Euler Characteristic Exploration
In 1750, the Swiss mathematician Leonhard Euler noticed a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron. It is now called the Euler characteristic, and is written with the Greek letter <math>\chi</math>:
The Euler characteristic is <math>\chi</math> = V - E + F
For the Platonic solids, the Euler characteristic is always 2.
The Euler characteristic of a shape is the value of V - E + F and is usually written as <math>\chi = V - E + F</math>.
- Escher's Ivory Ball Study shows a Rhombic dodecahedron, with twelve rhombic faces. Compute V, E, F and <math>\chi</math> for this polyhedron.
- The dual of the Rhombic dodecahedron is the Cuboctahedron, which is a cube with its corners cut off, or equivalently, an octahedron with its corners cut off.
Compute V, E, F, and <math>\chi</math> for the Cuboctahedron.
- Compute V, E, F and <math>\chi</math> for the tessellation by 45°-60°-90° triangles in Concentric Rinds.
- Compute V, E, F and <math>\chi</math> for the Deltoidal Icositetrahedron.
The Euler characteristic of a Polyhedron is always 2.
Here, you will compute the Euler characteristic of tessellations that are not on the sphere: