How to make an edge-coloured origami dodecahedron
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- čas přidán 25. 12. 2016
- Check out Skillshare: skl.sh/standupmaths
Using the code STANDUPMATHS you can get two months of the premium version for free (which is normally from $8 USD per month).
Go straight to the origami star lesson: skl.sh/origami-class
Download my two colourings of the dodecahedron graph.
www.dropbox.com/s/x1sn5tqi4ar...
www.dropbox.com/s/3rpsbitya3n...
The site “What’s on my blackboard?” has a great post on colouring dodecahedra, which I found very useful.
whatsonmyblackboard.wordpress...
Here is the Petersen Graph on wikipedia.
en.wikipedia.org/wiki/Peterse...
So far I have been sent ZERO solutions to the “Why those six pentagon edge-colourings” challenge. Watch this space!
CORRECTIONS
- I sometimes say things like “three colouring” without specifying that I’m talking about colouring the edges, not the vertices (which are the default option). But I hope it is clear from context.
- Let me know if you spot any other errors!
Music by Howard Carter
Design by Simon Wright
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
Maths book: makeanddo4D.com/
Nerdy maths toys: mathsgear.co.uk/ - Zábava
I liked how the macro camera scaled down your hands ;)
+schogaia That's the quality you get from a top-end macro camera.
7:04 - And the "mouth mute" button must be handy, preventing your mouth from becoming a distraction when viewing the "macro" mode.
Only the best for this channel!
schogaia I thought his hands were shrinking 😂
schogaia me too, I was wondering how that possible?
You see how big the pen looks
13:59 Lol the giant yellow pencil. I love your dedication.
*dodedication
"It's a square so you can fold it in half whichever way you want."
*folds it diagonally*
trejkaz yes... or infinitely many other ways, if you don't over define what half is (as hasn't been done here) as long as the fold passes through the Centre of the paper.
Wait, that's illegal
Loving the macro camera! Haha.
+Enzo Grispo It's cutting-edge close-up technology.
Well played, Mr. Parker. Well played.
PS: My reaction at "0:20": 'No, you don't, your hands are still in scale and not skewed to hell and beyond.'
More than the actual origami dodecoration to be honest :D
I love the pen close-up.
especially the way it makes the dodecahedron look bigger and Matt's hands look smaller
I don't know what's funnier, the "macro" camera gag, or the fact that so many people are so confused and don't understand what's happening haha XD
Yeah, it's a wonderful invention that magnifies only inanimate objects, like the paper squares; not animate ones, like Matt's hands.
I don't know how that escapes so many people.
it's because the editing is so well done XD it's great
D. C. Lol I know first I see he holds the paper with his fingers and next clip he holds the paper with his wole hand Lol I was like o.o all the time
it got me the first time, but once I saw his hands were tiny compared to the second dodecahedron shot I realized that I had been gaffed, but at least not all the way through the video, I find it hilarious.
Or the table.
Really like your zoom-macro solution. I gotta say it's genius, no, GINORMOUS.
Wow, the close ups... just amazing. Standupmaths really is at the pinnacle of Macro Lens technology!
Hi Matt! I was actually playing with this a few weeks ago.
It turns out the symmetry group (all the rotations) of the dodecahedron is the same group (isomorphic) as the "Alternating group on five letters" (A5) - which is taking a list of 5 things i.e. {red, yellow, blue, green, white} and swapping one element for another an even number of times. i.e. {yellow, red, blue, white, green} is another element under this rule (2 swaps - 1st,2nd and 4th,5th - OK). but {yellow,red,blue,green,white} isn't (1 swap - 1st,2nd - Not OK).
All the other even number of swaps you can do will be elements of this group.
And now, if you look at any pentagon face (in one of it's 5 orientations) - the order of the colours of the edges (count them going round clockwise, from the bottom edge) will be elements of the group.
