So why are these the only (regular) dice you'll ever see?
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This is a follow up to the last video ( • The things you'll find... ) regarding higher dimensional shapes. Here we answer why there can only be 5 shapes in 3 dimensions that meet certain criteria (as listed in the video).
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damn wish i would ever be able to tell a pick up line as smooth as this proof
Có công mài sắt có ngày nên kim.
When you 🛑 STOP, 👁️👁️ LOOK, 👂 LISTEN 👂👍☯️ YOU NO LONGER NEED ANY PICK-UP LINES, & YOUR OUT OF THE ONLY PLATONIC SOLIDS ZONE. THE PROBLEM IS 197:1 ratio of (THINKING) 🤔 SKILLS isn't a skill at all & you don't have to have or use smooth pick up lines AT ALL, YOU GUYS JUST GOTTA BE YOU, as the brilliant & probably gay Oscar Wilde said....you CAN ONLY BE YOU, AS EVERYONE ELSE IS TAKEN !👍🤔👂👁️👁️👂🛑☯️
@@marcsalzman8082 What kinda crack
@@marcsalzman8082 Get help
@@marcsalzman8082 get some help
Platonic solids... Absolutely no displays of affection. Kissing is disgusting and hugging is totally forbidden!
Ouch, this comment got too real too fast in 2020.
Spheres though, they love kissing. They are quite polyamorous about is, I heard a sphere may kiss twelve other at the same time!
the cube can do it decent tho
@@ganondorfchampin Yes. That’s, why spheres are excluded from Platonic solids: They can’t keep things Platonic. 😅
@Tex Loch second joke? What is it?
Thank you, for not just this video, but for all the videos you put out.
3:20 me walking in on my parents fighting
I used to play D&D and remember all the dices, but it didn't occur to me until now that 1d10 is an outlier compared to the rest of the dices.
it's not a regular polyhedron but it is close: I have a 30-sided die for which the faces are rhomboids. So the faces aren't regular. Except that they are, in a way - if you look at each rhomboid as two triangles, co-planar and sharing an edge. Which means it is actually a 60-sided die but half of the faces are coplanar to each other so it just looks like a d30.
thats a rhombic triacontahedron. splitting the rhombic faces wont give regular triangles. plus it wont be strictly convex. and there is one more problem. on some vertices, 3 rhombuses meet up but on others, 5 meet up. so its not a regular polyhedron BUT it does make a fair die because of a property im too lazy to explain (like a "secondary" regular solid) in fact there are alot more fair dice than platonic solids. but sadly the rhombic triacontahedron isnt platonic but its still a really cool solid :)
It IS a fair dice because it's face-transitive
@@themobiusfunction ik but if a die is coplanar it wont be fair anymore cuz it will always land on 2 faces at the same time
@@Qaptyl a rhombic triacontahedron doesn't have any coplanar faces. Though you can create a convex version of the shape he mentioned (each rhombus split into 2 triangles), and that gives you the pentakis dodecahedron (a dodecahedron with a shallow pyramid attached to each face, and while the convex version's triangles look regular at a glance, they actually aren't. Some corners have 6 triangles, and some have 5 triangles. Each triangle has two 6-corners and one 5-corner.) Or the triakis icosahedron (icosahedron with a shallow triangular pyramid attached to each face. Each triangle has 2 narrow angles that meet in groups of 10, and a wide angle that meets in groups of 3.). Both make for fair 60-sided dice because of all of their faces being identical, but irregular polygons.
I saw a spherical 6 sided die last week. Play around with it a bit to see if it's fair. Nothing conclusive, but it seems close to fair based on preliminary testing. Forgot to mention, the die is weighted so that it will clearly come to rest 1 of the 6 sides.
Yes, I think the only way to make a die like that fair is to mess with it in some way like weighting the sides
@@insertname252 Actually the weight is only necessary to reduce momentum, so the sphere will stop rolling in a sensible timespan. Using a cube and blowing up the faces so the whole thing is a sphere is naturally fair.
Those have an octaheadral hole inside, such that the vertices of the octahedron match up to the numbers on the sphere. Since an octahedron is fair, so is this. (sorry if i misspelled anything).
0:27
I'm a medical student and have nothing to do with maths but since I took maths until my 12th grade, I really enjoy your videos. Thanks.
More than 360 as never mentioned, would also leave it concave.
Amazing! Your channel is one of the best and most entertaining channels related to mathematics.
I literally had this proof 2 weeks ago in my topology course :O
Please hit us with a video on how differential equations model the world.
Beautiful, elegant proof. I have a question. How do you explain to someone that a mathematical proof, unlike a scientific experiment, is absolute?
It isn't actually per say... axioms make up a theory and theorems are proved from axioms. In science we observe general truths and then prepose axioms that are always true (sometimes they are not true any of the time) while in math we arbitrarily choose our axioms on our own creative whim, there is no right or wrong. (there can be wrong but that would be a fallacy in the mathematician, not the mathematics i.e. redundancy or inconsistency in the axioms) So in that regard a mathematical proof is always absolute but only to your chosen theory (set of axioms) and so not actually absolute. One might come to say it is absolute because the vast majority of mathematics is done under a single generally accepted but hardly derived theory. Some of the theorems of the theory of euclidean geometry hold and others break down when making the switch to the theory of hyperbolic geometry. Those "rock solid" proofs unravel as they are only as useful/valid as the proofs of their components (theorems used to prove another theorem), all the way back to the validity of the axioms used to prove theorems used to prove theorems used to, etc., which as I said, is arbitrary and based on the mathematician. So in math, it is all about finding new axiomatic systems and studying old ones (sometimes unifying two theories such as algebra + geometry -> analytic geometry, i.e. applying the cartesian product to the axiomatic concept of points, in synthetic geometry we say the point is undefined while in analytic geometry we give it the algebraic definition of the coordinate pair, triplet, or in general n-tuplet) while in science it is about empirically discovering all of the unchanging unchosen axioms of reality and uniting all of scientific phenomena under a SINGLE theory, a very hard, very messy, thing to do in practice but theoretically achievable absolutely speaking. I would honestly argue that the scientific theory -> hypothesis -> experiment -> theory if done correctly, is more absolute then the mathematical axioms -> conjecture -> proof -> theorem -> revised axioms process. The less wrong we are about science, the more science we can do, which will in turn make us less wrong about science. Sounds really freaking dumb but its true. Darwin had trouble being a perfect biologist because he understood so little biology. Given more science, he could have derived even more science and accurate science. As for what you might be actually getting at... math can be broken down into its component logic. Mathematician's do not tend to do this, (they can and have before, look at the book "Principia Mathematica", it took them 350 pages to prove 1+1=2 from just logic) as it isn't a part of their profession, they are creative individuals working a step above logic and petty logicians :D. Natural science on the other hand isn't based on logic but instead observation. This makes results SEEM less inherent when in fact because the axioms of the universe never change, they are always and more so "inherent or absolute" then in mathematics, which more accurately are relatively absolute to any particular theory. In conclusion, both natural science and mathematics are absolute, with errors being not of the subject or process but of the mathematician or scientist. The nuance of why mathematical proof isn't necessarily more absolute comes in how mathematical and scientific theories operate differently, in terms of goals and reach. What I find particularly interesting is that these are about the only two absolute academic subjects with legitimate theoretical (theory) structure that makes for absolute arguments. I prefer mathematics as you can be creative and work where and with what you please while also demanding the same absolute consistency of natural science but through logic instead of the fundamental laws of nature. Have a good day!
Joseph Cocchiola Science can be proven. Not with absolute certainty but still with a HECK OF A LOT MORE certainty than faith.
Joseph Cocchiola, you can’t prove anything with absolute certainty. Of course, faith is almost certainly real. We have overwhelming evidence of faith, including, but not even nearly limited to holy wars, terrorism, certain prejudices, MLM schemes such as essential oils, the Holocaust, etc.
