rotatopes as partitions

[PWrong]

I've found a simple technique for counting rotatopes in each dimension. I'm not sure if someone's noticed this before, since I don't read the geometry forum very often, but here it is.

The trick is to consider a rotatope as the product of several solid spheres from lower dimensions. Remember that since a 1D sphere is two points, a line is a 1D "ball".

Cube = product of three 1-balls = (1,1,1)
Cylinder = a 2-ball and a 1-ball = (2,1)
Sphere = 1 3-ball = (3)

The important thing is that if you add up the dimensions of all the balls, you get the dimension of the rotatope.
Cube: 1+1+1 = 3
Cylinder: 2+1 = 3
Sphere: 3 = 3

Here's the five 4D rotatopes in this notation.
Tetracube: 1+1+1+1
Cubinder: 2+1+1
Duocylinder: 2+2
Spherinder: 3+1
Glome: 4

Each N-dimensional rotatope corresponds to a partition of the number N. This gives us a simple way to count rotatopes in any dimension, and a few different ways to notate them.
Anyone think this could be useful?

[Marek14]

Well, this was the first thing that came to my mind when I read about the rotatope concept here. But I thought that is too simple, so I "expanded" the definition to the point where it now has rotatopes in N dimensions corresponding to graphs on N vertices.

But yes, the basic concept is about partitions.

[wendy]

The current working definition for rototopes, is to allow one to freely mark the edges of a simplex with a circle or square section, eg

AB, BC, CD = square , CA, AD, DC = circle => longdome.

That's as far as Marek 14 got.

I got as far as considering the nested product of two radiant functions, ie [] for prism, () for cylinder

[x,y,z] = cube
[x,(y,z)] = cylinder
(x,[y,z]) = dome
(x,y,z) = sphere

In four dimensions, this gives 10 of the 11 known examples, the missing example being the longdome.

W

[PWrong]

Hmm, that's interesting. I'll have another look at the thread about domes, longdomes e.t.c.

What about the different kinds of torus in 4D? They have a similar construction to a sphere, except you can rotate around an arbitrary line/plane, as opposed to one through the origin.

Before I do this, I'm introducing a simpler way of describing rotations. Rather than rotating "about" a line/plane that isn't moving, you rotate "through" the plane that is moving, around a point in the plane. So in 4D, rotating around the xy plane is the same as rotating through the zw plane. Note that in N dimensions, you rotate "about" an (N-2)-hyperplane, but you always rotate "through" a 2D plane.

Now here's the "toratopes" in each dimension. Using the torus construction
I'm also including a number of "hollow" rotatopes

2D
We have a new kind of torus, a "hollow circle". It's formed by rotating a line around a point outside the line.

3D
1. Torus
Take a circle in the xy plane, with centre O, radius r.
Rotate it through the xz plane, around the point (R,0,0).
That is, rotate it about the line parallel to the y axis and containing the point (R,0,0).

2. Hollow Cylinder
Take a square, and do the same as above.
This is the extension of a hollow circle.

We also have a hollow sphere and a hollow torus, formed by rotating the hollow circle.

4D torachora
1. Spherical torus
Take a sphere in the xyz realm, with centre O, radius r.
Rotate it through the xw plane, around (R, 0, 0, 0).
This is the analog of the hollow circle and hollow cylinder.

2. extended torus
Take a cylinder with it's base on the xy plane, radius r, height z.
Again, rotate it through the xw plane, around (R, 0, 0, 0).
This is the same as linearly extending a 3D torus, hence the name.

3. cylindrical torus
Same thing, but rotate through zw, around (0, 0, R, 0)
I have a feeling this might be similar to a duocylinder.

4. duotorus
Take a 3D torus. Call the two radii r1 and r2.
Rotate through xw around (r3, 0, 0, 0).
I think this is only real torus I've found. It has a hole

5. psuedo-torus
Same 3D torus, but rotate through zw around (0, 0, r3, 0). I can't seem to visualise this one.

6. Cube torus
Take a cube in xyz, rotate through xw around (r3, 0, 0, 0).

There are many more 4D torachora, including many hollow rotachora. I'm working on a notation for them. I'm thinking about using {} for a torus-like rotation.

[Keiji]

I've found a simple technique for counting rotatopes in each dimension. I'm not sure if someone's noticed this before, since I don't read the geometry forum very often, but here it is.

What you described there is practically the same as Method 2 of this page...

[wendy]

The general thinking from me is that you can take a torus-product of spheres, as long as the surface adds up to the surfaces of the elements. The torus-product is not associative.

Adding to the front is like connecting a hose: that is, one bends the thing so that the "inside" of the figure remains inside.

