## I need an explanation of notations for rotopes.

Discussion of shapes with curves and holes in various dimensions.

### I need an explanation of notations for rotopes.

I've gotten a new interest in toratopes. Unfortunately, I do not know what a cartesian product, and how common notation actually works. Can anybody help?

ubersketch
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### Re: I need an explanation of notations for rotopes.

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.
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wendy
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### Re: I need an explanation of notations for rotopes.

I was talking about rotopes, which also use tapering operation.

ubersketch
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### Re: I need an explanation of notations for rotopes.

ubersketch wrote:I was talking about rotopes, which also use tapering operation.

Ah, I think you're talking about Keiji's old notation : (II)'I , II' , (II)'I , etc? It combines toratope notation (incl open/closed) with the taper operator as the apostrophe symbol : '

Well it pretty much works in a linear way (reading left to right), along with the nested toratope symbols. It helps describe all of the various types of n-cube pyramids, n-simplex prisms, n-cylinder pyramids, n-sphere cones, and any possible hybrid. I think Keiji abandoned the notation on the premise that it was describing impossible geometric figures, like the cyltrianglintigroid ((I')(II)) , circle over the cartesian product of the 1D surfaces of a circle and triangle. But I stumbled upon the equation one day, which proved the existence. So, it's not that bad after all!

This was the notation that inspired me to create my own (STEMP notation), where I further reduced the circle into two operators, the extrude : I , followed by bisecting rotate ; O , which reads IO instead of (II) . The taper symbol is > , and the cartesian product is x[y] .
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