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5 Platonic Solids - Numberphile
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- Опубліковано 5 лис 2012
- Why are there just five platonic solids (and what are platonic solids!?)
More links & stuff in full description below ↓↓↓
The solids are the tetrahedron, hexahedron (cube), octahedron, icosahedron and dodecahedron.
Featuring Katie Steckles and James Grime - / stecks and / jamesgrime
Our thanks to the kind people at Polydron for giving Katie some pentagons - www.polydron.com/
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"What is this?"
"That is a cube"
"Yes it's a hexahedron well done"
+Enlightened Penguin That was brilliant
+Enlightened Penguin Yeah well that's exactly what he said, so he was technically right.
+
Well a cube is a regular hexahedron. You could count the pentagonal pyramid and triangular bipyramid as well
IKR!!! IKR!!!
"That's a cube."
"Yes! A hexahedron! Well done!" I found that highly amusing for some reason
Yeh you are right
Made me think of the term "quadrangle". Used to get maps from the U.S. Geological Survey, and that's what each map was called. Because when you divide up the surface of the globe, you can't get perfect rectangles.
It's the same you can refer to it by both of them ^^
It just made me cringe for some reason haha
it's the English folks , 2 polite 4ever🤣
So basically:
tetrahedron: D4
Cube: D6
Octahedron: D8
Icosahedron: D20
Dodecahedron: D12
Yup!
Someone rolled a natural 20 on their Int check. :)
Rhombicosadodecahedron: D30
IraTheSquire that was a weird way to spell hexahedron
d10 is a lie
you don't have to be an artist to draw a pentagon, just an organic chemist
+Chris Letchford-Jones And even then, I've still got a lot of room for improvement.
Organic chemists get a lot more practice drawing hexagons, though.
Can confirm
I'm so glad they sent her pentagon's that actually made me really happy
Maybe - but not ALL viewers have reached the level of education, eloquence and command of the written word that you seem blessed with! ;)
-"That is a cube"
-"Yes! A hexahedron! Well done"
"One of my favourite platonic solid" You don't have a lot of choices lol.
*The 5 platonic solids*
jan Misali viewers: There's 48, actually.
There’s 5 platonic solids, 48 regular polyhedra
@@palatasikuntheyoutubecomme2046 Why do these people just stay stuff without knowing the truth. It's really annoying.
@@praneethmashetty591 well, using the definition they gave in the beginning, op is correct
@@palatasikuntheyoutubecomme2046 what is the difference ?
@@octavylon9008 i would guess that Platonic solids need to enclose an area
I love how ive actually learned things from these videos and have started caring about math, so stuff may be elementary but some of us are getting our daily dose of education
I usually miss half of what they're saying because I've never studied math in english, but the videos still make me more excited about the subject than anything else.
what about romantic solids?
Peepo romantic solids are, by definition, fluid
Let's just be friends solids.
"Well Brady What's That?"
"That Is A Cube"
'Yeah, It's A Hexahegron"
"That is a cube."
"Yea, it's a hexahedron, well done!"
I can see it now: Indiana Jones and the Sixth Platonic Solid
Legendary Dairy
Indiana Jones and the √-1
There is a 6th platonic solid in 4-space. It's called the hyper-diamond.
Ah you mean the rhombic dodecahedron
TO BE FAIR
Its names is rhombohedron
"There are 5 Platonic Solids"
*angry jan misali noises*
Well, these are the only 5 that can actually be represented in real space. The Kepler-Poinsot polyhedra self-intersect, and the rest aren't really "solids". Plus, it is true that there are only 5 solids discovered by Plato.
You ALWAYS ask the questions I want to ask I almost feel like I am having the conversation with your guest instructors... and for that, I can't thank you enough.
Each of these solids can have their faces and vertices interchanged. I would love to see an animation of this.
5:23 - Quote: "A sphere isn't a Platonic Solid, because it technically doesn't have faces."
What about the fact that the surface area of a sphere can be divided into exactly four times its shadow?
Wouldn't it be more accurate to say that a sphere is not a Platonic Solid, because there are no straight edges to divide its surface into a whole number of faces of equal size?