This also means rotating about one of the pentagons is a group operation, i.e. {red, yellow, blue, green, white} -> {yellow, blue, green, white, red}
Another operation you can do is rotate around a vertex, which will permute three of the colours. (e.g. 1st, 3rd,5th) -> {blue, yellow, white, green, red}
With these two operations, you will be able to get to any other orientation of the dodecoration (they are the generators of the group)
So you're right that any single swaps don't appear, as they are elements of the larger group, which allows for both even and odd swaps ("symmetric group" - S5) - but the dodecahedron rotations only allows you to do even swaps.
If you do a single 'odd' swap on the list of colours, and then only do even swaps afterwards, you'll have an odd swap in total. So if you build the dodecahedron out the 'missing' pieces, it won't share any of the face colourings of the original.
The descriptions says there is still no answer to the coloring question, but I scanned the comments and suggest that everyone "like" this thread to get it higher for those looking for it.
Coloring the edges are equivalent to coloring the faces of a rhombic triacontahedron
www.mi.sanu.ac.rs/vismath/zefiro/_polyhedra_colouring_2007_08_10.html#Rhombic%20triacontahedron
www.georgehart.com/virtual-polyhedra/five-cube-intersection.html
They say there that it's only possible to color it one way. Either that's wrong or you would find that the two color schemes found are actually one and the same, just differing in the mapping of color to each letter. Haven't verified myself, but there is actually a different suggestion below, found if you're not blinded by symmetry.
Contradicion, yes here they seem to make sure to avoid mirror faces and find another color scheme.
czcams.com/video/TV7sbaffuNo/video.html&lc=z12ry51hululc5c3y22wi5q5ommqhhini
Explaining the colors Matt found and missed however relies on a symmetric choice. Best explanation, I think, is this thread based on group theory
czcams.com/video/TV7sbaffuNo/video.html&lc=z12izlsqak22innxp04cj3goskvqwlyavnc0k
but these say the same
czcams.com/video/TV7sbaffuNo/video.html&lc=z131tpkorlz5cvm5m04cftvw1wbmex1qkfw
czcams.com/video/TV7sbaffuNo/video.html&lc=z12hdt2yfrewz5hw104cirjw3uvkjfn4v1g
czcams.com/video/TV7sbaffuNo/video.html&lc=z12kyx1jfxrxfvbta22hfnmz2ujetxwjl04
czcams.com/video/TV7sbaffuNo/video.html&lc=z12iuzaimvenvx50f233tbkpcxnoxrusw
czcams.com/video/TV7sbaffuNo/video.html&lc=z12zvjoiiun1hp2ur23dc1artzvkzjilm
czcams.com/video/TV7sbaffuNo/video.html&lc=z12kzlazytqainumu23gyzjwsy2ie5aef
czcams.com/video/TV7sbaffuNo/video.html&lc=z13ewn0omp32w5zll222jnvqptahvfstj
czcams.com/video/TV7sbaffuNo/video.html&lc=z13msz1zvpeyexmk523wynebjkivvn01s
and it's also the property mentioned in the video.
czcams.com/video/TV7sbaffuNo/video.htmlm12s
Some point at the vertices and chirality
czcams.com/video/TV7sbaffuNo/video.html&lc=z13li5baeve4c5lii04cc3jrnojoxzabvzw0k
czcams.com/video/TV7sbaffuNo/video.html&lc=z12cw1k4uxyqxfxgi04ce3cwjv3bz5yizsw0k
other focus on geometrics and mirror faces to find same conclusion.
czcams.com/video/TV7sbaffuNo/video.html&lc=z13ecnopsvyrcf5ht04ci35aqruzzzwyqn00k
czcams.com/video/TV7sbaffuNo/video.html&lc=z12ijn0yvrr0x5euc23mxrwxeny2ddfl5
czcams.com/video/TV7sbaffuNo/video.html&lc=z130xdibznmkcj1km223c5tqvsu2e3sbd04
czcams.com/video/TV7sbaffuNo/video.html&lc=z12bu15gnzb5fxent23jvnih0om5v1t1i
czcams.com/video/TV7sbaffuNo/video.html&lc=z120hzeapnv3s15ic04cinxibq2guhersxo0k
czcams.com/video/TV7sbaffuNo/video.html&lc=z135fv3gtwnyzpdec22efz5awt2edfj02
Bonus mention for additional math that is worth looking at.
