Solids, graphs and paths

chris cambré, Nov 14, 2017

Leonardo da Vinci made perspective drawings of solids. Solids can also be represented by planar graphs. In fact they are very useful to investigate the number of steps to go from a vertex of the solid to the opposite vertex. Direct connections between vertices of solids can be represented by a connectivity matrix, which we can multiply. I started this book reusing the applets with the da Vinci drawings from the GGbook [url=https://ggbm.at/Pr3bvbx3]Graph Theory for kids[/url] by [url=https://www.geogebra.org/voracova]Sarka Voracova[/url]. Starting with these applets I took an investigation of all 5 the Platonic solids and of the 13 Archimedean solids as well.

Table of Contents

  1. Schlegel diagrams
    1. Schlegel diagram
    2. Schlegel diagram tetrahedron
    3. Schlegel diagram hexahedron
    4. Schlegel diagram octahedron
    5. Schlegel diagram dodecahedron
    6. Schlegel diagram icosahedron
  2. Solids, graphs and paths
    1. representing a prism
    2. connections and paths
    3. connectivity table
    4. opposite points
    5. connectivity matrix
  3. tetrahedron
    1. tetrahedron as planar graph
    2. connections between points
    3. connectivity table
    4. connectivity matrix
  4. hexahedron
    1. hexahedron as planar graph
    2. connections between points
    3. connectivity table
    4. connectivity matrix
  5. square pyramid
    1. square pyramid as planar graph
    2. connections between points
    3. connectivity table
    4. connectivity matrix
  6. octahedron
    1. octahedron as planar graph
    2. connections between points
    3. connectivity table
    4. connectivity matrix
  7. dodecahedron
    1. dodecahedron as planar graph
    2. connections between points
    3. connectivity table
    4. connectivity matrix
  8. icosahedron
    1. icosahedron as planar graph
    2. connections between points
    3. connectivity table
    4. connectivity matrix
  9. truncated tetrahedron
    1. truncated tetrahedron
    2. planar graph
    3. connectivity matrix
  10. cuboctahedron
    1. cuboctahedron
    2. planar graph
    3. connectivity matrix
  11. truncated cube
    1. truncated cube
    2. planar graph
  12. tuncated octahedron
    1. truncated octahedron
    2. planar graph
  13. rhombicuboctahedron
    1. rhombicuboctahedron
    2. isometric view and planar graph
    3. paths to opposite points
  14. other archimedean solids
    1. truncated cuboctahedron
    2. snub cube
    3. icosidodecahedron
    4. truncated dodecahedron
    5. truncated icosahedron
    6. rhombicosidodecahedron
    7. truncated icosidodecahedron
    8. snub dodecahedron

1.

Schlegel diagrams

  • 1. Schlegel diagram
  • 2. Schlegel diagram tetrahedron
  • 3. Schlegel diagram hexahedron
  • 4. Schlegel diagram octahedron
  • 5. Schlegel diagram dodecahedron
  • 6. Schlegel diagram icosahedron

1.1.

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1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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2.

Solids, graphs and paths

What's a planar graph, how and why would we use them?

  • 1. representing a prism
  • 2. connections and paths
  • 3. connectivity table
  • 4. opposite points
  • 5. connectivity matrix

2.1.

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2.2.

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2.3.

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2.4.

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2.5.

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3.

tetrahedron

  • 1. tetrahedron as planar graph
  • 2. connections between points
  • 3. connectivity table
  • 4. connectivity matrix

3.1.

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3.2.

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3.3.

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3.4.

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4.

hexahedron

  • 1. hexahedron as planar graph
  • 2. connections between points
  • 3. connectivity table
  • 4. connectivity matrix

4.1.

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4.2.

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4.3.

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4.4.

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5.

square pyramid

  • 1. square pyramid as planar graph
  • 2. connections between points
  • 3. connectivity table
  • 4. connectivity matrix

5.1.

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5.2.

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5.3.

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5.4.

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6.

octahedron

  • 1. octahedron as planar graph
  • 2. connections between points
  • 3. connectivity table
  • 4. connectivity matrix

6.1.

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6.2.

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6.3.

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6.4.

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7.

dodecahedron

  • 1. dodecahedron as planar graph
  • 2. connections between points
  • 3. connectivity table
  • 4. connectivity matrix

7.1.

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7.2.

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7.3.

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7.4.

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8.

icosahedron

  • 1. icosahedron as planar graph
  • 2. connections between points
  • 3. connectivity table
  • 4. connectivity matrix

8.1.

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8.2.

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8.3.

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8.4.

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9.

truncated tetrahedron

  • 1. truncated tetrahedron
  • 2. planar graph
  • 3. connectivity matrix

9.1.

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9.2.

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9.3.

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10.

cuboctahedron

  • 1. cuboctahedron
  • 2. planar graph
  • 3. connectivity matrix

10.1.

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10.2.

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10.3.

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11.

truncated cube

  • 1. truncated cube
  • 2. planar graph

11.1.

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11.2.

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12.

tuncated octahedron

  • 1. truncated octahedron
  • 2. planar graph

12.1.

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12.2.

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13.

rhombicuboctahedron

  • 1. rhombicuboctahedron
  • 2. isometric view and planar graph
  • 3. paths to opposite points

13.1.

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13.2.

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13.3.

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14.

other archimedean solids

The next eight archimedean solids grow more and more complex in number of verteces end edges. With up to 120 verteces I won't make connectivity matrices, but I think even just showing the solid and it's planar graph is worthwhile.

  • 1. truncated cuboctahedron
  • 2. snub cube
  • 3. icosidodecahedron
  • 4. truncated dodecahedron
  • 5. truncated icosahedron
  • 6. rhombicosidodecahedron
  • 7. truncated icosidodecahedron
  • 8. snub dodecahedron

14.1.

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14.2.

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14.3.

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14.4.

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14.5.

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14.6.

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14.7.

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14.8.

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