Adding to the end makes a connexion like a sock. That is, the surface rolls down so that the outside becomes the new interior.

The trick is: h=1, c=2, t=3, p=4, ..... You add to you get N-1 as a sum, and at least two terms. Order is dependent, so chc is different to cch or hch. For something like hct [a 7d torus], each of these are different:

hct, cht, htc, cth, thc, tch.

2d none

3d

hh = circle ## circle

4d

hc = circle ## sphere
ch = sphere ## circle
hhh = circle ## circle ## circle

5d

ht = circle ## glome
th = glome ## circle
cc = sphere ## sphere
hhc = circle ## circle ## sphere
hch = circle ## sphere ## circle
chh = sphere ## circle ## circle
hhhh = circle ##*4

usw.

All of these have proper holes. For example, hhh can be formed thus.

Start with a page in 4d. This is a prism x.y.z

Make the top and bottom [z-margins] of the page join. You then have a square-circle prism (which does not bound)

Make the y-margins join. You then have a 3d torus * line prism.

Make the x-margins join. You then have a 4d torus, with a hhh hole. This hole reduces dimensionally to a composition of hc and ch holes. but i have not figured out the general arithmetic.

Wendy

[wendy]

One sees that there are 2^(n-1)-1 possible torii by this construction, from the simple ruse, eg

x0x0x1x th (ie 3.1)
x0x1x0x cc (ie 2.2)
x0x1x1x chh (ie 2.1.1)
x1x0x0x ht (ie 1.3)
x1x0x1x hch (ie 1.2.1)
x1x1x0x hhc (ie 1.1.2)
x1x1x1x hhh (ie 1.1.1)

Let's call the process of "spherate" to make a surface r distant from an already existing structure. This gives a figure equivalent to replacing all the points by spheres of radius r from the point.

For example, a unit sphere is a point, spherated by 1/2 diam.
A line might be made into a pipe by spherating it.

hch then consists of a triple spheration, done thus.

h = point, spherated by 1 in 2 dimensions (h = hedrix = 2d_
hc = h, [a 2d thing], spherated by 1/3 in [2+3-1] = 4d, giving a hc.
hch = hc, [4d] spherated by 1/9, in [4d+2d-1d] = 5d.

Note that instead of using spheration, you could use a torus. This equates to making a long tube of a torus, and then bending said thing to a circle.

eg c = chorix -> sphere, One takes a long column of sphere-surface, and bends it into a ring. This gives a hc.

One must always remember that composite holes, like hch etc, can be dismantled topologically into simple (or 2-element) holes. The sequence of external holes defines the genus of the solid.

The genus of the exterior is the complement of this. In three dimensions, these are the same, so one speaks of surface genus. In 4d and higher, the exterior and interior generally are different if not palidromic.

[PWrong]

Let's call the process of "spherate" to make a surface r distant from an already existing structure. This gives a figure equivalent to replacing all the points by spheres of radius r from the point.

I suppose you could use vector notation to make this a bit more precise. A sphere is the set of vectors u, such that u is a linear combination of (i, j, k), and|u| = r.

We could write this as {u| u E(i,j,k), |u| =r}.
E is the "element of" symbol, and (i,j,k) represents the span of the vectors i, j and k, which are the basis vectors in R^3.

Since we're working in N dimensions, we'll use e1, e2, ... en for the basis vectors in R^n, instead of i,j,k.

A torus would be {u + v| u E (e1,e2), |u| = R, v E (u,e3), |v| = r}
Note that u describes the large circle, and v describes a smaller circle in the same plane as u.

Now, given an object S, that exists in n dimensions, we can spherate this by m dimensions, to get a new (n+m-1) dimensional object.

{u + v| u E S, v E (u, e_n+1, e_n+2, ... , e_n+k), |v| = r}

Thus, if S is a circle in the xy plane, with radius R, then the torus is
{u + v| u E S, v E (u, e3), |v| = r}
where S = {u| u E (e1, e2), |v| = r}

We don't have to limit ourselves to spheres though. We could replace every point in one object with any other object. We have to be careful about the dimensions, because we don't want vectors overlapping everywhere.[/b]

[wendy]

The equation for a torus is quite hard for me. This is why i use a series of "distances from a previous figure". For example, the three-dimensional torus is points in xyz equidistant from a circle in xy.

In any space, x is the direction of torric collapse, and one can put a break after any combination of letters y,z,...., For example, the 5d torus formed from 123/45 is a sphere in x1,x2,x3, and then go some distance in x1,x2,x3,x4,x5 from the sphere-surface. It is the toric product of two spheres.

[PWrong]

It's all well and good to describe objects and use combinations to classify them, but if you want to actually use them you really need a precise definition.

For example, the three-dimensional torus is points in xyz equidistant from a circle in xy.

This suggests that we put a sphere at every point on the circle. This would give a solid torus, but it seems like an inelegent construction. Most points in the torus correspond to two unrelated points in the circle. It's also difficult to define the surface of a torus like this.

I'll explain my construction in a bit more detail. We start with a circle in x,y.
That is, the set of all vectors u in (x,y), such that |u| = R.
{u| u E (x,y), |u| =R}

Now we put a small circle at each point in the larger circle. The circle has to be in the plane parallel to both u and the z-axis. We effectively rotate each circle so that one axis is pointing back towards the origin. So we have the set of all v in the plane spanned by (u, z), with |v| = r.
{v| vE(u,z), |v| = r}

Now, a point on the torus is the vector sum of u and v, so we have
{u + v| uE(x,y), vE(u,z), |v| = r}

By the way, when I say x, I mean a vector parallel to the x-axis, usually called x-hat, i, or e1

If G(x1, x2, ... xk) is a kD object in nD space (where x1, x2... are vectors) then G(u1, u2, ... uk) is the same object, but rotated into the span of (u1, u2, ... uk).
For instance, if C(x,y) is a circle in the xy plane, then C(2x, z+y) is the same circle (same size and shape), in the plane parallel to 2x+y and y+z

We can still define a toric product with this, although it's a bit difficult.

Let A(x1, x2, ... xm) and B(x1, x2, ... xn) be mD and nD objects respectively. The torus product of A and B gives a k= (m+n-1) dimensional object.

A#B ={u+v| u E A(x1, x2, ... xm), v E B(x_m+1, x+m+2, ... x(m+n-1) }

Basically you replace every point in A with B, but each B is rotated so that one of the axes is always pointing back towards the origin.

[wendy]

The idea of spheration is that one takes points at a distance R from a different body. The motif of using spheres is to provide a shell at distance R.

In practice, the points at distance r from a circle R, correspond to the solid shell fromed by allowing a sphere radius-r travel around the circle R. But the spheration is always the surface, not "any point distant r from any point in the set". It's more "a point exactly r from the nearest member of the set".

Correspondingly, if you spherate a point, you get a sphere (in N space).

To make a general torus, you need a set of R's that don't clash with each other. Successive division by 1/3 does here. Since the primitive must be at least a circle, you can then freely insert spheration-markers in the dynkin symbol, as long as the first node (o, x) is an o, and at least two nodes are marked. So

oOxOo normal circle * circle torus 3d
oOxOoOo circle * sphere 4d
oOoOxOo sphere * circle 4d
oOxOxOo circle * circle * circle 4d
oOxOoOxOo circle * sphere * circle 5d
oOoOxOoOo sphere * sphere 5d
oOxOxOoOo circle * circle * sphere
oOoOxOxOo sphere * circle * circle
oOxOoOoOo sphere * glome
oOoOoOxOo glome * sphere

I replaced the last x by an o, which makes it easier to see how the inversion works. If you do a radial inversion to swap inside to outside, the symbol is reversed. This is why symmetric symbols appear the same, while assymetic symbols become a different solid, but the surface is topologically the same.

This of course makes nonsense of the claim that surface has a genus.

W

[PWrong]

I just figured out that the parametric equations for torii are just the sums of sphere equations.

For instance, the equation for a circle in the xy plane is:
x = R cos a
y = R sin a
z = 0

For a sphere:
x = r cos b cos a
y = r cos b sin a
z = r sin b

Now the equation for the 3D torus is simply the sum of a circle and a sphere.
x = R cos a + r cos b cos a
y = R sin a + r cos b sin a
z = 0 + r sin b

Similarly, the 4D object you describe as oOxOoOo , or "circle * sphere" is the sum of the equations for a circle and a glome.

Now, let S1 be the equations for a circle, S2 be a sphere, S3 be a glome. e.t.c.

The 3D torus is S1 * S1, but it's equation is S1 + S2. We have

S1 * S1 = S1 + S2
S1 * S2 = S1 + S3
S2 * S1 = S2 + S3
S1 * S1 * S1 = S1 + S2 + S3

The S's are unneccessary, so I'll leave them out.
Here's all the torii up to 5D. For completeness, I've also included the spheres.
*(abc) means S_a*S_b*S_c

2D
*(1) = +(1)

3D
*(2) = +(2)
*(11) = +(12)

4D
*(3) = +(3)
*(12) = +(13)
*(21) = +(23)
*(111) = +(123)

5D
*(4) = +(4)
*(31) = +(34)
*(13) = +(14)
*(22) = +(24)
*(211) = +(234)
*(121) = +(134)
*(112) = +(124)
*(1111) = +(1234)

Clearly, each sequence on the right is the cumulative sum of the one on the left. So not only can we classify any torus with a simple list of integers, we now have an easy method to convert a sequence into a set of equations for the torus.

