Bees not only like math. They are also into physics. They developed sting theory.

@lakshaymd

6 жыл бұрын

Oh god 😂😂😂😂😂

@twogungunnar9456

6 жыл бұрын

WACKETY SCHMACKEDY DOO! *DOING!*

@robintaylor3713

6 жыл бұрын

They are also into theatre. Remember that famous line from one of their plays, "To bee or not to bee"

@e1123581321345589144

6 жыл бұрын

sting theory :))) caked me up

@brumm0m3ntum94

6 жыл бұрын

69 likes

Next up I'll be researching how bad it would hurt to be stung by a 4-D bee… so much fun doing this video with you guys!

@qwerty11111122

6 жыл бұрын

They wouldn't be bounded by your skin, as they can go around it in the 4th dimension. They could literally sting you in the heart :(

@jameskolby

6 жыл бұрын

a 4D bee might exist as a section of space and a section of time. It could hurt you for however large a section of time it would take up. If you could have a bee the size of five years, you would feel a sting for five years and however long it would take such a sting to heal. Sounds like you need to get the Doctor.

@TheVelvetTV_Riesenglied

6 жыл бұрын

That's a task for Coyote Peterson

@WillToWinvlog

6 жыл бұрын

If you want to know what it would be like to be stung by a 4d bee, the fact is you would seem flat to it, so it could sting you from the inside without actually going through the outside of you! In other words, it could sting your heart, for example lol. Consider a two dimensional object (like a flat coin) being stung by an object in 3d space. You can get to the middle without penetrating the edges.

@imkevliet5931

6 жыл бұрын

part of you will be send in the 4th dimention, thus you will be able to relocate yourself in the 4th dimention, its actually pretty awesome to get stung by a 4d bee, the movie "flatland" will explain the rest for you.

“Bees have six legs so they like six-sides shapes” I have two legs, my favorite shape is... *line?*

@comingbacksoon.8410

4 жыл бұрын

Nani? (Uhhh)

@jongxina4929

4 жыл бұрын

Maria Ross no you have 4 limbs therefore your favourite shape is squares Christ we even make massive buildings mainly made of 3 d squares known as cubes

@Qermaq

3 жыл бұрын

If you're a coke head, your favorite shape is line.

@igorjosue8957

3 жыл бұрын

i have 4 legs, but 2 of them we call arms, my favourite shape is a square

@smartart6841

3 жыл бұрын

Semi circles

4D bees are not existing at an alarming rate.

@spacejamgoliath

5 жыл бұрын

Yet

@AlineBourderau

3 жыл бұрын

I don’t understand.

@Ggdivhjkjl

2 ай бұрын

You've never been stung by one have you? 🐝

“Everything is the way it is because it got that way” lmao

A swarm of 4-dimensional bees sounds like something from Lovecraft's nightmares.

@Mysteri0usChannel

5 жыл бұрын

Nah. Bees are nice. 4-dimnesional WASPS however would be hell! I mean, they could just buzz through walls by moving around them in the fourth dimension. You wouldn't be able to run or hide from them. No 3-dimensional wall could stop them.

@silience4095

3 жыл бұрын

@@Mysteri0usChannel Good intuition!!! Congrats.

_"Philosophy is written in that great book which ever lies before our eyes - I mean the universe - but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth."_ -Galileo Galilei

Anyone else think Lord Kelvin sounds like a Sith?

Bees should just make hyperbolic heptagonal honeycombs

@peabrainiac6370

6 жыл бұрын

that even was my first thought for the best tiling in 2d... hyperbolic pentagons ^^

@iLoveTurtlesHaha

6 жыл бұрын

Geo Quag What is that even? XD

@recklessroges

6 жыл бұрын

Would work fine for storing pollen and making honey, but bees use the same space to make more bees. I don't think a hyperbolic heptagonal bee would fly as well as a normal bee.

@LukeLane1984

5 жыл бұрын

In 3D hyperbolic space one could make a dodecahedral honeycomb. But I don't think it would be very practical to use for hyperbolic bees.

I kinda always thought bees made hexagons because if you put circles (holes) as closely as possible, each circle is surrounded by 6 other circles, and the connections between them are inconsistent, and waste space, therefore, to un-waste as much space, you thin out the edges on all sides to the minimum, thus creating a hexagon

@yosefmacgruber1920

5 жыл бұрын

Isn't that then because bees have not invented closets, and have no clothes to hang up on hangers, and no stuff to hoard away somewhere?

9:09 Hey this structure is in Matt Parker's Book. I guess Kelvin's structure was a Parker square arrangement.

@qwerty11111122

6 жыл бұрын

I get that reference!

@want-diversecontent3887

6 жыл бұрын

Daniel Shapiro Are bubbles parker spheres

@marafolse8347

6 жыл бұрын

i cackled

How do honeybees know to make hexagons? According to the linked video, bees simply make circular honeycombs but as the wax hardens the surface tension molds/pulls the circles into hexagons. I may be spoiling it for some, but I don't like it when I'm forced to go watch another video when it takes 7 seconds to tell me the answer. So, I'm helping others like me.

@Icewind007

5 жыл бұрын

Bees make circular honeycombs. Those circles melt and expand into hexagons due to physics. Just like bubbles on a surface would be circular, but would become hexagons should there be at least 6 bubbles to surround another bubble. It is just the most stable structure for adjacent circles.

@deathbyseatoast8854

5 жыл бұрын

Icewind007 Thanks for literally explaining what the first guy said. We totally needed it.

@Archipelagoes

4 жыл бұрын

The thing is we're talking about the second part of the vid.. 4D bees

@williamm8069

4 жыл бұрын

That begs the question, how do they know how to make circles?

@cephasmartin8593

4 жыл бұрын

Actually, if you've ever looked at a bee's head you would see that the shape of the honeycomb cell has more to do with the bee's head than anything else.

Biology is applied Chemistry Chemistry is applied Physics Physics is applied Mathematics Conclusion: Maths is OP

@jameskolby

6 жыл бұрын

math is pure logic applied to the real world and hypothetical worlds. math is applied philosophy. philosopy is OPer.

@dcs_0

6 жыл бұрын

No philosophy sucks. Maybe maths is applied logic but philosophy is a wannabe science.

@davejacob5208

6 жыл бұрын

though i would always say philosophy is op, "true" philosophy is probably always governed by logic. so logic would be OPer.

@omri9325

6 жыл бұрын

Comments are applied XKCD

@mrmarten9385

6 жыл бұрын

And tautologies are tautologies.

I've always heard different from what she says in the beginning, I learned that bees actually just make circleish honeycombs and surface tension pushes them into hexagons the same way as when you have a bunch of bubbles intersecting

@nullpoint3346

2 жыл бұрын

They might melt a bit too.

What would the honeycombs of a 2-D bee look like?

@pbsinfiniteseries

6 жыл бұрын

Huh? Good question. Well, if we assumed they made "sheets" of 1-dimensional unit length combs, like in the other dimensions, then I guess they'd just be little lines...

@TheRMeerkerk

6 жыл бұрын

What if bees lived in a non-euclidean world?

@Private27281

6 жыл бұрын

TheRMeerkerk Taxi cab?

@Alinoe67

6 жыл бұрын

In two dimension, it was about 1 cm² pattern, and 1 cm^3 in three dimension. So it would need some 1cm line ? Is there a general math word to name what is length for 1-D, area for 2-D and volume for 3-D ?

@dcs_0

6 жыл бұрын

+TheRMeerkerk. Interesting. If that world was like a sphere (negative curvature i think), then i think there could be multiple shapes depending on the size of the shape relative to the sphere.

Watched both vids... nice way to approach the problem from both angles - good work :)

Love the collaboration! Seamlessly well done!

Ever notice how a hexagon is a 2D representation of a 3D cube? 😉

@kuakkauk.

5 жыл бұрын

Yep whenever i draw a cube

@alexwang982

5 жыл бұрын

Yes it’s a slice of the cube

@aqualung2000

5 жыл бұрын

@@alexwang982 But the optimal 3D shape is not a slice of a 4D cube? Is the optimal 3D shape a slice of a 4d anything?

@CookieSlime-tf8gq

5 жыл бұрын

The 😉 made me cringe.

@kuutti256

4 жыл бұрын

They are 1d hexagons

Hexagon is the bestagon

In a regular sphere packing structure each sphere will touch 12 spheres around it. The 12 spheres will form the shape of a cuboctahedron. When compressing the structure, each of the 12 'touching' points on the surface of each sphere will become a face / plane, and eventually those faces will touch each other with their edges and fill up all the space. When points of a shape become faces (by truncating) the resulting figure is the dual of the original figure. The dual of the cuboctahedron is the rhombic dodecahedron. So, in my opinion, when compressing a sphere packing structure you will get a space filling structure made of rhombic dodecahedra (that have 12 regular sides), and not truncated octahedra. Packing of truncated octahedra and the Weaire Phelan shapes do of course exist, and they might be more efficient (less surface area), but the regular rhombic dodecahedra structure is what you get if you equally compress (or expand) spheres that are regularly and closely packed. So I think this is what 4D bees would come up with. Besides that the rhombic dodecahedra structure will also be stronger than the other packing structures, and strength is another important property of a beehive. scholar.harvard.edu/files/sbabaee/files/1-s2.0-s1359645412000900-main_0.pdf

Hexagons are the bestagons

Wow. I thoroughly enjoyed this, due to how much match and proofs and history is involved in the video. I'm going to check out the rest of your channel and likely start following you.

Why don't we live in hexagonal houses?

@peabrainiac6370

6 жыл бұрын

Because we don't need to minimize street length.

@iLoveTurtlesHaha

6 жыл бұрын

Scott ... Yet, we don't fly yet. XD

@dragoncurveenthusiast

6 жыл бұрын

and our streets would not be straight

@Tuchulu

6 жыл бұрын

that actually sounds like fun

@edwardfanboy

6 жыл бұрын

Because gravity requires us to make vertical load-bearing walls and horizontal floors (so no vertical hexagons), and it needs to be possible to travel in a straight line through intersections (so no horizontal hexagons).

I love me some dimensions. Excellent trac. Keep it coming! tyvm

Beautiful, beautiful episode, I loved it!

stimulating video!

Wow, that's the best I've seen a mathematician and a biologist get along...

I watched the "It's Okay to be Smart" video first and immediately went "so how do bubbles arrange if they aren't confined to a plane?" So I can't say how happy I was when you then announced this video! 😄

that was nice 10 minute build up of expectation for a "nobody knows what the shape would be" answer. thanks for that.

Another way to approach it, and the companion video came very close, but never quite explained this way, is to just take the ideal circular (or spherical for 4D bees) packing order and expand the shapes outward until the spaces are gone, creating flat surfaces where the shapes overlap. I also was reminded of spreading centers in plate tectonics. They're almost always relatively straight segments connecting 120 degree junctions. Some are triple junctions, but in reality, they're all triple junctions; the ones with just two segments had a failed third section, called an aulacogen.

"What honeycomb would a 4-dimensional bee make" subscribed.

Circles are the most efficient shape, except in packing or tiling a space. A creature builds a circle to store a liquid, but finds space wasted between circles packed together. So the circles are extended into the wasted space, and viola!, hexagons emerge.

This verifies another video about bees using hexagons. This other video suggested the bees really build round honeycombs but the natural forces of nature pull it into the hexagonal structure, the same way soap bubbles naturally seek the most stable state. Interesting video and the part about the 4D bees was a nice addition.

since this video, i wonder, would a 5D bee have a 5D honeycomb, which is arranged into 4D Hexxeracts? A hexxeract is just a 4D version of the 3-D hexagon

If I had you as a math teacher my life would be quite different. Oh and the whole six legs six sides theory is just brilliant hahahahaha

I love her... Beautifully explained

What An Excellent Video of: The Bees Algorithm 👍👍 Thanks for Sharing the Bees🙏

What about pomegranates?

@doronq1847

4 жыл бұрын

Wow cool. How about orange fruits

@navjotsingh7360

3 жыл бұрын

i don't think much effort is put by plants into making the perimeter small or organising space efficiently ,so they are just basic random shapes plants evolved with.

Actually bees make circles and the surface tension pulls them together to make hexagons, you can do it yourself with bubbles. Physics loves hexagons. Thanks v sauce

Man am I glad I got a beer for this.

you may feel a bit of cringe when both kelsey and joe are on screen. i have a hunch they filmed separately and were just put together on the green screen, and that's why it feels like something's off.

@TheInterestingInformer

6 жыл бұрын

That's most likely the case, lol. It's a lot easier to email video clips to each other than to arrange meetings.

The twist, of course, is that honey bees make _circles_ and then build more circles _on_ those circles in a way that, as they tile more and more circular cells, _forces_ them into hexagonal shape because of the distribution of forces involved.

I giggled so hard within the first 15 seconds, the rest was just entertainment :D

two of my favorite people

Looking forward to some type theory and foundations of mathematics stuff! Sounds really exciting!

i honestly love how earnest this show is. it's just refreshing these days..

