Geometric shapes of truncated Platonic polyhedra

Roman Chijner, Mar 8, 2020

Geometrical shapes of truncated polyhedrons

Table of Contents

  1. The Platonic polyhedra. Geometric shapes of their truncations.
    1. Trisecting the sides of a polygon
    2. Truncated Tetrahedron.
    3. Truncated Tetrahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    4. Truncated Octahedron.
    5. Truncated Octahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    6. Truncated Cube.
    7. Truncated Cube. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    8. Truncated Icosahedron.
    9. Truncated Icosahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    10. Truncated Great Dodecahedron from Truncated Icosahedron
    11. Truncated Dodecahedron.
    12. Truncated Dodecahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    13. Truncated square antiprism
    14. Trancated Square antiprism. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    15. Truncated Cuboctahedron
    16. Truncated Cuboctahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    17. Truncated Icosidodecahedron.
    18. Truncated Icosidodecahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    19. Truncated Biscribed Pentakis Dodecahedron.
    20. Truncated Biscribed Pentakis. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
    21. Images. Polyhedrons and their truncated, critical truncated polyhedra
    22. Rhomben-Hexekontaeder
    23. Polyeder mit ganzzahligen Koordinaten
    24. Vier Hexaederstümpfe
    25. Vier Würfelkonstellation
    26. Disphenoide

1.

The Platonic polyhedra. Geometric shapes of their truncations.

  • 1. Trisecting the sides of a polygon
  • 2. Truncated Tetrahedron.
  • 3. Truncated Tetrahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 4. Truncated Octahedron.
  • 5. Truncated Octahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 6. Truncated Cube.
  • 7. Truncated Cube. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 8. Truncated Icosahedron.
  • 9. Truncated Icosahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 10. Truncated Great Dodecahedron from Truncated Icosahedron
  • 11. Truncated Dodecahedron.
  • 12. Truncated Dodecahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 13. Truncated square antiprism
  • 14. Trancated Square antiprism. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 15. Truncated Cuboctahedron
  • 16. Truncated Cuboctahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 17. Truncated Icosidodecahedron.
  • 18. Truncated Icosidodecahedron. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 19. Truncated Biscribed Pentakis Dodecahedron.
  • 20. Truncated Biscribed Pentakis. Dependence of the average distance between the vertices of a polyhedron on the truncation parameter. Extreme distribution.
  • 21. Images. Polyhedrons and their truncated, critical truncated polyhedra
  • 22. Rhomben-Hexekontaeder
  • 23. Polyeder mit ganzzahligen Koordinaten
  • 24. Vier Hexaederstümpfe
  • 25. Vier Würfelkonstellation
  • 26. Disphenoide

1.1.

Trisecting the sides of a polygon

[size=85] A Truncation is an operation applied to a polyhedron. Using this applet, you can explore the dividing sides of a polygon into 3 unequal parts: t, 1-2*t, t where 0<t<0.5. I have used this method to construct truncated polyhedra.[br] In the following applets, you can explore polyhedra at various stages of truncation.[br][url=https://www.geogebra.org/m/a6w4y8m8]Truncated Tetrahedron[/url]. [br][url=https://www.geogebra.org/m/peus7qcy]Truncated Cube.[/url] [br][url=https://www.geogebra.org/m/eusfhczd]Truncated Octahedron.[/url] [br][url=https://www.geogebra.org/m/uyt4cfxw]Truncated Dodecahedron.[/url] [br][url=https://www.geogebra.org/m/ntknhpec]Truncated Icosahedron.[/url] [br][url=https://www.geogebra.org/m/xhwfp8pb]Truncated square antiprism[/url] [br][url=https://www.geogebra.org/m/rf39hzpw]Truncated Cuboctahedron.[/url] [br][url=https://www.geogebra.org/m/ntknhpec]Truncated Icosidodecahedron.[/url] [br][url=https://www.geogebra.org/m/m36m3dkt]Truncated Biscribed Pentakis Dodecahedron.[/url] [br] Drag the slider [b]t[/b] to show the truncation the sides of the n-gon. [/size]
– Roman Chijner

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