Thursdays on a Boeing Dreamliner. This is a key extension of our network on the west coast and it will allow our customer to have additional connection between the west coast and Europe KLM’s flight connects passengers to Las Vegas via a major aviation hub — Amsterdam — which has connecting flights to Europe, Africa and the Middle East. This new service took a lot of planning and research. The teams from McCarran and the LVCVA as well as KLM have been working for six years to make this flight a reality, to make sure that it made sense, for KLM the demand was there. McCarran International Airport is now also the gateway to the U.S. Southwest for travelers from Israel. EL AL launched weekly nonstop service from Tel Aviv June 14th. The flight, operated on a Dreamliner, arrives on Friday mornings and departs Las Vegas on Saturday nights EL AL’s arrival marked the first scheduled nonstop service from Israel. It was a dream for us to come to Las Vegas We are operating today from Los Angeles and San Francisco, and it’s just so natural to put Las Vegas as a destination from Tel Aviv today. So we have the gold triangle — Los Angeles, San Francisco and Las Vegas. International travelers are desirable for Las Vegas because they tend to stay longer and spend more, a win-win for our tourism driven economy. Certainly the strength of the destination of Las Vegas is what attracts everyone. What we the airport tried to do is not make it difficult for a new entrant to come on in. So I think Terminal 3 is still a phenomenal terminal, very efficient, very smooth for an airline to come in and operate. Nonstop flights on KLM and EL AL also offer locals new ways to see the world, connecting Las Vegas with yet another city in Europe and the history and excitement of Tel Aviv. As more airlines expand their service offerings to include Las Vegas, McCarran will continue to do its part to provide a warm welcome to travelers from around the globe. Now you’re ready to pack your bags and fly the skies so have a great trip. Remember to check us out on Facebook and follow us on Twitter, and use all the resources of our website, McCarran-dot-com Thanks for joining us and we’ll see you next time on “Flight Path.”

American Airlines Flight 965 | Wikipedia audio article

American Airlines Flight 965 was a regularly scheduled flight from Miami International Airport in Miami, Florida, to Alfonso Bonilla Aragón International Airport in Cali, Colombia On December 20, 1995, the Boeing 757-200 flying this route (registration N651AA) crashed into a mountain in Buga, Colombia, killing 151 out of the 155 passengers and all eight crew members The crash was the first U.S.-owned 757 accident and is currently the deadliest aviation accident to occur in Colombia It was also the deadliest accident involving a Boeing 757 at that time, but was surpassed by Birgenair Flight 301 which crashed seven weeks later with 189 fatalities Flight 965 was the deadliest air disaster involving a U.S. carrier since the bombing of Pan Am Flight 103 in 1988 Five passengers, all seated within two rows of each other, survived the initial impact, but one died two days later of his injuries In addition to the four human survivors, a dog, who had been in a carrier in the cargo hold at the time of the crash, survived the incident The Colombian Special Administrative Unit of Civil Aeronautics investigated the accident and determined it was caused by navigational errors by the flight crew == Aircraft == The aircraft was a Boeing 757-223 registered N651AA Its first flight was on August 12, 1991, and was the 390th Boeing 757 built The aircraft was powered by two Rolls-Royce RB211 engines == Flight history == === Departure === At that time, Flight 965 mainly carried people returning to Colombia for the Christmas holiday, vacationers and businesspeople A winter storm in the northeast United States caused the airline to delay the departure of the airliner for thirty minutes to allow for connecting passengers to board the flight, so Flight 965 pushed back from Gate D33 in Miami at 5:14 pm, and then taxied to runway 27R, but seasonal congestion caused the 757 to take off at 6:35 pm, 1 hour 21 minutes late.The cockpit crew consisted of Captain Nicholas Tafuri, age 57, and First Officer Don Williams, age 39 Both pilots were considered to be highly skilled airmen Captain Tafuri had more than 13,000 hours of flying experience and First Officer Williams had almost 6,000 hours The cabin crew consisted of Purser Pedro Pablo Calle and Flight Attendants Magdalena Borrero, Rosa Cabrejo, Teresa Delgado, Gilberto Restrepo, and Margaret “Maggie” Villalobos All cabin crew personnel were born in Colombia and were veterans from Braniff International Airways who had moved to Eastern Air Lines and then to American Airlines, when the routes were transferred from one airline to the other They had voluntarily chosen the flight, as a prerogative awarded by seniority, to spend Christmas time with their families in Bogotá

=== Going off-course === Cali’s air traffic controllers had no functional radar to monitor the 757, as it had been blown up in 1992 by the terror group FARC Cali’s approach uses several radio beacons to guide pilots around the mountains and canyons that surround the city The airplane’s flight management system already had these beacons programmed in, and should have, in theory, told the pilots exactly where to turn, climb, and descend, all the way from Miami to the terminal in Cali.Since the wind was calm, Cali’s controllers asked the pilots whether they wanted to fly a straight-in approach to runway 19 rather than coming around to runway 01 The pilots agreed to approach straight-in, hoping to make up some time The pilots then erroneously cleared the approach waypoints from their navigation computer When the controller asked the pilots to check back in over Tuluá, north of Cali, it was no longer programmed into the computer, and so they had to pull out their maps to find it In the meantime, they extended the aircraft’s speed brakes to slow it down and expedite its descent.By the time the pilots found Tuluá’s coordinates, they had already passed over it In response to this, they attempted to program the navigation computer for the next approach waypoint, Rozo However, the Rozo NDB was identified as R on their charts Colombia had duplicated the identifier for the Romeo NDB near Bogotá, and the computer’s list of stored waypoints did not include the Rozo NDB as “R”, but only under its full name “ROZO” In cases where a country allowed duplicate identifiers, it often listed them with the largest city first In other words, the “ROZO” waypoint should have been at the top of the FMS, as it was the nearest one, but in the case of Flight 965, it was not Only several other waypoints that began with “R” were displayed By picking the first “R” from the list, the captain caused the autopilot to start flying a course to Bogotá, resulting in the airplane turning east in a wide semicircle By the time the error was detected, the aircraft was in a valley running roughly north-south parallel to the one they should have been in The pilots had put the aircraft on a collision course with a 3,000-metre (9,800 ft) mountain The air traffic controller, Nelson Rivera Ramírez, believed that some of the requests of the pilots did not make sense, but did not know enough non-aviation English to convey this == Crash == Twelve seconds before the plane hit the mountain, named El Diluvio (The Deluge), the Ground Proximity Warning System activated, announcing an imminent terrain collision and sounding an alarm Within a second of this warning the first officer disengaged the autopilot, and the captain attempted to climb clear of the mountain; however, neither pilot had remembered to disengage the previously deployed speed brakes, which reduced the rate of climb At 9:41:28 pm Eastern Standard Time it struck trees at about 2,700 metres (8,900 ft) above mean sea level (MSL) on the east side of the mountain The crash was 9.7 kilometres (6 mi) south of Tuluá VOR and 45 kilometres (28 mi) north

of the approach end of runway 19 at Alfonso Bonilla Aragon International Airport During the investigations, it was found that neither the Boeing fixed-base simulator nor the flight management system simulator could be backdriven with the data obtained directly from the accident airplane’s flight data recorder Because the 757 flight simulators could not be backdriven during the tests, it could not be determined with precision whether the airplane would have missed the mountain/tree tops if the speedbrakes had been retracted during the climb attempt However, the final report stated that if the flightcrew had retracted the speedbrakes one second after initiating the escape maneuver, the airplane could have been climbing through a position that was 46 metres (150 ft) above the initial impact point Because the airplane would have continued to climb and had the potential to increase its rate of climb, it might well have cleared the trees at the top of the mountain == Crash investigation and final report == The crash was investigated by the Special Administrative Unit of Civil Aeronautics (Aeronáutica Civil) of the Republic of Colombia, with assistance from the U.S. National Transportation Safety Board (U.S. NTSB) as well as other parties, including the U.S. Federal Aviation Administration, Allied Pilots Association, American Airlines, Boeing Commercial Airplane Group and Rolls Royce Engines The Aeronáutica Civil prepared a final report of its investigation in September 1996, which was released through the U.S. NTSB.In its report, the Aeronáutica Civil determined the following probable causes of the accident: The flight crew’s failure to adequately plan and execute the approach to runway 19 at SKCL and their inadequate use of automation Failure of the flightcrew to discontinue the approach into Cali, despite numerous cues alerting them of the inadvisability of continuing the approach The lack of situational awareness of the flightcrew regarding vertical navigation, proximity to terrain, and the relative location of critical radio aids Failure of the flightcrew to revert to basic radio navigation at the time when the Flight management system (FMS)-assisted navigation became confusing and demanded an excessive workload in a critical phase of the flight.In addition, the Aeronáutica Civil determined that the following factors contributed to the accident: The flight crew’s ongoing efforts to expedite their approach and landing in order to avoid potential delays The flight crew’s execution of the GPWS escape maneuver while the speedbrakes remained deployed FMS logic that dropped all intermediate fixes from the display(s) in the event of execution of a direct routing FMS-generated navigational information that used a different naming convention from that published in navigational charts.The Aeronáutica Civil’s report also included a variety of safety-related recommendations to the following parties (number of individual recommendations in parentheses): U.S. FAA (17) International Civil Aviation Organization (3) American Airlines (2)Investigators later labeled the accident a non-survivable event == Aftermath == Scavengers took engine thrust reversers, cockpit avionics, and other components from the crashed 757, using Colombian military and private

helicopters to go to and from the crash site Many of the stolen components re-appeared as unapproved aircraft parts on the black market in Greater Miami parts brokers In response, the airline published a 14-page list stating all of the parts missing from the crashed aircraft The list included the serial numbers of all of the parts.In 1997, U.S. District Judge Stanley Marcus ruled that the pilots had committed “willful misconduct”; the ruling applied to American Airlines, which represented the dead pilots The judge’s ruling was subsequently reversed in June 1999 by the U.S. Court of Appeals in Atlanta, which also overturned the jury verdict and declared that the judge in the case was wrong in issuing a finding of fault with the pilots, a role which should have been reserved for the jury only.American Airlines settled numerous lawsuits brought against it by the families of the victims of the accident American Airlines filed a “third-party complaint” lawsuit for contribution against Jeppesen and Honeywell, which made the navigation computer database and failed to include the coordinates of Rozo under the identifier “R”; the case went to trial in United States District Court for the Southern District of Florida in Miami At the trial, American Airlines admitted that it bore some legal responsibility for the accident Honeywell and Jeppesen each contended that they had no legal responsibility for the accident In June 2000, the jury found that Jeppesen was 30 percent at fault for the crash, Honeywell was 10 percent at fault, and American Airlines was 60 percent at fault.An enhanced ground proximity warning system was introduced in 1996, which could have prevented the accident Since 2002, aircraft capable of carrying more than six passengers are required to have an advanced terrain awareness warning system.As of November 2017, American Airlines still operates the Miami-Cali route, but as American Airlines Flight 921 and using a Boeing 737-800 == Notable passengers == Paris Kanellakis, a computer scientist at Brown University, died with his wife and two children.The U.S. encountered difficulty while trying to distinguish Americans from non-Americans, as many passengers held dual citizenships == In popular culture == The events of Flight 965 were featured in “Lost”, a Season 2 (2004) episode of the Canadian TV series Mayday (called Air Emergency and Air Disasters in the U.S. and Air Crash Investigation in the UK and elsewhere around the world) The episode was broadcast with the title “Crash on the Mountain” in the United Kingdom, Australia and Asia The accident was also featured on Why Planes Crash on MSNBC, in an episode titled “Sudden Impact” The episode “Disastrous Descents” of the TV series Aircrash Confidential produced by WMR Productions and IMG Entertainment, featured the accident The Sound of Things Falling, a 2011 novel by Juan Gabriel Vásquez (Bloomsbury 2012 in English, translated by Anne McLean)

== See also == Air Inter Flight 148 Air New Zealand Flight 901 Prinair Flight 277 Crew resource management Ground proximity warning system (GPWS) List of accidents and incidents involving commercial aircraft

MIG's Report to Putin on New Fighter MIG 35 Flight Test

Good afternoon, friends, Let me start by congratulating you on this important event – the start of our new MiG-35 light fighter plane’s flight tests My congratulations go, of course, to the designers, engineers, workers and pilots to everyone who worked and continues to work on this big and important project to put a new aircraft in the skies I note that the new multipurpose MiG-35 fighter has enhanced flight and technical characteristics and is equipped with the very latest weapons systems You know this better than I It can follow from 10 to 30 targets at once, and can operate over land or sea This is a genuinely unique and promising aircraft, you could say, very close to being fifth generation I hope very much that this fighter will contribute substantially to bolstering our Air Force and Aerospace Forces The plane also has good export potential, given that more than 30 countries actively operate another model, the MiG-29, and these countries have a good infrastructure for using these fighters and have the trained personnel At the same time, of course, industry and everything related to these planes’ operation must be ready so that we can offer potential partners the best possible maintenance and servicing available in the world today Another point to which I want to draw attention is that one of the MiG facilities plans to manufacture a civilian aircraft for which there is great demand in the national economy and from the public This is the medium-haul propeller plane for use on domestic routes I hope that this work will progress according to schedule, all the more so as we decided on financing sources, and we will gain a modernised aircraft that will be widely used on domestic routes Please I’m listening to your report Mr President, colleagues, Director of United Aircraft Corporation (OAC) Yury Slyusar reporting Here at my side is Chief Designer of the OAK Sergei Korotkov and Chief Pilot of MiG Corporation Mikhail Belyayev Mr President, today, we started flight tests of the new light multipurpose MiG-35 fighter The fighter was designed specifically for combat in high-intensity conflict and dense air defence conditions The plane’s excellent results were achieved through using a new on-board defence system and new infrared search and track The plane’s radar visibility has been reduced by a several-sold factor We have increased from six to eight the number of suspension points, which will make it possible to use current and future airborne weapons systems, including laser weapons The plane’s range has been more than doubled This was achieved through bigger capacity of internal tanks and mid-air refuelling functions, which can be done in tanker regime with aircraft of the same family too All systems used in the MiG-35 are Russian designed and made, including the newest systems- the inertial system and helmet-mounted targeting system The plane was planned as part of the state arms programme with mass production starting in 2019 On January 27, here in Lukhovitsy, we will present this aircraft to prospective customers from other countries. We have great hopes that the plane will sell on markets abroad and we already see a lot of interest in it Of course, we hope too that the orders the Defence Ministry places and the orders coming in through military-technical cooperation, will keep the plant busy But, as you noted, here in Lukhovitsy, we are using the MiG production facilities to develop in parallel production of a regional propeller plane, the ll-114 We have started work on the decisions taken at the meeting you chaired six months ago

The resources have been allocated and we have received them At the same time, the design bureau is preparing documentation that will be sent to the plant, and here on the same production floor where the MiG-35 aircraft will be made, we will be building at least 12 ll-114 airplanes a year This will make it possible, among other things, in keeping with the tasks that have been set, to balance the corporation’s portfolio, increase the share of civilian products and meet our strategic targets to increase the number of civilian aircraft to 45 percent of the corporation’s portfolio by 2015 Mr President, we would like to show you video footage of today’s flight, if you want Please I would like to give the floor to the pilot who took this machine on its maiden flight: Mikhail Belyayev Comrade Supreme Commander-in-Chief, comrade members of the Military-Industrial Commission, As part of the MiG-35UB (two-seat version) trial programme, a crew of Russian Aircraft Corporation MiG – test pilots Mikhail Belyayev and Stanislav Gorbunov – have performed a flight to demonstrate its stability, controllability and manoeuvrability characteristics The objectives and results and been fully accomplished All on-board systems worked properly The engine and the integrated aircraft control system worked properly The crew’s evaluation is positive The quantitative evaluation will be made following the processing and analysis of the materials received from flight recording equipment That concludes my report Thank you, Mr Belyayev Mr Korotkov, how do you evaluate the work? (United Aircraft Corporation General Designer and Vice President for Innovations Sergei Korotkov)

Mr President, members of the Military-Industrial Commission, We have created a multi-spectrum system that was integrated into the armament system and additionally installed aboard the MiG-35, and along with other systems, we have developed the Generation 4++ complex We hope that after the trials, the Defence Ministry will buy this machine and that foreign customers will also come to sign contracts with us I hope this project will have success Thank you very much Please convey my warmest and sincerest congratulations to the entire staff that has worked on this product, on this machine All the best. Good luck

Departure runway 21L Athens International Airport (ATH LGAV).

CPT: Captain FO: First Officer CPT: Uh.. nice story… bye! FO: What ever… 😉 CPT: It is always nice being here CPT: I will set the trim at 6 units

FO: Yes, Check CPT: That sounds like a very complex clearance But we are on “I” D4 is straight ahead…. actually till the end left FO: Yes, it is clear for me CPT: There we go FO: Yes FO: Nice people those Greeks CPT: Indeed they are FO: Hello with Mike Purser: Cabin is OK FO: Cabin OK… thanks! CPT: Cheeeeck, I was listening as well FO: Nice CPT: Yes… mountains FO: Yes CPT: Departing aircraft It looks like an Airport FO: You would think so FO: It’s a pity it’s an Airbus CPT: It looks like an empty flight FO: Indeed, they went very early airborne CPT: Yes FO: So!… nice shadow The Airbus A350 is a nice aircraft CPT: I do agree FO: Especially the long version. This is the short version CPT: Can you please select your “legs” page on the FMC? FO: Off course! CPT: Nobody expects these kind of clearances FO: “Olympic”… was that the flag carrier of Greece? CPT: Yes CPT: They where flying with Boeing 727’s! CPT: Boeing 727… is a Boeing 737 with 3 engines FO: Ah yes… exactly FO: Actually we are ready CPT: Uh… yes, except for the checklist… but report ready first please FO: Thats also a way to say it… just Transavia CPT: Yes CPT: And there is a traffic on final FO: Yes CPT: Nice, we are next to the runway. We can’t get closer except for entering the runway The Before Takeoff Checklist CPT: EGO… EI-EGO

FO: I didn’t hear what the wind was CPT: SPCY… spicy FO: Sometimes they create a registration to mean something CPT: Yes indeed CPT: Well… welcome in Amsterdam… today we drove you to Amsterdam… Looks like the “Polderbaan” FO: Yes…. pretty far indeed CPT: I see some SAR helicopters… and a Boeing 747 FO: That aircraft doesn’t fly anymore CPT: I think so too! FO: From Olympic or… no CPT: No… another company I don’t know CPT: Nice… I’ll select terrein mode FO: Yes, then I’ll select the weather radar Is finished FO: Well CPT: Well.. nice! Lights… transponder… all on CPT: Then I say “Takeoff”! CPT: There is suddenly some wind! FO: Sorry? CPT: Thats why we climb so fast… LNAV is checked CPT: Is it OK with you if I fly the first two turns manually? FO: Off course! CPT: I didn’t brief it, thats why I ask CPT: Great! Thanks

CPT: The 2 turns have just been taken from me! 😀 Thats a pity! Then switching to CMD (Autopilot) CPT: Unrestricted climb, thats nice I feel some wind changes We receive some strange pushes CPT: She is teasing you! FO: I think so too FO: I’ve never been so busy during departure CPT: I can see ATC and you have a “click” FO: Yes? CPT: Yes FO: I think she is nice to us CPT: Indeed! FO: Yes CPT: Have a look!!! FO: She is just giving us a scenic tour! CPT: Super cool! FO: Yes CPT: The French Post had some brown beans last night for dinner, did you notice?? FO: Yes, exactly CPT: PEREN… selected and execute FO: Yes CPT: And LNAV FO: Check FO: Would you like to do the after takeoff checklist CPT: Shall we just do it FO: Uh… did you wanted to wait for a bit (Fasten seatbelt sign) CPT: That is fine with me, if only you can give the passengers some warm air FO: They would appreciate that CPT: Just in front of us is the Olympic park FO: Ah yes! But where are those famous ruins? FO: Yessss FO: It should be in the mountains CPT: Yes… you might be right…but which mountains Yes… there we go! Bad preparations.. But I never expected to depart over the city! FO: It should be a big building you can’t miss CPT: You would say so FO: Yes… I have no idea CPT: I see an arena… with gladiators! FO: You can see the gladiators! CPT: Ah… now I have a visual I think It is a big “rock” with those pillars FO: Oh, So you are able to see it CPT: Yes. FO: Nice! CPT: Now we know where to go to! FO: Yes FO: I am gonna make some notes now. 😉 CPT: I am happy to switch off the fasten belt sign FO: Me too! FO: So what do we request

CPT: In the flight plan it was initially FL270 but we are able FL360

ESTA VEZ SI ME PASE = DUBAI-PARIS (AIRBUS A380 EMIRATES) REAL FLIGHT SIMULATOR.

WELCOME ON BOARD THIS EMIRATES AIRBUS A380 TODAY WE WILL BE MAKING A FLIGHT FROM THE CITY OF DUBAI TO THE CITY OF PARIS FRANCE OUR FLIGHT WILL LAST A TOTAL OF 6 HOURS WITH 30 MINUTES (WHAT FOR YOU ARE ABOUT 20 MINUTES, I ALWAYS PUT 20 MINUTES HAHAHA) THE DURATION OF OUR FLIGHT IS PHENOMENAL I CANNOT COMPLAIN AS WE HAVE CLEARED ALL I INVITE YOU TO SUBSCRIBE AND GIVE IT LIKE AND SHARE THE VIDEO SO THAT MORE PEOPLE COME TO ENJOY THESE GREAT FLIGHTS THIS FLIGHT IF I HAPPENED 6 HOURS BEFORE WE DID 3 HOURS AND IF YOU LIKE IT, THEN SHARE IT (HAHAHA IT IS ALREADY REPEATED) ALSO IF YOU LIKE THE MUSIC IT WILL BE DOWN BELOW IN THE DESCRIPTION SO YOU CAN LISTEN AND ADD IT TO YOUR PLAYLIST NOTHING BUT TO SAY THIS IS MY REPORT TODAY. SO BRING YOUR PAPITAS BECAUSE TODAY WE WILL TAKE OFF AND I HOPE YOU ENJOY THE FLIGHT

Love at First Flight – Emergency Landing: Trust and Secrets (S1, E5) | Full Episode | Lifetime

Narrator: Thousands of singles, all hungry for real love and ready for marriage, were surveyed and evaluated by our expert matchmaker to create four perfectly matched couples Now it’s up to them to make it work while traveling across the country in a 30-day journey I think I will do this one packed with relationship challenges, rewards Ooh! [ Hissing ] and more baggage than they can carry Oh, no! I’ll come save you Their last destination? The altar Could he be someone I can marry? Aah, good job! For some couples, the lines of communication opened up You have great potential, and that’s all that I look for Some experienced static I’m just saying your age scares me Okay, well, that’s — I mean It just does And others found themselves in a dead zone The love totem, up in flames Let it burn like us, Cale Will tonight bring them one step closer to a walk down the aisle Yeah We’re gonna have some champagne I don’t know what’s gonna go on Tomorrow, I turn 30, so I’m going out with a bang Narrator: …or to walking away from it all? Ryan: I’m not going to sit here and just watch this I’m basically done Stop! What the [bleep] are you doing?! ♪ Michael N.: Wow Alma: Oh, this is gorgeous Oh, my God It’s a little bright We had a great stay here I mean, what if we miss our flight, though? It’s okay I wouldn’t mind staying here I wouldn’t mind a hot cocoa and a fireplace right now, either Right? It depends on where we’re going, where we’ll decide if we miss our flight or not. Yeah Alma: Mike has opened up to me When he told me about, uh — about his brother This is not very…easy for me to say at all My brother committed suicide I’m sorry It was super meaningful He has so much courage for having done that I’m just starting to build something more genuine and more of a deeper connection as opposed to just very superficial and shallow Gorgeous, right? That is lovely I’m talking about me Oh, wait. You want your left side, right? You’re a left-side girl? Thank you! He knows my good side Oh, man All right All right. Let’s go Off to bigger and better ♪ Maybe over there Okay ♪ Oh, my God [ Chuckles ] We have a little bit of a bump in the road Got it? Good Get it. Got it Good We have conflicting ideas of where we would potentially move to ♪ Where are we going? ♪ ♪ Where are we going? ♪ ♪ So just a little bump We’ll work through it, hopefully get through it, and it’ll make us stronger [ Cellphone dings ] Oh, I just got a text Did you just get — Aaaaaaaaah! Yeah! [ Laughs ] What? [ Laughing ] I got it! Yeah! What?! Are you serious? I swear to God Whoo! Oh, my God [Bleep] knew it Sin City! We’re going to Vegas! Whoo-hoo! Sin City We’re going to Vegas! Gonna have a time We’re going to Vegas! Hey Vegas, man, here we come Let’s go! ♪ Live from Denver airport Yeah Denver was great When was the last time you went to Vegas? I’ve never been I’m supposed to be there — You’ve never been to — Like, this is the first time? This will be the first time So, you know, we’re — we’re — we’re floating on cloud nine right now ♪ It don’t matter where we at ♪ ♪ We’re gonna have a fun attack ♪ ♪ Where there’s a will ♪ ♪ There’s a hood queen ♪ ♪ We’re going to Las Vegas How exciting And that is where I’ll be spending my 30th birthday I’m super excited Jenna says that my enthusiasm level is at a level 20 right now and I need to notch it down to about, you know, maybe like a 6 or a 7 Fingers crossed that it’s a good time and — It will be At this point, I think we’re able to maybe start laying the foundation for a friendship So I am happy to be moving on to the next place now Just new energy, new environment I think that’s — That could be really helpful for us ♪ Narrator: Couples were matched based upon overwhelming high compatibility scoring You got it? Okay In each destination, they’ll complete challenges designed to help them learn in 30 days what most couples take months to uncover so, at the end of the road, there’s only one question left to ask —

“Will you marry me?” We just landed here in Vegas I love this city It’s like my second home I come here all the time, so I couldn’t be happier to be coming here and then also sharing it with Alma ♪ Me and Stephanie are definitely out of the friend-zone level, so, you know, we’re actually moving and growing, you know, together So the text was, like, you need to — you got to trust her in Vegas? I-I-I saw trust I definitely saw trust I think they’re going to make us want to cheat on each other So they’re gonna give me a cake, and they’re gonna give you n-naked chicks You know I can’t resist the cake, now ♪ Welcome to Las Vegas Gimme, gimme, gimme, gimme, gimme, gimme [ Speaks indistinctly ] It’s interesting that we’re kind of put to the task of being in Sin City to learn to trust each other, but it’s known for “Nope. You’re not going there You’re not going there without me.” There. You get this one, and I’ll — I’ll get this one Although Denver had ended not the best way, I feel as if Las Vegas is going to be a fun experience for both of us, so I’m hopeful I’m optimistic Now, getting a ride Just make sure I have all of my stuff here is the next step Oh! Sun! [ Cellphone dings ] ♪ Oh, my God [ Laughing ] Read it out loud What does that mean? Okay I don’t know what that means, but I’m eager to find out Yep. Yep ♪ ♪ You can’t really see, but there’s Vegas! Morning Excited to see whatever we’re doing I’m gonna try to live in the moment and not, uh — not let any of these past conversations ruin a good time We’ll figure it out Vegas, day 1 ♪ Tomorrow, I turn 30 It’s a big — big day, so I’m going out with a bang The best part about traveling with Cale is that he maintains this very positive attitude The worst part about traveling with Cale is that he always maintains this positive attitude [ Cellphone dings ] Oh. Check it out You got the magic text? You ready for what we’re doing today? Yes I don’t know What do you think? I don’t know Carrying a load, that’s — that makes me a little nervous And trusting each other to carry the load I trust you to carry the load [ Laughs ] I trust me to carry the load, too [ Laughs ] Cool We kind of work well together, but there’s just an absence of trust there, and it’s definitely hindering our ability to appreciate each other’s company ♪ Stephanie J.: We’re — We’re in Vegas Officially excited to see Michael in the morning now It’s official now that Michael and I have identified that, yeah, we do like each other, and he does meet the qualifications that I have for someone that I would date and be courted to a-a marriage, um, so I need to jus and enjoy the rest of the process and journey, whatever that is Oh, it’s bright Did they call for a car or what? Not that I know of [ Cellphone dings ] This is not — Oh, I just got a text Mission order? [ Chuckles ] What? Okay, I’m — I’m totally confused, and there’s — How are we supposed to get there? ♪ Combat ops? There’s no — For real? Y’all playing [ Laughs ] Don’t just stand there and look! Get over here! Come on! Let’s go! Hurry up! Hell, nah, man Let’s go What’s your name? Michael. Yes Michael? Yours? Stephanie Oh, you’re just the people I’m looking for Are these zombies? Come on. Hurry up. Get in there Don’t worry about it. Go! [ Screaming ] Hurry up! Get in, get in, get in! Stephanie, are you gonna get in? Hurry up! Get in, get in, get in! I don’t even want to look! What does it say? What? What? [ Chuckles ] Oh, my God What? What? Is that — Is that supposed to be, like, dominatrix [bleep]? That is really cool if that’s what it is [ Chuckles ] I’m so, like, not into that but, like, so curious [ Chuckles ] Oh, my God [Bleep] about to get weird ♪ [ Knock on door ] [ Sighs ] Coming ♪ Hi Hey Come on in How are you? You look great Thank you Like, he looks different

Vegas is doing something to his eyes Like, they’re just popping out 10 times more Like, you can tell that this is his element This is his world Back in your natural habitat? Yeah, my second home Oh, yes, please Cheers Thank you. Cheers She’s not ready. Typical So, how much time do you need to get ready? Um, another 3 hours Oh, really? Yeah Oh, no No, not another 3 hours, but something like that Something? I’m gonna let you be alone I make you nervous Yeah. Yeah, I’ll speed it up, too, if we’re not talking Okay Can I go sit inside? Yeah, yeah. Go sit, please All right. Thanks I’ll be quick about it I’ll just be here ♪ Ready? ♪ Are you ready? Yes Tell me if I’m Vegas enough, okay? Show me, show me ♪ Yeah Yeah? Yeah Damn. This girl looks good You know, everything’s out, and if we’re gonna have some champagne, I don’t know what’s gonna go on Damn, you look good Okay, thank you I’m really happy He loves my outfit. [ Laughs ] How many times do you come to Vegas? I try and get out here, like, at least four times a year Four times a year? Yeah Is there any business, or is it just all fun? Oh, no, all fun I’m starting to get that Mike is not over the clubbing scene, so I’m trying to make sense of it So, okay, what’s fun? So, like, summertime, um, there’s a bunch of, like, awesome pool parties, and depending how crazy you get, if you can make it to the nighttime club after, if you want to continue, there is after-hours Like, you can walk out of the club at 3:00 a.m and continue partying until, like, noon the next day So you’re into clubbing? You do a lot of clubbing? Yeah “I come here, like, seven times a year I party a lot, daytime and night.” And I’m like, “What the heck? Like, who are you?” I think you rage here Oh, I definitely rage Mike is 31, just like me, and so if he’s doing what I did at 19 years old, I’m not down with that [ Knock on door ] Oh Are you expecting anyone? I’m not [ Chuckles ] ♪ Oh, thank you, sir What? What is that? I don’t know Do you want to open each other’s or your own? Yeah, please [ Cellphone dings ] All right It’s the itinerary [ Cellphone dings ] Okay. So, let’s see what’s in here Oh, hello. Hello $500 5? $500? So, we need to make $2,000 in order to stay in a suite Ohh! This is gonna be hard I’m not thrilled at the aspect of us having to gamble because I don’t like blowing money You are gonna know how to do this one, ’cause I don’t I’m not a gambler Well, we’re in this together Michael N.: I mean, it’s Vegas I’m in my element The odds should be in our favor I know she’s outside of her element, but we’re in this together, and we’re gonna achieve this Win big Vegas, baby Go big or go home! ♪ [ Laughing ] What the hell are you doing? This way. Aw, yeah Oh, my God This is [Laughing] insane [ Cellphone dings ] This looks fun You just got a text message [ Laughing ] Shut — [ Laughing ] Shut — [ Laughs ] [ Laughing ] No! [ Laughs ] [ Chuckles ] Hello! Hello, hello Oh, my God [ Laughs ] Oh, yes. Here we go These beautiful, amazing men walk out, and I just know someone’s — someone’s getting a dance [ Ting! ] Hi, you guys Hi Hello, hello How are you? Welcome to “Magic Mike Live.” We are so delighted that you guys are here with us My name is Chelsea, and today we’re gonna make sure that all your fantasies come true ♪ What?! We are gonna teach you how to truly please Stephanie If you complete this challenge, you will spend two nights in a beautiful suite here in Las Vegas Oh, yeah If you don’t, you’re going to spend two nights in a motel I am so excited All right We’re gonna do this We need a chair This is amazing Stephanie is in heaven That’s for sure She knows she’s gonna be getting special treatment today Chelsea: All right Now, Ryan, pay close attention This is what you need to learn, okay? All right, girlfriend Okay. I’m gonna be taking notes You’re right here in the hot seat Gonna be taking notes All right Okay. [ Laughs ] All right Now, guys, you know what time it is ♪ [ Laughs ] Dang it What?! Yeah Can I go work out real quick? These guys are making me look bad You can do a couple of push-ups ♪ It’s like a dream come true All right. Hit it ♪

This is amazing Is this real life? I’m in heaven No, I think I just died I literally think I just died ♪ Him watching without wanting to kill somebody is such an awesome, awesome feeling I mean, this is what it’s about It’s about learning to trust and communicate and be open and have fun with your life ♪ All right. So, like, piece of cake, right? Whew! You can cut the music ♪ Ryan: Stephanie is definitely not the first girl I’ve lap-danced [ Chuckles ] ♪ [ Speaks indistinctly ] [ Laughs ] Man: You’re gonna nail it Boom! Oh-oh-oh! Chelsea: Whoo! You did it! Yes! Okay, guys You got it! Yes ♪ [ Giggles ] Yay! Oh! Ohhhh! Ohhhh! Ohhhh! Ohhhh! Hang on to him Hang on to him Hold on tight! Oh, he’s free-styling Okay Stephanie V.: I am hot and not bothered Ryan’s literally picking me up, grabbing me by my ass, tossing me up on a ladder, grinding on me Okay. All right Did not expect that Whoa! Whoa! There’s kissing? Okay! Yeah! That’s cool “Magic Mike Live” Whoo! bringing people together All right Ryan, Stephanie, I think you guys did great, but it’s not up to me It’s up to the “Magic Mike Live” guys Each one is gonna whisper to me what they vote Okay Come on, guys. Come on Ryan: Be generous We love your show ♪ [ Both laugh ] Ryan: I just hope that the judges, these guys, are a little generous All right We have decided ♪ Chelsea: All right We have decided that you guys are gonna spend two nights in a beautiful suite in Las Vegas! [ Both cheering ] All right. All right And, just to sweeten it a little bit more, you guys have two VIP tickets to tonight’s performance of “Magic Mike Live” right here at the Hard Rock! Stop it! That’s amazing So we can’t wait to see you tonight Ryan: I’m excited for the show, but I definitely know that Stephanie is way more excited than I am That’s just not — not a question. [ Chuckles ] Stephanie V.: Ryan, do not forget the choreography ♪ All right, guys. Welcome to the aggression session Here is your challenge, okay? This is going to dictate your accommodations The object here is to just totally demolish this car This is all about you guys working together on a trust basis and get rid of your pent-up frustrations And by the time you guys are done, you guys will be best buds Is that how that works? That’s how it works All right. That’s good We’ve been going at this all wrong all this time That way, whenever you’re going to crush it, you can crush those things out of your memory Okay? All right Sounds good Can I go first? Here we go. Yeah Yeah, do it ♪ “Haters.” Haters gonna hate Rationale? People who prejudge and hate you before they know you [ Can rattles ] Cale: She’s trying to pick a fight with me Your turn ♪ All right What does that say? I wrote “Bad attitudes” because bad attitudes are lame Agreed They suck all the fun out of the room, and they kill my — my vibe Was that directed at anything, or that was just in general? Personal example. Uh Yeah whenever somebody has a bad attitude, I hate it Mine actually wasn’t directed to you, but I see where we’re going with this I don’t know I almost thought we were bonding and having a second, but that just felt very [hisses] For me, when I say, “Haters,” I don’t believe you’re a hater I think a lot of people are Um I feel that you felt that was directed towards you, though, and that’s why you did “bad attitude” directed towards me ’cause you — that’s something we’ve talked about I-I-I don’t like the back-and-forth of this at all So I’m not going to relate these back to you [ Can rattling ] I’ve got one, Jen ♪ All right It says, “Manipulation.” When I was in college, I dated a girl, on again, off again, for way too long, and I allowed her to manipulate me So I have some emotional scars from that, and maybe it’s difficult for me to get close, uh, because of that, with other people I felt like, for the first time, everything that he said was authentic and real, and, for the first time, I felt like we could build trust

♪ Drugs. [ Chuckles ] Drugs ruin relationships Um my last boyfriend was a drug addict, but he was sober when I was with him Then, when we broke up, he started doing drugs again, and, um it…kills me ♪ This definitely opened up some new stories that we haven’t shared with each other before, so, I mean, this is a good exercise I hope that we’ve now gotten to a point where we are willing to kind of put our guards down and move forward Who knows what’s to come? You guys ready to crush this baby? Are you ready? Let’s go. Yeah. Ha! Oh, my God. That’s awesome All right. Good hit That was a good hit Just crush it Keep going. Keep going Slam it! [ Chuckles ] There you go. Now Whoo! Oh, you got to get higher Now we’re gonna make Jenna get in there, too Get it, Jenna Good job. Okay, Jenna Now, right hand forward Oh! Oh-ho! Gently I thought we were crushing [bleep] Sorry [ Laughs ] Good job Nice job, Jenna Slam it down [ Chuckles ] Slam that sucker down Should I drop it? Should I just drop it? Drop it! ♪ Do you like it? Yeah, I-I love it Oh, my God. Jenna! Aaaaaaaaaaaaaah! ♪ Yeah! ♪ Anything calling your name? No? Anything? Oh, man ♪ I love this city. Excited to show Alma what it’s all about If we’re gambling someone else’s money, like, how could we go wrong? ♪ I’m gonna watch you first No, let’s do this together Let’s start with $100 You want to start with $100? Let’s start off with $100 Okay Alma: We have to double the amount of money we have in the envelopes Like, seriously, there’s no way, especially when I don’t gamble at all Can I just do $2 first? You — That’s fine. No, fine Let’s do $20 on red just so if I feel bad You want $20, red? Yeah Okay Wait, wait Is that good for you? Yeah. Let’s start there, just so you can ease into it Oh, my God ♪ This is your first time Luck is gonna be on your side I know, but that’s, like ♪ Croupier: 33 black, odd Oh, that hurts so bad, Michael I’m sorry No, it’s fine I’m sorry That’s what this is about, though You win some, and you lose some I don’t know I’m having a good time ♪ Would you feel worse about this if this was our money? I would not even be in here with you if this was our money I’d be like, “You’re insane, and I don’t know if I can be with you.” ♪ Everything off? You want to — What’s your favorite number? 7 7? I don’t want to do numbers, though You don’t want to do numbers? I want to stay the hell away from numbers Okay You like low or high? I like to be safe, Michael So, basically, I’m saying colors, evens I like to be safe Okay I don’t want — That’s all right I mean, you’re ballsy I got that Let’s do another $20 That is not the time to show off, but I’m pretty sure you’re not gonna nail the number We got to get this money high Like, we got to get this going So, you good with this? I — That’s your call I’ve been making other ones Yeah Croupier: You guys want to bet like this? We have a lot riding on this No problem [ Laughs ] Oh, gosh This sucks so bad Oh, my God. I can’t do this I hate this ♪ Uh, lost all No? Lost it all Just $100? No, we just lost Everything with this game? $20, $40, $60, $80, right? Yeah. 8– $120 I thought we said $100 each time I said $100 each time Why did you put the extra $20? I wasn’t paying attention to that Alma: I’m, like, realizing the more I get to know Mike, I can trust him on so many levels, but on so many other levels, I’m like, “What are you doing?” And that is so scary ♪ ♪ Croupier: Uh, lost all Alma: No? Michael N.: Lost it all Just $100? Well, we just lost Yeah. 8– $120 I just think it’s safer to always do just $100 each time, though Okay. That’s fine I think that’s the easier way to build I know you’re reliable, but we have — we have to make this Sure. Okay Alma: He knows where I’m coming from He got it pretty quickly “All right. You’re right Not the time to be risky I got you.” And he definitely wanted to make me feel more comfortable, and I appreciate that What are you feeling, girl? What you got? What do you think? This is $100, right? Well, I support you ♪ Good vibes, good vibes We got this Please, baby Jesus Christ, help us

♪ [ Chuckles ] Yeah! Yeah. It’s 34 Whoo-hoo! ♪ So see how quickly it’s gonna climb? Shut up, Michael ♪ ♪ I mean, I’m loving Alma’s smile Like, when we win, that smile is ear-to-ear You good? Oh, I’m good You good? You doing better? The closer we get to our goal, you can definitely feel, like, the tension dissipating This is — This is our shot right here ♪ Wow! Oh, my God! [ Both laugh ] Viva Las Vegas! [ Laughs ] $1,800, $1,900, $2,000 How do you feel? I feel like I won the lottery, but, when it comes to marriage, I’m not sure if I’m down with Mike’s clubbing, raving, or gambling I don’t want that ♪ Scared? I’ve been through worse You’ve been married before? No All right, then You evil, evil man Michael and I were just kidnapped by a strange man with a beard Bottom line is we’ve got to trust each other to watch each other’s backs, all right? Are there gonna be zombies coming out today? Yes Stephanie J.: They’re telling us that we have to go hunt out zombies and kill them I’m terrified of zombies, and I think they’re real Here’s what we’re gonna do We’re gonna get you suited up, get you your weapon, and then we got to go out and we got to accomplish these missions. You ready? Yeah Yeah All right Follow me. Come on Michael S.: I don’t know what we’re about to experience, but, um, as long as I have a gun and good instructions, we’re good All right. Sometimes it’s dark out there There’s gonna be bombs going off and everything [ Siren wailing ] Michael! Panic, panic, panic I’ve already passed out, mentally Rumor has it that there’s a case out there that, if you guys find it and you guys can open it, you guys are gonna be staying in a better room than you’re currently staying in right now Remember, this is teamwork and trust, all right? We’ve got to trust each other while we’re out there I’m just hoping Michael is paying attention so that he can kind of [clicks tongue] carry me on through this Come on. Let’s go, let’s go, let’s go, let’s go. Come on [ Wailing continues ] Come on. Come on, girl Michael! [ Chuckles ] I’m behind you I’m behind you Go kick that door open, all right? Go kick it open? Yeah. That’s our first house We’ve got to get in there By myself?! No! We’re right behind you, silly! Okay! Go with me! Wait, wait, wait. Hold on We talked about trust, right? You got to trust us You ready? Come on. Let’s go, let’s go, let’s go, let’s go Come on, Detroit! There you go Let that inner Detroit out you were talking about Come on. Get in there Go. Come on. Come on! Get in there Go, go, go, go, go, go, go Michael, I got your back I got your back I got your back [ Gasps ] Is that a zombie?! [ Laser fires ] Okay! I got you! I got him! Okay All right. Get in the house [ Screaming ] Get in the house! Get in the house! Get in the house! [ Screaming indistinctly ] Get in the house! Hurry up! We’re safe! Get in that house! Come on, come on, come on Oh, this alive right there, boy Yeah! Hold on, cuz! I thought I was gonna die from zombies, but Michael just picked up the slack, which is huge for me because I’m usually the slack– I’m usually taking all the slack in relationships So it’s nice I can have a bad day, and he would still make sure we didn’t die God! Oh, my God! I think Michael has done a great job at pr-protecting me Go, go, go! He was what I would picture a husband doing Oh, my nail All right Both of you guys go on in there Get in there together Teamwork There’s a — There’s a case in here Start searching for intel Hey, can I just shoot this? Michael, let me try Come on. All right You guys got it Go ahead Oh, no! No way! [ Laughs ] “You have defeated the zombie and trusted each other through the opposition Ohhhhhh! You are being rewarded with luxurious com– accommodations So great! for the stay in Sin City.” You guys have survived! What’s up? Sheesh! Whoo! Michael S.: Every activity like this definitely improves the trust between the both of us There was a point where, you know, she had to cover my back, man, and, uh, it was a good moment And she held her own, man, and that — that was definitely a-a good mark ♪ [ Cheers and applause ] Ladies, are you ready for some men?! [ Cheers and applause ] What’s going down, Las Vegaaaaaaas?! ♪ I’m so excited We’re in the VIP area It’s not only just a “Magic Mike” show, “Oh, my God, there’s hot guys.” These professional dancers do these amazing moves I appreciate the choreography and an amazing live performance that I’ve never seen I’m like, “Hell, yeah.” I’m just super excited just to see these professional dancers perform [ Ting! ] ♪

I want to live here I want to live on that stage It is the most amazing show I have ever been to in my entire life I want to never leave this [ Laughs ] I want each and every one of you to feel what I just felt [ Laughs ] I love her ♪ [Bleep] ♪ [Bleep] ♪ ♪ Ryan: I noticed Stephanie start to, you know, get more lively and have more drinks It seems like she’s completely consumed by the show, and that’s fine She’s having a good time All: [ Chanting ] Take it off! Chelsea: Louder! All: Take it off! I’m hoping that she’s having as good of a time as she appears to This is definitely for her, 100% So if I went to go get drinks, I’m just looking to basically have an okay time and let her have all the fun ♪ ♪ You know, I come back She’s getting a lap dance Okay. That’s fine All right One lap dance is whatever Then the second one happened Then the guy hands me her drink, basically, like, “Hold this while I dance all over your girl in front of you,” kind of thing I’m like, all right The second time, I try to make it, like, funny I’m putting dollar bills all over them [ Laughs ] And then the third time it happened You…deserve…a lap dance! Did the same thing, trying to keep my composure I was feeling pretty uncomfortable at that point, and then that’s when, I guess, the fourth time happened When they brought her down, I was like, “I’m done I’m not gonna sit here and — and just watch this, ’cause she’s obviously having a ball, and she wants to be doing this, so I’m — I’m basically done.” ♪ ♪ [ Cheers and applause ] ♪ I’m so excited because I love aerial artists [ Cheers and applause ] Like, I-I fantasize about aerial artists being at my wedding That’s such a cool thing to me ♪ [ Cheers and applause ] From my perspective, I thought he was watching ♪ Ry! Ryan! Ryan! ♪ I do something that disrespects you, you don’t tell me, and you lock yourself in the bathroom? ♪ It is not how you deal with problems in a relationship Ryan, stop! What the [bleep] are you doing?! Come, please She’s trying to talk to me, and — and she was starting to get emotional, and I felt like, knowing me and knowing the fact that I had been drinking, I didn’t want to say anything I would regret, and I think that’s the difference between her and I And so I told her, “If you want to talk, we’ll talk later.” But she kept trying to press it and tr– kept trying to talk ♪ And then that’s when she got upset, and she got out of the car, and that’s fine because I was done I would never have done that to her, like, ever I wouldn’t have done that to anyone That’s so [bleep] up ♪ ♪ I’m Mike And I’m Steph This is the “Mike And Steph Show.” So, what’s the topic for the day? The topic of the day is why do men cheat? Men cheat because — because we’re physical beings So if we’re — We are. We are Okay. As if — Are we not physical beings, as well? I mean, but you guys are more so emotional beings I got cheated on, you know, in a previous relationship So it took a while to really kind of let anybody, you know, in I liked hearing what you said about that That was good

Word But, um I’m actually reading the book “The Wait”? “The Wait,” yeah This is definitely a new, you know, um — a new situation You know, the fact that, you know, she’s definitely working on her celibacy walk, as well, and that’s something I’m interested in So there’s a lot of things that I, you know, may hold back on that are physical, you know, ’cause I don’t want to step on any toes, or I don’t want to cross any boundaries and things like that Since we’re just talking about sex in general Uh-huh? how does that work in a relationship where the guy is like, “I love sex Like, I love you, but I don’t think I can wait Like, I don’t know.” If she’s worth the wait, he’ll wait I want to make sure, like, that I love you, I love every bit of you Like, you wake up in the morning, like, what I see, I love that before I even get the cookie Like — Like, of course So you admit that sex does taint relationships Oh, it clouds it like a mug I didn’t think you felt that That’s good stuff What? Hell, yeah Give me — give me one. Mm It’s easier for, if a woman is, like, already there, and the guy is willing, it makes it easier for us just to do it together So, excited to help him stay celibate It’s all about that walk, man ♪ [ Cellphone dings ] Okay Well, I’m glad you have those muscles You’re gonna put them to work today [ Laughs ] And you might have to take off your shirt, too Take it off, huh? You might have to, Michael There are things about you that I like a lot, like, a lot, but you come to Las Vegas not for the sightseeing and coffee You — You still club Like, that’s why I’m like Right, but I come for — “Are we so different?” The way he lives his life, I’m starting to notice it doesn’t jive with my comfort level sometimes That, I’m — I’m not gonna be okay with continuing or furthering a relationship, let alone a marriage You know, I’m not married I don’t have anything tying me down, so I can do that But if I’m in a relationship and we want to go on vacation, that’s fine for me, too It’s not like I have to be like, “All right No, I’m only going to Vegas, and I’m only doing this and only doing that.” As a single person, you’re only worrying about how it’s going to affect you In a relationship, whether you’re married, have children, every decision that you make, it’s gonna affect everyone If it’s important to her, I would change whatever is going on in my life to make it work I mean, I understand that’s more about, like, seeing if I fit into what’s right for you, if I’m good for you or not Mm Yeah And I understand that Thank you Thank you for saying that It does make me happy to know that we’re having this conversation I feel like if we threw caution to the wind from Day One and jumped into it, I don’t think we would have built this relationship Yeah that we have together and opened these lines of communication Alma and I took things very slowly, and I think that’s important for both of us It solidifies a stronger bond So, are there feelings there? I mean, yeah Okay I mean, yeah You’re not crazy [ Both chuckle ] What I learned about Mike and I, at that moment, was I can trust him and trust and loyalty are the most important thing to me ♪ There’s, like, nothing out here This isn’t even a paved road Where are we going? What the hell? What is going — [ Gasps ] Oh, gosh Put my lawyer on the phone, please ♪ ♪ Oh, yes ♪ Cale: I think that trust is a-a pillar of any sort of relationship, and, right now, I-I think that there’s not a lot of trust between Jenna and I If Jenna and I can, you know, work things out trust-wise, uh, there is a chance Jenna really is the one for me ♪ Hi, guys Welcome to Trapeze Las Vegas Are there any past injuries or anything of that nature, bad attitudes that we should know about, stuff like that? [ Laughs ] I’m so glad that he brought up the bad-attitude thing because I just want him to keep Cale in check As long as you listen to me, everything’s gonna to be just fine All right. Let’s go Jenna Ladies first All right ♪ [ Sighs ] Mason: Ready? Yes I think that Jenna can do this I think she’s gonna have no problem Thanks, Cale! You need to lean out forward a little bit You’re gonna trust me ’cause I got you with this part Okay. Okay Here’s your bar Good You’re gonna hold me? Absolutely, girl I got you So, ready! Bend your knees Hup! Go ahead and slide your feet Ries: Good! And knees up! All the way in. Perfect! Hands off! Big arch! Look for him! Both: Whoo! [ Chuckles ] ♪ Ah! Cale: Yeah! Nice, Jenna! It is easy for me to trust the stranger that’s catching me because he looked like he knew what he was doing Um, had it been Cale trying to catch me, it would have been a different story

