₇3d shapes:Construction polyhedra. Coloring edges and faces

Roman Chijner, Oct 4, 2020

*From Book: Extended definitions of point location estimates [url]https://www.geogebra.org/m/hhmfbvde[/url] From: List of My Public Books on GeoGebra Topics: Constructing polyhedra -[url]https://www.geogebra.org/m/eabstecp[/url]

Table of Contents

  1. Construction polyhedra
    1. 1. The problem of extreme distribution of points on the surface of a sphere.
    2. 2. Generating Elements of mesh modeling the surfaces of convex polyhedrons and its dual images
    3. 3. Vertices n=24. Coloring the edges and faces of a polyhedron and its dual image.
    4. 3 in 1. Constructing, surface triangulation, visualizing polyhedron. New Version.
    5. Images. Constructing, surface triangulation, visualizing polyhedron
    6. Images. n=4: Tetrahedron -Platonic Solid and n=5: Triangular bipyramid (Triangular Dipyramid) -Johnson solid J₁₂ and their Dual Polyhedrons
    7. Images. n=6: Octahedron -Platonic Solid and n=8: Square antiprism and their Dual Polyhedrons
    8. Images. n=12: Icosahedron -Platonic Solid and n=20. Their Dual Polyhedrons
    9. Images. n=16: Two polyhedra with the same number of vertices but different extreme distributions on the described sphere
    10. Images. n=24: Biscribed Snub Cube (laevo) and n=32. Their Dual Polyhedrons
    11. Example of non-uniqueness of the extreme distribution of n=16 particles on the surface of a sphere.
    12. 3d shapes: n=72. Extreme distribution of points on the surface of a sphere and comparison with two known but not extreme ones having the same number of vertices
    13. Coloring of edges and faces of a polyhedron(n=72) extreme distribution and its dual image.
    14. Biscribed Pentakis Snub Dodecahedron (V=72) and its dual image: Coloring the edges and faces
    15. Coloring the edges and faces of a polyhedron(n=72) Pentakis Snub Dodecahedron (laevo) and its dual image.
  2. Coloring edges and faces
    1. Images. n=4: Tetrahedron -Platonic Solid and n=5: Triangular bipyramid (Triangular Dipyramid) -Johnson solid J₁₂ and their Dual Polyhedrons
    2. Images. n=6: Octahedron -Platonic Solid and n=8: Square antiprism and their Dual Polyhedrons
    3. Images. n=12: Icosahedron -Platonic Solid and n=20. Their Dual Polyhedrons
    4. Copy of 3 in 1. Constructing, surface triangulation, visualizing polyhedron. New Version.
    5. Truncated Tetrahedron, n=12. Polyhedra with extreme distribution of equivalent vertices.
    6. Example of non-uniqueness of the extreme distribution of n=16 particles on the surface of a sphere.
    7. Truncated Cube, n=24. Polyhedra with extreme distribution of equivalent vertices.
    8. 3. Vertices n=24. Coloring the edges and faces of a polyhedron and its dual image.
    9. Images. n=24: Biscribed Snub Cube (laevo) and n=32. Their Dual Polyhedrons
    10. Biscribed Truncated Octahedron, n=24. Polyhedra with extreme distribution of equivalent vertices.
    11. Icosidodecahedron, n=30. Polyhedra with extreme distribution of equivalent vertices.
    12. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
    13. Truncated Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    14. Rhombicosidodecahedron-c(V=60) and its dual image - Deltoidal Hexecontahedron(V=62): Coloring the edges and faces
    15. Biscribed Truncated Icosahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    16. Rhombicosidodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    17. Biscribed Snub Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
    18. Coloring the edges and faces of a polyhedron(n=72) Pentakis Snub Dodecahedron (laevo) and its dual image.
    19. Biscribed Pentakis Snub Dodecahedron (V=72) and its dual image: Coloring the edges and faces
    20. Coloring of edges and faces of a polyhedron(n=72) extreme distribution and its dual image.
    21. Rectified truncated icosahedron(V=90) and its dual image: Coloring the edges and faces
    22. Biscribed Truncated Icosidodecahedron, n=120. Polyhedra with extreme distribution of equivalent vertices.
    23. Finding and coloring edges and faces of a polyhedron and its dual image. V=180
    24. Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
    25. Images: Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
    26. Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
    27. Biscribed Hexpropello Dodecahedron (dextro) (V=140) and its dual image: Coloring the edges and faces
    28. Biscribed Orthotruncated Propello Icosahedron (V=120) and its dual image: Coloring the edges and faces
    29. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
    30. Truncated Dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
    31. Truncated Icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
    32. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
    33. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
    34. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
    35. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
    36. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
    37. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
    38. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
    39. Biscribed Pentakis Dodecahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 11th-order segments

1.