@Joseph Cocchiola I did too. I just had a discussion with someone and they didn't understand the difference between Maths and Science. Their point was that there's a lot we don't know and it can change. I said that was fine but a proof is true as long as the axioms are. No new proof can be derived from the same axioms that can unprove that proof.
Anime Banz Unless there‘s a contradiction in the set of axioms you‘re using.
Even your concave star-like 3D shape draws one of two of the platonic solids if you link together the inward or outward points.
Also; notice, how the faces that meet at 1 vertex of the tetrahedron, make up 3/4 of the tetrahedron; for the dual pair of the cube and the octahedron, the faces meeting at each vertex make up 1/2, or 2/4, of each entire object (3/6, for the cube; and 4/8, for the octahedron; respectively); and finally, for the dual pair of the dodecahedron and the icosahedron, the faces meeting at each vertex make up 1/4 of each object (3/12, for the dodecahedron; and 5/20, for the icosahedron). Continuing the pattern of subtracting 1/4 for each step of increasing complexity would result in 0/4 coverage for the next dual pair, which would correspond to a polytope that has infinite faces; in other words: a flat tiling. Indeed; the next dual pair is not of 2 closed objects; but rather, of 2 Euclidian 2D-tilings: the hexagonal tessellation, where the faces meeting at each vertex make up 3/∞ = 0/4 of the entire polytope, and the triangular tessellation, where the faces meeting at each vertex make up 6/∞ = 0/4 of the entire polytope. If you tried to go to the other direction, to a simpler-than-the-tetrahedron-polytope, you would get a 4/4 = 1 coverage; meaning that the faces that meet at a single vertex would make up the entire solid; which is also not valid for a closed, convex, 3D-solid, or a polyhedron; but would likely correspond to some kind of singular 2D-polygon, or the 3D-waterdrop-shape, with a single curved face, a single vertex (the apex), and no edges; which is obviously not a polyhedron. 🤔
I'd like to remark that the video only shows that there are at most 5 Platonic solids. It still remains to be shown that for each candidate, such a Platonic solid actually exists. I've only heard that it can be proved by explicitly calculating that the geometry works out, but if someone has more insightful proof, I'd love to hear about it. After all, it's quite curious that each candidate actually gives a Platonic solid. (Building a cardboard/computer model is not a proof, as there could be very small errors which are not visible for naked eye/floating point presition of the computer)
Calculate the angles lol
Tetraedri_ Or coord bash it with 3D coords maybe
@@alexwang982 This is the ''calculating the geometry works out" method, which of course works. However, I don't find it satisfying as it gives little to no insight why every topological Platonic solid has also geometrical counterpart. Especially if this holds also for higher dimensions and their respective Euler formulae (I may check this once I have time). Here I have to admit I haven't calculated the angles and whatnot myself, so maybe doing only that would give a good understanding why things works out. However, I just thought more insightful proof is an interesting (and admittedly ambiguous) question.
A cardboard model *can* be a proof. When 3 identical equilateral triangles are folded as in the video, the remaining face is a triangle and its edges are shared with the rest. QED
@@bookashkin Not really. Physical models always have a degree of imprecision in them accounting for material thickness and properties, and error associated with the constructor, be it human or machine. I could fold three triangles together that appear to be equilateral, but i would not be able to assure that those triangles are perfectly equilateral all the way even past the Planck limit and into infinity, nor could i assure that they were folded just as perfectly. Leave the edges of one triangle even a picometer too long and the proof falls apart. generally speaking, most proofs via straight-up modelling fall apart, as all models have some degree of error associated with them. Most of the times when this isn't the case is when the models are generated from mathematical rules prescribed to them, and in that case the proof is in the math, and the model is merely a visualization.
Good content! Really cool how you proved this.
Brilliant, mate! Thanks!
Was looking for this vid from your last vid lol
Was not expecting this to help me understanding my discrete math homework, where i had to use euler’s characteristic to prove that a given complete graph wasn’t planar. I found the characteristics online but I didn’t under why they worked.
Please make a video on William Lowell Putnam Mathematical Competition
I mean by the Euler's characteristic definition the Great stellated dodecahedron also satisfies all conditions, it's euler's number is 2, and there are 3 edges per vertex and 5 edges per face. Sadly it fails the first definition.
I am a high school who loves science and math; my passion actually evolved more as I watched a lot of your videos. Out of all STEM careers, I like computer science the most. I heard that a highly sought out after job is being a Data Scientist. Would it be possible for you to create a video on Data Science? Specifically, what kind of math you need to know and what to expect in college and post-college? Thank you for your help!
He has videos on statistics and neural networks, as far as I know, its not much the math needed, basically its linear algebra and calculus, and ofc statistics, which is not a branch of math :v Btw im a cs major, first year.
Donlans Donlans hello, what programming is needed?
Wow..!! Wonderful proof!!!
Thank you for showing the great stelated dodecahedron
Great video! Ly
I love this geometrical proof of there being only 5 platonic solids, done by exhausting all possibilities in a very simple way!! ♥ I wonder, if similar can be done in higher dimensions.
I'm not very good at math, I only know the basics.. but I can't wait to go back to college and learn all this.. I'm definitely subscribing to your channel the topics are little bit complicated for my level of math but you got a new subscriber lol
2:54 it doesn’t barely leave an angle of 36 degrees, it actually leaves an angle of exactly 36 degrees
Looks like we got mr./ms. smarty pants over here. xppp
I think about LoL’s gemstone when first looked at the thumbnail..
i have a d3 (its 3 sides making a triangle from a profile view, but from a frontal view it looks like a curved rectangle thing) and the numbers are similar to a d4 (the correct number is on the top)
amazing!! but I didn't get the result equation when you divided by 2E
Holy shit that's amazing
I just wan to say that even though I’m not interested in math or physics and I’m not necessarily good at this subjects (in fact, I’m a realistic pencil artist) I really enjoy your videos, you are one of those few youtubers that make actually good content.
okidoki
Yes, there are only five Platonic solids... but you already mentioned the rhombic dodecahedron, and noted that the term "dice" was only used for familiarity.
Can you do a video on engineering technology( such as electrical engineering technology, mechanical engineering technology etc...)
whats the name of the s ed shape at the 25 second mark
Snap he banned the pentagram platonic solids
Pentagrammic polyhedra are not platonic tho. The reason he made this vid was to see if anything besides triangles, squares, and pentagons could used as a face for dice. But a pentagram is just a pentagon with five thin triangles and four wide triangles, so it would not be fair dice since the faces are different. Sorry ik your comment is almost a year old now but I just started getting interested in this stuff so I had to reply
@@kiraoshiro6157 a pentagram is a singular face. It is a Pentagon with acute angles
@@greatestgrasshopper9210 ik, by pentagrammic polyhedra, I meant polyhedra that have pentagrams on them. If you cut up a pentagram what you'll get is a pentagon and triangles. Not saying pentagrams aren't legitimate shapes, just that they are also made up of simpler shapes already used in platonic solids so it wouldn't make sense to add it, especially since there isn't any polyhedron with only pentagram faces. Assuming it's the star shape by itself without an outer pentagon, because then all you could make is a dodecagon. In order for it to be platonic, it has to have each face be the same, so there cannot be a pentagram platonic solid. Sorry for the long explanation but I when I said I recently got into this I mean extremely recent like just yesterday I've been researching all I can about polyhedra. Def not because I'm making up a fake religion where ppl worship cognitive polyhedra and the more complex ones are more powerful as gods haha
@@kiraoshiro6157 just match up 3 or 5 pentagrams to each point of each pentagram, and you get a solid with intersecting faces, not different faces, but he banned that in the video.
Didnt think there would be a way to make d&d even nerdier
How thick is the edge and is it included in the area of the square ?