B.t.w. I have no background in math, so I hope my question makes sense.
I always a sphere as a hedron with infinite faces. I reason this based on how as you increase the number of faces the shape looks increasingly like a sphere. With millions of sides, it may indistinguishable from a true sphere. But since a true sphere has no faces that can be inspected, perhaps viewing it as having infinite faces is intuitively true.
Math is so amazing, thank you numberohile for showing me this again and again!
Hey Numberphile! You should do a follow up video on truncated/archimedian solids!
Kudos to James and Katie! Reckon this the best numberphile ever! I've always wondered why 5 and how it was certain there weren't others lurking in some platonic realm! Epiphany!
Does that mean that a plane is a platonic solid with an infinite number of hex agonal faces.
i reccomend looking up jan misali's "48 regular polyhedra" video for all the weird stuff that happens when you stretch the definition to its limits
"You can't add any more triangles"
Hyperbolic geometry: "Allow me to introduce myself"
The last question "Where can you get some of those magnetic things?" was EXACTLY what I was thinking. I kept googling them till they reached that part of the video! Well done brady!
Thank god Brady and the other dude pronounced it "pentagOn", not "pentag'n" like most Limeys :P
"That is a cube"
"Yeah, it's a hexahedron. Well done."
Watching this while sewing all platonic solids with felt - they’re gonna make wonderful ornaments!
This visual proof is easy to understand - and nearly impossible to forget. Thanks!
Haha, exact same over here! Love DnD! Especially that moment when you roll a 20 for initiative or a full dmg roll for a daily skill!
Several people have mentioned Dungeons & Dragons. I have my dice in front of me now. The d10 is a perfectly good die, but it is not a platonic solid. A sphere, in D&D terms, would be a d1, which would be pointless, which makes sense because a sphere has no points.
Sailor Barsoom The one that looks like a golf ball? Those are hard to read.
Yeah because that's what you need Brady, more youtube channels :) Joking, day just got so much better
I'm quite happy about the fact they sent her the pentagon, the thing in the end made me smile so much :D
what about the Kepler solids?
The are polyhedra but aren’t platonic
They’re talking about the Platonic solids in this video. Although there isn’t any detailed video talking about them besides jan Misali’s video, so seeing one would be very nice.
how many reguler hyper solid there are in 4-d
+oldcowbb the elements of cube-like regular polytopes can be given by the factors of x in the expansion of the equation (1+2x)^n where if m is the power x is raised to n-m is the dimension of the element
3
5:46
YES! So glad they sent you some! You earned it!
Halfway through the video I started looking for those cool magnet shapes so I could make shapes for fun, and lo and behold they anticipated my question at the end! Clever numberphile...
Three dimensonal bees makes hexagon shaped honeycombs. What about four dimensional bees? Which 3D shape would them honeycombs be?
+Anežka a Jiřík They would make 4D hexagon shaped honeycombs
+Anežka a Jiřík Honeycombs are 2d shapes projected through the third dimension, so I would imagine the 4d bees would make 3d shapes projected through the 4th dimenion, most probably one of those platonic solids ;p
+Anežka a Jiřík My guess would be truncated octahedron projected perpendicularly through the fourth dimension. The Truncated octahedron is the shape you get if you take the octahedron described in the video and cut off each of its vertices.
Scabbage
Well the reason why honeycombs are made of hexagons is because they tessellate the 2D plane and they have the greatest ratio of area enclosed to edge length (or material used).
So my thought process is that these 4D honeycombs would tessellate 3-space, and have the greatest volume to surface area ratio.
Since there are a lot of polyhedra that can tessellate 3-space I decided to limit my search to the polyhedra that have uniform edge length throughout the polyhedron. This leads to 7 choices that I found, five with regular faces(the triangular prism, hexagonal prism, cube, truncated octahedron, & gyrobifastigium) and two without the rhombic dodecahedron, and elongated dodecahedron.
