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.

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.

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.

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.