What if bees are actually 4D but we can't see it? Haha

@maksphoto78

4 жыл бұрын

4D objects would have a 3D cross-section (or "shadow") in our 3D world, so those beed would appear to us in some way.

@r.v.k.6932

4 жыл бұрын

@@maksphoto78 As a regular 3 D bee, maybe! Maybe we walk in the 3D world as shadows of our greater selves :-)

@igorjosue8957

3 жыл бұрын

what if we are 4D and ourself dont know?

"physics is doing math"... That got me thinking, maybe math is really just an invention of our brain because physics itself is doing the work that requieres the least amount of energy and that it is stable so that in itself translates to biology and somehow our brain found a language for that which is math and it has developed just like spoken language and concepts. Mindblowing stuff.

@hdwe1756

6 жыл бұрын

galesx95 Sounds like something I'd come up with.

the fact that the human has 4 limbs and prefers squares over lots of other shapes is such a massive coincidence LOL

You don't have this quite right. I remember writing a paper on this in one of my undergrad classes in architecture school. A lot of fun figuring it out. Bees aren't intentionally making extruded hexagons, they'd be perfectly content making sloppy extruded circles, for example--it's the adjacent bees tunneling away and the counterpressures involved that create a hexlike honeycomb. Look for example at the shapes other than hexes bees produce when they aren't adjacent to other bees--you guys should really look at actual honeycombs, in the wild. We built fabric structures out of the shapes suggested by soap bubbles, too, understanding minimal surface area and the like. It's a great project for kids. Mix soap liquid with a plasticizer and the soap shapes "set" very quickly, making it possible to study these shapes at length. You can generate very different shapes by using differently shaped wire loops that in turn generate all manner of saddles, for example.

hexagons are my favorite 2d shape and octahedrons are my favorite 3d shape and my favorite platonic solid.

Bees actually make circles first, however the soft wax forms itself naturally into hexagons

Ohhh fun! I have been ranting on twitter about nuclear core designs and why we use squares instead of hexagons....sadly, it looks like the reason is just we used squares first and didn't really care for optimal packing into (in our case) cylinders!

Bees make circles. Circles turn into hexagons. So wouldn’t it make sense for a morphing sphere?

Heh, it's interesting that when thinking about honeycombs, I have always assumed that the bees would have started by making circular or even spherical pockets, and then the natural optimization of trying to fit more stuff into the same space, would have pushed those pockets together, reducing the "dead space" and progressively getting more hexagon-like.

This is the only reason anyone in the history of geometry would take geometry in high school.

The honey has a role in choosing its own shape. Honey comes out of a bee in round tubes. Then the honey itself settles into hexagonal prisms.

A joke for all German speakers why bees are math geniuses: Natürlich sind Bienen Mathematik-Genies. Sie beschäftigen sich den ganzen Tag mit Summen.

when I was a child, one thing that helped me get interested in geometry was the soccer ball. I was fascinated by the fact that hexagons make a flat surface and hexagons combined with pentagons make a something close to a sphere.

@igorjosue8957

3 жыл бұрын

lol, add pentagons for icosahedrons trunked

If you actually look at honeycomb, the cells are essentially circular. The walls between them are a hexagonal grid because the wax itself packs together that way. The bees build circles, not hexagons - the hexagons are an emergent property of the comb due to the existence of many circles close together. The cells along the edge of a foundationless piece of comb aren't even particularly round.

Great video

@Platin_2004

6 жыл бұрын

Lordious you didnt even finish it in that time lmao

I felt in love

wow nice video!

Bees actually start by a circle, but the conditions inside the hive turn it into hexagons. When your a bee keeper you see it early in the year.

What about the Rhombic Dodecahedron? 11 years ago I began asking myself the same question: "The square is to the cube, as the hexagon is to the ____?" and the answer I came to was the Rhombic Dodecahedron. I'm surprised that it wasn't talked about in this video. It has the same length of edge for every edge, and every face is the same as well. It uniformly fills 3d space. It has a hexagonal lattice arrangement in one direction, and it has a checkerboard lattice structure in another direction. It looks like a cube when viewed from one direction, and it looks like a hexagon from another. It's a neat structure. If I were to make a modern version of minecraft, I would use the Rhombic Dodecahedron as my building block.

"A great example of how despite our *tendency to assume that making something complex requires special intelligence* a simple physical process or mathematical rule can form an ordered structure" I see what you did right there :).

Something uncomprehendable, you can't see it. But if you could, you would feel confused and possibly sick.