I then compared the volume : surface area ratios of all these solids and found that the truncated octahedron is the greatest
Of course there is a chance that the optimal 3D space filler does not have regular faces and that I missed some polyhedrons with uniform edge length that tessellate 3 space.
I just wasn't able to research these to the extent required to know for sure, but I'm pretty confident that the truncated octahedron fills 3space optimally.
Luke Doherty Funny this conversation came up because I was experimenting with cuboctahedrons, triangular orthobicupolas and truncated octahedrons for optimal 3D self organising maps a little while before I saw +Anežka a Jiřík comment here, lol
Uhh... Metatron's Cube anyone?
Plato was really clever to come up with this. He even did not have a smartphone.
Thank you so much for sharing!!
FIRST VIEW OMG!
OMFG
i came from the "there are 48 regular polyhedra" video to dislike this
jan Misali makes such varied content.
You’re disliking the video for no reason. Jan said there are 48 regular polyhedral, not Platonic solids
@ The only reason you disliked is because proving someone wrong makes you feel smart. You disliked without knowing the truth. Don't get me wrong, I loved jan Misali's video, I liked the video, it's just the fact that people like you ruin the premise of some great videos, like jan Misali's video.
That was genius!! I mean, that was a 6 minutes and 38 seconds long ad for Polydron. Awesome! I'm buying... :D
Well, I didn't finish elementary school successfully, so I found this video educational and rewarding. You never do get to old to learn!
Also, did you consider actual elementary school students might come here and learn this from this video? Go shit on some videos you find confusing instead!
I've noticed that the 5 Platonic solids have dualities that are very similar to the dualities of the 5 string theories. The heterotic SO(32) and the heterotic E8xE8 are dual, the type IIA and IIB are dual, with the type I being dual to itself. I don't know if this has any significance, but it's interesting.
You're definitely on to something. I've seen criticisms from philosophy teachers. About string theory having peculiar similarities with Plato's philosophy about his solids.
I love simple visual proofs which are intuitive and just as valid as formula-laden written proofs. This explanation of why there are only 5 such solids is much easier to understand than the two page mathematical proof in the appendix of Carl Sagan's book of "Cosmos", and yet just as effective. Thanks for a great video
Why am I still awake? Adderall, thank you for keeping me awake to watch all the numberphile videos.
Nice touch putting the Phi song right after they mentioned pentagons!
Also, you can fit more polygons around a point in hyberbolic space; in fact, you can fit as many as you'd like. Essentially, the platonic solids are tilings of a sphere (positive curvature), triangles, squares, and hexagons can tile a plane (zero curvature), and anything more can fit in a hyberbolic plane (negative curvature).
The most fascinating part of it to me is that infinitely many regular polygons can be inscribed in a circle, but only five regular solids can be constructed in a sphere, which is the three dimensional correlate of a circle. I like the video and don't know much, but believe that's sort of the main thing about this that is significant.
Fabulous vid!
Oh yeah, the Phi metal song at the end got me pumped up!
Absolutely. The amount of times I've wanted to bite my own tongue off when talking to my lecturers kills me!
Inspiring video! Recently learned that 4 tetrahedron and one octahedron, with an edge length of 1", can all fit together to form a larger tetrahedron, of edge length 2". Is the study of combining platonic solids called something? Also, are there any other combinations of platonic solids that form larger platonic solids?
"I'm a mathematician not an artist" best line ever
When ur math teacher jus doesn’t want to give you a math lesson, so we get some random Australian🥱
The fact they sent her the gift made this video get the like. I'm so happy for her😂
The video was very informative.
I did not know about platonic solids. I learned something new.
Good to see I'm not the only one who has a difficult time drawing pentagons.
If I'd watched this video yesterday, I would have got one more maths challenge question right. Damn pentagons and their 108 degree angles!
The TARDIS on her T-Shirt! I'm so excited! :D
I'm glad Katie got her pentagons
Yay! Then talk about the duals!
btw, a great way to teach the kids about the shapes is to buy (dollar store, super cheap) stirring straws that have one hole in them and then have them string/thread them together using long needles or wire and make them themselves. This teaches alot about their math such as threading patterns. You could also theoretically nest them using some careful measuring and cutting. That's always been a mathematical obsession.
