The Angle Deficit Constant

To build a closed polyhedron, it is necessary to remove or add 'angle' to make vertices. The angle deficit constant theorem tells us that if the resultant polyhedron is simply closed, the sum of the angles missing at each vertex will total two Pi.

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Decomposing Deltahedra: the Icosahedral Family

Using the subdivision (skewed or straight) of an equilateral triangle into smaller equilateral triangles, and applying that to the net of an icosahedron, what sort of polyhedra result?

Well, there are results such as the triangulated, dimpled soccer ball, the dimpled dodecahedron, the Endo-Pentakis-Icosi-Dodecahedron, and many others.

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Decomposing Deltahedra:
the Tetrahedral and Octahedral Families

And what shapes do you get if you do the same thing with a tetrahedron (see left), or octahedron?

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Deriving the Archimedean or Uniform Tilings and Solids

In a uniform or Archimedean tiling or solid, all faces or tiles are regular polygons, and all vertices are equivalent. There are 11 tilings and excluding prisms and antiprisms but including the Platonic Solids and mirror images, there are 20 solids that obey these conditions. But how do we know there aren't more?

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From the tiled tetrahedron to the regular dodecahedron, explorations in regular coloring

Trying to color a regular or semi-regular polyhedron with symmetrical distribution can be a challenge. Studying the result can give you clues about their symmetries—and their relationship to other polyhedra.

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Vertex-Centric Nets of Semi-Regular Deltahedra

Starting with a folded regular triangular grid, and using notions from the Angle Deficit Constant theory, deltahedra can be built by removing or adding angle at each vertex. Starting in the middle of the hexagonal face in the example illustrated, the vetices are 1x6, then a row of 6x6, a row of 6x4, another row of 6x6 and finally a closed 1x6.

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Which Delta-Polyominoes are Nets
of Closed Deltahedra?

An equilateral triangle subdivided into four triangles of half the linear size is a net of the regular tetrahedron. What other polyominoe made of equilateral triangles assembles into a closed polyhedron?

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