The Pizza Theorem
ฝัง
- เผยแพร่เมื่อ 26 ก.ย. 2024
- We’re all familiar with standard pizza slices - one center slice and two 45-degree cuts and you’re ready to party with 6 perfect slices. Throw in another cut and you’ve got 8.
But what if I told you that you can cut a pizza anywhere on its surface -- not just across its diameter -- and get a perfectly fair, shareable pizza for two people by assigning alternating slices? Prepare to witness the most delicious, unexpected presentation of mathematical beauty: a proof without words. And bring napkins.
** LINKS **
The Pizza Theorem
web.maths.unsw....
Mathematics Magazine, Volume 41, 1968: "Problem 660" by L.J. Upton, Solution by Michael Goldberg: www.jstor.org/...
Mathematics Magazine, Volume 67, 1994. "Proof without Words: Fair Allocation of a Pizza" by Larry Carter and Stan Wagon: www.jstor.org/...
Dividing Thin Crust, Calzones, and Stuffed Crust Pizzas:
lsusmath.rickm...
Exotic Mathematical Pizza Slicing: www.newscienti...
Infinite Families Of Monohedral Disk Tilings: arxiv.org/pdf/...
Pizza Theorem Proof Without Words Calculator by Christian Lawson-Perfect: www.geogebra.o...
***********************************
Vsauce Links
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Hosted, Produced, And Edited by Kevin Lieber
Instagram: / kevlieber
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Website: kevinlieber.com
Research And Writing by Matthew Tabor
/ matthewktabor
VFX By Eric Langlay
/ ericlanglay
Huge Thanks Paula Lieber
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Select Music By Jake Chudnow: / jakechudnow
The book is perfectly fine. No damage whatsoever. Although it tastes delicious now...
you know I was just about to comment that. First like, thanks for the info
NO YOU MONSTER!!!
I was about to flip
No u
Hey there! How did you find yourself a part of this vsauce thing? I'm really curious. Maybe you have made a video about it? But if not, or, either way, I'd love to know about it. Thanks for everything. brotha!
can't wait to hand someone a monohedral disk tile of pizza at my next party
TierZoo um tierzoo amazing.....
TIERZOO SENPAI!
Woah when did you get here
MrAubery This video was about cutting pizza slices. What exactly do your ridiculous conspiracy theories have to do with it?
MrAubery The earth isn't flat, you dummy. It's just a fake conspiracy made up by the government to figure out who's actually onto them and who's just pretending. You're in the latter group.
My lifespan shortened significantly when you placed the book on the pizza
I stopped the video in that exact moment
When you thought pineapple was the worst thing o put on a pizza 😂
Same
I came here for this comment.
I'll have nightmares about it tonight...
First Hawaii with the pineapple pizza, then San Francisco with the broccoli pizza, and now vsauce with the book pizza
Clark Good XD
Actually Hawaiian pizza originated in Canada
@@isabellewolvetang66 it originated in hell
Clark Good omg I wanna try a broccoli pizza
Ehm... I'm italian, and "broccoli e salsiccia" it's a classic combination of toppings here. It's actually very good
Puts book on pizza:
Pizza Lovers: HERESY!
Book Lovers: HERESY!
_Everyone Disliked That_
Chaotic neutral: why not?
I put mayo, fish, and pineapple on my pizza...
@@borisborisvichborislav4225 I'm sorry, but I had to report you for putting lettuce on your pizza. You shd be ashamed of urself
paper eaters: ( ͡° ͜ʖ ͡°)
It hurts to see the book on the Pizza :P
I felt the same, man! It hurt so much! I love books and he's making it all greasy! So disrespectful.
I actually felt it
I'm not sure if the soiled book or the ruined pizza made me more upset...
Ben V ditto
Man i was hurt
That moment when people complain about pineapple on pizza while there are people who put books on pizza ...
People who complain about pineapple pizza never tried it
hawaiian pizza is life, but pizza with books on them, no thanks
Chef Gordon Ramsay doesn't like Hawaiian Pizza... Its his loss.
KiemPlant Don't forget about the alphabet letters!
I immediately went to the comment section when he did
Friend: What's your favorite pizza topping?
Me: Magnetic letters and math
Made me go haha
dont forget the book!
Science go brrrrrr
@@kcldnx3485 and the Green and yellow peppers!
Probably better than pineaple
Comment section :
10% Actual video
90% That book on the Pizza.
amy davy 10% video, 45% book on pizza, 45% the way he sliced them in the beginning.
BUT WHAT ABOUT THE PIZZA SLICE HE GRABBED FROM UNDER THE TABLE NO ONE IS TALKING ABOUT THIS
I just died on the inside when he placed
*THE BOOK ON THE PIZZA*
Kioerrr I thought I was the only one 😂😂😂
I was fine. It was interesting. I was living my life and then I witnessed with my own two eyes this man place a BOOK on a PIZZA and now
ikr
Proof that vsauce is run by sociopaths
when he did that, it gave me ebola.
Or you can just get more pizza.
Or twelve.
Just in case.
Or more pizza for me
@@_rr_-qm5og
bUt I kNoW sOmEtHiNg FuNnIeR tHaN 24....
25!!!
U kno wat iz funnier than 25 and a half
25 and a 1/3
Expert gig
2:00 Him: Puts book on pizza
Me: Screams on the inside
Memento mori
what is a book?
@@stoppled what is life?
@@magnuslkke3564 i dont know i think it is a subscribtion or something?
@@stoppled no, that would be Lyfe.
The person who isn't as hungry? You mean the person who didn't have the chance to pick up a third slice and is now depressed?
Daniel lol I'm fine with 2 slices, have at it 😉
There are plenty of people on a "diet"
He means the guy who had his own entire pizza to himself before and now just takes slices off your pizza because it's pizza.
Nope? No, I promise you less-hungry people really do exist in the world.
I live in a house of 3. I sometimes only eat one slice never more than 2. I literally feel like I will explode if I eat another bite after 2 slices.
*Me* : Um yes hello is this Domino's?
*Domino's* : Yes.
*Me* : Hi, I'd like a Pepperoni pizza.
*Domino's* : Is that all?
*Me* : Yeah, but can you cut it into Monohedral disk tiles?
*Domino's* : ~Hangs up~
Domino's : Sir/Ma'am we only serve REAL pizza.
Domino's : Monohedral Disk pizza is a legend, in fact. Because of how LONG it takes to cut all this for just a few people.. If you want it, it's worth 2.4 Million U.S dollars
And 1 cent.
Uñkøwn Dãißy шнат??? І dфит наvе а сеит...
@@saraahmed5072 then I guess you cant get it unless you pay 1 dollar more
Yes, and may I have the new book topping please?
@@Felandian we ran outta books... And we have to pay 10 dollars because we were supposed to give that book to the library a long time ago...
Due on 1998
i cringed harder than i should have when the book touched the pizza, just why??
@@zenzeb
You don't know anything about grammar, girl.*
1:46 that’s how I cut my pizza everytime...
Wow
Are you ok?
Same why wouldn’t you?
Really, or am I getting whoooshed
@@adeliew4230 XD R/WOSHOOOOOOOOOOOOOOO LOLLL XDDD I AM REDIT XDDDDDDD R/WOOOOSH R/MEMES R/JOKES R/FFFFFFFFFFUUUUUUUUU R/IHAVEREDDIT XD IM HAVE REDDIT
Or I could just eat the whole pizza
You can't stop me!
Alexis Pallette same
Most reasonable person hear
Or he could have just ordered 2 pizzas
Alexis Pallette I once ate an entire large pizza due to peer pressure, this is my story...
But that doesn't make sense because you wouldn't get the right crust to pizza ratio.
Thank you.
No one eats that anyway
Exactly what I thought
Haha who eats the crust
The people in the replies are sociopaths
I wonder if I was the only one who cringed when he *PLACED THE BOOK ON THE PIZZA* jesus christ
It is a three day old pizza by his own admission and it was probably room temperature given that the cheese was not all melted at worst a little grease probably got on the book cover.
thank you! i hope he will never do it again...
