Imagine, Geometry transitions - haptihedron and vector equilibrium

These pages are to give flight to your imagination. Transitions of the transition: from haptihedron through the vector equilibrium. I use the term inverse loosely, rather than technically correct terms*, because the forms just seem to be opposites visually. I named the first, the haptihedron, technically known as a rhombicuboctahedron (see links). The second was referred to by R. Buckminster Fuller as the vector equilibrium, but is technically known as a cuboctahedron or cubo-octahedron. Call it VE for short. You can see how the two transitional structures are interrelated in the middle graphic, "Inverses". Click it to see an animation and more, a QTVR movie.

HaptihedronInversesVector Equilibrium
haptihedron Between VE and haptihedron vector equilibrium

The VE is outside the haptihedron. Start with the top square face of a haptihedron. Each corner is the center point of a larger surrounding square, which is the top face of a VE. If you find the outer square for each face of the haptihedron, you describe a VE. The eight triangles between the square faces of the VE, contain the eight triangles of the haptihedron.

If you start with a VE, you could see the relationship in a different way. Imagine a cube within the VE. Expand the faces of the cube so the corner points bisect each face of the VE. Connect these corner points with triangles within each triangle of the VE. You have created a haptihedron within the VE.

The six square faces of the haptihedron (that are an expanded cube) are the ones that are contained in the six square faces of a VE. The haptihedron has 12 more square faces. They each form the base of a pyramid with the 12 vertexes of a VE.

Technically, perhaps the rhombicuboctahedron could be called a snub cuboctahedron, because all the corners are cut off. In my visions, the haptihedron is the evolutionary form.

What next? See the missing link

* In dual polygons, each vertex is centered in the face of the opposite form but technically, the haptihedron is a rectification of the VE.

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