To make a general torus, you need a set of R's that don't clash with each other. Successive division by 1/3 does here.

That's not general at all. If you don't want them to clash, all that's required is that r1 < r2 < r3 < ... < r_n.

If you do a radial inversion to swap inside to outside, the symbol is reversed. This is why symmetric symbols appear the same, while assymetic symbols become a different solid, but the surface is topologically the same.

This of course makes nonsense of the claim that surface has a genus.

So does this mean that *(112) and *(211) are the same on the outside but not on the inside?

Unfortunately, I don't know much about topology. I do know that topologically, a cube, cylinder or anything without a hole is equivalent to a sphere. But I'm not sure how to actually define a hole in 4D. Is a duocylinder equivalent to a glome? Or what about the linear extension of a torus?

[Keiji]

I don't know much about topology.

It's actually quite simple: imagine you made a shape out of blutack. Now a shape is topologically equivalent to it, if you can make that shape from the original one without sticking parts of the shape together, or making holes in it.

And by the way, a solid cube isn't topologically equivalent to a hollow cube...

[PWrong]

It's actually quite simple: imagine you made a shape out of blutack. Now a shape is topologically equivalent to it, if you can make that shape from the original one without sticking parts of the shape together, or making holes in it.

Yes, but how do you do that mathematically? If you have a set of parametric equations describing a surface, can you prove that it has a hole, without looking at the graph?

[Keiji]

Hmm, that would seem impossible to me.

[PWrong]

Ok, mathworld isn't very helpful on topology, since every word is defined in terms of another word I don't understand. But I found something that looks a bit like a definition.

http://mathworld.wolfram.com/ConnectedSpace.html
A shape is "connected" or "0-connected", if every two points (a 0-sphere) in the shape can be connected by a curve (a 1-ball) inside the shape. So 0-connected means it's not made up of separate parts.

A shape is "simply connected" or 1-connected, if every curve (1-sphere) can be extended to a disk (2-ball) inside the shape. Another way of saying this is that every curve can be shrunk to a point

For example, a solid torus isn't simply connected, because if you choose a large circle around the outside, you can't "fill it in", without going outside the torus. The surface of a torus isn't simply connected either, because you can choose a curve wrapped around the "arm".

Now, a shape is 2-connected if every 2-sphere can be extended to a 3-ball. Similarly, a shape is 3-connected if every 3-sphere can be extended to a 3-ball, and so on.

It's important to see the difference between solid and hollow torii. The last n-sphere is always the only solid one. For instance, consider a circle spherated by a sphere, (the 1*2 torus). When solid, the sphere is filled in, but not the circle. Otherwise you'd have points being specified in more than one way. Let's specify a solid toratope with a 's' at the beginning, and a hollow one with an 'h'. So s12 is the solid 1*2 torus.

s12 : 2-connected, not 1-connected.
s21 : 1-con, not 2-con.
s111: 2-con, not 1-con.

h12: 2-con, not 1-con
h21: not 1-con or 2-con
h111: 2-con, not 1-con.

I think we can get more information than this though. On the hollow 3D torus, you can draw three different circles that can't be transformed into each other. It depends on how you wrap them. So we might extend this idea to higher dimensions as well.

[thigle]

so a 2-torus (topological notation for '3-d hollow torus', 2 after dimensionality inherent to object in question, regardless of the space of embedding) has 3 circles ('1-spheres' in topological notation?), not transformable into each other. 2 are its generative geodesics, and one is any circle not geodesic to the surface ?

so analogically, 3-torus (geometrical 4d-torus), would have 3 2-spheres, not transformable topologically into each other ? then that would be somehow connected to the fact that Dupin's cyclides are families of spheres tangent to 3 fixed 2-spheres? and torus is a kind of cyclide ?

somewhere i read, that a glome can be considered as a family of nested tori. how is that kind of foliation made, and according to what do different tori correspond to different whats of glome ?

[PWrong]

so a 2-torus (topological notation for '3-d hollow torus', 2 after dimensionality inherent to object in question, regardless of the space of embedding) has 3 circles ('1-spheres' in topological notation?), not transformable into each other.

Yes, it has 3 different circles. You can wrap around the larger circle, or around the small circle, or you can just draw on the surface without wrapping around anything. Only the last circle can be shrunk to a point.

so analogically, 3-torus (geometrical 4d-torus), would have 3 2-spheres, not transformable topologically into each other ?