Construction polyhedra

  • 1. 1. The problem of extreme distribution of points on the surface of a sphere.
  • 2. 2. Generating Elements of mesh modeling the surfaces of convex polyhedrons and its dual images
  • 3. 3. Vertices n=24. Coloring the edges and faces of a polyhedron and its dual image.
  • 4. 3 in 1. Constructing, surface triangulation, visualizing polyhedron. New Version.
  • 5. Images. Constructing, surface triangulation, visualizing polyhedron
  • 6. Images. n=4: Tetrahedron -Platonic Solid and n=5: Triangular bipyramid (Triangular Dipyramid) -Johnson solid J₁₂ and their Dual Polyhedrons
  • 7. Images. n=6: Octahedron -Platonic Solid and n=8: Square antiprism and their Dual Polyhedrons
  • 8. Images. n=12: Icosahedron -Platonic Solid and n=20. Their Dual Polyhedrons
  • 9. Images. n=16: Two polyhedra with the same number of vertices but different extreme distributions on the described sphere
  • 10. Images. n=24: Biscribed Snub Cube (laevo) and n=32. Their Dual Polyhedrons
  • 11. Example of non-uniqueness of the extreme distribution of n=16 particles on the surface of a sphere.
  • 12. 3d shapes: n=72. Extreme distribution of points on the surface of a sphere and comparison with two known but not extreme ones having the same number of vertices
  • 13. Coloring of edges and faces of a polyhedron(n=72) extreme distribution and its dual image.
  • 14. Biscribed Pentakis Snub Dodecahedron (V=72) and its dual image: Coloring the edges and faces
  • 15. Coloring the edges and faces of a polyhedron(n=72) Pentakis Snub Dodecahedron (laevo) and its dual image.

1.1.

1. The problem of extreme distribution of points on the surface of a sphere.

[size=85] [b]The Problem:[/b] [b] Find a uniform distribution of particles on the surface of the sphere of radius R.[/b][br][i][color=#333333] It is assumed[/color][/i] that this distribution must satisfy the [i]Principle[/i] of [color=#ff0000]Maximum[/color] [b]Distance Sum[/b]. The sum of distances([b]Distance Sum[/b]) is measured by summing all the segments connecting each possible combination of 2 points. In this case, the "measure" of the distribution is the [color=#1e84cc][i]average[/i] distance between the particles on the unit sphere([/color][color=#9900ff]p[sub]n[/sub][/color][color=#1e84cc]).[/color] The problem comes down to finding distributions that [color=#ff0000]maximize[/color] the selected "measure". So what is the configuration (set of locations) of n points on the sphere so that the sum of the distances is maximal for n=1,2,3,...?[br][b]a[/b]. My solution to this problem started with directly maximizing the Distance Sum. The disadvantages of the proposed algorithm are slow convergence [b]([[url=https://www.geogebra.org/m/dVXk2Gbq]1[/url]],[[url=https://www.geogebra.org/m/aeqJmSdH]2[/url]][/b]).[br][b]b[/b]. Then, using the [b]Lagrange method ([[url=https://www.geogebra.org/m/stfexhyj ]3[/url]],[[url=https://www.geogebra.org/m/d9ytv4wg]4[/url]] or [[url=https://www.geogebra.org/m/tnut2f9y ]5[/url]])[/b] it turned out that the problem [i]has a simple geometric solution[/i]. The iterative procedure does not require calculating all the constantly changing distances between particles. Using GeoGebra Script, the programmed iterative [url=https://help.geogebra.org/topic/geogebra-windows-portable-zip-for-december]procedure[/url] could not cope with a large array of numbers: [url=https://www.geogebra.org/m/b5zcy52h]Generating an extreme arrangements of points on a sphere[/url].[br][b]c. [/b]Finally[b],[/b] I achieved my goal. A programmed task using JavaScript is fast and reliable! You can explore this with this applet. Of course, offline computing is even faster. I calculated here distributions for 72 (maybe more!) particles on a sphere .[br] *[size=85]You can find simultaneous worksheet of this and other applets in [url=https://www.geogebra.org/m/wsj6hdrs]https://www.geogebra.org/m/wsj6hdrs[/url] .[/size][/size]
[size=85]*[url=http://dmccooey.com/polyhedra/BiscribedLsnubCube.html]Biscribed Snub Cube (laevo)[/url][table][tr][td][url=http://dmccooey.com/polyhedra/BiscribedLsnubCube.html][/url][/td][/tr][/table][table][tr][td][/td][td]biscribed form[/td][/tr][/table][table][tr][td]Vertices:  [/td][td]24  (24[5])[/td][/tr][tr][td]Faces:[/td][td]38  (8 equilateral triangles + 24 acute triangles + 6 squares)[/td][/tr][tr][td]Edges:[/td][td]60  (24 short + 24 medium + 12 long)[/td][/tr][/table][/size][table][tr][td][/td][/tr][/table][table][tr][td][/td][/tr][/table]
– Roman Chijner

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2.

Coloring edges and faces

  • 1. Images. n=4: Tetrahedron -Platonic Solid and n=5: Triangular bipyramid (Triangular Dipyramid) -Johnson solid J₁₂ and their Dual Polyhedrons
  • 2. Images. n=6: Octahedron -Platonic Solid and n=8: Square antiprism and their Dual Polyhedrons
  • 3. Images. n=12: Icosahedron -Platonic Solid and n=20. Their Dual Polyhedrons
  • 4. Copy of 3 in 1. Constructing, surface triangulation, visualizing polyhedron. New Version.
  • 5. Truncated Tetrahedron, n=12. Polyhedra with extreme distribution of equivalent vertices.
  • 6. Example of non-uniqueness of the extreme distribution of n=16 particles on the surface of a sphere.
  • 7. Truncated Cube, n=24. Polyhedra with extreme distribution of equivalent vertices.
  • 8. 3. Vertices n=24. Coloring the edges and faces of a polyhedron and its dual image.
  • 9. Images. n=24: Biscribed Snub Cube (laevo) and n=32. Their Dual Polyhedrons
  • 10. Biscribed Truncated Octahedron, n=24. Polyhedra with extreme distribution of equivalent vertices.
  • 11. Icosidodecahedron, n=30. Polyhedra with extreme distribution of equivalent vertices.
  • 12. Pentakis Dodecahedron(V=32) and its dual image -Truncated Icosahedron(V=60): Coloring the edges and faces
  • 13. Truncated Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 14. Rhombicosidodecahedron-c(V=60) and its dual image - Deltoidal Hexecontahedron(V=62): Coloring the edges and faces
  • 15. Biscribed Truncated Icosahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 16. Rhombicosidodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 17. Biscribed Snub Dodecahedron, n=60. Polyhedra with extreme distribution of equivalent vertices.
  • 18. Coloring the edges and faces of a polyhedron(n=72) Pentakis Snub Dodecahedron (laevo) and its dual image.
  • 19. Biscribed Pentakis Snub Dodecahedron (V=72) and its dual image: Coloring the edges and faces
  • 20. Coloring of edges and faces of a polyhedron(n=72) extreme distribution and its dual image.
  • 21. Rectified truncated icosahedron(V=90) and its dual image: Coloring the edges and faces
  • 22. Biscribed Truncated Icosidodecahedron, n=120. Polyhedra with extreme distribution of equivalent vertices.
  • 23. Finding and coloring edges and faces of a polyhedron and its dual image. V=180
  • 24. Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
  • 25. Images: Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
  • 26. Canonical Truncated Snub Cube (laevo)(V=120) and its dual image: Coloring the edges and faces
  • 27. Biscribed Hexpropello Dodecahedron (dextro) (V=140) and its dual image: Coloring the edges and faces
  • 28. Biscribed Orthotruncated Propello Icosahedron (V=120) and its dual image: Coloring the edges and faces
  • 29. Rhombicosidodecahedron from Biscribed Pentakis Dodecahedron for the case of trisection of its 1st-order segments
  • 30. Truncated Dodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 2nd-order segments
  • 31. Truncated Icosahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 3rd-order segments
  • 32. Truncated icosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 4th-order segments
  • 33. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 5th-order segments
  • 34. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 6th-order segments
  • 35. The Great Rhombicosidodecahedron (V=120) from Biscribed Pentakis Dodecahedron for the case of trisection of its 7th-order segments
  • 36. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 8th-order segments(Variant1)
  • 37. Rhombicosidodecahedron (V=60) from Biscribed Pentakis Dodecahedron for the case of trisection of its 9th-order segments(Variant2)
  • 38. Polyhedron(V=120) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 10th-order segments
  • 39. Biscribed Pentakis Dodecahedron (V=32) from Biscribed Pentakis Dodecahedron for the case of a trisection of its 11th-order segments

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– Roman Chijner

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