## Flight Night: Israël Campain Day 1 TASMO

zo doe uw bril weer omhoog kwam jawel jawel klein jointje mobile hoe lang de wereld hoogspannings rl stine kreeg naast verf ook wel heel lang de wereld houdt van heel lang voor leven voor god

dus gerrie productie cobra heel veel power echt goede roman stel ik voor de duivel bal op advocaat als ik te kunnen geven doe geurolie

oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi goede zo moeilijk is x-two lekker fit de cellen vlucht yo vince toen viel cm chapman context over klok in de maas ja sha-la-li sha-la-la ik spreek my ride a ons academic ontdek uw stem

jamie lie sjala lie echt een verhaal dat het vredig het is de big bang eventueel zou al lang had hij gevangen afval weg nee blijf en mijn pink spijt [Muziek] [Muziek] vervanging [Muziek] was look ik ga voor twee de toe scheppen ja oké wat ty lang by everyone nee op spreekt eigenlijk nog dik sharla het eigenlijk spreek bluft de times streak denk ik legt ja dit zijn schip check elk object op ik bang een splash en hadden van sha-la-li sha-la-la

thatcher clare met zou ik wil het jullie nu zo toe ja ik zit een beetje is heel mooi heel mooi eventjes vieren f16 zwijgen white zijn toch niet in misselijk zelf de makkelijkste trends toe de context at stip point to op stip point youtube context bood ja pak jij de ligger toe lees jij bent page charlotte was fictie met go to lie sjala lie patcher claire hi hi hi hi

hoi ollie boy oh boy oh boy oh boy oh boy oh waarom zij diy je krishna schok all you wat stil je vind je niet op optie reden om atmosfeer dus erg splash ja klopt ik zag zijn daar niet op zien wat voor korra op waar we waren moest ik hem opzetten a copy niet om een tweede kans op impact in die van mij wel maar ik hoor mezelf manier geprikkeld ons terecht want toen was ik niet wim hof is toen hij ik zou niet doen small arms christ maybe jai lai lai la la la nature claire om het job is naar alles hoi ja is les toe ja hoe the bottom of the sea door die pot jam of the theme ebitda

sha-la-li sha-la-la en turismo tot nu toe voor 5 miles kwaliteit hoe ze dat lucht oh sorry miles miles to tell ie gek de lucht ara is niet eenvoudig de fights voor nu no pages to we tellen het is een gadget is uw em all star van de klok bieden succes housing beneden hulp iets eten talia metaal verder precies en telling enkele pcm en het is ongeveer 60 maal dus we moeten moet het kunnen overleven ja een beetje staat hotel ik ja we zijn er al zeker

ik ben a hoi mayday mayday er een dag of twee miles maar ik wil om naar te bidden nep when you’re lending duidelijk je kunt ook de stream bekijken nee en ik ben dood hij kan lopen telefoon kijken altijd we ja hoor je zelf wel maar ja duurloop [Muziek] ja dat waren geen liever bootjes die schafte terug naar teller hij ik had het eigenlijk wel ik gun ik zag toen ik die patriot dan checken was toen kwam de wel en dan een boot symbooltje ook op mijn had dus eigenlijk had ik kunnen weten dat het dangerous was en ik zie hem nou ook op me harder your maar is wel nasty wat hij goed dus niet op het goede moment gooi je die ze eerder dan hoe heet ie beluistering maar youtube chino’s gewoon rap komt 97 vol zorg centric leven ik zou link sturen op wordt sjala lie aanraakscherm heb hem al trek ik de wat is er [Muziek] [Muziek] ik zie geen afspeellijsten kanalen waar aan [Muziek] zich een livestream een probleem wat ik ook al we kijken ik geen raad staat er knap nee ik kijk hier op de youtube mijn eigen streamers heel lullig dat ik lui ben maar wat heb ik heb hier een kopie pers per view heb ik nog weinig aan het whatsapp ja mag dat ik een window moord is gelukkig kan hem altijd [Muziek] het is het engel incubator weg ja ja dat hebben we nog weinig aan werk is nee ik heb een nieuwe subscribers video’s of scriptuur channel ah koekoek voor wortel deel hangend aan de show ja dan gaan er om een geduren ding is gekkenwerk ja hoe leuk kijken kom mezelf nu ook tegen een zeer goedkope

[Muziek] oudste bal vertragingen ja zeker maar je moet er wel mee uitkijken want maar deze de deze samen die blijft in repeat eigenlijk is en waar jouw geluid via telefoon die komt dan weer industry en dan wordt het via wifi de stream wat geluid uitgezonderd en blijf silo groeien maar hij zeer infinite guido’s en nee precies [Muziek] bosstraat [Muziek] na ik heb zich reeds koken oh boy oh boy oh boy oh boy oh boy oh boy oh boy oh boy oh boy oh af job search feature to launch a boy and girl in a major op de piemel high-pass filter jai lai lai la jai lai lai la jai lai lai la miles hoi terrasje te pakken todesco toe grigio with de geur die miles wist dat de fielt in maat van over inbreker en wie won voor oh zo zo zo zo zo veel essentieel voor 1019q van jonas en werken ring ze had haar man heel lang voor lef oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oké hier heel en hier ook research and future there more i hope you will enjoy

my life or less mackenbach door te fancy oud keer was ik pull up hi hi hi hi hi hi maar dan geeft nu volgens mij deed er naar mij jij bent alles wat het boseind zoiets waarvan het wanneer miles pak bergen maar ik zie niks op de radar nou als jij gaat landen dan vraag ik alvast een nieuwe schroef rol over drie kyouma hij is zo groot ja de roppongi pongau nou dat zijn dat zijn chokers [Muziek] maar waarom zit je dan pilot sport een ja dat zal ik zijn natuurlijk weer net gesneden het water gehad doordat zijn apaches ze dan is er één die spritesheet nee egyptische in mij heb wat waar hoe hard je wat zo in een showroom worden voor nou weer tot leven [Muziek] [Muziek] mooi oké even veel herrie en kamer werd verlaten bedenken hoe ik beter dan gaat de lasten goede ja dat zeg je goed welkom bij open deuren intrappen wonnen we wat welkom bij open deuren intrappen koud geworden welk ze zou kunnen hebben wordt dan lowlevel flights [Muziek] we manpad zijn altijd gaan we alleen nog niet echt wat hij ons een [Muziek] 100 sessies op tikken oh wel heel hoog van mijn wang voor led wat een wat ik me kan bedenken is iets met de lava doen waar ik het even niet meer wat de zesde typte van de meeuwen in deze heel moeilijk hoeft het niet te zijn bagger [Muziek] oh een fijn tank zodat ik dat is leuk [Muziek] morning oké veel meer volume van uw iphone app sorteert lang ik heb echter geen idee hoe je zo’n pages er naar het kanaal

[Muziek] paula van de passer cobra heel veel power echt goede roman you feel a very young kamer het is een oud [Muziek] john green power je wat meer voor ook al heel lang voor jij hoe dat is wel leuk al zijn we had to one one druk op ziet zoiets van achter 16 zitten nummer 8 en 15 zeer te laat zijn voor chips maar [Muziek] slot tip h [Muziek] [Muziek] er is meer

## Flight Report: BHX-DXB EK038

Our Ride for today A6-ENO Delivered 24-02-2014

## American Airlines Flight 965 | Wikipedia audio article

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&#39;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

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

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 ♪

## (X-Plane 11) Full flight- Kent State to Portage Co.- Cessna 172 Skyhawk

*garage door opening*

He sounds like he just woke up

## 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