Small Stellated Dodecahedron

The other day my four-year-old daughter asked me to draw her something on her Magna Doodle^TM. Happy faces, hearts, flowers, and trees being beneath me, I accordingly drew her a small stellated dodecahedron, which is a Kepler-Poinsot solid. The next morning she came to me, fighting to hold back tears, to tell me that she'd accidentally erased half of it, and to ask me to redraw it for her. So I said, sure, let's do one on paper this time. She went and got me a piece of cardstock and her box of markers, and I whipped out the diagram shown here. I'm posting it for no other reason than to celebrate how awesome I am for being able to draw a small stellated dodecahedron without making any mistakes, just off the top of my head, in the space of a few minutes while I was on the way out the door for work. There are many, many years of doodling during classes and meetings behind that skill, my friends.

The regular or Platonic solids are those convex polyhedra that can be formed each from one type of regular polygon in such a way that each vertex is congruent to every other vertex. There are exactly five: the tetrahedron (formed from four equilateral triangles), the octahedron (eight equilateral triangles), the icosahedron (twenty equilateral triangles), the cube (six squares), and the dodecahedron (twelve regular pentagons). They bear Plato's name because they were mentioned by him in his dialogue Timaeus, but it's his colleague Theatetus who's credited with proving that there are only five. Euclid considered them each in turn in Book XIII of his Elements.

If we relax our definition of "polygon" to allow the sides to intersect one another, then "stars" are also regular polygons, the most familiar being the five-pointed star, or pentagram. It's therefore reasonable to ask if we can form regular "polyhedra" out of pentagrams. The answer is that exactly two are possible: the small stellated dodecahedron (shown) and the great stellated dodecahedron (which has pentagrams meeting in threes instead of fives). They were discovered (or recognized, anyway) by Johannes Kepler, hence are known as the Kepler solids.

Allowing the faces of regular polyhedra to intersect opens up two more possibilities we didn't consider before: the great dodecahedron (assembled from twelve ordinary pentagons) and the great icosahedron (assembled from twenty equilateral triangles). They were discovered by Louis Poinsot, and are dual to the Kepler solids.

So, with these relaxed definitions, there are nine regular solids rather than five.