The regular pentagon is a curious thing. It doesn’t tile the plane, but we can use twelve of them, three around each vertex, to tile the sphere, obtaining the dodecahedron.
This is one of the five Platonic solids. Their symmetries have intrigued mankind back way before Plato and any written history, but today’s story is contemporary. Johannes Kepler needed more than five regular shapes, because he had set his mind to explain the universe. In his Harmonice Mundi, he analyzed regular polygons, star polygons, polyhedra, and (re-)discovered star polyhedra, two of which I will look at today.
The Small Stellated Dodecahedron as conceived by Kepler does not have 60 triangles but rather 12 star pentagons as faces. It also has only 12 vertices and 30 edges. This leads to the annoying observation that this polyhedron has Euler characteristic -6, meaning it is topologically not a sphere, but a surface of genus 4. Similarly, his Great Stellated Dodecahedron
has 12 usual regular pentagons as faces, but is only immersed. To unwrap these, we need the hyperbolic plane, tiles by regular hyperbolic pentagons whose interior angle is 72 degrees so that five of them fit around a corner.
That is not part of Kepler’s story but that of William Richard Maximilian Hugo Threlfall, who was probably the first who understood the hyperbolic nature of Kepler’s polyhedra, and their group theoretic implications. So we can tile the hyperbolic plane with regular pentagons, five around each vertex. One of the surprising features of the hyperbolic plane is that shapes do not scale as in the Euclidean plane. Pentagons half the area have actually right angles, so that four of them fit around a vertex, as indicated by the reddish grid in the picture above.
Curiously, there also is a uniform polyhedron where four pentagons fit around each vertex, the so-called Dodecadodecahedron (yes, these names are odd).
It has as faces both pentagons and star pentagons.
There is another connection between Kepler and Threfall. Kepler begins the introduction of his Astronomia Nova from 1609 with the sentence Durissima est hodie conditio scribendi libros Mathematicos, praecipue Astronomicos. In 1938, Seiffert and Threlfall published a book (Variationsrechnung im Großen) that has as its motto the shortened quote Durissima est hodie conditio scribendi libros Mathematicos.
That was a risky thing to do back then.
Kepler was an interesting personality. It must have been maddening for him to believe himself on the verge of unraveling the universe and be constrained by earthly powers that threatened to burn his mother as a witch. There is a biographical novel about him by John Banville (whom I generally like for his affinity to bizarre characters). In this case, I am afraid, he falls short. Maybe only a scientist can truly understand scientific obsession.