Euler Characteristic Exploration

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Objective: Compute the Euler characteristic for some polyhedra and some other surfaces.

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.

Polyhedra

The Euler characteristic of a shape is the value of V - E + F and is usually written as <math>\chi = V - E + F</math>.

  1. Escher's Ivory Ball Study shows a Rhombic dodecahedron, with twelve rhombic faces. Compute V, E, F and <math>\chi</math> for this polyhedron.
  2. 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.
  3. Compute V, E, F and <math>\chi</math> for the tessellation by 45°-60°-90° triangles in Concentric Rinds.
  4. Compute V, E, F and <math>\chi</math> for the Deltoidal Icositetrahedron.
Cuboctahedron as a spherical tessellation
Deltoidal icositetrahedron as a spherical tessellation.

The Euler characteristic of a Polyhedron is always 2.

Surfaces

Here, you will compute the Euler characteristic of tessellations that are not on the sphere:

  1. Compute V, E, F and <math>\chi</math> for this square lattice: 4x3-sqaure-lattice.png
  2. Compute V, E, F and <math>\chi</math> for this graph: Graph of 26-fullerene 5-base w-nodes.svg
  3. Compute V, E, F and <math>\chi</math> for this picture of a torus: Torus.png

Handin: A sheet with answers to all questions.