9:33 A regular hexagon is a consequence of a regular triangle. So you will get 3D hexagons by filling with tetrahedrons.

Welp, time to breed some 4D bees

Intrestingly enough this hexadonal space closely matches the natural boundries found in electromagnetic interference patterns in space.

It took 1963 years from Marcus Terentius Varro stated his conjecture before Thomas Hales proved it. That surely has to be the longest time between conjecture and proof in the history of mathematics.

If you pause the video at 6:55 and you count the bubbles in the photo, all the ones that you can see all complete sides. Only about half of them are hexagons. (Depending on how you see some of the bubbles interact with the tiny ones; because this video makes so much of a point to pay attention to equal distribution I count alot of pentagons and heptagons even where there aren't tiny bubbles next door)

Yep. I'll vote for a second time for videos on Foundations of mathematics. Have read a little round this subject, and not being much good at maths, my head did tend to ache a bit. Some clarification in the PBS stylee would be great. For what it's worth, the book that got me interested in this topic is The Foundations of Mathematics" By Ian Stewart and David Tall, but some elucidation from Kelsey (and of course the team behind the scenes as well!) would be fascinating. Here's hoping!

I didn't get the math, although I have never thought about it either, but I like the concept!

Bees have six legs that's why they are the best dancers

Quick proof that hexagon is the most optimal shape. First we need (or at least I needed to), find an expression to represent the area of the equailaterial triangle. We know the area of the triangle is b*h/2, so we need to find the height. So we construct a right triangle inside the equilaterial triangle by bisecting a line. By the Pythagorean Theorem (a/2)^2 + h^2 = a^2, so c^2 = 3a^2/4 or h= sqrt(3)a/2, so the general expression is a * sqrt(3)a/2*1/2 = sqrt(3)a^2/4. The hexagon formula is likewise obtained by multiplying this formula by six, as a regular hexagon is just six combined regular triangles. For a given a perimeter, say K. The individual length of side of triangle tile is K/3. Using our formula we get sqrt(3)/36 * K^2 or about .048k^2 For squares we use K/4, so the area is simply K^2/16 or .0625K^2 For Hexagons we get 6*sqrt(3)*(K/2)^2/4 = 3sqrt(3)/72 * K^2 or about .072K^2, Therefore for a given non-Zero perimeter K, a Hexagon maximizes the area.

@ocarinafan100

6 жыл бұрын

Skipped some algebra steps, since I didn't want to type them all in a youtube comment haha

@NoNameAtAll2

6 жыл бұрын

Jonathan A. That's proof of hexagon>square and triangle, not "best of all"

@ocarinafan100

6 жыл бұрын

Absolutely a true statement. I didn't show that only Hexagon, Square and Triangles will work (which is true), but I think it's implicit in context of the video. Still a completely valid point.

@stevethecatcouch6532

6 жыл бұрын

+Jonathan A You did what she asked for, you proved that of a triangle, square and hexagon with the same perimeter, the hexagon has more area. What she did not expressly request, but could be inferred, is a proof that of all shapes that can tile the plane, the hexagon is most efficient. If you limit the question to regular polygons, you proved that as well, because of the regular polygons, only triangles, squares and hexagons tile the plane. A circle of circumference K has an area of k^2/(4*pi), about .0796K^2. The area of a regular octagon will be between that and .072K^2. A regular octagon will not tile the plane, but irregular ones will. Any irregular polygon of perimeter K will have an area less than a regular polygon of the same type. The question is, is there an irregular polygon that will tile the plane and whose area is greater than .072K^2?

@butanestove1514

6 жыл бұрын

dude you wrote "Quick proof that hexagon is the most optimal shape" but that's not what you did.

Did anyone else first come up with the solution of the pentagonal dodecahedron honeycomb? It seemed to be the intuitive thing to me as the hexagonal pattern arises from projecting tangent lines from the contact points of close-packed circles, so you could just do the same with tangent planes on close packed spheres. It at least gets you close to the most efficient answer, plus it keeps all the surfaces and edges flat.