I’m sorry I take book hygiene very seriously
Lucas Preti i cringed at the fact that i wasn’t the only one cringing about the book on the pizza
That grease will never come off of the book now
The second pizza theorem states that a pizza of radius z and thickness a has a volume of pi . z . z . a
A Friend of Vsauce: hey our pizza slices aren't equal
Vsauce: Hold my beer
Zamista X *hold my book or just put it on the pizza
Vsauce: hold my V
Guy: so you are just sauce now?
Sauce: yeah
Guy: THATS AWESOME SAUCE
Hold my protractor
Nobody understands that *WE* are Vsauce.
wrong its hold my slice
Genius cuts pizza into perfect slices and then people just argue about how much crust they get
Don't you think it's dumb to post the same comment twice under the same video only slightly different?
Who else immediately checked the comments as soon as he put that book on the pizza?
ACE 777 so true😂
😂😂😂😂😂
Yup
ACE 777 me 100%
Me I thought I was the only one who noticed
Michael: i got pizza
Jake: they arent cut equally
Kevin: *Yes they are*
But what says joe?
lil bendY r/woooosh
lil bendY you don’t get the joke he meant Vsauce 1, 2, and 3 years
@lil bendY: Your Mom
lil bendY who the hell is joe
I work at dominos, i should cut someones pizza like that one day
if by any chance you cut mine like that I'll tip you big.
i wouldn't count on it tho, i almost never eat pizza, let alone Domino's and we probably live in different countries.
Sam Running Don't put a book on it though. Angry customers..
If you *wok* at Domino's expect to get fired soon, because I think you misunderstood how to cook and prepare a pizza 🤔
Sam Running YES YOU SHOULD DEFINITELY DO IT
Dew it
Why would you out that book on the pizza, that actually annoyed me lmao
It triggered me how he put the book on the Pizza
Yeah
I was just about to type that.
2:00
Him: puts book on pizza
*Me: CONFUSED SCREAMING*
You put the book on the pizza . Silly, learnings not a topping.
But learners are toppers
-_- learning is the only topping in this house, now go eat your math homework, no and's if's or buts about it.
Its the only way to absorb the learnings.
But it could be tho
That pizza has tomato sauce on it
I wonder if it’s the special v sauce
lmao 😂
I wanna punch you realy bad!
Yeah, the "v" sauce, eh?
Kiwi_ Dont_fly There is a door over there and I suggest that you use it.
thats better than tomato sauce
Rip whoever gets slice *_E_*
TΛBBY lmao
No
Maybe
^ Above me are all the answers ^
Can we take a moment to appreciate that he has probably the coolest pizza-slicer ever
If you call the radius "z" and the hight "a" the volume of a Pizza is pizza
you opened my eyes
What the actual hecc
For anyone who needs an explanation:
The volume of a cylinder (the pizza is a cylinder. Any discs with height are.) is pi * radius squared * height, or pir^2h.
r^2 can be written as rr instead, or pirrh. Now, if we replace the r with z and h with a....
pi zz a. Pizza.
I prefer this use of math over calculus any day.
Lmao
Brendan Risney Going to nod like I understand...
2:00 How dare you stain that book???
The pizza is like 3 days old
THANK YOU! That was all I could think too. "GET THAT BOOK OFF THE PIZZA YOU SAVAGE!"
That was the moment where i thought, that the pizza must be fake, but at the end he said its 3 days old, such old pizza might still stain the book!
Couldn't focus on anything after that tbh Oo
How dare him stain that pizza
Well I guess i'm not the only one that died inside when he place *THAT BOOK ON THE PIZZA*
_G O D D A M M I T K E V I N_
His name is Kevin.
@@williamsledge3151 thx for that
placed*
*When he put the book on pizza I went straight to the comments*
Him laying the book on top of the pizza really disturbed me
Steven Nagle I was just about to comment that lol
Jfran2013 why everyone had commented this already
Ze Kraken you didn't... Checkmate
Jfran2013 brah that book on the pizza is pretty damn gross man
Please do not put your book on a pizza.
