CONVEX EQUILATERAL POLYHEDRA WITH POLYHEDRAL SYMMETRY

• US 20150037766A1
• Filed: 07/29/2014
• Published: 02/05/2015
• Est. Priority Date: 08/02/2013
• Status: Active Grant

• Current Assignee: Regents of the University of California (University of California)
• Original Assignee: Regents of the University of California (University of California)
• Inventors: Schein, Stanley Jay, Gayed, James Maurice
• Granted Patent: US 9,720,881 B2
• Time in Patent Office: Days
• US Class Current: 434/211
• CPC Class Codes: E04B 1/32 Arched structures; Vaulted structures; Folded structures E04B1/345 takes precedence; vaulted roofs E04B7/08
• CPC Class Codes: E04B 1/3211 Structures with a vertical rotation axis or the like, e.g. semi-spherical structures
• CPC Class Codes: E04B 2001/3223 Theorical polygonal geometry therefor
• CPC Class Codes: E04B 2001/3235 having a grid frame
• CPC Class Codes: G06F 17/10 Complex mathematical operations function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544
• CPC Class Codes: G09B 23/04 for geometry, trigonometry, projection or perspective for surveying G09B25/06
• CPC Class Codes: G09B 23/24 for chemistry
• CPC Class Codes: G09B 25/04 of buildings
• CPC Class Codes: G09B 27/08 Globes celestial globes G09B27/06

Abstract

A new class of polyhedron is constructed by decorating each of the triangular facets of an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” A unique set of internal angles in each planar face of each new polyhedron is then obtained, for example by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, where the independent variables are a subset of the internal angles in 6 gons. Alternatively, an iterative method that solves for angles within each hexagonal ring may be solved for that nulls dihedral angle discrepancy throughout the polyhedron. The 6 gon faces in the resulting “Goldberg polyhedra” are equilateral and planar, but not equiangular, and nearly spherical.

Claims

1. A method for designing a convex equilateral cage structure comprising:

selecting a Goldberg triangle comprising an equilateral triangle having three vertices that are each positioned on a center of a hexagon in a hexagonal tiling such that the equilateral triangle overlies a plurality of vertices from the hexagonal tiling, wherein the Goldberg triangle further comprises the plurality of vertices and each line segment connecting any two of the plurality of vertices;

transferring the Goldberg triangle to each of the twenty faces of an icosahedron;

adding connecting line segments that connect corresponding vertices across adjacent Goldberg triangles such that the Goldberg triangle line segments and the connecting line segments define a non-polyhedral cage, wherein the non-polyhedral cage comprises only trivalent vertices; and

transforming the non-polyhedral cage such that the transformed cage comprises a plurality of hexagons and a plurality of pentagons, and the transformed cage is equilateral and convex.

2. The method of claim 1, wherein the step of transforming the non-polyhedral cage comprises solving for interior angles in the plurality of hexagons that produce a zero dihedral angle discrepancy throughout the transformed cage.

3. The method of claim 2, wherein the interior angles in the plurality of hexagons are solved for by identifying all independent interior angles in the non-polyhedral cage, and determining the independent interior angles by solving a system of equations that enforce planarity in the plurality of hexagons.

4. The method of claim 3, wherein the system of equations that enforce planarity in the plurality of hexagons solve for interior angles that produce zero dihedral angle discrepancy throughout the transformed cage.

5. The method of claim 2, wherein the interior angles in the plurality of hexagons are solved for by finding a set of interior angles within each of the plurality of hexagons that zeroes the dihedral angles within that hexagon.

6. The method of claim 1, wherein the plurality of pentagons are regular pentagons.

7. The method of claim 1, wherein the Goldberg triangles comprise at least four vertices from the hexagonal tiling.

8. The method of claim 1, further comprising constructing the convex equilateral cage structure by interconnecting a plurality of elongate struts to form the plurality of hexagons and pentagons.

9. The method of claim 8, wherein the plurality of elongate struts are interchangeable.

10. The method of claim 1, further comprising constructing the convex equilateral cage structure by interconnecting a plurality of planar members that define the plurality of hexagons and pentagons.

11. The method of claim 1, wherein the convex equilateral cage structure comprises a dome-shaped structure.

12. The method of claim 1, wherein the convex equilateral cage structure has tetrahedral, octahedral, and icosahedral symmetry.

13. A convex equilateral cage comprising a plurality of interconnected elongate members that define regular pentagons and a plurality of hexagons, wherein at least some of the plurality of hexagons are not equiangular.

14. The convex equilateral polyhedral cage of claim 13, wherein the dihedral angle discrepancies throughout the cage are zero.

15. The convex polyhedral cage of claim 13, wherein the cage comprises a plurality of interconnected struts.

16. The convex polyhedral cage of claim 13, wherein the plurality of interconnected struts are interchangeable.

17. The convex polyhedral cage of claim 13, wherein the convex equilateral cage comprises a plurality of planar hexagonal and pentagonal plates.

18. The convex polyhedral cage of claim 13, wherein the convex equilateral cage structure has tetrahedral, octahedral, and icosahedral symmetry.

22. A framework comprising at least a portion of the cage at least a portion of a nearly spherical equilateral cage that is designed in accordance with the method of claim 18.

23. The framework of claim 22, wherein the cage comprises a plurality of interchangeable elongate struts.

19. A method for designing a nearly spherical equilateral cage comprising:

selecting a Goldberg triangle constructed as an equilateral triangle having three vertices that are each positioned on a center of a hexagon in a hexagonal tiling such that the equilateral triangle overlies a plurality of vertices from the hexagonal tiling, wherein the Goldberg triangle comprises the plurality of vertices and each segment from the hexagonal tiling connecting any two of the plurality of vertices;

forming an icosahedron comprising twenty of the selected Goldberg triangle;

forming a preliminary cage by adding segments that connect vertices across adjacent faces of the icosahedron, wherein the preliminary cage comprises a plurality of hexagons and a plurality of pentagons; and

transforming the preliminary cage to define a nearly spherical equilateral cage by setting all of the segments to the same length, and setting interior angles in the plurality of hexagons to angles that null dihedral angle discrepancies throughout the transformed cage.

20. The method of claim 19, wherein the selected Goldberg triangle is sized such that it includes at least four vertices from the hexagonal tiling.

21. The method of claim 19, wherein the interior angles in the plurality of hexagons that null the dihedral angle discrepancy throughout the transformed cage are determined by identifying all independent interior angles in the plurality of hexagons, and solving a system of equations that enforce planarity in the plurality of hexagons.