Difference between revisions of "Euler Characteristic Exploration"

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{{Time|30}}
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{{Objective|Compute the Euler characteristic for some polyhedra and some other surfaces.}}
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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.
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It is now called the Euler characteristic, and is written with the Greek letter <math>\chi</math>:
  
In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron:
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<center>
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{{{define|The Euler characteristic is <math>\chi</math> = V - E + F}}}
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</center>
  
He found that the Euler characteristic of the surface V - E + F depends on the surface
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For the Platonic solids, the Euler characteristic is always 2.
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==Polyhedra==
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The Euler characteristic of a shape is the value of V - E + F and is usually written as <math>\chi = V - E + F</math>.
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# 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.
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# 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.
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# Compute V, E, F and <math>\chi</math> for the tessellation by 45°-60°-90° triangles in [[Concentric Rinds]].
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# Compute V, E, F and <math>\chi</math> for the [[wikipedia:File:Deltoidalicositetrahedron.jpg|Deltoidal Icositetrahedron]].
  
Let's check this formula on some of the shapes below.
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[[Image:spherical-cuboctahedron.png|thumb|left|Cuboctahedron as a spherical tessellation]]  
 
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[[File:Spherical_deltoidal_icositetrahedron.png|thumb|center|Deltoidal icositetrahedron as a spherical tessellation.]]
===Tetrahedron===
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{{clear}}
 
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{{boxed|The Euler characteristic of a Polyhedron is always 2.}}
A tetrahedon is a simple shape that is made up of 4 triangles. Below you see a picture with labels on the vertices (V) and edges (E).
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==Surfaces==
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Here, you will compute the Euler characteristic of tessellations that are not on the sphere:
[[Image:Tetrahedron-labeled.svg|300px]]
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<ol start="5">
 
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<li>Compute V, E, F and <math>\chi</math> for this square lattice: [[Image:4x3-sqaure-lattice.png|200px]]</li>
Number of vertices = V = _______
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<li>Compute V, E, F and <math>\chi</math> for this graph: [[Image:Graph_of_26-fullerene_5-base_w-nodes.svg|200px]]</li>
 
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<li>Compute V, E, F and <math>\chi</math> for this picture of a torus: [[Image:Torus.png]]</li>
Number of edges E =  ______
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</ol>
 
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{{handin}}
Number of faces F =  ______
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[[category:Non-Euclidean Geometry Explorations]]
 
 
Now find V - E + F = 
 
 
 
 
 
 
 
 
 
===Octahedron===
 
 
 
A tetrahedon is a simple shape that is made up of 8 triangles. Below you see two pictures, the one on the left is given with labels on the vertices (V) and edges (E).
 
 
[[Image:Octahedron-labeled.svg|500px]]
 
 
 
Number of vertices = V = _______
 
 
 
Number of edges E = ______
 
 
 
Number of faces F =  ______
 
 
 
Now find V - E + F = 
 
 
 
 
 
 
 
===Cube===
 
 
 
A cube is a simple shape that is made up of 6 squares. Below you see a picture with labels on the vertices (V) and edges (E).
 
 
[[Image:Cube-labeled.svg|300px]]
 
 
 
Number of vertices = V = _______
 
 
 
Number of edges E =  ______
 
 
 
Number of faces F =  ______
 
 
 
Now find V - E + F
 
 
 
 
 
==Dodecahedron==
 
 
 
A dodecahedron is made up of pentagons (5-gons). There are 12 pentagons in one dodecahedron.
 
 
 
[[Image:Dodecahedron.png|200px]] [[Image:Dodecahedron-tombstone.jpg|200px]]
 
 
 
Above you see a drawing of a dodecahedron and a tombstone in the form of a dodecahedron.
 
 
 
Number of vertices = V = _______
 
 
 
Number of edges E = ______
 
 
 
Number of faces F =  ______
 
 
 
Now find V - E + F = 
 
 
 
 
 
==Icosahedron==
 
 
 
An icosahedron is made up of triangles. There are 20 triangles in one icosahedron.
 
 
 
[[Image:Chem star.png|200px]] [[Image:Icosaedro.jpg|200px]]
 
 
 
Above you see a drawing of a icosahedron and a game piece (like dice) in the form of a icosahedron.
 
 
 
Number of vertices = V = _______
 
 
 
Number of edges E =  ______
 
 
 
Number of faces F =  ______
 
 
 
Now find V - E + F =
 

Latest revision as of 10:34, 16 November 2015


Time-30.svg

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.