The “Diagonals in SPACE” interjection might be the best highlight for this channel in a while. And I’m glad to be a part of it when it becomes a happy meme. :)
@@robinsparrow1618 Jean-Luc Godard was a famous French filmmaker. One of the many things he was noted for was having English subtitles that were frequently different from the spoken French soundtrack of his films. These were not slight differences in the translation, they contained different storylines, conversations, and descriptions of what was happening. They were frequently written by literary authors he'd invited and they were told just to view the movie and write their own script that went along with the visual film and not to worry about what the original French film was about. He was a legendary innovator and invented the "jump cut" film transition among many other things. I didn't mean to imply this was actually stolen, this was meant as a humorous joke.
If you want Steve's subscribers, you need to fill that thing with water. You were so close! 😄 I really dig how your personality and style come through even in the bits other people help you with these days. Been a fan of yours for many years, you always bring me smiles, quite a few belly laughs and a ton of inspiration
13:34 the convex hull of the 5 octahedron compound is the icosidodecahedron. I know this video is about regular dodecahedrons, but I was a little sad when you brushed it off. It’s my favorite compound, my favorite stellation, and my favorite faceting. It also looks a whole lot like my one of my favorite polyhedra, the disdyakis triacontahedron!
It goes even deeper than that. As you may know, the octahedron is dual to the cube. As you mentioned, the convex hull described by the compound-5 octahedron is an icosadodecahedron. Well, the _interior space_ described by the intersection of the cubes comprising a compound-5 cube, is a rhombic tricontahedron, which is dual to the icosadodecahedron. Pretty neat--as above, so below..
Fun fact: Both the small stellated dodecahedron and the great stellated dodecahedron can be thought of as 3D versions of a pentagram. They are both very cool shapes.
I absolutely adore the visual representation of the see through dodecahedron with the tape to show the square, pentagons etc! it's just such a satisfying visual proof of the lengths of the space diagonals
This was so much fun to watch and in my opinion one of Matt’s best in terms of pure enjoyment and entertainment. Matt’s enthusiasm is totally infectious and a delight to watch. The brilliant choice of music only added a new dimension (!) and I cannot praise this video enough!
@@needamuffin oh yes, in fact, that is also a result of same connection between the symmetry groups. (the icosahedron is duel to the dodecahedron, and three orthogonal planes have an associated cube)
Every d12 I own (which is 2, I'm not a weird dice hoarder) has the cube shape drawn on in sharpie, it's so satisfying to look at. I also like how the pieces you'd have to "cut off" to make the dodecahedron into a cube are shaped like little rooftops.
The smaller solids left behind by the shapes discussed are super satisfying in their proportions too, both the sort of flattened, obliquely truncated triangular prism you get from cutting along the square/cube and the frustrum of a pentagonal pyramid cut off by the near-equatorial pentagon...
I like several of the integrated shapes. Discovering the square within the dodecahedron reminds me of the end of this demonstration of the Cross Sections app: th-cam.com/video/2uHfFp1XCPc/w-d-xo.html
4:25 This diagram is the longest space diagonal, not the medium-sized one Matt is talking about in this segment. But the length (phi^2 = phi + 1) is correct for the medium-sized one!
Matt, you are honestly a master educator. I'm in my thirties and failed nearly every math class I ever took (and whatever I did manager to learn, I promptly forgot when I graduated high school). Every time I watch one of your videos I learn something and I'm able to truly understand and retain concepts that boggled my mind before.
When I started learning 3D modeling and animation on the Amiga circa 1988, one of the bigger challenges I set for myself was modeling and animating regular pentagonal dodecahedron with a raised star on each face (similar to the Chrysler logo) -- thus the dodeca-deathstar was born. A couple years later, working in high end video post production, I used the mathematical precision of the amazing Ampex digital optics device with a precise pentagon matte to layer a spinning dodecahedron with different video on each face -- calculating exact angles and depth offsets with an HP-15c calculator was a wonderful challenge that grew my maths skills considerably. Sorry Matt, but the platonic dodecahedron is, and always will be, the BEST dodecahedron.
I almost audibly gasped when you taped that square on. This was a really cool way of showing everything, better even than a 3d animation or something I think.
I've always loved the regular compounds and their beautiful symmetry. When I built my first raytracer and figured out how to raytrace cylinders, the compound of 5 tetrahedra (which is my favorite) was one of the first things I made a render of. The regular compounds were the first things I printed when I first got my hands on a 3D printer. Great video as always!
I had to make a geometric solid out of paper as a highschool project. I chose a dodecahedron and it was pretty wild finding out that the whole net can be constructed (with straightedge and compass) using just a unit side and like 3 or 4 powers of the golden ratio. Imagine unfolding one half of the dodecahedron into a flower shape. That flower is bounded by a pentagon that's phi^2 larger than the faces.
Matt thanks for running the only math channel I've found that will always explain things in a way that makes sense and makes me laugh every time! I've been watching your videos for a long time and you've only gotten better with time!
Discovering the square within the dodecahedron reminds me of the end of this demonstration of the Cross Sections app: th-cam.com/video/2uHfFp1XCPc/w-d-xo.html Let me know if a tool like that could aid in visualization. You could possibly do a follow-up on the hexagon within the hexahedron.
Fun fact: if you take that hexagon found within a cube, and replicate it three times by rotating it 90 degrees each time along one of the three four-fold axes (through the center of opposing faces), you'll have four hexagons whose edges describe those of the cuboctahedron..
One thing to do would be to also tape the insides, but wait, there's more... you could have taped each cube with a tape (or drew with a marker) that reacts to a different wavelength of UV. Then by switching different blacklights you could switch between the cubes instead of having them on all at the same time.
That's a lovely, easy to visualize, and excellent way to explain these conceps. Absoulytely a great example of how to teach a concept really well. Good job, Matt.
6:45 the coolest part of this was that this was wholely unsurprising thanks to your previous videos on the rhombic dodecahedron It's lovely when one maths investigation is helpful in understanding a completely unrelated one
The subtle joke for diagonals in space about 4 minutes in was really good. I imagine you were thinking, this is a bit silly, no one’s gonna even care. I care. That caught me off guard.
Matt, I'm pretty sure the dodecahedron mesh at 1:07 is wrong - aren't the opposite faces supposed to have opposite orientation? All the front faces are also meeting the rear faces on two edges!
another fun way to build a dodecahedron take a bunch of inflatable tubes (innertube , donut , torus) and lash them together i managed to build all of the solids except for the icosahedron it collapsed on itself you end up with some very large pool toys
The first time Matt had a collab with Adam, he referred to him as "Adam Savage from Mythbusters". In return, Adam Savage referred to Matt as "Matt from Numberphile".
@@wierdalien1 Numberphile is a collection of math. Was Matt in the first video? Are they friends? (those are rhetorical) Numberphile is a channel. Matt is a guest on their channel.
I love the ones where you can tell how much fun he had with it, and also where the concepts don't fly too far above my head. Also I can see myself making a shitty scaled down version of this in the future.
I have to be honest, I really like the stella octangula (the compound of two tetrahedra) just because it has a simplicity that a lot of the other regular compounds don't have. You can take a single glance at it and instantly know how it's constructed.
This video made me realize why I’m not a mathematician. I can fully understand why a square being a integral part of a dodecahedron is fascinating to some people, but i literally said out loud in a room by myself “oh, I don’t like that”. I find it supremely uncomfortable.
There is an error at 13:33 regarding the five intersecting octahedra: they do have a convex hull (every bounded set has) - it is just not a Platonic solid. It’s an Icosidodecahedron.
What a lovely coincidence I'm building nested platonic figures in bamboo sticks (up to 3m) with my students at the moment and analyzing this video is their homework. Thanks, Matt.
The compound polyhedron made of a pair of intersecting regular tetrahedra, is aka the "stella octangula." It was a favorite of Johannes Kepler, the guy who fiddled around with the 5 Platonic solids to try to explain the relative sizes of the planetary orbits, and the guy who formulated the famous "3 Laws of Planetary Motion" that bear his name. Anyway, the 8 vertices of the stella octangula are the vertices of a cube. Which also explains the 10 regular tetrahedra in the regular dodecahedron, once you've highlighted the 5 cubes in it. Fred PS. Also interesting to note, is that the main (longest) diagonal of an n-dimensional hypercube of unit edge, is √n.
