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Higher-dimensional commutative laws

If you click on the link, you'll see +Scott Carter's picture of the two sides of the Zamolodchikov tetrahedron equation, a tongue-twisting and brain-bending equation that shows up in topology.

My blog article explains it, with pictures. But in simple terms, the idea is this. When you think of the commutative law

xy = yx

as a process rather than an equation, it's the process of switching two things: in this case, the letters x and y. You can draw this process using two strings that switch places: that is, a very simple "braid" like this:

\ /
/
/ \

It turns out that this braid obeys an equation of its own, the Yang-Baxter equation. This is easy to explain with pictures, but it's hard to draw pictures here, so visit my blog article.

If you then think of the Yang-Baxter equation as a process of its own, that process satisfies an equation: the Zamolodchikov tetrahedron equation. This equation really wants to be drawn in 4 dimensions, but you can get away with drawing it in 3 - just as I drew that simple braid on the plane.

This goes on forever: whenever you reinterpret a equation as a process, that process can (and usually should) obey new equations of its own. As you do this, you naturally go to higher dimensions. The Zamolodchikov tetrahedron equation is a nice example of how this works!

#topology #4d

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The hypercube of duads

The British mathematician James Joseph Sylvester, who lived from 1814 to 1897, was one of the first to dig deeply into the beautiful patterns you can form using finite sets.  But he got into lots of trouble.

For example, he entered University College London at the age of 14.  But after just five months, he was accused of threatening a fellow student with a knife in the dining hall!  His parents took him out of college and waited for him to grow up a bit more.

Later, at the age of 27, he went to the United States and became the chair of mathematics at the University of Virginia in Charlottesville.  After just a few months, a student reading a newspaper in one of Sylvester's lectures insulted him.  Sylvester struck him with a sword stick.  The student collapsed in shock.  Sylvester thought he'd killed the guy!   He fled to New York where one of his brothers was living. 

Later he came back.  According to the biography I'm reading, "the abuse suffered by Sylvester from this student got worse after this".  Soon he quit his job.

One thing I like about Sylvester is that he invented lots of terms for mathematical concepts.  Some of them have caught on: matrix, discriminant, invariant, totient, and Jacobian!  Others have not: cyclotheme, meicatecticizant, tamisage and dozens more.

Sylvester defined a duad to be a way of choosing 2 things from a set.  A set of 6 things has 15 duads.   A hypercube has 16 corners.  This picture by Greg Egan shows a hypercube with 15 of its 16 corners labelled by duads.   The bottom corner is different.

This may seem just cute, but in fact it can help you visualize a rather wonderful fact: the group of permutations of 6 things is isomorphic to the symmetry group of a 4-dimensional symplectic vector space over the field with 2 elements.

For details, read this:

http://blogs.ams.org/visualinsight/2015/09/01/hypercube-of-duads/

I quoted this biography of Sylvester:

http://www-history.mcs.st-and.ac.uk/history/Biographies/Sylvester.html

#4d
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A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

http://math.ucr.edu/home/baez/mathematical/dodecahedron_with_10_tetrahedra.gif

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

You can also do something like this starting with a cube.  A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  But you can do this in 2 ways.  If you choose both, you get a cube with 2 tetrahedra in it:

http://math.ucr.edu/home/baez/mathematical/cube_with_2_tetrahedra.gif

All this is just the start of a much more elaborate and beautiful story which also involves the golden ratio, the quaternions, and 4-dimensional shapes like the 4-simplex, which has 5 tetrahedral faces, and the 600-cell, which has 600 tetrahedral faces!   You can read it here:

http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/

I learned some of this story from Adrian Ocneanu at Penn State University.  Greg Egan and I figured out the rest... or most of the rest.  There's an unproven conjecture here, which needs to be true to make the whole story work.  Can you prove it?

Puzzle: If you take a regular 4-simplex whose vertices are unit quaternions, with the first equal to 1, can you prove the other 4 vertices generate a free group on 4 elements?

Hmm, I see that this puzzle has been solved by +Ian Agol and someone else on Mathoverflow:

http://mathoverflow.net/questions/204464/do-unit-quaternions-at-vertices-of-a-regular-4-simplex-one-being-1-generate-a

I don't understand the solution yet, because I don't know what a 'Bass-Serre tree' is... but I'll try to learn about this.  Math is infinite, there's always more to learn.

#geometry #4d  
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Hypergears!

A while ago I showed you a sphere covered with gears.  Some of you wondered if a 4-dimensional version was possible.  Here's a 4d version, created by Greg Egan!

It's hard to draw a hypersphere covered with 3d balls.   Luckily, half a hypersphere can be drawn as the inside of a big ball, just as half a sphere can be drawn as the inside of a circle - imagine flattening out the northern hemisphere of the Earth and drawing it as a disk.

So, that's what Egan shows here: half a hypersphere full of rotating 3d balls, touching at points.  And so you can see different views, the whole thing is rotating, too.

If we could see the whole hypersphere, it would contain one ball for each vertex, edge, square and cube in a 4-dimensional cube! 

Here is his description:

Take a 4-cube and project the centres of its k-cubes (k=0,1,2,3) onto a circumscribing 3-sphere.  At each projected k-centre, place a 2-sphere, with radii depending only on k, and chosen so that the sphere at each k-centre is tangent to the ones at the nearest (k+1)-centres.  There’s one degree of freedom in choosing the radii, which we fix by making the spheres at the 4-cube’s vertices the same size as the ones at the projected edge centres.

Give all the 2-spheres angular velocities such that they roll against each other at their points of tangency, like ball bearings.  The animation shows a solution where the angular speed depends only on k for the spheres at k=0,1 and 3, but there are two different speeds for k=2, the centres of the squares.  This image shows half the 3-sphere projected orthogonally into a solid ball in three dimensions (with some of the 2-spheres bisected at the boundary).

The periods of the spheres are not in rational ratios to each other, so this motion could not be performed exactly with gears in place of friction between the spheres.

The spheres are coloured:

• red at the vertices,
• green at the edge centres,
• blue at the 2-face centres, and
• cyan at the 3-face centres.

Just for fun, here are some counts of things ...

• There are 16 vertices, 32 edges, 24 2-faces and 8 3-faces in the complete hypercube.
• Every red sphere at a vertex touches 4 green spheres (edge centres).
• Every green sphere at an edge centre touches 2 red spheres (vertices) and 3 blue spheres (2-face centres).
• Every blue sphere at a 2-face centre touches 4 green spheres (edge centres) and 2 cyan spheres (3-face centres).
• Every cyan sphere at a 3-face centre touches 6 blue spheres (2-face centres).
• This means that altogether there are 16×4 + 32×3 + 24×2 = 8×6 + 24×4 + 32×2 = 208 points of contact between spheres.

• Lying on the boundary of the solid ball in the animation are 8 edge-centres (green), 12 2-face centres (blue) and 6 3-face centres (cyan), along with 48 of the 208 points of contact.

#gears   #4d
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This is a truncated 4-dimensional cube, drawn in an artistic curved style by +Jos Leys.

You can take an ordinary 3-dimensional cube, cut off its corners, and get a polyhedron with 2×3 = 6 octagonal faces and 2³ = 8 triangular faces. It’s called the truncated cube.

Similarly, you can take a 4-dimensional cube, cut off its corners, and get a shape with 2×4 = 8 truncated cubes as faces and 2⁴ = 16 tetrahedral faces!  It’s called the truncated 4-cube.

For more on the math, and for links to even fancier pictures of 4d shapes by Jos Leys, visit my Visual Insight blog:

http://blogs.ams.org/visualinsight/2013/12/15/truncated-hypercube/

#4d  
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This is 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 honeycomb is {6,3,3} because 3 hexagonal tilings meet along each edge.

3 hexagonal tilings meeting at each edge!   That's a bit hard to visualize.  But if you stare carefully at this picture, and look at one of the big fat edges near the top, you can see 2 hexagonal tilings meeting at that edge - one in front that's easy to see, and one in back.  The third, not shown, goes upward.