Man you’re so cool
I want to see a dice with all hexagonal sides that is not cylindrical-ish.
What if we consider 2.5-gon as the base of a hedron?
You get the Kepler-Poinsot solids.
if you split the cube up to have 24 edges it would also double the faces as now you can see the inside of the faces as it is split
Can someone please tell me the name of the red shape at the 25 second marker. Ive been trying to find the name of it all day but cant find it and then i see it in this video!
Great stellated dodecahedron
There’s also a physical and only true explanation - perfectly distributed pressure in space. Same goes for hexagons, tho there’s no perfect shapes in nature, therefore Platonic solids, sphere and all equilateral n-gons are nothing but concepts, which sometimes could be used practically because of the approximation.
What about a d100, or is that not a regular shape?
2:36 if it is greater than 360 degrees , we can fold but it becomes hyperbolic. I learned this thanks to Henry Segerman and Daina Taimina :)
And a 3D shape that consists of 14 triangles and I call it the sopodelhedron
Hi everyone, how would you solve the system of inequalities at the end? As in what method.
Theres 2 unknowns with one equation so theres not a good method, all we have is to fix an arbitrary value for one and try to solve for the other. Then combined with the criteria that both are larger than 3 it fixes it to these values
WHAT ABOUT THE OTHER 43 REGULAR POLYHEDRA
Give us link to Numberphile's and Vsauce's videos, thank you.
What with the d10?
It's faces aren't regular polyhedra. (of course it is still a fair die, and symmetric over its faces.)
The number of edges to a vertice is not constant either
I'm mad about hexagons not having a platonic solid.
don't worry they have 2 apeirohedra
"Your Next Mission, should you decide to accept it Humynity, is to 'discover' which 'dimensional configuration' ALLOWS x-number of Platonic Hexagons to become a Platonic Solid, eh." just a message in my courier packet. I do not write this stuff. I simply read the label and post it where necessary to rescue Humynity from its suicidal tendency as a whole.
I understood everything and then he got to the equations and threw me all off lol
Wouldn't a Mucube be a platonic solid because it's not convex nor concave? Same for all other Mu shapes.
Just realized that this is about dice and stuff but a mucube could and couldnt be a die because it has no interior or exterior
Because the other 43 regular polyhedra are abominations.
just like ur mom OK IM SORRY I HAD TO SAY IT kidding tho, hope you have a nice day man Read more
Can you prove that 1/ef + 1/ev > 1/2? I think that I understand the intuition. But I can't prove it myself.
I want to know if there exists such a 3d shaped where any surface is a prime number of times/reciprocal of another surface & all surfaces are similar (not inclusive of 1).
Meme man must take note of this
I made a almost flat icosdohedron
How am I supposed to figure out which number my concave star dice landed on?
Dice?
what if we put a bunch of them together
God, I wish the Pentagonal trapezohedron ( D10 ) did it to the list... I kinda love percentile rolls...
HI
Uh oh! Where's the pentagram?
or 3 i meant
If you live in a non euclidean space then you might have more platonic solids in 3d
yeah but not limiting it to euclidian space basically just makes hell
Hey I mailed about writing english captions for your videos.plz reply
But you can fold two square planes to make a new face or four squares you just move one over...the waybyiu said it doesnt make sense..and you have an open cube not a closednsurface when you do it..
2:34 Refer to a cube folding diagram before spreading this.
Commenting for KZhead Algorithms5
Did anyone watch the rest of the video that brought us here?
i thought there we're only 4
you should change the channel name
Going of off ?
The notion of "regular" I think is poorly defined in 3d. For example, all faces being equal seems regular to me. Though not every apex is equal, they share a symmetry group.
third
I think you forgot the 10 sided dice kzhead.info/sun/bNekdJSvr5WJkpE/bejne.html
Last
Up-vote if you think that we are more dumber than this stuff
true bhai
i hate that you call them by their shape names and not the dice names
why can't it be stellated? so non-inclusive
They said it has to be convex, so no stellated ones.
they said it,so it must be what?