# Difference between revisions of "Euler Characteristic Exploration"

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+ | {{Time|30}} | ||

+ | {{Objective|Compute the Euler characteristic for some polyhedra and some other surfaces.}} | ||

+ | In 1750, the Swiss mathematician [[wikipedia:Leonhard Euler|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>: | ||

− | + | <center> | |

+ | {{{define|The Euler characteristic is <math>\chi</math> = V - E + F}}} | ||

+ | </center> | ||

− | + | 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>. | ||

+ | # Escher's [[Ivory Ball Study]] shows a [[wikipedia:Rhombic dodecahedron|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 [[commons:File:Cuboctahedron.gif|Cuboctahedron]], which is a cube with its corners cut off, or equivalently, an octahedron with its corners cut off. <br /> 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 [[wikipedia:File:Deltoidalicositetrahedron.jpg|Deltoidal Icositetrahedron]]. | ||

− | + | [[Image:spherical-cuboctahedron.png|thumb|left|Cuboctahedron as a spherical tessellation]] | |

− | + | [[File:Spherical_deltoidal_icositetrahedron.png|thumb|center|Deltoidal icositetrahedron as a spherical tessellation.]] | |

− | + | {{clear}} | |

− | + | {{boxed|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: | |

− | [[Image: | + | <ol start="5"> |

− | + | <li>Compute V, E, F and <math>\chi</math> for this square lattice: [[Image:4x3-sqaure-lattice.png|200px]]</li> | |

− | + | <li>Compute V, E, F and <math>\chi</math> for this graph: [[Image:Graph_of_26-fullerene_5-base_w-nodes.svg|200px]]</li> | |

− | + | <li>Compute V, E, F and <math>\chi</math> for this picture of a torus: [[Image:Torus.png]]</li> | |

− | + | </ol> | |

− | + | {{handin}} | |

− | + | [[category:Non-Euclidean Geometry Explorations]] | |

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## Latest revision as of 10:34, 16 November 2015

**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>.

- 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.

## Surfaces

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

- Compute V, E, F and <math>\chi</math> for this square lattice:
- Compute V, E, F and <math>\chi</math> for this graph:
- Compute V, E, F and <math>\chi</math> for this picture of a torus:

**Handin:**
A sheet with answers to all questions.