We know that we can express any rotatope as the Cartesian product of spheres, unless it is a sphere in itself. It seems much like multiplication using the point as the multiplicative identity. In Polyhedron Dude's number series notation, the Cartesian factorisation of a rotatope is evident. First, we can define a prism as any figure having 1 (the line segment) as one of its Cartesian factors.
I have classified the rotatopes into prisms, duisms, trisms, tetrisms, pentisms... for easier counting. The classification is based on the least Cartesian factor (LCF) of a rotatope. If a rotatope's least Cartesian factor is 1, it is a prism. Eg: Line segment, square, cube, cylinder, tesseract, cubinder etc. If its LCF is 2 (Circle), it is a duism. Eg: Circle, duocylinder, cylspherinder, triocylinder, cylglominder etc. If its LCF is 3 (Sphere) it is a trism. Eg: Sphere, duospherinder, spherglominder etc. This goes on forever. (A rotatope with LCF = x is called x-ism).
Also, I am calling all rotatopes with LCF > 1 hyperprisms (This is where I fear mixing of names), those with LCF > 2 hyperduisms etc.
The following are the results I have obtained:
The Cartesian product of any rotatope with 1 gives a prism. (This is already very well known).
The Cartesian product of any hyperprism with 2 gives a duism.
The Cartesian product of any hyperduism with 3 gives a trism.
The Cartesian product of any hypertrism with 4 gives a tetrism.
And so on.
We know that R(0) = 1 and R(1) = 1. This is obvious. Now, the 1D rotatope, the line segment, is a prism. Among the rotatopes, I denote the number of n-D prisms as R1(n), the number of n-D duisms as R2(n) etc. I also denote hyperprisms as R>1(n), hyperduisms as R>2(x) etc. Also, where [x] denotes the greatest integer function of x, we can say that x-isms do not exist in n-D, where [n/2] < x < n and when x > n. The latter condition is obvious, as any x-ism requires at least x dimensions. The former condition holds because x is the least Cartesian factor of any x-ism. If [n/2] < x < n, then (n - x) < [n/2]. Since (n - x) < x, x would not be the least Cartesian factor. And we can see for ourselves that there are no duisms in 3D, no trisms in 4D but there is a duism in 4D - The duocylinder.
Now comes my formula.
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R[sub]1[/sub](n) = R(n - 1)
R[sub]2[/sub](n) = R[sub]>1[/sub](n - 2)
R[sub]3[/sub](n) = R[sub]>2[/sub](n - 3)
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R[sub]x[/sub](n) = R[sub]>(x - 1)[/sub](n - x)
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R[sub]n[/sub](n) = 1
Adding all these, we get R(n) = R(n - 1) + R[sub]>1[/sub](n - 2) + R[sub]>2[/sub](n - 3) + ... + 1
We can verify it:
R(1) = 1
R(2) = 1 + 1 = 2
R(3) = 2 + 0 + 1 = 3
R(4) = 3 + 1 + 0 + 1 = 5
R(5) = 5 + 1 + 0 + 0 + 1 = 7
R(6) = 7 + 2 + 1 + 0 + 0 + 1 = 11
R(7) = 11 + 2 + 1 + 0 + 0 + 0 + 1 = 15
R(8) = 15 + 4 + 1 + 1 + 0 + 0 + 0 + 1 = 22
R(9) = 22 + 4 + 2 + 1 + 0 + 0 + 0 + 0 + 1 = 30
R(10) = 30 + 7 + 2 + 1 + 1 + 0 + 0 + 0 + 0 + 1 = 42
We can continue this process further.
This is useful in number theory also, as the number of rotatopes in n-D is the number of ways in which n can be expressedas the sum of natural numbers.