Tegum product

The tegum product is an operation that can be applied on any two polytopes. The tegum product of two polytopes A and B is formed by placing them in hyperplanes with a single point as intersection, and lacing them together. The resulting polytope can be called a duotegum. Its facets will be the pyramid products of the facets of a polytope times the other polytope. of which the orthotope or cross-polytope is the regular form.

The word "tegum" is derived from the Latin word "tegere", meaning "to cover". This alludes to how tegums seem to "wrap about" the polytopes that define them.

Tegum products extend the notion of a bipyramid (also sometimes called a tegum), which results from the special case when one of the factors is a dyad. As such, they're particularly applicable to orthoplices. The pyramid product of an m-orthoplex and an n-orthoplex is an (m+n)-simplex. In particular, an n-simplex can be seen as the pyramid product of n dyads.

Tegum products are closely related to pyramid products, but are differentiated mainly by the additional dimension required by pyramid products. They're also related to the prism product, given that the dual of the tegum product of two polytopes is the prism product of their duals. Like both of these products, the tegum product is both commutative and associative.

The product was discovered by Wendy Krieger in 1977.

Properties
The product is a drawn product of surfaces. This means that the product has a surface made from lines drawn from point A on the surface of A, to point B on the surface of B. In essence, one takes an orthotope of the required dimension (say 5), and replace three of the axies with a 3d space containing a dodecahedron, and the remaining two by a polygon (or circle),

The surtope consist of a tegum product is derived from the tegum-form (which is nulloid + vertex-edge-hedron..., but not including the content), the Tegum-form of the product is the product of the tegum-forms. One sees that tegum-form Tg(line) = (1n+2v) taken to the third power gives third power gives the tegum-form of an octahderon (1n+2v)^3 = (1n+6v+12e+8h).

The tegum-product is coherent, in that the tegum-volume of a tegum or pyramid products of the bases (and altitude), it the product of the volumes of the individual elements. The tegum-volume of a measure-polytope or n-cube, is n!. The tegum-volume can be derived by recursively taking the formula of V=r·dS, where r is the radial vector, and dS is a normal to the surface. Taken from the vertex of a line, square, cube, this amounts to the number of opposite faces to the point, times the content of each face, so it gives 1*2*3*.. or n! units.

The tegum-product is radiant, in that the surface of the product of elements on the cartesian axies, is the sum of that of the elements. The radiant function is set up based on the standard sphere (radius 1 at the origin), defines a radial function where the surface is 1, the interior less than one, and the exterior greater than one. If this is taken as a direction-dependent function, then any solid can be defined as radial. Several different elements completely orthogonal to each other, define in x1, x2, x3... the radial X, in y1, y2, y3,... the radial Y etc. For the point x1, x2, x3, ... y1, y2, y3, ... z1, z2, z3, ... the simple radials are X, Y, Z, .. The tegum-product is given by sum(X,Y,Z...)=1, the prism product by max(X, Y, Z...)=1, and the crind-product by rss(X, Y, Z...)=1, where rss is root-sum-square, or the square root of the sum of squares.