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A 5-dimensional Analogue of the Rubik's Cube

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Originally shared by Erno Rubik

A 5-dimensional Analogue of the Rubik's Cube

You can now try to solve a 5-dimensional Rubik's Cube.
How many permutations of the 3x3 cube in 5-dimensions? 
Approx. 7.0 x 10^560
Good luck.

Visit here to download the program and learn more:
http://www.gravitation3d.com/magiccube5d/index.html

#mathematics   #geometry   #rubikscube

A more accurate rendering of the {6,3,3}.

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A more accurate rendering of the {6,3,3}.  This shows the geodesic edges more properly, as arcs orthogonal to the "plane at infinity" of hyperbolic space (vs. straight lines).

This the {6,3,3} honeycomb, as drawn by Roice Nelson.

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Originally shared by John Baez

This the {6,3,3} honeycomb, as drawn by Roice Nelson.

A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3d analogue of a tiling of the plane.  But besides honeycombs in ordinary Euclidean space, we can also have them in hyperbolic space.  This is a curved 3d space.  The {6,3,3} honeycomb lives in hyperbolic space.  That's why it looks weirdly distorted.  Actually all the hexagons are the same size... but we have to warp hyperbolic space to draw it in ordinary space.

You can learn more about all these concepts by going to my new blog, Visual Insight:

http://blogs.ams.org/visualinsight/2013/09/15/633-honeycomb-in-upper-half-space/

But let me just answer one obvious question: why is it called the {6,3,3} honeycomb?  

{6,3,3} is a Schläfli symbol.   The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. Similarly, the symbol for the hexagonal tiling hone…

Feels good to see a past project pop up unexpectedly.

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Feels good to see a past project pop up unexpectedly.  The 5D Rubik's cube was born in 2006 in the 4D_Cubing yahoo group, though it was conceived much earlier!  See this 1982 paper by Kamack and Keane:
http://helm.lu/cube/tesseract.pdf

Here's the 4D_Cubing page:
http://groups.yahoo.com/neo/groups/4D_Cubing/info

And you can browse the messages from March to May 2006, when all the excitement was happening:
http://groups.yahoo.com/neo/groups/4D_Cubing/conversations/messages?messageStartId=215&archiveSearch=true

This is the first time I've seen the new yahoo groups interface, so hopefully those links will work.

Originally shared by annarita ruberto

MagicCube5D

In the spirit of taking things too far, here is a fully functional 5-dimensional analogue of Rubik's cube.

The 3D cube has 43252003274489856000 permutations.

Go here to learn more and  download the program:
http://www.gravitation3d.com/magiccube5d/index.html

Here you find  Erno Rubik, the inventor of  famous Rubik's Cube: