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van der Vegt A.K.'s Order in space PDF

By van der Vegt A.K.

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The next double-pyramid is the ((4 4 4)), which is nothing else than the (3 3 3 3) or {3,4}, the regular octahedron. The cube appeared also to be the second in the series of the Archimedean prisms! The top-angle of the triangles is, of course, 60°. 52 Order in space Then ((4 4 5)) and ((4 4 6)) etc. 875) etc. The general formula for the top-angle αm of the faces of a ((4 4 m)) is: 1 cos αm = cos θ + 2 sin2 θ with θ = 360°/m ; this relation follows in a simple way from the foregoing paragraph. It is clear that with increasing m the top-angles become smaller and smaller and approach to zero.

875) etc. The general formula for the top-angle αm of the faces of a ((4 4 m)) is: 1 cos αm = cos θ + 2 sin2 θ with θ = 360°/m ; this relation follows in a simple way from the foregoing paragraph. It is clear that with increasing m the top-angles become smaller and smaller and approach to zero. While the Archimedean prism approaches to a flat disk, the doublepyramid becomes a two-pointed needle. The coordinates of ((4 4 m)) can, in the most simple way, be expressed by locating the vertices on the z-axis, and the common base, {m}, in the x-y plane.

As a first illustration of this principle we consider the ((3 6 6)); here obtuse triangular pyramids have been placed on each of the four faces of a tetrahedron in such a way that the dihedral angles are equal. The solid thus has 12 faces in four groups of 3, which explains its name “triakistetrahedron” (3 * 4). Of its eight vertices four form a trihedral angle and four a hexahedral angle; these groups of four coincide with the vertices of two regular tetrahedra, whose edges are perpendicular to each other and have a length ratio of 3 : 5.

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Order in space by van der Vegt A.K.


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