The small stellated dodecahedron is made of 12 pentagrams, or 5-pointed stars, with 5 pentagrams meeting at each vertex. It is one of four nonconvex polyhedra with regular polygons or stars as faces, called Kepler–Poinsot polyhedra.

The small stellated dodecahedron was studied by Kepler. It appears in his Harmonice Mundi, published in 1619. However, it also can be found in a floor mosaic in St Mark’s Basilica in Venice, which seems to have been created by Paolo Uccello as early as 1430.

Later it was used by Escher in two of his prints: Constrast: Order and Chaos and Gravitation.

The small stellated dodecahedron and other Kepler–Poinsot solids posed a challenge to early topologists. If we treat the small stellated dodecahedron as having 12 five-sided faces, two meeting along each edge, then it should have \(12 \times 5 / 2 = 30 \) edges. Since it has 12 vertices as well, its Euler characteristic should be

$$ \chi \, =\, V – E + F \, =\, 12 – 30 + 12 \, = \; – 6 $$

Since the genus \(g\) of a surface is related to its Euler characteristic by \(\chi = 2 – 2g \), the small stellated dodecahedron should have genus \(4\)! This was first noted by Poinsot, and it caused some confusion about the validity of Euler’s formula \(V – E + F = 2\), which we now realize holds only for convex polyhedra.

However, we can in fact treat the small stellated dodecahedron as the image of a continuous map from a surface \(\Sigma\) of genus 4 into three-dimensional space. This map has branch points of order 2 at the center of each pentagram, hidden from view in the picture!

Indeed, if we think of the stellated dodecahedron as defining a branched cover of the sphere, this branched cover becomes a Riemann surface. The orientation-preserving symmetries of the icosahedron clearly act as conformal transformations of this Riemann surface. It thus has at least the alternating group \(\mathrm{A}_5\) as symmetries.

This was observed by Klein in 1877:

• Felix Klein, Weitere Untersuchungen über das Ikosaeder, Math. Annalen 12 (1877), 321–384.

The Riemann surface \(\Sigma\) can be tiled by 12 pentagons, 5 meeting at each corner. These correspond to the 12 pentagrams of the small stellated dodecahedron. Thanks to this tiling, \(\Sigma\) can also be thought of as a quotient of the hyperbolic plane \(\mathbb{H}^2\) by a discrete group \(\Gamma\) acting as isometries. This group preserves a tiling of the hyperbolic plane by regular hyperbolic pentagons, 5 meeting at each corner:

This tiling of the hyperbolic plane is called the {5,5} tiling.

The small stellated dodecahedron is the image of a certain map from the Riemann surface \(\Sigma\) into \(\mathbb{R}^3\). But this surface can also be mapped into \(\mathbb{R}^3\) in a different way, giving another Kepler–Poinsot polyhedron, the great dodecahedron:

This should not be surprising, because the great dodecahedron has 12 pentagonal faces, 5 meeting at each vertex. However, the ultimate explanation is that the great dodecahedron is the dual of the small stellated dodecahedron — the vertices of one lie at the centers of the faces of the other — and the {5,5} tiling is its own dual!

For more on the great dodecahedron see this post, which explains its connection to an exceptionally symmetrical code:

• Golay code.

The Riemann surface \(\Sigma\) also has an interesting connection to the quintic equation. A Riemann surface is also called a ‘complex curve’, since points on it can be locally described by a single complex number. Klein showed that \(\Sigma\) is isomorphic to the complex curve described by these homogeneous equations:

$$ \sum_{i=1}^5 z_i = 0, \qquad \sum_{i=1}^5 z_i^2 = 0, \qquad \sum_{i=1}^5 z_i^3 = 0 .$$

These three equations in five complex variables pick out a set of complex dimension two, but when we ‘projectivize’, identifying solutions that differ by a complex multiple, we obtain a complex curve embedded in the projective space \(\mathbb{C}\mathrm{P}^4\).

This is called as Bring’s curve, and it is famous because it has the largest possible symmetry group of any complex curve (or Riemann surface) of genus 4. In fact, this group is not just \(\mathrm{A}_5\) but the symmetric group \(\mathrm{S}_5\).

We can see that \(\mathrm{S}_5\) acts as symmetries by relating Bring’s surface to the quintic equation. Consider a quintic of the form

$$ Q(z) = (z- z_1) \cdots (z – z_5) $$

If the three equations

$$ \sum_{i=1}^5 z_i = 0, \qquad \sum_{i=1}^5 z_i^2 = 0, \qquad \sum_{i=1}^5 z_i^3 = 0 $$

hold, then \(Q\) takes the special form

$$ Q(z) = z^5 + p z + q .$$

Conversely, if \(Q\) takes this special form, its roots \(z_1, \dots, z_5\) obey the three equations listed. Thus, Bring’s curve is the set of ordered 5-tuples \((z_1, \dots, z_5)\), modulo scalar factors, that are roots of some quintic of the form \(z^5 + p z + q\).

For more on these topics, see:

• Matthias Weber, Kepler’s small stellated dodecahedron as a Riemann surface, Pacific J. Math. 220 (2005), 167–182.

Abstract. We provide a new geometric computation for the Jacobian of the Riemann surface of genus 4 associated to the small stellated dodecahedron. Starting with Threlfall’s description, we introduce other flat conformal geometries on this surface which are related to holomorphic 1-forms. They allow us to show that the Jacobian is isogenous to a fourfold product of a single elliptic curve whose lattice constant can be determined in two essentially different ways, yielding an unexpected relation between hypergeometric integrals. We also obtain a new platonic tessellation of the surface.

The featured picture of the small stellated dodecahedron was created using Robert Webb’s Stella software and placed on Wikicommons. The same is true of the picture of the great dodecahedron. Robert Webb allows anyone to use these pictures for any purpose, provided that the copyright holder is properly attributed. The picture of the floor mosaic in the Basilica of St Mark is from this page:

• George Hart, Paolo Uccello’s Polyhedra.

but is available on Wikicommons, where it is listed as being in the public domain. The picture of the {5,5} tiling was drawn by Tom Ruen using Jeff Week’s KaleidoTile software and placed in the public domain on Wikicommons. The rotating image of the small stellated dodecahedron was created by Cyp and placed on Wikicommons under a Creative Commons Attribution-Share Alike 3.0 Unported license.

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