To roughly answer 1:37 (correct me if I'm wrong) A: area P: perimeter s: side length (note regular polygons have equal side length) n: number of sides The area of a regular polygon is given by the following formula (which I leave out the proof of; something something margin something): A = ( n * s^2 ) / ( 4 * tan(pi/n) ) The perimeter of a regular polygon is simply: P = n*s To rewrite the side length in terms of perimeter, we have: s = P / n Substituting out side-length in the above area equation gives: A = (P^2 / n) / ( 4 * tan(pi/n) )= P^2 / ( 4n * tan(pi/n) ) We do this because side length varies depending on the perimeter, which is held constant. We want to maximize the area-to-perimeter ratio (A / P): A / P = P / (4n * tan(pi/n)) Now we have an equation for the area-perimeter ratio in terms of the perimeter and the number of sides. At this point, we can plug in 3, 4, and 6 for the number of sides (n) to see that 6 has the maximum area-to-perimeter ratio for these three regular polygons which can tile the plane. What's interesting to note is that n*tan(pi/n), for n>=3 (as in the case of regular polygons) decreases (monotone) as n increases, so A / P increases as n increases. Of course, our intuition tells us that at eventually this area-to-perimeter ratio reaches a maximum. This is because n*tan(pi/n), while decreasing, is bounded below, and actually approaches pi. Thus, the upper bound for an area-to-perimeter ratio (in terms of perimeter) is P / (4 * pi). Of course, this is achieved for a circle, with: A = pi * r^2 P = 2 * pi * r To see these agree: (using known formula): A / P = r / 2 (using our upper bound): A / P = P / (4 * pi) = 2 * pi * r / (4 * pi) = r / 2

The honeycombs top, always P point to the north pole... If you move the position, the bees rebuild the honeycombs!!!

Very interesting.

6:04 I find Thompsons argument comical, considering we are animals who through evolution have started to jump past the inefficiencies of trial & error and use math & physics to our advantage, and because of that in part we're the most dominant species on the planet

hexagon seems to be just the more sided shape and converge the most compared to square or triangle, and it is closer and closer to circle which was similar to how people estimated pi with integration.

BUMP P=NP & Kolmogrov Complexity

A computer simulation could solve? Set the algorithm to behave like bubbles. Would be a minimization problem solver.

Yay for joe

We should build space craft around this shape, then the core could be used as a safe zone. Protection against radiation, and the vitals of the ship in the middle as well.

At 6:58 there actually are three bubbles in a pentagonal shape as well as two septagonal bubbles :D Of course they don't fill the plane (except for weird quasi-crystal structures that can kinda fill it more or less), I just thought the irregularity was funny

Flowers often come in five petals, apples have small five branched stars in their section (as do many fruits and things derived from flowers). A researcher posited that this pentagonal stuff came from the optimal shape if you had to bulge from a plane. Think soccer balls, fullerene, bucky balls..

I always thought the idea of honeycomb being "hexagonal" was misguided and misleading. I think it's more apt to think of honeycomb as stacked cylinders. In this context, the "hexagonal" shape of honeycomb seems more coincidental than intentional. Just a hypothesis! Great work PBS!

Bees do not make hexagonal honeycomb cells. They make round ones! The wax then gets soft and the walls of the circular cells get pushed outward by their contents. This is what forms the hexagonal pattern, much the way groups of soap bubbles will form hexagons.

@yasinomidi7525

6 жыл бұрын

Isaac Hanson you learned that the video before this one

@spacejamgoliath

5 жыл бұрын

Yea we just watched that dickhead

@polyphorge6621

5 жыл бұрын

I think that too

When a mathematician and a biologist meet, what they talk about is physics.

Now to figure what honeycombs 5-dimensional bees would make.

Good crossover

“ Mathematics is life!” Is what my classmate say to me when I was grade 5... XD

I'll never look at bubbles in the same way again

Now, what's the most optimal shape for tiling non-euclidean spaces?

@xenontesla122

6 жыл бұрын

AFastidiousCuber I think it depends on the area of the cells if we're talking about constant curvature.

@NathanK97

6 жыл бұрын

have a shape share all sides with itself

@yosefmacgruber1920

5 жыл бұрын

You have more than just squares and hexagons, if you remove the restriction that all shapes must be the same exact shape. Oh, you asked about non-euclidean geometries? I doubt that euclidean-based regular polygons will work very well for that.

@SandaaaGW2

4 жыл бұрын

Order-8 Aperiogonal tiling.

The best part is: Bees have a mathematical brain, like Fibonacci number embedded through out nature, like the mind of god.

Kelsey's got that groove, wanna dance?

I'm thinking the "bulging sides" tile would be the same as a square. The area gained by the sides that bulge "out" or offset by the sides that bulge in. Plus it's taking more perimeter than a square anyways, since the sides are bent rather than straight.

I love this, but my head hurts.