by wendy » Wed Feb 27, 2019 11:03 am
Rototopes and Toratopes are different things, and there are several notations for each.
A cartesian product is the repetition of content, or the 'prism product'. For each point a, for each point b, there is a point ab in the product. The repetition of surfaces is the comb product, it makes toruses, and for honeycombs (which are a hyper-surfaces), the product thereof.
The pyramid product is the draught of content, for each point a, for each point b, there is a line ab. The draught of surface is the tegum product.
Rototopes use the various radiant products. In a radiant product, the centre is taken as 0, and the surface as 1. The usual distribution for the three radiant regular solids are powers of the line -1 to +1, with a centre at 0.
cube = max(x,y,z) = [x,y,z]
octahedron = sum(x,y,z) = <x,y,z>
sphere = rss(x,y,z) = (x,y,z). rss = root-sum-squares, ie sqrt(x²+y²+z²).
The trick is then to use these functions, to form other figures. For example, the cylinder is [(x,y),z]. This is a prism [ A, z] with a circular base (x,y). A duocylinder is formed by [(x,y)(z,w)] or [(i,i)(i,i)] where 'i' stands for an axis.
A figure that covers (tegum) two orthogonal circles (bi-circular) gives <(ii)(ii)>, the surface here is sum(rss(w,x),rss(y,z)) = 1, is the version where one rotates a 3d cone around its base, so the apex forms a second circle.
The fourth product is a pyramid product, which we might represent by &. The pyramid product adds an extra dimension each time it is applied, and without an argument, a point is implied. So (x,y)& is a cone, and [x,y]& is a square pyramid. Note that x&y is a tetrahedron, as is &&&.
This product was discovered in terms of the octahedron-face, so in the sense of X, Y, Z, we create a point xX, yY, zZ, such that x+y+z=1. This means we produce copies of X, Y, Z at the verticies of an orthogonal triangle, and the relative heights of this same triangle is occupied by a prism-product of xX, yY, zZ.
In i&i, the two i's are orthogonal lines, seperated hy some height. As one moves 30% up the height, there is a 30% x × 70 % y prism at that height.
This leads to things in 4D, like i&(x,y) which is a circle-line pyramid, its mid-sections being cylinders, and &(iii), a cone with a spherical base. Note the final dimension here is to count the i's and &. So &&&& is a 4d simplex, ie the fifth power of a point (lit .&.&.&.&.). Note that i&(ii) can be written as &&(ii), that is the pyramid of a pyrimid of a circle.
Torotopes
The torotope notation is written as a nested series of sphere-products, the implication is that the radius of the spheres increase as one goes inwards. Since this is ultimately based on the comb-product, we replace an axis of the sphere with a sphere itself.
(ii) = circle, (iii) = sphere, (iiii) = glome. Sometimes the i's are counted up, so (2)=circle, (3)=sphere, (4) = glome.
If you take a circle (ii), and replace a point on the boundary with a smaller circle (sharing the same radial direction), you get ((ii)i). This is a 3d torus.
If you take two orthogonal circles in prism product, and replace a point on the prism surface with a smaller circle, you get a tiger ((ii)(ii)).
If you take a spheric hose in 4d, and connect the ends to a circle, you get a big circle (ii), replacing the radius of a little sphere (iii), ie ((ii)ii).
If you take a spheric tube in 4d, and roll the top outwards, like taking off a sock, it will cover a spheric-ring, to make a small circle around a big sphere, ie ((iii)i).
Note in all cases, the surface of these figures is the prism-product of the surfaces of the brackets, so ((ii)i) > (2)(2) = circle . circle
In 4D, ((ii)(ii)) gives (2)(2)(2) a "tritorus" in math speak, but also (((ii)i)i) gives (2)(2)(2).
The next two ((ii)ii) and ((iii)i) are topologically different solids, but the surface in both cases is (2)(3), ie E1E2 in prism-product.
This is because the comb-product is a repetition of the surfaces, but the order is specific (ie it's not communitive): a×b <> b×a.