czcams.com/video/TV7sbaffuNo/video.html&lc=z121sjpwdseagd0qz04cj1v4nyrqijn5kto0k
czcams.com/video/TV7sbaffuNo/video.html&lc=z133ypbzzqbctjxsh23luxy5xwzsupnxc04
czcams.com/video/TV7sbaffuNo/video.html&lc=z13csfbiwrzpipcvq23hvnuy4kzszfth504
czcams.com/video/TV7sbaffuNo/video.html&lc=z12dz3j5toqkgjlhk04cjrnp1kjbvd0gq14
@@JensGulin The picture in czcams.com/video/TV7sbaffuNo/video.html&lc=z12ry51hululc5c3y22wi5q5ommqhhini does not work anymore. I believe at i.imgur.com/9zS6kWW.png is a colouring that avoids mirror faces. Also look at pastebin.com/tk2nBeLu for a write up I did.
I tried doing this but I can't get the paper to change size =/
Maybe you need a higher end zoom camera.
watching stand up maths:
1) yah, i know
2) yah, i know
3) yah, i know
4) oooh, that's cool
5) *jaw drops*
Man, you can make anything look bigger with different angle and lens!
I think that the paper was actually bigger, considering the relationship to a hand :)
Yeah, everything except Matt's hands. :P
That opening joke almost made me spit out my tea. Well done, Mr. Parker, for that Parker square of a pun. :p
Instruction not clear enough; I made a hyperdodecahedron.
Topsoil Depletion Awareness haha most other people won't understand this
I think on a channel like this many people should understand what it is.
Absolutely, there were imprecise. I folded in half diagonally and ended up with a rhombic dodecahedron.
Instructions not clear; penis caught in ceiling fan.
Topsoil Depletion Awareness
hyperdodecaration
We can always count on you to take the sight gags to the next dimension on your videos - absolutely loved the macro bit.
Love the macro lens. Somehow your hands aren't affected ;-)
It is the angle. He is a mathematician, so he knows how to fool our perspective
Marcos Vinícius Petri no he just used bigger objects... there is no angle that changes the size of things that drastically
bloodhiybrid Man, don't be a party pooper. I know this. He obviously used two sizes of paper and made two objects of different sizes. We are all just getting into the fantasy.
He also used a giant fountain pen lol.
OMG, I died now!! I hadn't watch that part, thanks!
Dude, its killing me that the macro shots use large paper than the regular shots. That confused the hell out of me for the first couple minutes.
He also has huge identical pens and pencils for those shots. Cannot unsee.
If you look closely enough it's not a micro shot. He just made a 2nd one but then about 2x larger. You can see that at 6:16 When the larger one he is making is the scale of his table. :D
Scy nah, those were just funny tbh, thanks to their stubby writing ends it helped make the larger version look clearly distinct from the smaller version, letting me enjoy the humour. :P
the best is when he has a giant version of the pen
You'd think that people who watch this channel appreciate angles...
The "Macro Cam" was brilliant. The totally normal sized pen was what sold me on it, hilarious stuff.
I love the macro camera! I hope you use it more often. It really helped me to see the detail.
The giant yellow colored pencil is my favorite thing XD
For anyone wondering why this works, the obtuse angle he made when folding it like that at 3:15 turns out to be about 108.4°, which is only about 0.23% away from the 108° angle of a regular pentagon. A slight drawback is that these little errors do eventually rack up and the vertices of the dodecahedron have little holes in them.
If I ever end up in government I'm making this guy official secretary of math
senororlando2 i would too even though im american
Some Random Fellow me too, in the commonwealth I think it'd be Maths
The chief executive decorative mathematician
I made five of these a few years ago after watching a James Grime video, all out of post-it notes. I did one with one color, one with two, one with three, one with five, and one with six colors. I made a plan to do all the other factors of 30, but then I realized I would need a lot more colors of post-it notes and then I got bored and did something else.
Lovely! Perhaps next time you do origami you could try a stellated dodecahedron, or maybe a tetrahedron like the Five Intersecting Tetrahedra.
I'm so happy you did this! I made so many very recently. My friends and I even made a snowman with dodecahedra of decreasing sizes!
If you swap two colours throughout the 5-coloured dodecahedron you made, you will obviously still have a dodecahedron with a valid 5-colouring. Every face will have exactly 2 colours swapped by construction. So there are two ways of 5-colouring it - one containing all even permutations of the 5 colours, and one with all the odd permutations. What is even more interesting is that rotating a 5-coloured dodecahedron is equivalent to permuting the colours - by an even permutation cause otherwise you get the other one. There are 60 even permutations of the 5 colours, corresponding exactly to the 60 rotations of the dodecahedron.
Was going to explain this myself, but I checked first to see if any comments had already explained it better than I would. I knew I found one just by recognising the username :)
On the 5-colored version each color of edges basically makes a cube - if you look at, say, all the white edges they line up exactly with the faces of a cube. This is true of all the colors
indeed, the faces on the five colored one appear to be even permutations of an original face (as in, an even number of edge swaps). This is supported by the opposite faces, because they are sequences of length 5 (which is odd), the reverse of the sequence is an even permutation of the original.
There's somthing about a macro camera that you just cant beat!
Matt doesn't have a macro lens -- he just made giant versions of everything.
Shh! Don't spoil it for the kids! ;-)
This macro camera thing is trippin' me out.
Maybe the 6 permutations of colours on the pentagons are the even permuatations, and the other 6 not on there are the odd permutations? If that's the case then swapping two of the colours should give you all of the other permutations.
love the sneaky scale changes
Your channel brings me so much joy. And I'm still recovering from seeing that giant yellow pencil in your macro cam. Truly a stroke of genius. Thank you.
I loved the macro cam and its marvelous magnifying features.
I love that he smiled at his own use of the over sized colored pencil.
The fact that Matt has a giant yellow pencil is the best of the video
I just found this in my watch later where is has been lurking for 11 months. Much to my surprise it contained an amazing macro photography effect! Well done.
the macro shots are great, very slick
origami on standupmaths? hell yeah!
Y'see... Now that I'm looking at it... I think, and stick with me here, the macro shots MIGHT actually be just bigger pieces of paper.
+xXjesperoXx I don't understand what you're trying to say.
No, it's one of those new 'reverse perspective' macro lenses,, The hands (nearer the camera) are projected onto a smaller area than the paper. something to do with non-euclidian geometry, I'm sure +Standupmaths can explain, maybe do a video about it?
I wish I could add an extra like for the macro cam, such a casual (but awesome) use of the jumbo-sized coloured-pencils and multi-coloured pen! :)
"and I have a macro scale camera..."
Oh god. I didn't notice on the first and second transitions to the macro camera. But when I did it gave me such a giggle. Well played.
i was really expecting "first of all, last christmas, i gave you my heart"
If you use 6 colors, you can place the colors in a way that makes the dodecoration look like 6 intersecting rings, each of one color.
Loving the comments about the macro bits. Great idea Matt!
I believe the reason that the opposite faces wind up mirrors instead of the other 6 combinations is because of the same properties that cause the opposite vertices to be mirrors.
For example, I happen to have the video paused at 17:46 and I can see the Yellow-Green-White vertex facing me, and through the whole directly under it I can see the opposite vertex as well. And I can see that the face with the white and green sides of that vertex (top-right of that vertex), is opposite of the face with the white and green edges of the opposite vertex as well.