Pause at 4:06 You need to cut the pizza outside of the pizza.
eyewars s
you also need to cut B inside of C, and D inside of B???
bamboozled
*cuts the table
gotta just
*_-NEGATIVE-_* *ŠÜBTRĀÇT*
Explanation please!
The human brain after drug abuse:
4:48
But is there pizza grease on the pages of the book... big, greasy fingerprints... spreading thru the pages... leaking onto the next page... seeping... seeping thru the book... attracting mites... mites that nibble at the grease... nibble at the page with grease... slowly nibbling... nibbling...
eek
Are you ok? Do you need help?
I feel like ive heard this before
Lol u tried to make it dramatic but u just look like a downer
I divide a pizza between my friends and I by eating all the pizza and letting them decide how to split the cheese stuck to the box.
My concern with this is that people would complain about crust distribution. This was a perennial pizza argument that happened in my home as a kid whenever the shop cut pizza into squares, and probably the reason why I now like the crust better than inside pieces - there were never any inside pieces left.
Lol I don't like inside pieces cuz there's nothing to hold onto
they’re all inside pieces, only some are outside pieces also.
1:01 If anyone is wondering; it says "what's wrong with this person"
Actually its: *what's wrong with this person?
You didn't put an asterisk at the beginning or a question mark at the end
@Gab Bouzen wtf how is that a wooosh the original comment wasn't even a joke
@Gab Bouzen r/thatisnothowwhooooshesworkbuddy
@@Ziegenhainy r/SubsIFellFor
@@edgyvalen4470 r/breadstapledtotrees
*_WHY DOES THIS EVEN EXIST_*
Vsauce: Places the book on the pizza.
Chainsmokers: Stopped smoking.
Alan Walker: Stopped walking
Marshmallow: Stopped eating marshmallows.
Me: Trivago isn't my hotel anymore....
Duolingo:stopped stalking me
Chicken: wont cross the road anymore
Was debating whether to order pizza, Vsauce2 to the rescue with a miraculous notification. The God's have declared, "Yes, today thou shall have pizza!".
Lol
Now that is some food for thought
Chaitanya Singh damn
Chaitanya Singh Yeah including the book
Pizza flavored books
That poor book
Justin Y. hey mate
Everyone is concerned because he is getting pizza on the book and I'm sitting here like you're getting book on the pizza.
Go away, bot.
Justin Y is my father and he beats me
If only there was a way to downvote TH-cam comments.
Shopkeeper:
Aww, are you teaching your kid the alphabet?
Vsause:
Don't be silly, I'm providing how pizza can be cut evenly into weird shapes to a couple million people 😁
2:00 What teachers wand us to eat
wan *d*
wan *d*
im fun at parties
*want (not trying to be mean, just giving you a tip)
The pizza might be the same size for all cut, but they are still not equal. Some slice has more crust than others.
Bryan Nguyen ikr? I thought of that too
Anyone else notice you cut the pizza in the exact shape of a cannabis leaf?
Top 10 Archive Hi I'm a subber
Wait a minute how do you know ;)
Did*
**takes hand roll out of mouth**
@2024: Andrew Boucaud tbf your kind aren't hard to find...
2:50 this how Chuck E. Cheese be have their pizzas
Did Kevin seriously lie that book directly on that Pizza? :D
Putting the book on the Pizza really made me uneasy... Hardback book with oil...Ahhhh
Tinseltopia lololololololol
Is no one bothered that at 2:21 he literally put a book right on top of a pizza?
Judging by the comments, LOTS of people are bothered :-p
I don't know, if only we could read all the other comments saying the exact same thing.
[places my greasy fingers on your book page... forever staining your book... no way to remove the grease from the paper... seeping slowing thru the book...]
cloudkitt
I did went through many of the comments just to see if someone had posted the same thing so either I missed some comments or I am the few early people who posted this before the masses did
Zom Bee Nature
I am more bothered about the pizza
Am I the only one that is a little angry he put a book on a pizza 2:13
Nope
Umm, If you notice, he still basically cut the pizza in half first. Take all of the slices on one side and you still got 1/2 area.
