It never occurred to me that those were pineapple slices, although now that you say it, they clearly are. I just saw them as yellow backgrounds so the numbers would be more visible.
Truly a mathematician's sort of pizza; reducing it to a bare minimum set of ingredients that have been known to make up a pizza (even if they're in an unconventional form), calling it a previously solved problem, and not even bothering to bake it :p
Original pizza didn't have tomato. Tomato was not known in Italy (or anywhere outside America) until the early 1500's, after the Spanish discovery / conquest. Pizza has existed in Italy since before there was a Italy, or a Roman empire.
Love how Brady jumped right in for -1/12, implying that if you take this algorithm to infinity you wind up with a pizza cut into 11/12 regions. Get the Sixty Symbols squad on this, I smell a Nobel prize for topological defects in higher dimensional pizzas!
Well that proofs then, if you cut through the middle, that infinity is the same as 6 cuts, since that leaves you 12 regions. With the lazy cutting technique infinity using the -1/12 answer is impossible since you either have 11 regions or 16, but no 12. 8-)
@@Nemelis0 it's not that you get 11 of 12 regions, it's that you get 11/12ths of "the concept of a region". Think of it in terms of countries, if I cut a country in half I wind up with two countries, I can move the line wherever but I still get out 2 whole units. If I cut an infinite number of times though, I get 11/12ths of a country, 11/12ths of whole indivisible unit.
I'm afraid you need a blade or pizza roller that can cut over 1.00 efficiency. In other words, it needs to cut into non-parallel non-curved spacial direction while being a curved-space object.
@@Yezpahr @Yezpahr I mean, moving this from the imperfect doughy world of pizza and into the mathematically pure realm of infinitely thin cuts and infinitely flat planes the algorithm still holds. The algorithm is just a greedy method of partitioning a flat region (doesn't have to be a circle) into as many sub-regions as possible by bisecting it over and over. But in math, you can make the cuts arbitrarily close to one another and the regions arbitrarily small but greater than 0 in area.
Now that we know that there is a cut pattern that makes 1+n(n+1)/2 pieces out of n cuts, we could ask about fairness in several ways: - What is the pattern that makes the smallest piece the biggest? - What is the pattern that makes the biggest piece the smallest? - What is the pattern that minimizes the variance of the area of the pieces? - Are some of those questions equivalent?
The thing to remember is that you don't want to come even close to parallel lines, because each line must intersect each other line. So change the angle the smallest possible amount on each cut, staying just shy of the limit an infinitely small (but not zero) amount. Once you pass 180 degrees you get a parallel line that cannot possibly cut all lines, so your range of motion is half a circle. So between (exclude the limits) 0 and 180 degrees you'll be making cuts with the two outermost cuts as close to perpendicular as you can get. The closer the first cut for both of these is to the far edge of the circle, the less long their adjoining pieces are. And since we're talking about cuts with an angle, we're talking triangular pieces for these, two so the longer they get, the bigger they get. If you plot your intersections you should get a half circle with anywhere between zero or infinite surface area. of intersections. So the question becomes: When I draw half a circle inside a circle, how close away can the start and beginning of that half circle be from the edge? The closer they are, the smaller their pieces. Then again, that's only true if the half circle of intersections is for a smaller circle than the pizza. If the intersection half-circle is infinitely bigger, the intersection half-circle is effectively a straight line, so the closer the line is to the middle, the fairer the cuts. So depending on the difference between the pizza's radius and the intersection half-circle's radius, the best option is either as close to the edge or as far from the edge. I wonder what the middle point would be where the pizza and intersection radius are the same. This is probably basic, but math was a long time ago for me.
@@bartolhrg7609After you make all n cuts you see what's the size of the smallest piece. Now you consider if you can cut differently such that the smallest piece is larger.
I have not attempted to learn mathematics or work with mathematics since I last studied it a couple years ago and this is my first time in a long time I have had to think mathematically. It really feels uncanny but awesome how all of this kind of stuff just *makes sense*. Thank you for presenting these kind of simplified versions for people like me to get back into mathematics!
