ARCHIMEDEAN CAGES, POLYHEDRA, AND NANOTUBE STRUCTURES AND METHODS

• US 20150315775A1
• Filed: 07/14/2015
• Published: 11/05/2015
• 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, Yeh, Alexander James
• Granted Patent: US 9,863,136 B2
• Time in Patent Office: Days
• US Class Current: 1/1
• CPC Class Codes: E04B 1/3211 Structures with a vertical rotation axis or the like, e.g. semi-spherical structures
• CPC Class Codes: E04B 2001/0061 Buildings with substantially curved horizontal cross-section, e.g. circular
• CPC Class Codes: E04B 2001/0069 Prismatic shaped buildings with substantially triangular vertical cross-section
• CPC Class Codes: E04B 2001/3223 Theorical polygonal geometry therefor
• CPC Class Codes: E04B 2001/3235 having a grid frame
• CPC Class Codes: E04B 2001/3294 with a faceted surface
• 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 7/00 Electrically-operated teaching apparatus or devices working with questions and answers mechanically operated G09B3/00; computing arrangements G06F

Abstract

A method for designing a structure includes selecting an initial cage, defining a secondary cage by positioning a plurality of tiles, reduced tiles, or larger patches obtained or derived from a selected one of the uniform Archimedean tilings over the faces of the initial cage, resizing edges of the secondary cage such the cage is equilateral, and planarizing the cage faces. In an embodiment the patch comprising a network of edges and vertices from a uniform tiling decorates the faces of a polyhedron to define a non-polyhedral cage that is transformed by planarizing the faces. In an embodiment the secondary cage comprises tiles derived from an Archimedean tiling that decorate faces of the initial cage comprising a polyhedron. In an embodiment the secondary cages resemble a nanotube.

Claims

1. A method for constructing a convex equilateral structure comprising:

selecting a triangular or square patch defining edges and vertices derived from a uniform tiling having 4fold or 6fold symmetry;

defining a cage corresponding to the edges and vertices defined by the selected patch applied to each face of a polyhedron selected from an icosahedron, an octahedron, a tetrahedron, a truncated tetrahedron, and a cube, and connecting patches on adjacent faces of the selected polyhedron, wherein the defined cage edges define a plurality of polygons;

resizing at least some of the edges such that all of the edges have the same length, the resized cage defining a modified plurality of polygons, wherein at least some of the modified plurality of polygons are not planar;

solving for a set of interior angles or vertex coordinates for the plurality of modified polygons that cause the plurality of modified polygons to be planar;

constructing a convex equilateral cage structure comprising interconnected members having a uniform length and oriented according to the set of interior angles or placed according to the vertex coordinates.

2. The method of claim 1, wherein the members comprise elongate struts having the uniform length, and the convex equilateral structure comprises a cage.

3. The method of claim 1, wherein the members comprise equilateral planar members, and the convex equilateral structure comprises an enclosure.

4. The method of claim 1, wherein the uniform tiling comprises an Archimedean tiling.

5. The method of claim 1, wherein the uniform tiling comprises a tiling having 6fold symmetry, the patch comprises a triangular patch, and the polyhedron comprises a tetrahedron, an octahedron, or an icosahedron.

6. The method of claim 1, wherein the uniform tiling comprises a tiling having 4fold symmetry, the patch comprises a square patch, and the polyhedron comprises a cube.

7. The method of claim 1, wherein the uniform tiling comprises a compound tiling having a hexagonal portion and a triangular portion, and the polyhedron comprises a truncated tetrahedron.

8. The method of claim 1, wherein the set of interior angles are solved for by determining the set of interior angles that produce zero dihedral angle discrepancy throughout the edges of the cage.

9. The method of claim 1, wherein the set of vertex coordinates are solved for by determining the set of coordinates that produce zero dihedral angle throughout the polygonal faces of the cage.

10. The method of claim 1, wherein constructing the convex equilateral structure comprises interconnecting a plurality of elongate struts.

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

12. A method for designing a dome structure comprising:

defining at least one planar patch comprising a network of edges and vertices that correspond to a contiguous section of a uniform tiling having 4fold or 6fold symmetry;

defining a non-polyhedral cage corresponding to (i) a plurality of the at least one planar patch disposed on the faces of a selected polyhedron, and (ii) joining patches from the plurality of patches across adjacent faces of the selected polyhedron, wherein the non-polyhedral cage defines a plurality of polygonal faces, each polygonal face defining a set of internal angles;

solving for new values for the sets of internal angles of the plurality of polygonal faces that result in the polygonal faces being planar, such that transforming the non-polyhedral cage to incorporate the new values for the sets of internal angles produces a polyhedral cage; and

constructing a dome structure having polygonal edges that correspond to the polyhedral cage.

13. The method of claim 12, further comprising transforming the non-polyhedral cage such that all of the edges are of the same length.

14. The method of claim 12, wherein the dome structure comprises a plurality of struts that are joined to define a cage corresponding to the dome structure.

15. The method of claim 12, wherein the selected polyhedron comprises one of a tetrahedron, an octahedron, an icosahedron, a truncated tetrahedron, and a cube.

16. The method of claim 12, wherein the uniform tiling comprises a regular or a semi-regular tiling.

17. The method of claim 12, wherein the uniform tiling comprises a compound tiling having a hexagonal portion and a triangular portion, and the polyhedron comprises a truncated tetrahedron.

18. The method of claim 12, wherein the new values for the set of interior angles are solved for by determining the set of interior angles that produce zero dihedral angle discrepancy throughout edges of the cage.

19. The method of claim 12, wherein the new values for the set of vertex coordinates are solved for by determining the set of coordinates that produce zero dihedral angles throughout the polygonal faces of the cage.

20. A method of defining a convex equilateral cage comprising:

selecting an initial cage comprising a plurality of edges extending from vertices, wherein the edges define a plurality of faces;

defining a secondary cage comprising edges and vertices from a plurality of tiles, reduced tiles, or larger patches obtained or derived from a selected one of the uniform Archimedean tilings, wherein the plurality of tiles, reduced tiles, or patches are positioned over the plurality of faces of the initial cage;

resizing at least some of the edges of the secondary cage such that all of the edges have the same length, the resized cage defining a plurality of polygons, wherein at least some of the plurality of polygons are not planar;

solving for a set of interior angles for the plurality of polygonal faces that transform the plurality of polygonal faces to be planar or solving for a set of vertex coordinates for the plurality of polygonal faces that transforms the plurality of polygonal faces to be planar; and

constructing a convex equilateral cage structure comprising members defining edges having the same length and oriented according to the set of interior angles or spanning the set of vertex coordinates.

21. The method of claim 20, wherein the initial cage comprises one of a tetrahedral cage, an octahedral cage, an icosahedral cage, and a hexahedral cage.

22. The method of claim 20, wherein the plurality of tiles, reduced tiles, or larger patches comprise Archimedean tiles obtained from an Archimedean uniform tiling.

23. The method of claim 20, wherein the plurality of tiles, reduced tiles, or larger patches consist of a plurality of tiles from a selected Archimedean uniform tiling and a plurality of reduced tiles derived from the tiles from the selected Archimedean uniform tiling.