Francisco Lara-Dammer

Professor of Mathematics
Escuela Politécnica Nacional
Quito - Ecuador

This is a Klein diagram (named after the great German mathematician Felix Klein) of the alternating group A5, the group of symmetries of the icosahedron. This diagram emphasizes the subgroups of A5 whose order is a multiple of four — namely, the Klein subgroup and the tetrahedral subgroup A4, which have 4 and 12 elements, respectively. It also emphasizes the left cosets of these subgroups. The diagram is obtained by centrally projecting an icosahedron onto a sphere (with the center of one edge projected onto the north pole), then making a stereographic projection of the sphere onto a horizontal plane. If one considers horizontal and vertical flips and a 180-degree rotation about the center, the diagram’s own symmetries form the Klein group.

A visualization of A5 with emphasis in its Klein and A4 subgroups.
50 x 50 cm
The Geometer's Sketchpad, Adobe Illustrator

This artwork is related to group theory and Galois theory, and was inspired by the diagrams of Felix Klein and by his book Lectures on the Icosahedron. It was also influenced by the book Groups and their Graphs by Wilhelm Magnus and Israel Grossman.