So continuing this pattern around means that each opposite face is forced to be a mirror, so you wind up with 6 combinations and their mirrors.
Thank god (or Matt), finally something to watch during these boring days.
+Pav Phone I'll take that credit. :]
Cried when I saw the huge yellow pencil. Entertainment factor is off the chart
As for why you only get half of the possible sets of 5, it's a bit of group theory. The symmetry group is A5, the even permutations on 5 points, which are here represented as the 5 colors. Note that, with the nice way you've placed the 5 colors, any way you move the dodecoration you are permuting some or all of the colors. Since the symmetry group is only even permutations, you can't have a single transposition, which would be odd. Yet you need a transposition to get from some of the sets of 5 to the others. In other words, there are two conjugacy classes of elements of order 5.
Notice how the centers of every edge group of some color forms the centers of the 6 sides of a cube (or the 6 vertices of an octahedron), this shows how the symmetry of a dodecahedron relates to the symmetry of a cube.
cool stuff!
I love the "macro" camera. ;D
usually i wait for a really funny part in a video before i give it a thumbs up.
8 seconds - "dodecoration" must be a new record! gj, matt!
Love the scaling you used. Definitely gotta try to build one of these
so there ARE perks to staying up till 5:22AM for me
+Relish Relisher It's a civilised 13:22 here.
damn you timezones!
Relish Relisher when you try to say a time and it turns into a time stamp
Whatsapp makes me so angry with it's number highlighting. it's totally unnecessary!
13:59 should be the thumbnail for this video. Hilarious! And how is Matt not cracking up laughing at 14:01? He plays it ALMOST straight-faced.
i love the "macro" camera, I haven't seen it done the way you did it before
i have used a similar module to make a few different shapes. the dodecahedron is one of the smallest. the one most people like most is the torus,10 heptagons all connected in a line to form the inner ring, hexagons to fill space, and 10 pentagons along the outer edge to curve in the surface. it can be done with just 5 but then it loses its roundness similar to how the buckyball(90 pieces) is much more round than the dodecahedron
The timing with the cut to the parallel larger version is just surreal 😂😂 top quality visual effect and it looks hilarious
Asabyavideo
You know when matts playing a prank, he smorks his way through the whole video!
Smirks* it's a word apple I swear!
The change in perspective to the larger items was blowing my mind the entire time.
I made about ten of these as centerpieces for an event and have become really familiar with the method. I was therefore very excited to find that Matt had a tutorial on the exact same design.
The consistency between the normal and macro camera is great! Who would've guessed that optic physics can be trusted? :P
The fact you bought an huge yellow pencil makes the "macro" cemare so much better. To bad you didn't buy yourself a huge table and a set of huge foam hands.
bought your book! loving it already!
where can you get those giant pens and pencils? They would be fun
Woah, you really align with my nerdy interest, cubing, origami.
I'm in love with the macro came, keep using it,
Also Great channel (one of my favourites ) I mean the way you present Maths, origami as in this video or everything else is great, keep up, just please try to increase your viewer interactions, anyway love your channel, great work.
I think it has to do with the "parity" of the different pentagon colourings. So there are 120 different permutations of the five colours, but there are only 24 essentially different permutations because of rotation. If you count the mirror image, there are only 12. Both rotation and reflection are "even" permutations, because you have to do an even number of switches to rotate or reflect with 5 colours. So of the 12 different permutations of the pentagon colouring, 6 of them have one parity, and 6 of them have the other parity. That is why there were only 6 different colourings on your dodecahedron (up to rotations and reflections).
I'm not exactly sure if you can make a dodecahedron with all 12 permutations, something tells me it might be impossible...
that editing was so perfect, i was so confused at first. congrats, it was amazing
gotta love that macro cam, and matt's cheeky grin x
The "macro" pen really impressed me. Thanks Mat
lovin' your close up camera
The editing on this video is brilliant!