Ries: You ready? Ready Thanks, Shane Show us what we’ve got ♪ Well, I did gymnastics for about six years when I was a kid ♪ I have a little bit of a-an acrobatic background, so, going into the challenge, I’m feeling pretty good ♪ The hardest part about being a trapeze artist is just trusting each other It takes a while for me, I guess, to, uh, learn how to trust people, and, when it comes to love, I certainly have, like, a lot of walls up I want to get your right hand out there, and you’re gonna reach out for the bar I’ve held bigger guys than you, okay? Okay I’m really strong [ Chuckles ] Ooh, this bar is far [ Laughs ] It’s okay Wait right there I’m not sure why it takes me a while to trust people, but I’m doing the best I can to follow the directions Go ahead and grab it with your other hand I have you. I promise You have me? Yes. Suck it up Are you sure you have me? Positive I weigh a lot more than I look No, no. I got you, man I am a little surprised that Cale hesitates ’cause he’s such a, like, “Let’s do this! Whoo!” I got you, promise All my heart Yeah, I just — I wish Cale had a better attitude about this Ready! Bend your knees Hup! Go ahead and slide your feet off Birthday jump! Ries: Good! Knees up, all the way in! You got it. You got it Good! Hands off Big arch. Look for him There you go ♪ Whoo! I think it’s important for Cale and I to, you know, keep growing in some type of relationship, but we just aren’t close enough to the point where we trust each other yet, and that makes me sad We killed it. Nice! We did it! Thank you! Thanks, gang! Whew! Got to get some ice [ Chuckles ] ♪ What the hell is this? Alma: Mike and I had a really good conversation in the car I finally felt like we were in sync That’s — That’s it We were in sync That’s what we felt like Welcome to Area 51 Dreamland You’re gonna learn the unique, crazy sport of land sailing [ Record scratches ] Is it parasailing? I have no idea I’m thinking, like, we’re gonna be up in the sky That’s what I’m thinking now This is the land yacht that you’re gonna operate It’s called Big Mike If you take a look over here, there’s a red flag blowing It’s a marker flag That’s what you’re gonna be aiming for, okay? And this is our imaginary finish line right here with checkered flags and the van pointing into the wind We’re gonna have to communicate We’re gonna have to trust each other and make sure that we know what each other is doing ’cause there are no brakes on these things Let the wind fill the sail up all the way so that you’re almost at the end of your rope You want to straighten the — straight the direction out with your foot there Oh, what? There we go. There we go Okay. Go that way Go that way. You’re going — You’re gonna stop it We’re going to the wind Oh, there you go Oh [bleep] Michael! I got it Straighten out! [Bleep] [bleep] it Okay. [Bleep] Michael, stop! Michael, no! Michael! [ Laughing ] No, Michael, seriously pull the [bleep] thing! This is definitely more of Mike’s world again, so it’s kind of just like, “Oh, geez I’m really gonna have to trust him again, like, a lot.” Oh, my God. Oh, my God Oh, my God. Oh, my God! Whoo! Whoo! Whoo! Good Lord, Michael! [ Chuckles ] Oh, my God. Oh, my God Oh, my God, Michael Bring it in Bring it in, please Oh, my God. Pull it in Are you scared? Yeah! Okay. I’m not doing this with you Do you want to stop this [bleep] thing?! I’m letting the air out Ohhhh Oh, my God, Michael We almost flipped! [ Chuckles ] Michael! Oh! Oh! I got it. We’re good We’re good Humina, humina, humina! Oh, Michael! [ Laughs ] Whoo! Whoo! Whooooooooo! Yeah! Whoo! [ Laughs ] [ Laughs ] Ohh! We get across the finish line, and Mike puckers up and kisses me, and Mike kissing me feels right And I would have given it to you if I felt like it [ Laughs ] But I didn’t Boom, boom, boom Boom, boom, boom I’ve given Mike chance after chance to prove to me that he could hurt me or abandon me, and he hasn’t left my side, not one time Mike is not someone that I would have to fear his intentions or think that he would try to hurt me in any way, and that means a lot, and it says a lot about who he is and what I want in a husband [ Speaking Spanish ] [ Laughs ] You got it! ♪ Jenna: It is important to me that Cale has a good birthday ’cause I know it means a lot to him, so I got a table for us tonight, a nice dinner It’s never my intention to fight or have a confrontation Obviously, me pushing Cale isn’t gonna go anywhere, so I just need to stay on that comfortable surface level, and then, hopefully, he’ll just continue to open up ♪ [ Exhales sharply ] ♪ Oh, hey You look nice Happy birthday. So do you Thank you

Thank you I hope that he realizes that I do care and I am trying So, how do you feel after the trapeze today? That was fun That was really fun I was really impressed with how you got up there and just, like, did that reverse flip It was awesome, right on the first try Thanks, Cale. You, too Whenever I was on top of that platform on the trapeze Yeah I was seriously having some trust issues I had no reason not to trust Heather, though Right She looked like she’s done this for years, and she was fully equipped to handle it Yeah, exactly And she was even saying, like, she’s handled guys much bigger than me Totally I-I think I’ve just been so self-reliant for so long Yeah? that it’s — it’s really hard for me That’s — Yeah Maybe I need to trust people more I’m very similar, Cale Yeah I’ve always been very, very self-reliant I think I had a lot of, like, earlier childhood experiences that just, like, skewed my view on things and, like, made me like, “I don’t need anyone I can do this myself.” Same Who knew how much we had in common? Right? Cale: I-I feel like this is a good conversation, and it’s kind of like, “Gosh, I wish we would have had this on the first day.” You know, why — why couldn’t we have had this back in New York? Um…it feels good to finally have a moment where we’re not at each other’s throats and we’re…listening to each other and finding common ground, which is what we’ve been trying to do all along The things that I thought about myself that were, like, things that I was proud of, you know, are suddenly, like, becoming, like, uh, blatantly obvious that, like, maybe I need to work on these things Cale, are we connecting? Oh, my God Maybe ♪ Look what this is! Oh, you’re full of surprises tonight I-I’m — I am full of them always, Cale You should know that by now [ Laughs ] This is beautiful I truly wish you the happiest birthday ever 30 is a big one Thank you ♪ Cheers to you Thanks Jenna, you’re the best Thank you for spending my birthday with me Even if we’ve had issues, it’s like I hope this was just as fun as a 30th as you would have had with your friends This has actually been really great Good I think this has been, like, our best moment together [ Gasps ] this whole trip ♪ Ahhhhh! ♪ Um, I got you a very tiny gift Jenna! Here. Big bag, little gift Aw, thanks, Jenna ♪ [ Gasps ] It’s a rose quartz, the stone of gentle love It is a stone of love When we were in Sedona Aww I went to the crystal shop and got you one I appreciate you thinking of me Yeah That’s so nice It was the first stone gifted to me It’s one that I found to be most powerful and meaningful in my life I don’t know I hope that it will ♪ I-I don’t know, just have meaning to you, also ♪ That’s very sweet [ Voice breaking ] Thank you very much Of course If I don’t keep it together, I’m gonna start bawling [ Crying ] Yeah You know, I’ve — I’ve felt in the past that Jenna has been rude and — and mean and ill-tempered, but feelings can change Oh, man Yeah And now I’m feeling that Jenna is kind and gentle and fun-loving, and, you know, it’s — it’s interesting how people can surprise you Give me that Cale double high-five Yes! Yeah! What have we been doing, you know? Like, we could have been enjoying each other’s company this entire time, and this could have been every day, you know, and It can be every day, Cale It could be ♪ Cale’s making me feel emotional I’m happy he’s happy Yeah ’cause I know it meant a lot Let’s go win $10 Let’s go And then we can say that we’re walking out of Vegas as winners, then We are, regardless ♪ Stephanie V.: I don’t really know what happened last night I understand that I disrespected him, and I understand that he felt that way, but I don’t agree with how it escalated I think that it could have been handled better ♪ Last night was just, like, kind of, like, “What the hell?” kind of situation I’m not trying to defend, you know, the — the lap dances over and over and over again That’s disrespectful to you I get where you’re coming from I’m not trying to belittle your feelings, so — But I understand where you’re coming from I find it pretty absurd that she’s unable to keep in the front of her mind that, “Oh, what I do may affect him worse than it would affect a normal person who hasn’t had my dating history.” The first thing in my mind

is always the perception of how you’re feeling Got it So that’s where I was very hurt, because it’s like it wasn’t — that wasn’t in the front of your mind You’re right You — You weren’t consciously thinking, “How is this making Ryan feel?” To me, that’s, like, a reflection of your character I…have good character I’m a good girl Actions speak louder than words That’s the only problem ♪ I think I’m most upset that I really had high hopes in this whole process that I was gonna find the one, but when I see her actions do what they do, that’s the sad part When it comes to, like, a relationship or marriage, I have no — There’s no confidence in, like, that at all right now ♪ Just know that I came here for the right reasons, too I wish that I could believe that ♪ Narrator: Next week on “Love At First Flight,” the couples pick up some new passengers Ohhh! Oh, my God! So, I’m Cale’s mom No! Narrator: Will the matches pass the family’s initial screening So, you like her? Uh ♪ Is he marriage material for you? Yeah You guys just met Narrator: or will they be subjected to uncomfortable background checks? You want to tell us what happened in Vegas? Uh I don’t want to talk about Vegas Kind of got a little Yeah It got a little too much ♪