I can solve a megaminx, so I've spent a long time looking at dodecahedra. I noticed long ago that at a certain orientation you can find a set of 6 edges that are parallel to one of the 3 axes and you can therefore make those edges all line up with a face of a circumscribed cube. It's really cool to see the numbers behind an inscribed cube. 14:00 I think I know how to create all of them. You can make the double tetrahedron with the medium space diagonal of the dodecahedron, aka the face diagonal of the cube. You might even be able to get the fifth using cube-octahadron duality. I believe this would be drawing diagonals between the edges of the dodecahedron, specifically the 6 edges mentioned in paragraph 1
Instead of taping the inside of the dodecahedron you can also tape the back of the tape with a dotted line pattern, and that way when you turn on the blacklight you get dotted lines whenever you are looking through it!
A cube with roofs on the faces? So that the roof planes of one face continously match up with the roofplanes of the neighbour faces. Or the triangle part of one roof matches up with the trapezium part of another roof to make the pentagon without any kinks. LOVE IT!
00:14 Also interesting , if you take the unit-square _down_ a dimension, the diagonal from endpoint to endpoint is the square root of 1. And if you go down yet another dimention, the diagonal becomes the square root of 0. Starting to see a pattern here. Question then is: if you go up a dimension from the unit-cube... does the pattern hold?
Yes it does. You can do this with any unit-n-cube, with n being a positive integer. Consider the case for n = 4. We have a hypercube centred at (0,0,0,0). The two opposite points are (-0.5,-0.5,-0.5,-0.5) and (0.5,0.5,0.5,0.5). The distance would just be sqrt(1²+1² +1²+1²) = sqrt(4). Generalise this to any positve integer n and you get that the space diagonals (or *diagonals in space* if you prefer) are sqrt(n). Q.E.D.
I never really liked dodecahedrons... until now. They may not be as fun as a Truncated ditrigonary dishecatonicosachoron or as endearing as a Gyrotunnelled truncated cube, but I absolutely love how the ratios work out.
at the boring end of the range, there are two intersecting tetrahedra in a cube - the edges of the tets being the diagonals of the cube faces. The centers of thee faces of an octahedron are the corners of a nested cube - and vice versa.
9:20 "The shape we were trying to made was the compound-5-intersecting tetrahedra. Here is a picture [...] and I've actually got a little print out over here" - am I the only one who was a bit sad that it wasn't a 3D print?
1:10 a Cube and a Dodecahedron. ??? look closer : the dodecahedron is wrong! Two facing sides are never parallel but rotated over 36 degrees or 108 or 180 .
the autotuned "but i couldnt be bothered" cracked me up, this is why matt is the best
4:29 ϕ×ϕ this is my new favourite emoticon!
Fun fact: by definition, φxφ=Φ+1
🤯
фхф
ϕwϕ
classic cat eyes
¯\_(Φ×Φ)_/¯ - PARKER DIAGONAL IN SPACE!
The “Diagonals in SPACE” interjection might be the best highlight for this channel in a while. And I’m glad to be a part of it when it becomes a happy meme. :)
Matt still trying to make us forget about the Parker Square.
But we will never forget. :D
I was expecting a more Piiiiigs iiiiiiin Spaaaaaace vibe.
Clearly Science Asylum inspired, if you ask me. Not that I'm complaining.
I love dodecahedrons but our relationship will always be platonic
Groan... but also cute.
Perfect pun
nice
What a shame, I thought things were just golden.
That would break plato's heart, he thought the dodecahedron would always be your everything
D I A G O N A L S
I N
S P A C E
sadly youtube bitrate compression messes with my full enjoyment of D I A G O N A L S I N S P A C E
🛸👾
It is just a blatant theft from Science Asylum, but I'm not even mad. Well done by Parker-man.
@@Yezpahr nah, clearly it's blatant theft from The Muppets
I'm sure Adam Savage is a man. I'm tired of the misuse of my language by an elite few, who are trying to spread the misuse.
The quiet echoey "space" at 3:45 killed me 😭😭😭💝💝💝
My condolences to your family.
@@Elesario than you, its so sick how you still have access to TH-cam in the afterlife, didn't expect that 💝💝💝
Was about to comment this hahaha great attention to detail
1:31 Alex, do you want to give me a hand with this?
Alex: Sure
Caption: No
made me think I was insane since I had to scroll so far to find this 😭
Shamelessly stolen from Jean-Luc Godard.
@@2ndfloorsongs who?
@@robinsparrow1618 Jean-Luc Godard was a famous French filmmaker. One of the many things he was noted for was having English subtitles that were frequently different from the spoken French soundtrack of his films. These were not slight differences in the translation, they contained different storylines, conversations, and descriptions of what was happening. They were frequently written by literary authors he'd invited and they were told just to view the movie and write their own script that went along with the visual film and not to worry about what the original French film was about.
He was a legendary innovator and invented the "jump cut" film transition among many other things.
I didn't mean to imply this was actually stolen, this was meant as a humorous joke.
@@2ndfloorsongs oh ok, this is actually really interesting and cool to know about. and it's a good joke with this context, thank you
4:42: maybe tropic would be a better word than equator, as there are two of them parallel and equidistant from the central plane.
Great job on emulating the old educational film aesthetic for those insert animations, really sent me back...
I think a reference to Look Around You
It reminded me of the animated sequences from the classic BBC Hitchhiker's Guide to the Galaxy TV series
I see that VFX department got a raise recently
Timing department getting their budgets slashed
If you want Steve's subscribers, you need to fill that thing with water. You were so close! 😄
I really dig how your personality and style come through even in the bits other people help you with these days. Been a fan of yours for many years, you always bring me smiles, quite a few belly laughs and a ton of inspiration
Excellent. I especially liked the D I A G O N A L S I N S P A C E.
13:34 the convex hull of the 5 octahedron compound is the icosidodecahedron. I know this video is about regular dodecahedrons, but I was a little sad when you brushed it off. It’s my favorite compound, my favorite stellation, and my favorite faceting. It also looks a whole lot like my one of my favorite polyhedra, the disdyakis triacontahedron!
The icosidodecahedron _is_ its convex hull. I don't know what Matt was talking about, maybe he meant that the convex hull is not regular.
@@galoomba5559It really sounds like he accidentally skipped a word.
It goes even deeper than that. As you may know, the octahedron is dual to the cube. As you mentioned, the convex hull described by the compound-5 octahedron is an icosadodecahedron. Well, the _interior space_ described by the intersection of the cubes comprising a compound-5 cube, is a rhombic tricontahedron, which is dual to the icosadodecahedron. Pretty neat--as above, so below..
@ VERY cool, thanks for sharing!!!
Correction: graphic at 4:22 is longest diagonal = φ √3.
replying to boost the correction
8:00 the rotation due to parallax and the actual rotation cancel out briefly. very cool to see
Everyone taking about "diagonals in space" but 11:41 is the best voice sample for an EDM song.
And what about the "but I couldn't be bothered" from 7:47?
@@wyattstevens8574they couldn't be bothered to mention it
Hot take: _Howard Carter's entire soundtrack_ for Matt's entire channel is, like, the only _good_ EDM I've ever heard.
12:00 a good excuse for drawing 12 pentagrams on a dodecahedron
Matt is summoning something in the exact center of the dodecahedron so that he can trap it
Fun fact: Both the small stellated dodecahedron and the great stellated dodecahedron can be thought of as 3D versions of a pentagram. They are both very cool shapes.
All hail Satan^12.
@@Bluesine_Rwell, duh!
How else would you trap a demon in the centre?
Oh god the SFX budget went sky high for this video!
One might even say it is IN SPACE
I absolutely adore the visual representation of the see through dodecahedron with the tape to show the square, pentagons etc! it's just such a satisfying visual proof of the lengths of the space diagonals
This was so much fun to watch and in my opinion one of Matt’s best in terms of pure enjoyment and entertainment. Matt’s enthusiasm is totally infectious and a delight to watch. The brilliant choice of music only added a new dimension (!) and I cannot praise this video enough!
agreed
I was not prepared for the joke at the end. Well done.
Ben: Hey Matt! I've made a spinning dodecahedron in Geogebra!
Matt (after this video): I don't need you anymore! I can make my own spinning polyhedra!
The cube dodecahedron relationship is like, my favorite thing about 3d geometry, its so beautiful
Mine is the three orthogonal golden rectangles forming the verticies of the icosahedron.
@@needamuffin oh yes, in fact, that is also a result of same connection between the symmetry groups. (the icosahedron is duel to the dodecahedron, and three orthogonal planes have an associated cube)
Every d12 I own (which is 2, I'm not a weird dice hoarder) has the cube shape drawn on in sharpie, it's so satisfying to look at.
I also like how the pieces you'd have to "cut off" to make the dodecahedron into a cube are shaped like little rooftops.
The smaller solids left behind by the shapes discussed are super satisfying in their proportions too, both the sort of flattened, obliquely truncated triangular prism you get from cutting along the square/cube and the frustrum of a pentagonal pyramid cut off by the near-equatorial pentagon...
I like several of the integrated shapes. Discovering the square within the dodecahedron reminds me of the end of this demonstration of the Cross Sections app: th-cam.com/video/2uHfFp1XCPc/w-d-xo.html
the editing on this is impeccable
4:25 This diagram is the longest space diagonal, not the medium-sized one Matt is talking about in this segment. But the length (phi^2 = phi + 1) is correct for the medium-sized one!
Thanks, I paused and looked for this comment.
phi^2 = phi + 1
Golden ratio quadratic equation.
7:43 It's the "Parker Fluorescent Embedded Cube", he's done it again!
Matt, you are honestly a master educator. I'm in my thirties and failed nearly every math class I ever took
(and whatever I did manager to learn, I promptly forgot when I graduated high school). Every time I watch one of your videos I learn something and I'm able to truly understand and retain concepts that boggled my mind before.
When I started learning 3D modeling and animation on the Amiga circa 1988, one of the bigger challenges I set for myself was modeling and animating regular pentagonal dodecahedron with a raised star on each face (similar to the Chrysler logo) -- thus the dodeca-deathstar was born. A couple years later, working in high end video post production, I used the mathematical precision of the amazing Ampex digital optics device with a precise pentagon matte to layer a spinning dodecahedron with different video on each face -- calculating exact angles and depth offsets with an HP-15c calculator was a wonderful challenge that grew my maths skills considerably. Sorry Matt, but the platonic dodecahedron is, and always will be, the BEST dodecahedron.
8:24 matt's mental maths is on point
I couldn't be bothered marking the insides translates to "I gave it a go". Totally a parker cube.
I almost audibly gasped when you taped that square on. This was a really cool way of showing everything, better even than a 3d animation or something I think.
Are you THE Matt Parker from the Parker Square? What an honor!
Oh, yeah! I have Steve's last video on the "watch later" list but I always forget that list.
Thanks for reminding me, Matt!
Novice sorcerer: Pentagram on the floor, demon flies away.
Experienced Warlock: PENTAGRAM DODECAHEDRON!
8:32 it tickles me no end to learn that Matt is a Look Around You fan
I've always loved the regular compounds and their beautiful symmetry. When I built my first raytracer and figured out how to raytrace cylinders, the compound of 5 tetrahedra (which is my favorite) was one of the first things I made a render of. The regular compounds were the first things I printed when I first got my hands on a 3D printer. Great video as always!
7:40 this is next level editing XD
The dodecahedron is slowly de-throning the icosahedron from being my favorite platonic solid, thanks to crazy fun stuff like this.
Once again, you're knocking it out of the Parker with these videos!
"Lots of ridiculous maths things" .... is possibly the best description of this channel I have heard ....
I had to make a geometric solid out of paper as a highschool project. I chose a dodecahedron and it was pretty wild finding out that the whole net can be constructed (with straightedge and compass) using just a unit side and like 3 or 4 powers of the golden ratio. Imagine unfolding one half of the dodecahedron into a flower shape. That flower is bounded by a pentagon that's phi^2 larger than the faces.
I love the over-the-top editing style.
Just goes to show how you can't please everyone. I hate it.
There is something SO satisfying about those taped models. Thank you.
4:24 oops wrong graphic
Legend says that he still says "Diagonals in Space"
Matt thanks for running the only math channel I've found that will always explain things in a way that makes sense and makes me laugh every time! I've been watching your videos for a long time and you've only gotten better with time!
Discovering the square within the dodecahedron reminds me of the end of this demonstration of the Cross Sections app: th-cam.com/video/2uHfFp1XCPc/w-d-xo.html
Let me know if a tool like that could aid in visualization.
You could possibly do a follow-up on the hexagon within the hexahedron.
Fun fact: if you take that hexagon found within a cube, and replicate it three times by rotating it 90 degrees each time along one of the three four-fold axes (through the center of opposing faces), you'll have four hexagons whose edges describe those of the cuboctahedron..
One thing to do would be to also tape the insides, but wait, there's more...
you could have taped each cube with a tape (or drew with a marker) that reacts to a different wavelength of UV. Then by switching different blacklights you could switch between the cubes instead of having them on all at the same time.
Are we sure there is such a product?
That's a lovely, easy to visualize, and excellent way to explain these conceps. Absoulytely a great example of how to teach a concept really well. Good job, Matt.
3:30 I immediately know where this video is going and I love it!
your videos are capable of pulling one out of depression and make them fall deeper in love with mathematics. Thanks a lot for your work, sir.
6:45 the coolest part of this was that this was wholely unsurprising thanks to your previous videos on the rhombic dodecahedron
It's lovely when one maths investigation is helpful in understanding a completely unrelated one
Matt Parker: "There are 5 regular polyhedra."
Me, a jan Misali enjoyer: "there are 48 regular polyhedra"
The subtle joke for diagonals in space about 4 minutes in was really good. I imagine you were thinking, this is a bit silly, no one’s gonna even care. I care. That caught me off guard.
I love your visualization, it makes the whole thing insanely well understandable for me
Loved the book! I love how you were able to invent *time traveling* with trig! Mark my words, This is going to be the best-selling book in history!
The editing is genius
Matt, I'm pretty sure the dodecahedron mesh at 1:07 is wrong - aren't the opposite faces supposed to have opposite orientation? All the front faces are also meeting the rear faces on two edges!
oh yeah, that looks really weird now that i see it
I noticed this and immediately started looking for these comments. I'm glad I'm not alone!
It's a doParkerdecahedron.
A masterpiece of maths and editing
Turning obscure math into real world objects, keep up the good work Matt!
I aspire to enjoy my work as much as Matt
another fun way to build a dodecahedron
take a bunch of inflatable tubes (innertube , donut , torus)
and lash them together
i managed to build all of the solids
except for the icosahedron
it collapsed on itself
you end up with some very large pool toys
You should be honest - "You might know me from Numberphile video with the Parker Square"
The first time Matt had a collab with Adam, he referred to him as "Adam Savage from Mythbusters". In return, Adam Savage referred to Matt as "Matt from Numberphile".
Adam was with mythbusters. Matt isn't with numberphile...
@CBWP I mean he is, he has been doing videos since the start
@@wierdalien1 Numberphile is a collection of math. Was Matt in the first video? Are they friends? (those are rhetorical) Numberphile is a channel. Matt is a guest on their channel.
@@CBWP yes and yes and yes.
I love the ones where you can tell how much fun he had with it, and also where the concepts don't fly too far above my head. Also I can see myself making a shitty scaled down version of this in the future.
I have to be honest, I really like the stella octangula (the compound of two tetrahedra) just because it has a simplicity that a lot of the other regular compounds don't have. You can take a single glance at it and instantly know how it's constructed.
Pedantic error: Where Matt said the shapes "have mirror symmetry" I'm pretty sure he meant to say they DON'T have mirror symmetry.
This video made me realize why I’m not a mathematician. I can fully understand why a square being a integral part of a dodecahedron is fascinating to some people, but i literally said out loud in a room by myself “oh, I don’t like that”. I find it supremely uncomfortable.
So good, Matt!
Isn't the diagonal shown in 4:28 the longest diagonal, or am I seeing that wrong?
Still enjoying ur vids
Had the same thought!
Good ol' small stellated dodecahedron and the great stellated dodecahedron
@12:53 seven swans a-swimming,
six geese a-laying,
FIIIIIVE INTERSECTING CUUUUBES
the editing is amazing!!!
“Seemed clever at the start, I regretted it immediately”… that can basically be the theme of my life 😂
I always was a fan of the Icosahedron, but this video made me appreciate the Dodecahedron.
There is an error at 13:33 regarding the five intersecting octahedra: they do have a convex hull (every bounded set has) - it is just not a Platonic solid. It’s an Icosidodecahedron.
What a lovely coincidence I'm building nested platonic figures in bamboo sticks (up to 3m) with my students at the moment and analyzing this video is their homework. Thanks, Matt.
This is one of my favorite Stand-Up-Maths video!!!!!
The compound polyhedron made of a pair of intersecting regular tetrahedra, is aka the "stella octangula."
It was a favorite of Johannes Kepler, the guy who fiddled around with the 5 Platonic solids to try to explain the relative sizes of the planetary orbits, and the guy who formulated the famous "3 Laws of Planetary Motion" that bear his name.
Anyway, the 8 vertices of the stella octangula are the vertices of a cube.
Which also explains the 10 regular tetrahedra in the regular dodecahedron, once you've highlighted the 5 cubes in it.
Fred
PS. Also interesting to note, is that the main (longest) diagonal of an n-dimensional hypercube of unit edge, is √n.
This was such a good visual demonstration!
I can solve a megaminx, so I've spent a long time looking at dodecahedra. I noticed long ago that at a certain orientation you can find a set of 6 edges that are parallel to one of the 3 axes and you can therefore make those edges all line up with a face of a circumscribed cube. It's really cool to see the numbers behind an inscribed cube.
14:00 I think I know how to create all of them. You can make the double tetrahedron with the medium space diagonal of the dodecahedron, aka the face diagonal of the cube. You might even be able to get the fifth using cube-octahadron duality. I believe this would be drawing diagonals between the edges of the dodecahedron, specifically the 6 edges mentioned in paragraph 1
It hits better when you can visualize it in real life. Thanks Matt
Instead of taping the inside of the dodecahedron you can also tape the back of the tape with a dotted line pattern, and that way when you turn on the blacklight you get dotted lines whenever you are looking through it!
A cube with roofs on the faces? So that the roof planes of one face continously match up with the roofplanes of the neighbour faces. Or the triangle part of one roof matches up with the trapezium part of another roof to make the pentagon without any kinks. LOVE IT!
This is one of the most awesome things I've seen. So much better than CGI
Thanks, Matt. Thatt.
There are, in fact, more than 5 regular polyhedra! jan Misali has a great video on this, titled "There are 48 regular polyhedra" if I recall correctly
i will never look at a megaminx the same way after this
00:14 Also interesting , if you take the unit-square _down_ a dimension, the diagonal from endpoint to endpoint is the square root of 1. And if you go down yet another dimention, the diagonal becomes the square root of 0. Starting to see a pattern here.
Question then is: if you go up a dimension from the unit-cube... does the pattern hold?
Yes it does. You can do this with any unit-n-cube, with n being a positive integer.
Consider the case for n = 4.
We have a hypercube centred at (0,0,0,0). The two opposite points are (-0.5,-0.5,-0.5,-0.5) and (0.5,0.5,0.5,0.5). The distance would just be sqrt(1²+1² +1²+1²) = sqrt(4).
Generalise this to any positve integer n and you get that the space diagonals (or *diagonals in space* if you prefer) are sqrt(n).
Q.E.D.
I never really liked dodecahedrons... until now. They may not be as fun as a Truncated ditrigonary dishecatonicosachoron or as endearing as a Gyrotunnelled truncated cube, but I absolutely love how the ratios work out.
It is a very beautiful shape, I'm not sure whether I like the visualization of the single nested cube or the 5 cubes making pentagram faces more.
7:45 Not necessarily tape the insides, as that would have messed with the presentation, but prepare strips of effectively doublesided tapes.
great vid, and looking forward to your book and, as a UK resident, can't wait for the 6th day of 20th month to get it
We want a t shirt of this with text "Diagonals IN SPACE"
The graphic at 4:22 is wrong, it shows the long space diagonal when it should be the medium space diagonal!
The compound of 5 octahecdrons absolutely does have a convex hull. It's convex hull is the icosidodecahedron, an archimedean solid.
The cube inside the dodecahedron is actually how Euclid himself constructed the dodecahedron! Check out Book 13 of his Elements, it’s proposition 17!
at the boring end of the range, there are two intersecting tetrahedra in a cube - the edges of the tets being the diagonals of the cube faces. The centers of thee faces of an octahedron are the corners of a nested cube - and vice versa.
the stella octangula
You are on fire, so many videos
9:20 "The shape we were trying to made was the compound-5-intersecting tetrahedra. Here is a picture [...] and I've actually got a little print out over here" - am I the only one who was a bit sad that it wasn't a 3D print?
1:10 a Cube and a Dodecahedron. ??? look closer : the dodecahedron is wrong! Two facing sides are never parallel but rotated over 36 degrees or 108 or 180 .