The 120-cell encodes the symmetries of the dodecahedron.

The 120-cell encodes the symmetries of the dodecahedron. This is probably my favorite mathematical surprise, and amazed and shocked me when I learned it (still amazes and shocks me). I always thought it'd be nice to write a blog post about this.

This week, John Baez did a lovely post about this very thing, and I learned even more about the connection by reading it.

He's been building up to that post with his #4d series. Here's a link to that series, which should hopefully filter out all the reshares you get by just searching on the hashtag.

https://plus.google.com/u/0/s/%234d%20baez

Originally shared by John Baez

So here's the climax of the #4d story, though not the end. See this 'belt' of 60 dodecahedra that makes up half the 120-cell when you curl it up into a torus? Let's understand it! It might help if you first watch the 120-cell video again.

So, what is this belt, and why is it made of 60 dodecahedra?

First: each point in a ball of radius π describes a rotation in 3d space. The center of the ball means 'no rotation'. Going out a distance θ in some direction from the center means 'rotate clockwise by θ along the axis pointing in that direction'. We only need to go up to θ = π to get all possible rotations. See why? And opposite points on the surface of the ball describe the same rotation.

Now stick two balls together, each point on the surface of one touching the corresponding point on the surface of the other. The result is a 3-sphere. See why? So, each point on the 3-sphere describes a rotation, but two different points on the 3-sphere describe the same rotation.

The dodecahedron has 60 different rotational symmetries: rotations that leave it looking the same way it did at first. We counted them earlier in this story! Corresponding to these 60 rotations, there are 120 different points on the 3-sphere. These are the centers of the dodecahedra in the 120-cell!

These 120 dodecahedra come in 12 circular strands of 10 each - watch the movie to see what I mean. Why? Because if you take a dodecahedron and rotate it 5 times around an axis between opposite pentagons, it comes back where it was. But that means we need to go twice as many steps - 10 steps! - to march along dodecahedra in the 3-sphere and get back to where we started. These 10 steps form a circular strand of dodecahedra. See why?

If you paid careful attention to my talk about the Hopf fibration, you'll see I'm talking about a discrete version of that, where instead of circles we have circular strands of 10 dodecahedra. And just as the Hopf fibration lets us chop the 3-sphere into two solid tori, now we can chop the 120-cell into two solid tori made of dodecahedra: two 'belts' of 60 each. See why?

This was fairly intense; don't be upset if it doesn't make sense right away. It'll help if you ask questions, or look at recent posts in the #4d series.

But I hope you notice what's cool here: each of the dodecahedra in the 120-cell also stands for a rotational symmetry of a dodecahedron! There's something self-referential about that, which is very cool.

Comments

  1. Do you know what encodes the symmetry of 120 cell?

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  2. That's an interesting question.

    The group of symmetries of the dodecahedron is the Coxeter group H₃, aka the Binary Icosahedral Group.  The group of symmetries of the 120-cell is the Coxeter group H₄.

    http://en.wikipedia.org/wiki/Binary_icosahedral_group#Relation_to_4-dimensional_symmetry_groups

    So I think the question is whether there is a polytope that would correspond to H₄ in a similar way to how the 120-cell corresponds to H₃.  I don't know if there is, and I haven't had luck searching yet.

    H₄ has 14400 elements, so it would have a lot of cells!

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  3. ok, Nan Ma, I think there is no polytope which encodes the symmetries of the 120-cell.

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

    From that page:

    "Well, there are very few dimensions in which the unit sphere is also a group. It happens only in dimensions 1, 2, and 4! In 1 dimensions the unit sphere is just two points, which we can think of as the unit real numbers, -1 and 1. In 2 dimensions we can think of the unit sphere as the unit complex numbers, exp(i theta). In 4 dimensions we can think of the unit sphere as the unit quaternions.

    Only in these dimensions do we get polytopes that are also groups in a natural way."

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