Rotatopes: The Link Between Geometry and Number Theory

Discussion of shapes with curves and holes in various dimensions.

Rotatopes: The Link Between Geometry and Number Theory

Postby Prashantkrishnan » Thu Jan 29, 2015 7:49 am

I think I have found out how to compute the number of rotatopes in a given number of dimensions, and as I searched the entire forum of toratopes, I did not find any thread similar to this, other than http://hddb.teamikaria.com/forum/viewtopic.php?f=24&t=409. I am denoting the number of n-D rotatopes by R(n). Before proceeding to my method, I would like to introduce my nomenclature. (I would be glad to know if there already exists a nomenclature for these, because some words I use might be misleading).

:arrow: 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.

:arrow: 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).

:arrow: 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:

:arrow: The Cartesian product of any rotatope with 1 gives a prism. (This is already very well known).
:arrow: The Cartesian product of any hyperprism with 2 gives a duism.
:arrow: The Cartesian product of any hyperduism with 3 gives a trism.
:arrow: 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.

Code: Select all

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)
.
.
.
R[sub]x[/sub](n) = R[sub]>(x - 1)[/sub](n - x)
.
.
.
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.
Last edited by Prashantkrishnan on Fri Jan 30, 2015 9:23 am, edited 1 time in total.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby ICN5D » Thu Jan 29, 2015 5:52 pm

You will find similar things here:

http://hddb.teamikaria.com/forum/viewtopic.php?p=20768#p20768

and, here:

http://hddb.teamikaria.com/forum/viewtopic.php?p=20521#p20521

There are an equal number of rotatopes to toratopes, per dimension. Since the notation follows the combinatorics of Rooted Trees with Nested Leaves, it corresponds to integer sequence A000669

Basically, the number of Toratopes or rotatopes (including n-spheres/n-cubes), per dimension is:

1D : 1
2D : 1
3D : 2
4D : 5
5D : 12
6D : 33
7D : 90
8D : 261
9D : 766
10D : 2,312

The number of possibilities grows by nearly three-fold, with each added dimension.
in search of combinatorial objects of finite extent
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Prashantkrishnan » Fri Jan 30, 2015 10:13 am

ICN5D wrote:Basically, the number of Toratopes or rotatopes (including n-spheres/n-cubes), per dimension is:

1D : 1
2D : 1
3D : 2
4D : 5
5D : 12
6D : 33
7D : 90
8D : 261
9D : 766
10D : 2,312

The number of possibilities grows by nearly three-fold, with each added dimension.


Aren't these the number of toratopes? There doesn't seem to be any simple relation between the number of toratopes and the number of rotatopes. Your list matches with mine only for 4D where the number of open/closed rotatopes and the number of rotatopes both are 5. Otherwise, the rotatopes come like this

Code: Select all
Dimension          Number of Rotatopes          Number Series Notation          Name                    Type
                1                               1                                   1                             Line Segment            Prism
                2                               2                                   11                            Square                     Prism
                                                                                      2                              Circle                      Duism
                3                               3                                   111                            Cube                      Prism
                                                                                      21                            Cylinder                   Prism
                                                                                      3                                Sphere                  Trism
                4                               5                                   1111                           Tesseract               Prism
                                                                                      211                             Cubinder               Prism
                                                                                      22                            Duocylinder              Duism
                                                                                      31                            Spherinder               Prism
                                                                                      4                              Glome                     Tetrism
                5                               7                                  11111                         Penteract                Prism
                                                                                     2111                          Tesserinder             Prism
                                                                                     221                            Duocyldyinder          Prism
                                                                                     311                            Cubspherinder         Prism
                                                                                     32                              Cylspherinder          Duism
                                                                                     41                              Glominder               Prism
                                                                                     5                                Pentasphere           Pentism
               6                               11                                 111111                        Hexeract                Prism
                                                                                     21111                         Penterinder             Prism
                                                                                     2211                           Duocyltriinder         Prism
                                                                                     222                             Triocylinder            Duism
                                                                                     3111                           Tesserspherinder     Prism
                                                                                     321                             Cylspherdyinder      Prism
                                                                                     33                               Duospherinder       Trism
                                                                                     411                             Cubglominder         Prism
                                                                                     42                               Cylglominder          Duism
                                                                                     51                               Pentaspherinder     Prism
                                                                                     6                                 Hexasphere           Hexism
               7                               15                                 1111111                       Septerect              Prism
                                                                                     211111                        Hexerinder            Prism
                                                                                    22111                           Duocyltetrinder      Prism
                                                                                    2221                            Triocyldyinder         Prism
                                                                                    31111                           Penterspherinder    Prism
                                                                                    3211                             Cylsphertriinder     Prism
                                                                                    322                               Duocylspherinder   Duism
                                                                                    331                               Duospherdyinder   Prism
                                                                                    4111                             Tesserglominder    Prism
                                                                                    421                               Cylglomdyinder      Prism
                                                                                    43                                 Spherglominder     Trism
                                                                                    511                               Cubpentaspherinder    Prism
                                                                                    52                                 Cylpentaspherinder     Duism
                                                                                    61                                 Hexaspherinder     Prism
                                                                                    7                                   Septaspher            Septism
              8                                22                                11111111                        Octeract               Prism
                                                                                   2111111                          Septerinder           Prism
                                                                                   221111                            Duocylpentinder    Prism
                                                                                   22211                              Triocyltriinder       Prism
                                                                                   2222                                Tetrocylinder        Duism
                                                                                   311111                             Hexerspherinder  Prism
                                                                                   32111                               Cylsphertetrinder Prism
                                                                                   3221                                Duocylspherdyinder   Prism
                                                                                   3311                                Duosphertriinder   Prism
                                                                                   332                                  Cylduospherinder  Duism
                                                                                   41111                               Penterglominder   Prism
                                                                                   4211                                 Cylglomtriinder    Prism
                                                                                   422                                   Duocylglominder  Duism
                                                                                   431                                   Spherglomdyinder  Prism
                                                                                   44                                     Duoglominder      Tetrism
                                                                                   5111                                 Tesserpentaspherinder    Prism
                                                                                   521                                   Cylpentaspherdyinder   Prism
                                                                                   53                                     Spherpentaspherinder   Trism
                                                                                   611                                   Cubhexaspherinder       Prism
                                                                                   62                                     Cylhexaspherinder        Duism
                                                                                   71                                     Septaspherinder   Prism
                                                                                   8                                       Octasphere          Octism


