**The small stellated dodecahedron**

What if you let yourself make shapes that are as symmetrical as Platonic solids, but where all the faces are

*stars?*Then you get things like this.

If you look carefully, you'll see lots of 5-pointed stars. Each one is a

**regular pentagram**- a 5-pointed star whose corners are a regular pentagon. Each one touches 5 others at each corner, in exactly the same way. So, it's as regular as you might want.

But it's funny in some ways. First, the faces are stars instead of regular polyhedra. Second, the faces intersect each other: that's why you don't see

*all*of any star.

There are 4 polyhedra whose faces are all regular stars, with each face just like every other and each vertex like every other. They're called

**Kepler-Poinsot polyhedra**.

This particular one is called the

**small stellated dodecahedron**because if you remove all the pyramid-shaped pieces you're left with a dodecahedron! Each star lies in the same plane as one of this dodecahedron's faces. So, there are 12 stars in this shape.

On the other hand, the sharp points of this shape form the corners of an invisible icosahedron! So, there are 20 sharp points.

**Puzzle:**how many edges does this shape have?

This shape can be seen in a floor mosaic in the Basilica of St. Mark in Venice, built in 1430. It was rediscovered by Kepler in his work

*Harmonice Mundi*in 1619. This book, about the "harmonies of the world", is an amazing mix of geometry, astronomy and music theory - a mystical warmup for his later breakthroughs on the orbits of the planets.

Much later, Escher made himself a wood model of the small stellated dodecahedron, which he drew in two woodcuts called

*Order and Chaos*.

While the Kepler-Poinsot polyhedra are beautiful, I've avoided studying them because I don't see how they fit into the theory of Coxeter groups - the study of discrete symmetries that connects Platonic solids, Archimedean solids and hyperbolic honeycombs to deeper strands of math like Lie theory, the study of

*continuous*symmetries. I've been afraid these shapes are merely cute, not deep.

Maybe it's time to find out.

For more, see:

http://mathworld.wolfram.com/SmallStellatedDodecahedron.html

https://en.wikipedia.org/wiki/Small_stellated_dodecahedron

The Mathworld page has a much better picture of the mosaic in the Basilica of St. Mark.

#geometry

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4 comments

John Baez

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+Layra Idarani - fun questions! If we ignore the self-intersections and treat it as an abstract space, is it a topological manifold? I.e.: is it locally homeomorphic to the plane?

Is it orientable?

If the answers to both these questions are "yes", and I momentarily take your word that it has Euler characteristic -6, then it's homeomorphic to a 4-holed torus.

And that makes it interesting to seek a 4-holed Riemann surface on which the symmetry group of this shape - the group you're calling H3, or more precisely its orientation-preserving part, with 60 elements - acts as conformal transformations!

For a similar example, see the picture here:

https://en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem#Examples_of_Hurwitz.27s_groups_and_surfaces

Is it orientable?

If the answers to both these questions are "yes", and I momentarily take your word that it has Euler characteristic -6, then it's homeomorphic to a 4-holed torus.

And that makes it interesting to seek a 4-holed Riemann surface on which the symmetry group of this shape - the group you're calling H3, or more precisely its orientation-preserving part, with 60 elements - acts as conformal transformations!

For a similar example, see the picture here:

https://en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem#Examples_of_Hurwitz.27s_groups_and_surfaces

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