@@aimeeriverswrong. Pi times z squared (which is the same as z times z) times a is correct. So it of course does not matter how long z is. It can be 1cm or 349388383 lightyears. 😮
@@kennygeheim4230 oh i thought the equation was the diameter times pi (the diameter being 2 times the radius) ... but it is a very long time since i went to school 😂
@@aimeerivers Diameter times pi gives you the circumference. The surface (and then volume) requires the radius squared. Hopefully that clears up the confusion :)
Based on this I now understand that the total size of a cut pizza is 11/12 pizzas. I've weighed the crumbs left on the pan after cutting and believe this to be accurate.
@plackt this is what is beautiful about mathematics, such an unintuitive answer that it seems impossible but then apply it to a real world situation, and the surprising result confirms that what we once thought of as nonsense is indeed innovation.
I bet the dialogue of that teacher with the young Gauss went like this: "Can you calculate the sum of all integers from 1 to 100" - "Ahh, well, fifty-fifty"
My intuition is that the intermediate value theorem would allow you to make equal areas for any given solution by rotating and translating the cuts, without changing which overlap which.
I highly doubt that it could possibly be fair even for the 3-cut case. An interesting query is how fair can it get for a number of cuts, in terms of narrowing the range of slice areas? Also, how many fair slices can be achieved? (For that, we have the lower bound of 2n for the normal radial cutting method and the upper bound provided here. I don't expect the actual upper bound to be better than linear).
The slices look a lot like life, though. The "normal"/"fair" way is the PC version. This version is how life really is. Mine is the the tinny tiny one in the left corner.
2:10 There should be a follow up video on if it’s possible to “Lazy Cut” a pizza into equal area slices. Intuitively, this seems possible with 7 slices and 3 cuts, but with 4 or more cuts, it becomes a lot less clear whether or not it can be fairly divided.
After watching this I realized making the least number of pieces is incredibly simple, since you just keep cutting parallel lines to add one strip at a time.
I found an easy way to cut the pizza into the maximal number of pieces from n cuts (I don't know if this is how they do it in the paper or if this is even correct): if n is odd, construct a regular n-gon, and extend all the sides as infinite lines. Then, draw your circle big enough around the n-gon so that all the line intersections are contained. Each line will intersect every other line in the circle, as needed. If n is even, do the same with n+1 and remove a line. Kinda cool!
This is a greedy algorithm, you try to maximize the number of new regions in each step, but the greedy approach does not guarantee optimal solution in general, so its use should be justified in a case by case basis, right?
You are right, the greedy algorithm is not always correct, but I think it is proven in the video, that the greedy approach works here, perhaps not explicitly enough 0) f(n) is defined "The maximum number of slices using n cuts" 1) You have to do exactly n-1 cuts before making the nth cut 2) (1) => on the nth cut you can intersect no more than n-1 lines, making no more than n new slices 3) (1) => before nth cut there can be no more than f(n-1) slices 4) (2), (3) => f(n) = f(n-1) + n or less 5) f(n-1) + n slices is achievable by doing the greedy algorithm => f(n) = f(n-1) + n or more (or it would not be maximum) 6) (4), (5) => f(n) = f(n-1) + n (6) states, that the formula at 7:20 is correct and the rest is certainly in the video
2:05 "but obviously this one's quite small and no one wants it" I'd say the size isn't even the biggest issue; after all you could rearrange the cuts to make that middle piece bigger if you want. But it'd still be a bad slice of pizza since it doesn't have any outer crust, and thus cannot be easily picked up to eat without getting toppings all over your hands
if you don't want the middle triangle from the third cut to be too small, don't make the first two cuts meet near the center! that minimizes the possible size of that triangle!
I would quite enjoy that "pizza" as a snack, despite what haters say. Tortilla bread is delicious. Amazing video by the way. I had taken a hiatus from math, but this reminded me why I love the subject.