I stopped here with 8D as some of the names were getting troublesome to type, and the number increases drastically, but you can see that I have not missed a single rotatope. The toratopes upto 8D that are not in this list are not rotatopes. My counting matches with the formula and I think it is foolproof. The only problem with this method is that if you want to find, say R(85956) then you would have to start from R(1) and keep on counting till 85956. I can't see any direct formula.
Last edited by Prashantkrishnan on Sat Jan 31, 2015 5:17 pm, edited 3 times in total.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Prashantkrishnan » Fri Jan 30, 2015 10:14 am

The code tag seems too narrow to hold the list in a proper manner.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Marek14 » Fri Jan 30, 2015 11:23 am

Well, for the rotatopes and toratopes to match, the toratopes have to play their part in the rotatopes. For example, in 4D, there is a torus prism and in 5D you get torus x line x line, torus x circle and 4D toratopes x line.

Then you get parallel structures for both, for example in 4D:

IIII - tesseract <-> (IIII) glome
(III)I - spherinder <-> ((III)I) torisphere
((II)I)I - torinder <-> (((II)I)I) ditorus
(II)(II) - duocylinder <-> ((II)(II)) tiger
(II)II - cubinder <-> ((II)II) spheritorus

Basically, this creates one bigger category of shapes that contain toratopes and all their cartesian products. The distinction between rotatopes/toratopes is basically whether they contain any lower-dimensional elements or not.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Prashantkrishnan » Fri Jan 30, 2015 11:52 am

Marek14 wrote:Then you get parallel structures for both, for example in 4D:

IIII - tesseract <-> (IIII) glome
(III)I - spherinder <-> ((III)I) torisphere
((II)I)I - torinder <-> (((II)I)I) ditorus
(II)(II) - duocylinder <-> ((II)(II)) tiger
(II)II - cubinder <-> ((II)II) spheritorus


From what I know, among these, the torinder, the torisphere, the ditorus, the tiger and the spheritorus are not rotatopes. Are you using a wider definition of rotatopes? I am talking about the rotatopes as mentioned here.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Marek14 » Fri Jan 30, 2015 2:41 pm

Yes, but that page is twelve years old. Currently, I think that open and closed toratopes more or less replaced it.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Prashantkrishnan » Sat Jan 31, 2015 5:11 pm

It seems to me from your previous posts that what I know as open toratopes are what you are calling rotatopes and what I know as closed toratopes are what you are calling toratopes. My formula is for calculating the number of shapes that are either spheres or Cartesian products of spheres.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Marek14 » Sat Jan 31, 2015 6:20 pm

Prashantkrishnan wrote:It seems to me from your previous posts that what I know as open toratopes are what you are calling rotatopes and what I know as closed toratopes are what you are calling toratopes. My formula is for calculating the number of shapes that are either spheres or Cartesian products of spheres.


Yes.
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby Keiji » Thu Jun 18, 2015 7:41 pm

Marek14 wrote:Yes, but that page is twelve years old. Currently, I think that open and closed toratopes more or less replaced it.


I'm late to the party, but I want to point out that the 2003 definition of rotatopes stands firm: the set of rotatopes is strictly limited to Cartesian products of line segments and hyperspheres.

If you want to talk about toratopes, use the word toratopes. :)
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Re: Rotatopes: The Link Between Geometry and Number Theory

Postby PWrong » Fri Jun 19, 2015 3:10 pm

Open toratopes are not the same as rotatopes. Example: the toracylinder ((II)I)I is an open toratope, but not a rotatope. The sphere (III) is a rotatope and a closed toratope, not open.

The rotatopes in n dimensions correspond directly with integer partitions of n.

Every open toratope has a corresponding closed toratope, created by putting brackets around it. For example, (II)II becomes ((II)II). The open and closed toratopes in n dimensions correspond directly to series and parallel networks with n edges.
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