Difference between revisions of "Euler Characteristic Exploration"
From EscherMath
Jump to navigationJump to search (created exploration) |
(Adjusting position of images.) |
||
| (17 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{exploration}} | {{exploration}} | ||
| + | {{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]] | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | [[ | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | [[Image: | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | [[Image: | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
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.