What on earth is with the random size changes? It looks like Matt is playing with us by swapping in larger/smaller items during the close-ups and wide shots, just to get one over on us :P
If you number the colors 0, 1, 2, 3, and 4, and then write down the numbers around a face (starting at zero) you'll get the even permutations of the numbers 1 to 4 (or odd, based on how you number them).
I love this sort of practical video! This and the hexastix are awesome! Would love it if you could do more videos along these lines! :)
Matt, can I just say that your 'macro' lens was great to watch. Loved this video.
I was watching this video with my nephew, who does not speak English but was interested in the origami side of it. So I muted the sound, it's very funny to watch you with no sound!
that macro camera has some interesting properties :P
All colourings of the pentagrams are even permutations among each other, that means, you get from one pentagram with an even number of transpositions to any other pentagram, but no odd permutations are on the dodecahedron. The even permutations are a subgroup from all permutations.
love the camera work
The Best Macro Ever!
It makes sense that you can't have all 12 combinations of faces as the graph for the dodecahedron only had 6 faces, with it being flipped when you look from the "bottom". Same reason it works for the corners, rather than thinking of the 3d object it may be clearer to look for the face/edge/vertex patterns through the graph
I see what you did there Mr Parker, Bravo!
As a compsci guy I love the lists starting from 0, makes me smile everytime.
Hey Matt, I got your book for christmas. It's awesome!
You had WAY too much fun with that macro camera
"I'm not gonna buy that the paper at 2:03 is the same of the one at 2:06"
A few minutes later (13:49): "Oh c'mon!"
My favorite thing about this video is how the "macro close-up camera" is actually just him folding massive squares of paper on a separate video
Waaah, I needed this video two days ago. Need to bookmark it for the next Christmas:D
Giant props Matt.
I am confused by how the graph helps with the origami but love this video...the giant pens threw me for a second. Haha well done!
I've never notice how tiny your hands are, Matt!
Almost Trump-like in their tinyness!
That's one cool shape, my friend! 👍
wow!! loving the macro lens- i laughed every time it was shown!!😂😂
Regarding not all 12 cycles of 5 colours appearing in the 5-edge-colouring, this is because of a few things combined:
1) For each colour, the six edges of that colour lie on the faces of a cube,
2) If a rotation of the dodecahedron leaves a blue edge where a red one used to be, then all six blue edges end up where all six red edges used to be, so each rotational symmetry of the dodecahedron naturally results in a permutation of the five colours.
3) No non-trivial rotation leaves three of these cubes where they are, and the trivial rotation leaves all five where they are, so no rotation merely swaps two cubes.
4) If a cycle ABCDE appears in the edge-colouring, BACDE can't also appear, otherwise the rotation mapping the ABCDE face to the BACDE face would contradict #3.
This embedding of five cubes in a dodecahedron is a nice visual proof that the group of rotational symmetries of the dodecahedron is A5, the alternating group on 5 letters (or 5 colours). "Alternating" in this context means you can't swap two and leave the rest fixed.
Interesting how the macro cam miniaturizes the hand. ;)
Your close up thing reminds me of this Aussie children's show that was on when I was a kid, called Johnson and Friends. All the proportions were slightly off and it made me feel strange and uneasy when I watched it but I didn't know why. I figured out, when I was older, it was because it was actually a giant set that was made to look like it was in regular proportions even though it wasn't.
One interesting thing about the 5-color one is that whatever edge you use on the face is opposite a piece that's part of the vertex of the same color. (like at 12.44, you can see the two yellow ones). Maybe that has something to do with how you only get 6 (+ mirror images) because once you pick one vertex, you can only build out from that? If you've only made one, that could be why you didn't see the other 6 faces. If you made another version with a different starting vertex, I think that
should give you the 'other' dodecahedron!
That's some impressive macro lens technology. Rick Moranis would approve.