How to make an edge-coloured origami dodecahedron
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- Опубликовано: 6 окт 2024
- Download my two colourings of the dodecahedron graph.
www.dropbox.co...
www.dropbox.co...
The site “What’s on my blackboard?” has a great post on colouring dodecahedra, which I found very useful.
whatsonmyblack...
Here is the Petersen Graph on wikipedia.
en.wikipedia.o...
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/
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
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
13:59 Lol the giant yellow pencil. I love your dedication.
*dodedication
Really like your zoom-macro solution. I gotta say it's genius, no, GINORMOUS.
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.
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.
ruclips.net/video/TV7sbaffuNo/видео.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
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12izlsqak22innxp04cj3goskvqwlyavnc0k
but these say the same
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z131tpkorlz5cvm5m04cftvw1wbmex1qkfw
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12hdt2yfrewz5hw104cirjw3uvkjfn4v1g
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12kyx1jfxrxfvbta22hfnmz2ujetxwjl04
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12iuzaimvenvx50f233tbkpcxnoxrusw
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12zvjoiiun1hp2ur23dc1artzvkzjilm
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12kzlazytqainumu23gyzjwsy2ie5aef
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z13ewn0omp32w5zll222jnvqptahvfstj
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z13msz1zvpeyexmk523wynebjkivvn01s
and it's also the property mentioned in the video.
ruclips.net/video/TV7sbaffuNo/видео.htmlm12s
Some point at the vertices and chirality
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z13li5baeve4c5lii04cc3jrnojoxzabvzw0k
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12cw1k4uxyqxfxgi04ce3cwjv3bz5yizsw0k
other focus on geometrics and mirror faces to find same conclusion.
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z13ecnopsvyrcf5ht04ci35aqruzzzwyqn00k
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12ijn0yvrr0x5euc23mxrwxeny2ddfl5
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z130xdibznmkcj1km223c5tqvsu2e3sbd04
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12bu15gnzb5fxent23jvnih0om5v1t1i
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z120hzeapnv3s15ic04cinxibq2guhersxo0k
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z135fv3gtwnyzpdec22efz5awt2edfj02
Bonus mention for additional math that is worth looking at.
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z121sjpwdseagd0qz04cj1v4nyrqijn5kto0k
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z133ypbzzqbctjxsh23luxy5xwzsupnxc04
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z13csfbiwrzpipcvq23hvnuy4kzszfth504
ruclips.net/video/TV7sbaffuNo/видео.html&lc=z12dz3j5toqkgjlhk04cjrnp1kjbvd0gq14
@@JensGulin The picture in ruclips.net/video/TV7sbaffuNo/видео.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.
"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
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
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!
We can always count on you to take the sight gags to the next dimension on your videos - absolutely loved the macro bit.
watching stand up maths:
1) yah, i know
2) yah, i know
3) yah, i know
4) oooh, that's cool
5) *jaw drops*
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.
I love the macro camera! I hope you use it more often. It really helped me to see the detail.
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 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.
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!
I tried doing this but I can't get the paper to change size =/
Maybe you need a higher end zoom camera.
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
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 :)
The giant yellow colored pencil is my favorite thing XD
Lovely! Perhaps next time you do origami you could try a stellated dodecahedron, or maybe a tetrahedron like the Five Intersecting Tetrahedra.
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
I loved the macro cam and its marvelous magnifying features.
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.
love the sneaky scale changes
The change in perspective to the larger items was blowing my mind the entire time.
There's somthing about a macro camera that you just cant beat!
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
the macro shots are great, very slick
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 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! :)
I love that he smiled at his own use of the over sized colored pencil.
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 love the "macro" camera, I haven't seen it done the way you did it before
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.
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.
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
Loving the comments about the macro bits. Great idea Matt!
Thank god (or Matt), finally something to watch during these boring days.
+Pav Phone I'll take that credit. :]
Woah, you really align with my nerdy interest, cubing, origami.
"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.
The consistency between the normal and macro camera is great! Who would've guessed that optic physics can be trusted? :P
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.
origami on standupmaths? hell yeah!
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!
The timing with the cut to the parallel larger version is just surreal 😂😂 top quality visual effect and it looks hilarious
Asabyavideo
gotta love that macro cam, and matt's cheeky grin x
Cried when I saw the huge yellow pencil. Entertainment factor is off the chart
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.
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.
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.
The fact that Matt has a giant yellow pencil is the best of the video
The "macro" pen really impressed me. Thanks Mat
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.
i was really expecting "first of all, last christmas, i gave you my heart"
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!
Matt doesn't have a macro lens -- he just made giant versions of everything.
Shh! Don't spoil it for the kids! ;-)
As a compsci guy I love the lists starting from 0, makes me smile everytime.
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...
You know when matts playing a prank, he smorks his way through the whole video!
Smirks* it's a word apple I swear!
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?
Love the scaling you used. Definitely gotta try to build one of these
that editing was so perfect, i was so confused at first. congrats, it was amazing
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
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.
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
"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!"
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.
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 the "macro" camera. ;D
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.
The editing on this video is brilliant!
This macro camera thing is trippin' me out.
bought your book! loving it already!
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.
I made one, yay... now I've coloured it up and plan on trying to make one that's 6 colours each as a single interweaved band around the edge. Each one should be 5 pieces and 5x6=30 so in theory it works.
The Best Macro Ever!
lovin' your close up camera
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
Matt, can I just say that your 'macro' lens was great to watch. Loved this video.
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!
On top of being cubic and 4-colorable, snarks also have to be connected and bridgeless. Also, I want one of those amazing macro cameras.
If memory serves (it's been a few years since I worked closely with this sort of thing), the edge colouring of the faces and why you get 6 of the 12 possible plus their mirror images has to do with the symmetry of the shape. One way to visualize what's happening is to cut the dodecahedron along the plane between two opposing faces and flip one half over to overlay the other half. Another way to consider it is that once you pick the first face's colouring, you've automatically restricted the possible colourings of the adjacent faces because of the restriction of 5-coloured faces and 3-coloured vertices (eg: a blue-green corner of a face means the third edge on that vertex only has 3 possible colourings - red, white, or yellow). As you continue to fill those faces under the given restriction, you find that eventually you're forced into using the mirror images of the original 6 faces rather than continuing to exhaust all 12.
I know this isn't a rigorous proof, but it's hopefully enough to get you on the right track to seeing why it works. If I have a chance later, I'll try to work out a full proof and provide it for you.
Unless I am mistaken, and I've gone cross-eyed double checking it, I have found a coloring using all 12 permutations. See plus.google.com/u/0/b/105076799072526075058/photos/photo/105076799072526075058/6368669072548148978?icm=false
Sorry about the two shades of green. They looked better in Paint.
So I was working out some constructions (starting with a symmetric construction which generated Matt's colouring) to better understand what was going on before moving into the group theory. I came across the same one you have when I realized that my first intuition when I made my original post was forgetting a degree of freedom. Of note, this colouring has some permutations at the vertices which are duplicated, so it uses 14 permutations once and 3 twice, leaving 3 unused. For example, the right-most outer pentagon vertex (red - light green - purple) has the same colouring as the left-most inner pentagon vertex. The three pairs are also located opposite each other on the dodecahedron (that is, at pairs of vertices which are farthest apart on opposite faces).
Now that I've built up some more intuition I'm working on the group theory behind all this. Since it's possible to consider each face as a permutation of four elements (ie: number the colours and enumerate the edges of a face, starting with the position labelled 1 so edges 2-5 are permuted), this looks like a subset of S(4) - possibly even a subgroup.
I'm currently taking the colouring Matt posted and the colouring you posted and looking at the groups they represent to work out the generators and see if I can come up with a complete proof of the possible colourings. I'll come back in a day or two once I've (hopefully) worked it out but this is a particularly busy week for me both at work and home so time is limited (and bouncy buses aren't very conducive to writing much during my commute). Hopefully I can find out if it's possible (or why not) to colour using 12 non-mirrored faces while still using all 20 vertices, or some other nice constructions, and why we lost 3 vertices to double another 3 when we made our construction.
mestiarcanus Here's a coloring I made directly. The previous one was the result of recursive edge swapping. It too has at least 3 duplicated vertex patterns. This coloring resulting from starting with Matt's coloring and swapping 3 pairs of edges, specifically the grey and purple in the outer pentagon, the grey and purple edges that connect the outer pentagon to the decagon and the uppermost grey and purple edges in the decagon. I see now that each of those swaps created a duplicate vertex coloration pattern.
plus.google.com/u/0/b/105076799072526075058/photos/photo/105076799072526075058/6368716594148316578?icm=false&authkey=CIG9ycXK_PLNLA&pageId=105076799072526075058
The edge swapping is actually how I had made my non-Parker colouring (or is this the Parker colouring because it doesn't use all the vertices). I found that by enforcing 3-colouring of vertices and 5-colouring of faces, once I made one edge pair swap it forced the rest and I suddenly had the 12 unique face colourings.
Great video Matt, this particular model is made using 30 x 108° modules, which were independently discovered by Robert Neale and Lewis Simon.
I always find it difficult to create a perfect 3 colouring, but I find the 5 much easier. Of course you need 5 colours on a face which is easy enough, but a simple rule can be used which ensures a perfect colouring every time. Take 5 units of different colours and assemble as a sort of saw horse shape, then for the next two in the pentagonal face, look at the colour opposite. You will end up with a ring of 5 and 5 struts standing up. Using the rule as before, create a ring of 10. Then use 5 units to make the top struts and finally a ring of 5 to make the top face.
Also, another way to visualise it is to imagine the edges being flat planes. All edges of the same colour will appear at 90° to each other, forming a cube shape.
Interestingly too, 6 colours works nicely and can be viewed as bands of colours running around the model.
Finally, if you look at the vertices of the Dodecahedron, you can inscribe 5 separate Tetrahedra, which forms the construction of a Compound of Five Tetrahedra. This is a particularly useful concept if colouring an origami model which uses 'verticies' modules. This model uses edge modules and there are of course origami models which use face modules too.
I haven’t scanned all 723 comments but it occurred to me that, as I haven’t any square paper, could I use any rectangular shaped such as a sheet of A4? Then I though about the geometry, which lead me to calculate the angle created by the folding. I used GEOMETRY, which is a branch of maths as I was taught, back in the old days. It turns out that the angle created by the folding, is 108.44 deg, which is so close to the ideal of 108.0 deg, that I thought it extremely clever. I found a drawing construction for a pentagon many years ago, back in the 70’s, and it astonishes me even to this day.
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!
I made one out of five different-colored 15cm square origami papers cut into 9 squares each. I had to employ a toothpick to get the last piece in since the thing was quite small.
3 hours well worth it.
Now I have the 15 small squares left over with which I should do something . . .
That's some impressive macro lens technology. Rick Moranis would approve.
the dodecahedron is my favorite Platonic solid :D thanks Matt
I see what you did there Mr Parker, Bravo!
that macro camera has some interesting properties :P
Waaah, I needed this video two days ago. Need to bookmark it for the next Christmas:D
I love how matt shrunk to do the close up shots
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.
You sir got a thumbs up purely for the dodecoration!
That macro lens is pretty sweet.
The most interesting way to colour it in my opinion is using 6 colours going along the equators of the dodecahedron.