#4d
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Inside the 120-cell

Take 4 dodecahedra.  You can fit them together at a corner, but there's some room left over.  To fix this, use your superpowers and curl them up into the fourth dimension - just like a fold-up paper model of a cube, but in one more dimension!  With 120 dodecahedra, you can build a nice shape this way: the 120-cell.  

Since this shape lives in 4 dimensions, it's hard to visualize.  But just as an ant can crawl around the surface of a paper cube and try to understand it, we can move around inside the surface of the 120-cell and try to understand that!  That's what this movie shows.

This movie was made by Jason Hise.  Like a true hero, he put it on WikiCommons with this copyright:

You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

I'm going to do something like this for my column This Week's Finds and other webpages as soon as I find the time.

The 120-cell is connected to lots of other fun math.   For example...

You can get a Penrose tiling with approximate pentagonal symmetry - a quasiperiodic tiling that never quite repeats - by taking a perfectly repetitive pattern in 4 dimensions and slicing it at a funny angle.  

And the same idea works in higher dimensions!  You can get a quasiperiodic pattern with approximate 120-cell symmetry in 4 dimensions by cleverly slicing a pattern in 8 dimensions.  Start by taking the densest possible perfectly repetitive way to pack equal-sized balls in 8 dimensions.  The centers of these balls form the E8 lattice.  Then slice that...

But why is this related to the 120-cell?  Here's part of why: the symmetries of the 120-cell give symmetries of the E8 root polytope!

A rough explanation:

When you pack balls in the E8 pattern, each ball touches 240 others.  The centers of its 240 nearest neighbors are the corners of my favorite shape: the E8 root polytope.

Start with the 240 corners of the E8 root polytope.   There's a clever trick for grouping them into 120 ordered pairs.  Each symmetry of the 120-cell gives a permutation of its 120 dodecahedral faces - and thanks to this clever trick, we get a permutation of the 240 corners of the E8 root polytope.  And this permutation actually comes from a symmetry of the E8 root polytope!

If you want the details, go here:

• Veit Elser and Neil Sloane, A highly symmetric four-dimensional quasicrystal, J. Phys. A 20 (1987), 6161-6168. Also available at
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.1188

• Andreas Fring and Christian Korff, Non-crystallographic reduction of Calogero-Moser models, Jour. Phys. A 39 (2006), 1115-1131. Also available at http://arxiv.org/abs/hep-th/0509152

or my summary here:

http://math.ucr.edu/home/baez/week270.html

As you might expect, mathematicians have a dry, seemingly dull way to describe what I just said: "There's a homomorphism from the H4 Coxeter group into the E8 Coxeter group".  We talk this way so nobody realizes we're getting paid to have fun.  :-)

#spnetwork arXiv:hep-th/0509152 #quasicrystal #coxeter #E8 #4d  
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Suppose you live in 4 dimensions.  Draw the 4 coordinate axes as dashed lines.  Draw a dot one inch from the origin along each axis in each direction.  These dots are the corners of the 4d cross-polytope.  If you draw the corners and also the edges connecting them, it looks like this!  It's the 4-dimensional brother of the octahedron.

For more pictures and explanation, try this:

http://johncarlosbaez.wordpress.com/2013/07/05/symmetry-and-the-fourth-dimension-part-13/

#4d  
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In part 11 of this series, see pictures of the 4-dimensional cube, or tesseract:

http://johncarlosbaez.wordpress.com/2013/06/21/symmetry-and-the-fourth-dimension-part-11/

And read about Charles Howard Hinton, who made up the word 'tesseract'.  He was a very strange guy.  Even if you know about him, you might not know what he did for Princeton's baseball team.

#4d  
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In part 9 of Symmetry and the Fourth Dimension, I explain the work of this guy - Coxeter - on 3-dimensional geometry:

http://johncarlosbaez.wordpress.com/2013/06/02/symmetry-and-the-fourth-dimension-part-9/

Lots of pretty pictures.  And after this, we're ready to dive into the fourth dimension!

#4d
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