by wendy » Tue Jun 13, 2006 8:57 am
This is the definition i use for the four products.
A base is a figure in a space of n dimensions.
A centre (o) of product is the point somewhere in the same space as the base. For the normal products, this is the same as the geometric centre. But for "off-centre" products, this can be anywhere.
For any other point (p), it is defined such that a ray from o through p, crosses the surface s at 1, ie the value of the radial function at p is r = (OP/OS).
The three dimension-summing products are:
1. prism: R = max(r1, r2, ...)
2. tegum: R = sum(r1, r2, ...)
3. sphere: R = rss(r1, r2, ...) [rss = root sum square]
When these are applied to an edge (u) from +1 to -1, with a centre at 0, this gives, respectively,
max(u1, u2, ...) -> line, square, cube, tesseract, ...
sum(u1, u2, ...) -> diagonal, rhombus, octahedron, 16choron, ...
rss(u1, u2, ...) -> diameter, circle, sphere, glome.
When the edge is set to unit, these can be used as a coherent measure scale. The dimension of the product is the sum of dimensions in the elements.
We have, for two lines X, Y,
max(X,Y) -> rectangle sides X, Y
sum(X,Y) -> rhombus diagonals X, Y
rss(X,Y) -> ellipse diagonals x, y
These can be continued upwards to any dimension.
The fourth product is the pyramid product. It is defined in terms of an altitude A, in the shape of a simplex:
A = point, line, triangle, tetrahedron, pentachoron, ....
We then a point in the simplex that fits the equation,
a1+a2+... = 1
For a given point a, it is a1 of the height to vertex 1, a2 of the height to vertex 2, etc. This varies from a1 = 1, at the vertex, to a1 = 0 at the opposite side. If one wants a simple example, consider the simplex as a face of a cross-product, where the points form the axies, A is the face, being a1+a2+... = 1.
The simplex product is then defined in terms of
1 = sum(a1,a2,....) over A
R = max(a1*r1, a2*r2, ... ) dim = Na+N1+N2+...
When r1 is a point, we have r1 = 1, and have, eg max(a1, a2*r1, ...)
For a pyramid p, we have A in the z-axis, running from r1=apex to r2=base.
a point x,y,z then gives a radial function a1+a2*r2, which gives a tapering of height of a2 going from full size when a2 = 1, to a point at a2 = 0.
When one has a tetrahedron, where x= line, y = line, z= height, one gets a tapering of height, implemented by rectangles a1.x * a2.y as z varies over a1 to a2.
Wendy