₀Constructing polyhedron, triangulation, 3D Visualizing

Roman Chijner, Aug 14, 2020

[b]Construction of a polyhedron surface based on direct calculation of the maximum sum of distances [/b] The peculiarity of this applets is that 1. for a given number of n particles on the sphere, their extreme distribution for Convex Polyhedra is found. The [color=#c51414]maximum[/color] of the[b] sum of distances[/b] is found directly: maximization using the Maximize[]command and sliders. 2. Triangulation of polyhedral surfaces: this applets sorts and finds vertices, faces, and surface segments of a polyhedron and its dual image. 3. Visualization of polyhedral surfaces: the elements of the polyhedron and its dual image are directly translated into their 3d form. *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. Computer constructions of the vertices of polyhedra
    1. Computer constructions of a regular tetrahedron
    2. The largest volume tetrahedron whose points fit on a sphere
    3. Constructing, surface triangulation, visualizing polyhedron
    4. Computer constructions of polyhedra
    5. wa Construction the coordinates for Convex Polyhedra
    6. wb Triangulation of polyhedral surfaces
    7. wc Visualization of polyhedral surfaces
    8. Computer constructions of polyhedra 1
  2. Varianten: Computer constructions of polyhedra
    1. Computer construction of polyhedra using sleiders
  3. Set of polyhedra
    1. 1 Computer constructions set of polyhedra
    2. 2 Computer constructions set of polyhedra
    3. 3 Computer constructions set of polyhedra
  4. Visualization of polyhedral surfaces
    1. Polyhedron Computer constructions n=10
    2. n=12. Polyhedron Computer constructions
    3. n=24. Visualization of polyhedral surfaces
    4. n=32. Polyhedron Computer constructions
    5. n=40. Polyhedron Computer constructions
    6. n=48. Polyhedron Computer constructions
    7. n=48 Polyhedron Computer construction. Lowering the accuracy
    8. n=72. Polyhedron Computer constructions
    9. n=100. Polyhedron Computer constructions
    10. Polyhedron with 120 vertices and its dual Polyhedron-image
    11. Polyhedron with 120 vertices and its dual Polyhedron-image
    12. n=130. Polyhedron Computer constructions

1.

Computer constructions of the vertices of polyhedra

  • 1. Computer constructions of a regular tetrahedron
  • 2. The largest volume tetrahedron whose points fit on a sphere
  • 3. Constructing, surface triangulation, visualizing polyhedron
  • 4. Computer constructions of polyhedra
  • 5. wa Construction the coordinates for Convex Polyhedra
  • 6. wb Triangulation of polyhedral surfaces
  • 7. wc Visualization of polyhedral surfaces
  • 8. Computer constructions of polyhedra 1

1.1.

Computer constructions of a regular tetrahedron

Computer constructions of polyhedrons, whose vertices are defined from the maximum sum of the mutual distances among the polyhedron vertices lies on a sphere.
– Roman Chijner

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

2.

Varianten: Computer constructions of polyhedra

  • 1. Computer construction of polyhedra using sleiders

2.1.

Computer construction of polyhedra using sleiders

Computer constructions of polyhedrons, whose all vertices lie on a sphere and are defined from the maximum sum of all mutual distances (p_0) between the vertices. average bond length table from n=4 to n=40.[br]If convergence is weak:[br]-press the button of animation and stop the process[br]-reset the counter and start animation over again[br]All points except the first randomly each time placed on a sphere.
– Roman Chijner

3.

Set of polyhedra

  • 1. 1 Computer constructions set of polyhedra
  • 2. 2 Computer constructions set of polyhedra
  • 3. 3 Computer constructions set of polyhedra

3.1.

1 Computer constructions set of polyhedra

Animated Polyhedron Model.[br]Computer constructions of polyhedrons, whose vertices are defined from the maximum sum of the mutual distances among the polyhedron vertices lies on a sphere of radius r. Coordinates polyhedra are taken from: http://neilsloane.com/packings/
– Roman Chijner

3.2.

–

3.3.

–

4.

Visualization of polyhedral surfaces

  • 1. Polyhedron Computer constructions n=10
  • 2. n=12. Polyhedron Computer constructions
  • 3. n=24. Visualization of polyhedral surfaces
  • 4. n=32. Polyhedron Computer constructions
  • 5. n=40. Polyhedron Computer constructions
  • 6. n=48. Polyhedron Computer constructions
  • 7. n=48 Polyhedron Computer construction. Lowering the accuracy
  • 8. n=72. Polyhedron Computer constructions
  • 9. n=100. Polyhedron Computer constructions
  • 10. Polyhedron with 120 vertices and its dual Polyhedron-image
  • 11. Polyhedron with 120 vertices and its dual Polyhedron-image
  • 12. n=130. Polyhedron Computer constructions

4.1.

Polyhedron Computer constructions n=10

random oscillations on a spherical surface near the equilibrium positions
– Roman Chijner

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

4.10.

–

4.11.

–

4.12.

–

Information