Desarguesian plane | Wikipedia audio article

In mathematics, a projective plane is a geometric structure that extends the concept of a plane In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional “points at infinity” where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces Such embeddability is a consequence of a property known as Desargues’ theorem, not shared by all projective planes == Definition == A projective plane consists of a set of lines, a set of points, and a relation between points and lines called incidence, having the following properties: The second condition means that there are no parallel lines. The last condition excludes the so-called degenerate cases (see below) The term “incidence” is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression “point P is incident with line ℓ ” is used instead of either “P is on ℓ ” or “ℓ passes through P ” == Some examples == === The extended Euclidean plane === To turn the ordinary Euclidean plane into a projective plane proceed as follows: To each set of mutually parallel lines add a single new point. That point is considered incident with each line of this set. The point added is distinct for each such set. These new points are called points at infinity Add a new line, which is considered incident with all the points at infinity (and no other points). This line is called the line at infinity.The extended structure is a projective plane and is called the extended Euclidean plane or the real projective plane. The process outlined above, used to obtain it, is called “projective completion” or projectivization. This plane can also be constructed by starting from R3 viewed as a vector space, see § Vector space construction below === Projective Moulton plane === The points of the Moulton plane are the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, but other lines will remain unchanged. Redefine all the lines with negative slopes so that they look like “bent” lines, meaning that these lines keep their points with negative x-coordinates, but the rest of their points are replaced with the points of the line with the same y-intercept

but twice the slope wherever their x-coordinate is positive The Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the previous example, to obtain the projective Moulton plane. Desargues’ theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane === A finite example === This example has just thirteen points and thirteen lines. We label the points P1,…,P13 and the lines m1,…,m13. The incidence relation (which points are on which lines) can be given by the following incidence matrix. The rows are labelled by the points and the columns are labelled by the lines. A 1 in row i and column j means that the point Pi is on the line mj, while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige-Wexler normal form To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1’s appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1’s appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P1,P4,P5,and P8, for example, will satisfy the third condition. This example is known as the projective plane of order three == Vector space construction == Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each “point” of the real projective plane is the one-dimensional subspace (a geometric line) through the origin in a 3-dimensional vector space, and a “line” in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.Let K be any division ring (skewfield). Let K3 denote the set of all triples x = (x0, x1, x2) of elements of K (a Cartesian product viewed as a vector space). For any nonzero x in K3, the minimal subspace of K3 containing x (which may be visualized as all the vectors in a line through the origin) is the subset { k x : k ∈ K } {\displaystyle \{kx:k\in K\}} of K3. Similarly, let x and y be linearly independent elements of K3, meaning that kx + my = 0 implies that k = m = 0. The minimal subspace of K3 containing x and y (which may be visualized as all the vectors in a plane through the origin) is the subset { k x + m y : k , m ∈ K } {\displaystyle \{kx+my:k,m\in K\}} of K3. This 2-dimensional subspace contains

various 1-dimensional subspaces through the origin that may be obtained by fixing k and m and taking the multiples of the resulting vector. Different choices of k and m that are in the same ratio will give the same line The projective plane over K, denoted PG(2,K) or KP2, has a set of points consisting of all the 1-dimensional subspaces in K3. A subset L of the points of PG(2,K) is a line in PG(2,K) if there exists a 2-dimensional subspace of K3 whose set of 1-dimensional subspaces is exactly L Verifying that this construction produces a projective plane is usually left as a linear algebra exercise An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the set K3 ∖ {(0, 0, 0)} modulo the equivalence relation x ~ kx, for all k in K×.Lines in the projective plane are defined exactly as above The coordinates (x0, x1, x2) of a point in PG(2,K) are called homogeneous coordinates Each triple (x0, x1, x2) represents a well-defined point in PG(2,K), except for the triple (0, 0, 0), which represents no point. Each point in PG(2,K), however, is represented by many triples If K is a topological space, then KP2, inherits a topology via the product, subspace, and quotient topologies === Classical examples === The real projective plane RP2, arises when K is taken to be the real numbers, R. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.In this construction consider the unit sphere centered at the origin in R3. Each of the R3 lines in this construction intersects the sphere at two antipodal points. Since the R3 line represents a point of RP2, we will obtain the same model of RP2 by identifying the antipodal points of the sphere. The lines of RP2 will be the great circles of the sphere after this identification of antipodal points This description gives the standard model of elliptic geometry The complex projective plane CP2, arises when K is taken to be the complex numbers, C. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields (known as pappian planes) serve as fundamental examples in algebraic geometry.The quaternionic projective plane HP2 is also of independent interest === Finite field planes === By Wedderburn’s Theorem, a finite division ring must be commutative and so a field. Thus, the finite examples of this construction are known as “field planes”. Taking K to be the finite field of q = pn elements with prime p produces a projective plane of q2 + q + 1 points. The field planes are usually denoted by PG(2,q) where PG stands for projective geometry, the “2” is the dimension and q is called the order of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2,2). The third example above is the projective plane PG(2,3) The Fano plane is the projective plane arising from the field of two elements. It is the

smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small black balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the “lines” and the line segments and circle to be the “points” – this is an example of duality in the projective plane: if the lines and points are interchanged, the result is still a projective plane (see below). A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3,2) = PGL(3,2)) has 168 elements === Desargues’ theorem and Desarguesian planes === The theorem of Desargues is universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield as above. These planes are called Desarguesian planes, named after Girard Desargues. The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are called non-Desarguesian planes, and the Moulton plane given above is an example of one. The PG(2,K) notation is reserved for the Desarguesian planes. When K is a field, a very common case, they are also known as field planes and if the field is a finite field they can be called Galois planes == Subplanes == A subplane of a projective plane is a subset of the points of the plane which themselves form a projective plane with the same incidence relations (Bruck 1955) proves the following theorem Let Π be a finite projective plane of order N with a proper subplane Π0 of order M. Then either N = M2 or N ≥ M2 + M When N is a square, subplanes of order √N are called Baer subplanes. Every point of the plane lies on a line of a Baer subplane and every line of the plane contains a point of the Baer subplane In the finite Desarguesian planes PG(2,pn), the subplanes have orders which are the orders of the subfields of the finite field GF(pn), that is, pi where i is a divisor of n. In non-Desarguesian planes however, Bruck’s theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of order M in a plane of order N with M2 + M = N is an open question. If such subplanes existed there would be projective planes of composite (non-prime power) order === Fano subplanes === A Fano subplane is a subplane isomorphic to PG(2,2), the unique projective plane of order 2 If you consider a quadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called the diagonal

points of the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle) The name Fano for this subplane is really a misnomer. Gino Fano (1871–1952), in developing a new set of axioms for Euclidean geometry, took as an axiom that the diagonal points of any quadrangle are never collinear. This is called Fano’s Axiom. A Fano subplane however violates Fano’s Axiom. They really should be called Anti-Fano subplanes, but this name change has not had many supporters In finite desarguesian planes, PG(2,q), Fano subplanes exist if and only if q is even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for (in both odd and even orders) An open question is: Does every non-desarguesian plane contain a Fano subplane? A theorem concerning Fano subplanes due to (Gleason 1956) is: If every quadrangle in a finite projective plane has collinear diagonal points, then the plane is desarguesian (of even order) == Affine planes == Projectivization of the Euclidean plane produced the real projective plane. The inverse operation — starting with a projective plane, remove one line and all the points incident with that line — produces an affine plane === Definition === More formally an affine plane consists of a set of lines and a set of points, and a relation between points and lines called incidence, having the following properties: The second condition means that there are parallel lines and is known as Playfair’s axiom. The expression “does not meet” in this condition is shorthand for “there does not exist a point incident with both lines.” The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The order of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2,q) are denoted by AG(2,q) There is a projective plane of order N if and only if there is an affine plane of order N. When there is only one affine plane of order N there is only one projective plane of order N, but the converse is not true The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane. These statements hold for infinite projective planes as well === Construction of projective planes from affine planes === The affine plane K2 over K embeds into KP2 via the map which sends affine (non-homogeneous) coordinates to homogeneous coordinates, ( x 1 , x 2 ) ↦ ( 1 , x

1 , x 2 ) {\displaystyle (x_{1},x_{2})\mapsto (1,x_{1},x_{2}).} The complement of the image is the set of points of the form (0, x1, x2). From the point of view of the embedding just given, these points are the points at infinity. They constitute a line in KP2 — namely, the line arising from the plane { k ( 0 , 0 , 1 ) + m ( 0 , 1 , 0 ) : k , m ∈ K } {\displaystyle \{k(0,0,1)+m(0,1,0):k,m\in K\}} in K3 — called the line at infinity. The points at infinity are the “extra” points where parallel lines intersect in the construction of the extended real plane; the point (0, x1, x2) is where all lines of slope x2 / x1 intersect. Consider for example the two lines u = { ( x , 0 ) : x ∈ K } {\displaystyle u=\{(x,0):x\in K\}} y = { ( x , 1 ) : x ∈ K } {\displaystyle y=\{(x,1):x\in K\}} in the affine plane K2. These lines have slope 0 and do not intersect. They can be regarded as subsets of KP2 via the embedding above, but these subsets are not lines in KP2. Add the point (0, 1, 0) to each subset; that is, let u ¯ = { ( 1 , x , 0 ) : x ∈ K } ∪ { ( 0 , 1 , 0 ) } {\displaystyle {\bar {u}}=\{(1,x,0):x\in K\}\cup \{(0,1,0)\}} y ¯ = { ( 1 , x , 1 ) : x ∈ K } ∪ { ( 0 , 1 , 0 ) } {\displaystyle {\bar {y}}=\{(1,x,1):x\in K\}\cup \{(0,1,0)\}} These are lines in KP2; ū arises from the plane { k ( 1 , 0 , 0 ) + m ( 0 , 1 , 0 ) : k , m ∈ K } {\displaystyle \{k(1,0,0)+m(0,1,0):k,m\in K\}} in K3, while ȳ arises from the plane k ( 1 , 0 , 1 ) + m ( 0 , 1 , 0 ) : k , m ∈ K {\displaystyle {k(1,0,1)+m(0,1,0):k,m\in K}.} The projective lines ū and ȳ intersect at (0, 1, 0). In fact, all lines in K2 of slope 0, when projectivized in this manner, intersect

at (0, 1, 0) in KP2 The embedding of K2 into KP2 given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding ( x 1 , x 2 ) → ( x 2 , 1 , x 1 ) , {\displaystyle (x_{1},x_{2})\to (x_{2},1,x_{1}),} has as its complement those points of the form (x0, 0, x2), which are then regarded as points at infinity When an affine plane does not have the form of K2 with K a division ring, it can still be embedded in a projective plane, but the construction used above does not work. A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general “algebra” === Generalized coordinates === One can construct a coordinate “ring”—a so-called planar ternary ring (not a genuine ring)—corresponding to any projective plane A planar ternary ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring. They are called non-Desarguesian projective planes and are an active area of research The Cayley plane (OP2), a projective plane over the octonions, is one of these because the octonions do not form a division ring.Conversely, given a planar ternary ring (R,T), a projective plane can be constructed (see below). The relationship is not one to one. A projective plane may be associated with several non-isomorphic planar ternary rings. The ternary operator T can be used to produce two binary operators on the set R, by: a + b = T(a,1,b), and a • b = T(a,b,0).The ternary operator is linear if T(x,m,k) = x•m + k. When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane For example, Desargues’ theorem corresponds to the coordinate ring being obtained from a division ring, while Pappus’s theorem corresponds to this ring being obtained from a commutative field. A projective plane satisfying Pappus’s theorem universally is called a Pappian plane Alternative, not necessarily associative, division algebras like the octonions correspond to Moufang planes There is no known purely geometric proof of the purely geometric statement that Desargues’ theorem implies Pappus’ theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring and Wedderburn’s theorem that finite division rings must be commutative; Bamberg & Penttila (2015) give a proof that uses only more “elementary” algebraic facts about division rings To describe a finite projective plane of order N(≥ 2) using non-homogeneous coordinates and a planar ternary ring:

Let one point be labelled (∞) Label N points, (r) where r = 0, …, (N − 1) Label N2 points, (r, c) where r, c = 0, …, (N − 1).On these points, construct the following lines: One line [∞] = { (∞), (0), …, (N − 1)} N lines [c] = {(∞), (c,0), …, (c, N − 1)}, where c = 0, …, (N − 1) N2 lines [r, c] = {(r) and the points (x, T(x,r,c)) }, where x, r, c = 0, …, (N − 1) and T is the ternary operator of the planar ternary ring.For example, for N=2 we can use the symbols {0,1} associated with the finite field of order 2. The ternary operation defined by T(x,m,k) = xm + k with the operations on the right being the multiplication and addition in the field yields the following: One line [∞] = { (∞), (0), (1)}, 2 lines [c] = {(∞), (c,0), (c,1) : c = 0, 1}, [0] = {(∞), (0,0), (0,1) } [1] = {(∞), (1,0), (1,1) } 4 lines [r, c]: (r) and the points (i,ir + c), where i = 0, 1 : r, c = 0, 1 [0,0]: {(0), (0,0), (1,0) } [0,1]: {(0), (0,1), (1,1) } [1,0]: {(1), (0,0), (1,1) } [1,1]: {(1), (0,1), (1,0) } == Degenerate planes == Degenerate planes do not fulfill the third condition in the definition of a projective plane. They are not structurally complex enough to be interesting in their own right, but from time to time they arise as special cases in general arguments. There are seven degenerate planes according to (Albert & Sandler 1968) They are: the empty set; a single point, no lines; a single line, no points; a single point, a collection of lines, the point is incident with all of the lines; a single line, a collection of points, the points are all incident with the line; a point P incident with a line m, an arbitrary collection of lines all incident with P and an arbitrary collection of points all incident with m; a point P not incident with a line m, an arbitrary (can be empty) collection of lines all incident with P and all the points of intersection of these lines with m.These seven cases are not independent, the fourth and fifth can be considered as special cases of the sixth, while the second and third are special cases of the fourth and fifth respectively. The special case of the seventh plane with no additional lines can be seen as an eighth plane. All the cases can therefore be organized into two families of degenerate planes as follows (this representation is for finite degenerate planes, but may be extended to infinite ones in a natural way): 1) For any number of points P1, …, Pn, and lines L1, …, Lm, L1 = { P1, P2, …, Pn} L2 = { P1 }

L3 = { P1 } Lm = { P1 }2) For any number of points P1, , Pn, and lines L1, …, Ln, (same number of points as lines) L1 = { P2, P3, …, Pn } L2 = { P1, P2 } L3 = { P1, P3 } Ln = { P1, Pn } == Collineations == A collineation of a projective plane is a bijective map of the plane to itself which maps points to points and lines to lines that preserves incidence, meaning that if σ is a bijection and point P is on line m, then Pσ is on mσ.If σ is a collineation of a projective plane, a point P with P = Pσ is called a fixed point of σ, and a line m with m = mσ is called a fixed line of σ. The points on a fixed line need not be fixed points, their images under σ are just constrained to lie on this line. The collection of fixed points and fixed lines of a collineation form a closed configuration, which is a system of points and lines that satisfy the first two but not necessarily the third condition in the definition of a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or a degenerate plane Collineations whose fixed structure forms a plane are called planar collineations === Homography === A homography (or projective transformation) of PG(2,K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space. Using homogeneous coordinates they can be represented by invertible 3 × 3 matrices over K which act on the points of PG(2,K) by y = M xT, where x and y are points in K3 (vectors) and M is an invertible 3 × 3 matrix over K. Two matrices represent the same projective transformation if one is a constant multiple of the other. Thus the group of projective transformations is the quotient of the general linear group by the scalar matrices called the projective linear group Another type of collineation of PG(2,K) is induced by any automorphism of K, these are called automorphic collineations. If α is an automorphism of K, then the collineation given by (x0,x1,x2) → (x0α,x1α,x2α) is an automorphic collineation. The fundamental theorem of projective geometry says that all the collineations of PG(2,K) are compositions of homographies and automorphic collineations Automorphic collineations are planar collineations == Plane duality == A projective plane is defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines. As P and L are only sets one can interchange their roles and define a plane dual structure By interchanging the role of “points” and “lines” in C = (P,L,I)we obtain the dual structure C* = (L,P,I*),where I* is the inverse relation of I In a projective plane a statement involving points, lines and incidence between them that

is obtained from another such statement by interchanging the words “point” and “line” and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of “Two points are on a unique line.” is “Two lines meet at a unique point.” Forming the plane dual of a statement is known as dualizing the statement If a statement is true in a projective plane C, then the plane dual of that statement must be true in the dual plane C*. This follows since dualizing each statement in the proof “in C” gives a statement of the proof “in C*.” In the projective plane C, it can be shown that there exist four lines, no three of which are concurrent. Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure C* is also a projective plane, called the dual plane of C If C and C* are isomorphic, then C is called self-dual. The projective planes PG(2,K) for any division ring K are self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes The Principle of Plane Duality says that dualizing any theorem in a self-dual projective plane C produces another theorem valid in C == Correlations == A duality is a map from a projective plane C = (P, L, I) to its dual plane C* = (L, P, I*) (see above) which preserves incidence That is, a duality σ will map points to lines and lines to points (Pσ = L and Lσ = P) in such a way that if a point Q is on a line m (denoted by Q I m) then Qσ I* mσ ⇔ mσ I Qσ. A duality which is an isomorphism is called a correlation. If a correlation exists then the projective plane C is self-dual In the special case that the projective plane is of the PG(2,K) type, with K a division ring, a duality is called a reciprocity. These planes are always self-dual. By the fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography. If the automorphism involved is the identity, then the reciprocity is called a projective correlation A correlation of order two (an involution) is called a polarity. If a correlation φ is not a polarity then φ2 is a nontrivial collineation == Finite projective planes == It can be shown that a projective plane has the same number of lines as it has points (infinite or finite). Thus, for every finite projective plane there is an integer N ≥ 2 such that the plane has N2 + N + 1 points, N2 + N + 1 lines, N + 1 points on each line, and N + 1 lines through each point.The number N is called the order of the projective plane The projective plane of order 2 is called the Fano plane. See also the article on finite geometry

Using the vector space construction with finite fields there exists a projective plane of order N = pn, for each prime power pn. In fact, for all known finite projective planes, the order N is a prime power The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is the Bruck-Ryser-Chowla theorem that if the order N is congruent to 1 or 2 mod 4, it must be the sum of two squares. This rules out N = 6. The next case N = 10 has been ruled out by massive computer calculations. Nothing more is known; in particular, the question of whether there exists a finite projective plane of order N = 12 is still open Another longstanding open problem is whether there exist finite projective planes of prime order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order) A projective plane of order N is a Steiner S(2, N + 1, N2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes The number of mutually orthogonal Latin squares of order N is at most N − 1. N − 1 exist if and only if there is a projective plane of order N While the classification of all projective planes is far from complete, results are known for small orders: 2 : all isomorphic to PG(2,2) 3 : all isomorphic to PG(2,3) 4 : all isomorphic to PG(2,4) 5 : all isomorphic to PG(2,5) 6 : impossible as the order of a projective plane, proved by Tarry who showed that Euler’s thirty-six officers problem has no solution However, the connection between these problems was not known until Bose proved it in 1938 7 : all isomorphic to PG(2,7) 8 : all isomorphic to PG(2,8) 9 : PG(2,9), and three more different (non-isomorphic) non-Desarguesian planes. (All described in (Room & Kirkpatrick 1971)) 10 : impossible as an order of a projective plane, proved by heavy computer calculation 11 : at least PG(2,11), others are not known but possible 12 : it is conjectured to be impossible as an order of a projective plane == Projective planes in higher-dimensional projective spaces == Projective planes may be thought of as projective geometries of “geometric” dimension two. Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane. These turn out to be “tamer” than the projective planes since the extra degrees of freedom permit Desargues’ theorem to be proved geometrically in the higher-dimensional geometry. This means that the coordinate “ring” associated to the geometry must be a division ring (skewfield) K, and the projective geometry is isomorphic to the one constructed from the vector space Kd+1, i.e. PG(d,K). As in the construction given earlier, the points of the d-dimensional projective space PG(d,K) are the lines through the origin in Kd + 1 and a line in PG(d,K) corresponds to a plane through the origin in Kd + 1. In fact, each i-dimensional object in PG(d,K), with i < d, is an (i + 1)-dimensional (algebraic) vector subspace of Kd + 1 ("goes through the origin"). The projective spaces in turn generalize to the Grassmannian spaces It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes

that are contained in that space. Since there are projective planes in which Desargues’ theorem fails (non-Desarguesian planes), these planes can not be embedded in a higher-dimensional projective space. Only the planes from the vector space construction PG(2,K) can appear in projective spaces of higher dimension Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two == See also == Block design – a generalization of a finite projective plane Combinatorial design Incidence structure Projective geometry Non-Desarguesian plane Smooth projective plane VC dimension of a finite projective plane == Notes