This is the only numberphile video that I guessed anything. I had a feeling it was how many intersections there were. Im no math expert so that was exciting for me
That's the saddest "pizza" I've ever seen, but Professor guy gets a pass from Italy anyway - because of the interesting content as usual and the strikingly perfect free hand circle 😅
“I like to derive my own formulas” … this is why my grade went from 64% to 98% between my first and second semesters of calculus. We had a giant list of trig integrals to memorize for the first end of term 1, and I can’t memorize at all well. Term two, we learned integration by parts and I could derive them instead! I also did this for many years with the quadratic formula
I'm reminded of the peg-and-string thing where you start off with a pair of perpendicular axes (more than one axis, not more than one axe) and draw the lines (X,1), (X-1, 2), (X-2,3) and so on to (1, X). With a large enough X, you get the image of a quarter circle made out of many straight lines... however the intersections of each new line conform to this lazy algorithm too.
I propose a method to get the most pieces. (5 mins into video) 1) Initial cut 2) Almost parallel but intersects near edge of pie 3) Same thing but intersects the previous one 4) Same thing but intersects the previous two 5) Repeat til you get a dreamcatcher. Your intersection with the first cut changing location slightly each new cut that if linked form a bent chord.
Mammamia ! 😱 If I were an Italian mathematician (which I'm not), I'd challenge you to find: "The Lazy Way to Cut Christmas Pudding". In 3D, with planes instead of lines. ... and then label the slices of pudding with pieces of tomato or salami. 😝
I was confused for a moment since I vaguely remembered a similar topic in a 3b1b video that results in a different sequence but I realize my mistake now. This sequence is constructed by placing lines through a circle, while the 3b1b sequence was constructed by placing points on the circumference of a circle and drawing lines between those points
I had to solve this looong ago when I was a teenager. I did as in the video and I was rightfully happy with it. Another one did better, though somewhat out of the box. Cut the pizza in half, place one half on top of the other, repeat. We'll get 2^n equally sized pieces of pizza.
Also a classic for the whole "any finite sequence is the first n terms of infinitely many other sequences". Complete the sequence: "1,2,4..." you ask people and they invariably say 8, but if were using this, its 7!
Natural follow-up question: what pattern of cuts gives you the biggest small pieces/most even pieces. Like for one and two cuts, the centre is obviously the most even place, but for three cuts, it works better if the first two are not central.
Pausing at 6:47 to think about this going the other direction. If I get N new regions when I have made N cuts this suggests when I made 0 cuts I got 0 new regions. This means -1 cuts should ALSO have 1 region. But that would imply that when I made -1 cuts that I lost a region at that time. So if I had -2 cuts I should have 2 regions. And -3 cuts would have been 4 regions and so on.
real world pizza actually has thickness so you can for example make a cut parallel to the plane of the table, but then the solution to the number of pieces you get with n cuts is (n+1)(n^2-n+6)/6
I am a simple man with complicated pleasures. I enjoy math, science and cooking. Watching someone spread raw tomato paste on dough with a trowel then applying pre shredded mozzarella on top is devestating. 3:40 well with the hyperbolic way that you treated that beautiful piece of dough I think you can find a way.
Now the next step... Figuring out algorithm to maximize the cut area (of the final pieces) as much as possible, and then determining the distribution of area across all the cuts and see if it fits a pattern.
Fun fact: After Cheung Ka Long's victory in the olympics, the Italians were pissed. So Pizza Hut Hong Kong released a free pineapple topping to celebrate Cheung's victory, pissing the Italians even more.
There’s a really nice geometric derivation for that relationship. Take n 1x1 squares in a line. Stack on top of those n - 1 more 1x1 squares, then n - 2, etc. Align them all at the left, so you have what looks like stairs. What is the area of the stairs? If you draw a diagonal down the stairs, you can pretty easily see we have a triangle with base and height of n, so n^2/2 (you can also think of this as half an nxn square), but we are missing n half squares that were cut off by the diagonal. So, f(n) = n^2/2 + n/2 = (n^2 + n)/2 = n(n + 1)/2.
You should do a follow up video where you not only maximize the number of pieces, but you also figure out the cuts that would make the most equal sized pieces. In other words what cuts would result in the smallest difference between the biggest slice and the smallest slice?
I imagine Gauss' teacher not realising how he did it, so just kept giving him bigger and bigger values of n and Gauss kept coming back seconds later with the answer.