Those monohedral tiles are pretty cool, though!
That is what I though 😂 I didn't get all the shape-cutting mess
Exactly! I'm actually so frustrated about this whole video. Like, take a circle (pizza) throw a dart at it and wherever it lands connect that point to the center and make a cut and the pizza will be 100% even 1 to 1 then butcher it whatever way you like and mirror the damage to the other side and bam "even cuts" not very impressive.
He cut the pizza into eight slices, so that if each person got two opposite slices, four people would be able to share the pizza equally. If you put together any two shares, it would equal the sum of the other two, which allows two people to share it evenly as well.
MrFun So it would have been if he cut the pizza into quarters, why get so complicated?
Laowyn The point of the theorem is that it doesn't matter where the center point is. If you pick a spot not in the center, it could be difficult to quadrasect the pizza, but the theorem states that if you cut it into eight pieces, the sum of the areas of any two opposite pieces is the same.
1:02 what is wrong with this person?
AHH YOU SAW IT TOO **sighs of relief**
I saw it xD
I saw it too
I saw it too and paused it right when it got sent
What?
Is this obvious to anyone else? I mean, it's symmetrical. By definition, anything symmetrical has at least one equal half.
It works even if it isn't symmetrical. It would have been better if the video moved the "origin" a little off to the side. See the graphical tool referenced in the video to see how asymmetrical cuts still work.
You can't be "off to the side" on a circle... smh.
Supersonic Tumbleweed he should have said "if the cuts are off center" rather than talking about the point being off center. Regardless, it was clear what Ken meant
Daniel Ragsdale but you can't be "off center" on a circle...
+Daniel No, it was not clear. I was watching a video where he was showing pizza sliced in half, and jumps to a complexly sliced circle all of a sudden. I seriously don't understand what he was talking about.
When your friend gives you the smaller half:
*_insert spiderman 2 pizza theme here_*
StickMaster500 Hello from SMB
Anywhere you go. This guy will be here.
AYE Your the sheet music boss song! Your Famous now my dude!Congratulations on your Recognization of SMB!
I... did not expect to see you here. I remember you from Sheet Music Boss. How does it feel to be (somewhat) famous?
sTicKmaSTeR50oO henlo
The pizza kinda looks like the Super Smash Bros Logo
Pizza time
Natalie Alfera im not paying for that.
Cyka Blyat why?
Because you're late.
*a wizard is never late*
I hope someone gets that
CreepRoot can’t say I do. I am so sorry please forgive me for my sins
Plague Doctor aw, it's alright! Here's the peice of internet history for you: th-cam.com/video/cbq5r03Duj0/w-d-xo.html
CreepRoot thank you. I will remember you in my travels. You have opened a new chapter of my life. Here: th-cam.com/video/4feUSTS21-8/w-d-xo.html
2:08 Why you gotta put the book on the pizza though?
Kevin is a psychopath he put a book ontop of pizza
I put pineapple...
@@arvaborelius7269 absolute madlad
*EQUAL?*
☭
Ah, a *COMMIE*
Sickmer
You go to gulag
Would I be able to ask Dominoes to cut it like that?
You can ask
yes actually, you are able to ask dominoes to cut it like that but will they is up to them
Vexerux underrated comment lmao
2:22 that mans just put the book on the pizza BRUH
2:00 The cheese is dry people.
This illustrates the importance of commas, because if I believe your sentence I'll think I ate dry people when I ate cheese.
Falgor xD
Ew, eating cheese out of dry people
@@Falgor62 omg haha
Well, the pizza IS three days old.
Guess how im cutting my pizza next... dont even care if my family gets pissed off
Bradley Bell your hard aren’t you
Let's cut the pizza into perfectly equal pieces but not consider who gets the crust
Isn't it dumb to post the same comment twice under the same video only with slight alteration?
Pizza is so popular,it made it's own theorem.
*generic comment about the book on the pizza*
i love this synthwave outro these days aha :D
my fav music genre! :D
Do any of you know what song is it?
I feel... old with synthwave.
I will surely take the center non crusted parts... Yum...
Kevin *finishes episode* *grabs a piece of pizza, but the entire pizza is still there*
me: now thats a twist
he didnt take it from the pizza
0:27 Vsauce comfirmed for smash?
:0
wut
@@marcusdillem9678 it kinda looks like a smash ball
what about square pizza? this is only based on a circle
Fuzzman57 Then it’d just be as many squares as you want. Easy.
Are you talking about public grade school pizza?!
There are pan pizzas too, unless you're one of those dickheads who think those arent pizzas.
it works the same way, cut it 4 times, each time 45 degrees from the last cut and then eat the alternating slices
Nobody wants square pizza.
Impractical is correct
1980: i bet we will have flying cars in future!
2018: how to cut pizza smartly
YOU DROPPED A BOOK ONTO A PIZZA...........WUT?
at least vsauce 2 uploads
he's not a DonG!
But he ain't güd 🤔
There will be a big new Vsauce 1 video really soon
2030games what are ur sources
Bretton Johansen he made a post about it but to see it I think you need to have posts enabled on your sub feed
I want pizza now
The captions at the end say”Skelton eating noises”🤣
Skelton
Wow the captions are bad at doing their job
Skeleton* chewing noises
Now I can trick my friends to get more pizza for myself!
3:07 what does he mean by "each slice as two equal shapes"? later on some slices are clearly made up of three shapes
Karl Bernstein the larger shapes had the duplicate smaller pieces cut out then it was divided into two equal shapes, IE: 3 cuts
Ludus Machinae thanks m8
Karl Bernstein ya it was confusing, had to watch a few times just to get what he meant by duplicate
Still not clear. This video is confusing.
I got triggered when you put a book on the greasy pizza.
A 3-day-old room temperature pizza likely isn't very greasy
No pizza was harmed in the making of this video
And what about the crust?
thank you
Antroflux You mean the pizza peel?
No one eats it
SuperPenguin77664 yeah... whether you love it or not it won’t make all the pieces equal.
@@antroflux8969 Yeah, but we don't care, the equation is talking about the pizza only, not the crust
I dip the crust in tomato sauce. Its good!
He put the book on the pizza...
Finally a proper Vsauce video. Thank you Kevin.
When I worked at Pizza Hut 30 years ago, after you make the first cut to cut it in half, I’d make a second cut parallel to the first, about 1/2” away, slide the sliver out, push the two halves back together, continue cutting, and eat the sliver. When the pizzas were coming out of the oven, that’s how I got lunch. Just be sure to slide the pepperonis aside so you don’t split them. Then they can tell.
Ummm...what? That went way over my head lol.
Or you could just cut the pizza normally
Anyone else get triggered when he put the book on the pizza
2:00
I dont know.. do you see like every fourth comment?
The grease gets on the book xd
The visual proof is neat, but it wasn't really well presented in this video. As it stands, the proof in the video is incorrect - the capital A, C, and E need to be switched with their lowercase counterparts. This allows you to show that the *original* slices alternate between lowercase and capital, while also being of equal total size.
On top of this is the fact that this "visual proof" only really addresses this specific location of the lines' intersection point and not the general case. In addition, it doesn't actually demonstrate the fact that each lowercase section is the same size as its uppercase counterpart. Seems like less of a proof and more of a suggestion as to the fact that the theorem might be true.
Half Pixel yeah I don't get it. I bet I could get an easier explaination from Wikipedia.
*_WHAT DID I DO FOR YOU TO PUT THE BOOK ON THE PIZZA_*
PerSix watch the video lol
???
Pizza for one please
**puts book on pizza**
Me: well... now you have a greasy book and a booky pizza. Good job
_B O O K Y P I Z Z A_
HE SET THE BOOK ON THE FREAKING PIZZA.
sorr i cringed a little XD