I really enjoy this video.
This helpful video should increase the number of PLATONIC SOLID CITIZENS! Have a geometrically exuberant summer!
They can be paired where the numbers of vertexes and faces swaps.
Cube pairs with Octahedron: with 6 and 8.
Icosahedron pairs with Dodecahedron: with 20 and 12.
Tetrahedron has same number of faces and vertexes(4) so it's its own pair.
i love you brady, i was thinking "what about a sphere throughout the whole video"
Infinite sides
I love Katie's tshirt! With the TARDIS pattern. It's awesome!
Playing with magnet balls I discovered each of these shapes. I was going to argue for the ten sided die shape, but upon review, two of it's corners have five sides meeting, and the rest have three. Well played. I did discover that if you add a seventh triangle the seventh corner bows upward, you can't make an enclosed shape after that, but you can keep adding triangles until the shape bends far enough to meet itself. After that the triangles have to bend.
I like how they sent her some pentagons. That's totally cool.
- That is a cube.
- Yeah! It's a hexahedron, well done!
:'D
The visualization helped me realize why you had to have less then 360° because then it would be flat and there would be no concave.
Nice choice for 5. Has there been a video for 17 yet? I think Wallpaper Groups would be a great topic for a video
Wow. Good attention to detail.
That Polydron toy is really neat :D
The last theorem in the last book of Euclid's Element: there exists 5 and only 5 platonic solids. Beautiful...
loved the video, thanks... "liked" when Polydron gave Katie the pentagons... lol
I love platonic solids! xD I think you should explore more of the formulas around them, like the "V - E + F = 2" ^^
He, I ordered a set of poly-dice the other day and they arrived just before I watched this video so I've got models of each of these in front of me now :P
Im glad she got her pentagons :P
And now I want to play D&D. Thanks guys.
She wasn't talking about tetrahedrons at that moment, she was talking about the octahedron and how to make it. The tetrahedron was already discussed.
Those people at Polydron are so nice.
I would love that! There's so much potential!
It's all on the teacher. I had a physics teacher in high school that made me hate it. But now that I'm in college, I've got such a fun teacher that I'm unable to NOT pay attention just because how awesome it is.
The tetrahedron and hexahedron are easily generalized to higher dimensions, and done so on a regular basis in any introductory algebraic topology course. So there are at least two for any natural dimension > 2.
I found a fossil of a platonic solid. It's my prized possession.
Love the phi song at the end. Didn't see that coming
yes, they all are first because of the symetri and scound because they roll well (exept tetrahedron) the 10 sided dice has some sort of daimond shape for each side.
Cool. I've never thought that proving that those are the only regular solids would be so simple. And Now I wanna buy these magnets.
I can understand it. I was so angry the first time I saw this video that I flipped my desk and punched my goldfish. Damn platonic solids.
I only bring this up because of the link to that song in the end but if you want your mind blown do a search for "tool fibonacci sequence" and watch the first video. It's a great explination of how the tool song lateralus is written around the fibonacci sequence.
Now I know the shapes that attracted me into gift shops all the time
Is it only me? With a big box of those magnetic shapes I'd be kept quiet for hours.
What's the name of this one? A CUBE!
Well done! It's a Hexahedron!
That was funny
Nobody:
Mathematicians: Pentagons would be appreciated for my birthday
Euler's formula can also hold true for graphs or lattices of polytopes. The formula would have the form S0-S1+S2-S3+...=1 where Sn is the element in the polytopes of dimension n. Enclosed spaces in the lattices or graphs would have to be included as the last elements of the polytopes. For example, solving the equation 64-x+108-27=1 we can find that a regular Rubik's cube has 144 edges, counting shared edges only once.
Roughly at 3:15 concerning the six that lies flat, this is what is called "tiling the plane". There is another study of geometry that involves "tiles" of many different shapes.
By golly, you're right! Thank you for the simple explanation. :)
"a CUBE!" - "Yeah, it IS a hexahedron, well done (you noob!)"