No, that's not right. First, there are three different kinds of 3-torus. You can have circle*sphere, sphere*circle and circle*circle*circle. I've been calling them (12), (21) and (111) respectively. By the way, since it's confusing to say things like "a 1-sphere spherated by a 2-sphere", I'll use the word "torate" instead.

On the (12) torus, you have a large circle with lots of small spheres. You can wrap around the circle, but you can't really wrap around a sphere. So there's only two circles. However, there are two 2-spheres that can't be transformed to one another- one drawn straight on to the surface, and one wrapped around it.

then that would be somehow connected to the fact that Dupin's cyclides are families of spheres tangent to 3 fixed 2-spheres? and torus is a kind of cyclide ?

Mathworld has some information and pictures about cyclides. http://mathworld.wolfram.com/Cyclide.html
It looks like a cyclide is an ugly generalisation of the torus.

somewhere i read, that a glome can be considered as a family of nested tori. how is that kind of foliation made, and according to what do different tori correspond to different whats of glome ?

I'd never heard that before. I can't think of any interesting connection between a glome and a torus.

Here's a tricky question. Consider the 5D shape h22, that is, a large 2-sphere, torated by a smaller 2-sphere. Clearly, any circle drawn on this shape can be shrunk to a point. But if you draw a circle around the large 2-sphere, can it be transformed into a circle on one of the smaller 2-spheres?

[PWrong]

Yes, it has 3 different circles. You can wrap around the larger circle, or around the small circle, or you can just draw on the surface without wrapping around anything. Only the last circle can be shrunk to a point.

I tell a lie. There are five circles. Look at the picture about half way down this page.

http://mathworld.wolfram.com/Torus.html

They don't include the fifth trivial circle, which can be shrunk down to a point. I can't see any way to turn one of these into another. The two extra circles are equivalent to wrapping around the tube once, then going around the large circle before coming back to the start. Actually, you could also wrap around several times before coming back. So maybe there are an infinite number of different 1-spheres.

[thigle]

on the glome/torus connection, <google hypersphere foliation> brought these among others (in order of interest from my pov):

http://www.mathaware.org/mam/00/master/essays/SciAm/SA03.html
http://www.math.niu.edu/~rusin/known-math/99/reeb_foli
http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1030132589
http://mathworld.wolfram.com/ReebFoliation.html
http://www.cs.miami.edu/20_Announcements/50_PhD_Proposal/kocak-cv.pdf

so the 2 more circles on the 2torus that we missed were the famous Hopf circles. they are 1,1 torus-knots.
and there really is a heavy glome/torus connection, as one can get a foliation of 3-sphere as is stated on the mathworld, although i just imagine it pictorially rather than understand it rationally. two 2 toruses are joined with their surfaces, and their space between is the 4d-volume of 3-sphere. but i still don't understand what section of glome corresponds how to which tori. or if the meet of the 2 2-tori is at infinity, then its embedding (a plane to infinity with a handle) divides the 3-space of its embedding into two interlocked 3-volumes. the different glome-sections would spread (or 'foliate') both ways form this form of torus. this reminds me of wendy's statement that horosphere is just an ordinary euclidean plane extending to infinity ( i don't remember if infinity was included). so a horosphere is a zero-curvature 2-sphere - a 2-plane, E2. is then there such a thing as "horotorus" ? 2-horotorus would be this 2-plane with a handle mentioned above, a 2-torus of zero curvature ?

information is not energy !
http://www.zynet.co.uk/imprint/Tucson/4.htm#The%20topology%20of%20consciousness
:

'due to the similarities between toroids and twistors, mathematical topologies believed to act as a bridge between higher dimensional domains and 4D space/time reality, toroidal DNA can facilitate the cascade of higher dimensional information of consciousness into the electromagnetic (EM) domain. Since EM fields are well known to influence biological systems, they can carry the information of consciousness into the electro-chemical level. A mechanism is also proposed for the resonances between the quantum fields of consciousness and the energetic template of DNA. Such a mechanism is supported by recent experimental evidence for an energetic template of DNA using laser correlation spectroscopy (Poponin, 1998) and mathematical evidence that quantum fields can have a toroidal topology (Beltrami, 1889).'

[PWrong]

I'm still not sure what a foliation is, and mathworld doesn't help much, because I'm not very familiar with the notation.

two 2 toruses are joined with their surfaces, and their space between is the 4d-volume of 3-sphere.

That sounds like a duocylinder to me, but that's not the same as a 3-sphere.

information is not energy !
http://www.zynet.co.uk/imprint/Tucson/4 ... sciousness
:Quote:
'due to the similarities between toroids and twistors, mathematical topologies believed to act as a bridge between higher dimensional domains and 4D space/time reality, toroidal DNA can facilitate the cascade of higher dimensional information of consciousness into the electromagnetic (EM) domain. Since EM fields are well known to influence biological systems, they can carry the information of consciousness into the electro-chemical level. A mechanism is also proposed for the resonances between the quantum fields of consciousness and the energetic template of DNA. Such a mechanism is supported by recent experimental evidence for an energetic template of DNA using laser correlation spectroscopy (Poponin, 1998) and mathematical evidence that quantum fields can have a toroidal topology (Beltrami, 1889).'

It's a bit distracting when you post irrelevent links and quotes from other websites. That particular link looks like a list of abstracts, from pseudoscientific articles about consciousness. DNA and consciousness have very little, if anything, to do with quantum mechanics, and even less to do with 4D topology.

[thigle]

pw: DNA and consciousness has a lot to do with quantum mechanics, how can you state otherwise ? even if you omit DNA-quantum link, being a materialist you are, still, consciousness-quantum scale link is undenyable. i wouldn't argue about 4d topology link. at least not now. i would think it though.
sorry for too wide a scope of interest, i won't post links to as wide informational cross-overs as you found those last ones to be - so far you mistook it for thematic discontinuity.

[PWrong]

DNA and consciousness has a lot to do with quantum mechanics, how can you state otherwise ?

Well, if you think so, why don't you just start a new thread on the subject? Preferably in the theories or questions and answers section. I'd be happy to discuss it with you there.

For now, I'm going to try to understand where this cumulative sum thing comes from. Why is the equation for a torus the sum of a circle and a sphere, even though it's the product of two circles?

Or what about the connection between the torus and the duocylinder? In a way, a torus is like a duocylinder squashed into a 3D shape. Specifically, both the torus and the duocylinder are the product of two circles. Is every torii just a squashed rotatope like this?

I think our classification system is a bit vague, but my set-of-vectors definition is a bit cumbersome, so here's a slightly shorter way of defining the torus. It's just my old definition with some extra stuff removed.
u(x,y), v(u,z)

This simply means that u is a point on a circle in the xy plane, and v is a point on a circle in the same plane as u and z. The sum u+v is a point on the torus. If you want a solid torus, then you let the magnitude of v change.

Now, we can make all the toratopes and rotatopes in the same way.
For instance, the cylinder is
u(x,y), v(z)
Note the similarity to our old classification [(x,y),z]. The difference is that my new one gives us a precise construction for the object, and it also allows torii.

Normally I would use u, v, and w as vectors. But since w is also an axis in 4D, I'll use a, b and c from now on. Here's the 3D shapes.

Cube:
[x,y,z]
a(x), b(y), c(z)

Cylinder:
[(x,y),z]
a(x,y), b(z)

Sphere:
(x,y,z)
a(x,y,z)

Torus:
a(x,y) b(a,z)

Now the 4D shapes:

Tetracube:
[x,y,z,w]
a(x), b(y), c(z), d(w)

Cubinder:
[(x,y),z,w]
a(x,y), b(z), c(w)

Duocylinder:
((x,y),(x,y))
a(x,y), b(z,w)

Spherinder:
[(x,y,z),z]
a(x,y,z), b(w)

Glome:
(x,y,z,w)
a(x,y,z,w)

Torinder (linear extension of a torus):
a(x,y) b(a,z), c(w)

Circle*Sphere:
a(x,y) b(a,z,w)

Sphere*Circle:
a(x,y,z) b(a,z)

Circle*Circle*Circle:
a(x,y), b(a,z), c(b,w)

Now, we need some rules for this, to ensure that every point on an object is unique, and we don't get any ugly shapes.

One rule is that you can't use the same axis or vector twice. So these shapes aren't allowed.
a(x,y), b(x,z)
a(x,y), b(a,z), c(a,w)

As well as being ugly, these shapes have several points that exist twice.
Incidentally, you might notice that the second shape is made by spherating a circle by the second shape. We could make another rule that the product of an "illegal" shape and another shape is always an illegal shape.

There's a few shapes that I'm not sure about yet. For instance, is this an illegal shape? If it is, what kind of rule should we have to prevent it?
a(x,y), b(z,w), c(a,b)

[Marek14]

I usually look at 4D objects in a matter of cuts and layers, to get a better feel for them.

Now, I notice that the same way my extended rotatopes constructions are based in graphs, this has its base in binary trees or nestings:

Torus would be described as ((x,y),z) (where the parenthesis don't mean "spheric product" as in Wendy's notation, but just nesting)

Then torinder would be [((x,y),z),w)] circle*sphere would be ((x,y),z,w), sphere*circle would be ((x,y,z),w), and circle^3 would be (((x,y),z),w). Your undecided shape, however, would be ((x,y),(z,w)), so my gut feeling tells me it should be allowed.

Let's investigate the cuts. Every toratope (nice name ) is described by several numbers that correspond to number of vectors you describe them with.

In 3D, every toratope has three middle-cuts through coordinate planes:

Cube a*b*c: square a*b, square a*c, aquare b*c
Cylinder a*b (a is the diameter and b height): square a*b, square a*b, circle a
Sphere a: circle a, circle a, circle a
Torus a,b (a is the large diameter, b the tube diameter, right?): Here one cut are two concentric circles, of diameters (a+b) and (a-b). Two other cuts are two circles of diameter b and center separation a.

What can we deduce for 4D?

Torinder a,b,c: This is simple from torus. Based on your notation, x or y cut (in hyperplanes yzw or xzw respectively) will result in two parallel cylinders b,c with centre separation a. Cut through xyw will be cylinder with a hole of height c, with external diameter a+b and internal diameter a-b. Cut through xyz will be just the torus a,b

circle*sphere a,b
Putting x=0 we get that a would be in y direction and z would be a sphere in y,z,w translated through y - in other words, x-cut are two spheres of diameter b with centre separation a. Y-cut is the same.
Z-cut means that b(azw) simplifies to b(aw): the sphere is cut in circle. So, this cut will be torus a,b. The W-cut is the same.

sphere*circle a,b
Seems that x,y, and z-cut are all the same and look like torii (a,b). W-cut would be two concentric spheres, of diameters (a+b) and (a-b)

Circle^3 a,b,c
Here it gets a bit more complicated, but it seems that x-cut and y-cut are equal to two torii (b,c) whose large circles of diameter b lie in the same plane, and whose centres have the separation a. Z-cut looks like two concentric torii - torus (a+b,c) and torus (a-b,c). W-cut looks like two torii "concentric" on the same circle - torus (a,b+c) and torus (a,b-c).

Now let's investigate the a(x,y), b(z,w), c(a,b) shape. Symmetry suggests that all four cuts should look, if not the same, then at least similar. Unfortunately, I must go now, but I will return to ponder it.

[Marek14]

After careful thinking, it seems that the only reasonable cross-section for a(x,y),b(z,w),c(a,b) would be two torii separated in the z direction - i.e. one above the other instead next to it.

If we only count the glome, three 4D torii and this beast among "toratopes", each cross-section is either one 3D toratope or two toratopes which differ either in one of their dimensions or in one coordinate of their centers.

So, it's:

Sphere a(x,y,z) - occurs as cross-section of glome a(x,y,z,w)
Two displaced spheres a(x),b(a,y,z) - occurs as cross-section of circle*sphere a(x,w),b(a,y,z)
Two concentric spheres with differing diameters a(x,y,z),b(a) - occurs as cross-section of sphere*circle a(x,y,z),b(a,w)

Torus a(x,y),b(a,z) - occurs as cross-section of both circle*sphere a(x,y),b(a,z,w) and sphere*circle a(x,y,w),b(a,z)
Two torii displaced in xy plane a(x),b(a,y),c(b,z) - occurs as cross-section of circle^3 a(x,w),b(a,y),c(b,z)
Two torii displaced in z direction a(x,y),b(z),c(a,b) - occurs as cross-section of the beast a(x,y),b(z,w),c(a,b)
Two concentric torii with different outer diameters a(x,y),b(a),c(b,z) - occurs as cross-section of circle^3 a(x,y),b(a,w),c(b,z)
Two concentric torii with different inner diameters a(x,y),b(a,z),c(b) - occurs as cross-section of circle^3 a(x,y),b(a,z),c(b,w)

From this, we see that the torii in cross-section of the beast are a,c or b,c, and so if a>c and b>c, they are both normal and the beast should exist as a real object. So far I'm not sure how to extrapolate the global shape from these cuts, however it seems that I'm now able to create cross-sections in any dimension and also find toratopes there.

Let's look in 5D:

5 - petaglome: a(x,y,z,w,v) - all 5 CS are glomes.
4+1 - glome*circle: a(x,y,z,w),b(a,v) - 4 CS are sphere*circles, 1 CS are two concentric glomes of different diameters.
(3+1)+1 - sphere*circle*circle - a(x,y,z),b(a,w),c(b,v) - 3 CS are circles^3, 1 CS is 2 concentric sphere*circles with differing "sphere" diameters, 1 CS is 2 concentric sphere*circles with differing "circle" diameters.
((2+1)+1)+1 - circle^4 - a(x,y),b(a,z),c(b,w),d(c,v) - 2 CS are 2 circles^3 displaced in their first direction (x or y in their canonical form), 1 CS is 2 concentric c^3 differing in the largest diameter, 1 CS has them differ in second largest, and 1 CS has them differ in the smallest one.
(2+2)+1 - beast*circle - a(x,y),b(z,w),c(a,b),d(c,v) - 4 CS are 2 circles^3 displaced in z direction, 1 CS is 2 concentric beasts differing in diameter c.
(2+1+1)+1 - circle*sphere*circle - a(x,y),b(a,z,w),c(b,v) - 2 CS are 2 sphere*circles displaced in "sphere" direction, 2 CS are circles^3, 1 CS is 2 concentric circle*spheres with different "sphere" diameters.
3+2 - sphere/circle beast: a(x,y,z),b(w,v),c(a,b) - 3 CS are beasts, 2 CS are 2 sphere*circles displaced in the "circle" direction.
(2+1)+2 - torus/circle beast: a(x,y),b(a,z),c(w,v),d(b,c) - 2 CS are two displaced beasts, 1 CS is 2 concentric beasts differing in a or b (these two diameters are interchangeable), and 2 CS are circles^3 displaced in w direction.
3+1+1 - sphere*sphere: a(x,y,z),b(a,w,v) - 3 CS are circle*spheres, 2 CS are sphere*circles.
(2+1)+1+1 - circle*circle*sphere: a(x,y),b(a,z),c(b,w,v) - 2 CS are 2 circle*spheres displaced in "circle" direction, 1 CS is 2 concentric circle*spheres differing in "circle" diameter, and 2 CS are circles^3
2+2+1 - beast221 - a(x,y),b(z,w),c(a,b,v): 4 CS are 2 circle*spheres displaced in "sphere" direction, 1 CS is beast.
2+1+1+1 - circle*glome - a(x,y),b(a,z,w,v): 2 CS are 2 displaced glomes, 3 CS are circle*spheres.

For 6D, the valid toratopes should be:

6 - exaglome - a(x,y,z,w,v,u)
5+1 - petaglome*circle - a(x,y,z,w,v),b(a,u)
(4+1)+1 - glome*circle*circle - a(x,y,z,w),b(a,v),c(b,u)
((3+1)+1)+1 - sphere*circle*circle*circle - a(x,y,z),b(a,w),c(b,v),d(c,u)
(((2+1)+1)+1)+1 - circle^5 - a(x,y),b(a,z),c(b,w),d(c,v),e(d,u)
((2+2)+1)+1 - a(x,y),b(z,w),c(a,b),d(c,v),e(d,u)
((2+1+1)+1)+1 - circle*sphere*circle*circle - a(x,y),b(a,z,w),c(b,v),d(c,u)
(3+2)+1 - a(x,y,z),b(w,v),c(a,b),d(c,u)
((2+1)+2)+1 - a(x,y),b(a,z),c(w,v),d(b,c),e(d,u)
(3+1+1)+1 - sphere*sphere*circle - a(x,y,z),b(a,w,v),c(b,u)
((2+1)+1+1)+1 - circle*circle*sphere*circle - a(x,y),b(a,z),c(b,w,v),d(c,u)
(2+2+1)+1 - a(x,y),b(z,w),c(a,b,v),d(c,u)
(2+1+1+1)+1 - circle*glome*circle - a(x,y),b(a,z,w,v),c(b,u)
4+2 - a(x,y,z,w),b(v,u),c(a,b)
(3+1)+2 - a(x,y,z),b(a,w),c(v,u),d(b,c)
((2+1)+1)+2 - a(x,y),b(a,z),c(b,w),d(v,u),e(c,d)
(2+2)+2 - a(x,y),b(z,w),c(a,b),d(v,u),e(c,d)
(2+1+1)+2 - a(x,y),b(a,z,w),c(v,u),d(b,c)
4+1+1 - glome*sphere - a(x,y,z,w),b(a,v,u)
(3+1)+1+1 - sphere*circle*sphere - a(x,y,z),b(a,w),c(b,v,u)
((2+1)+1)+1+1 - circle*circle*circle*sphere - a(x,y),b(a,z),c(b,w),d(c,v,u)
(2+2)+1+1 - a(x,y),b(z,w),c(a,b),d(c,v,u)
(2+1+1)+1+1 - circle*sphere*sphere - a(x,y),b(a,z,w),c(b,v,u)
3+3 - a(x,y,z),b(w,v,u),c(a,b)
3+(2+1) - a(x,y,z),b(w,v),c(b,u),d(a,c)
(2+1)+(2+1) - a(x,y),b(a,z),c(w,v),d(c,u),e(b,d)
3+2+1 - a(x,y,z),b(w,v),c(a,b,u)
(2+1)+2+1 a(x,y),b(a,z),c(w,v),d(b,c,u)
3+1+1+1 - sphere*glome - a(x,y,z),b(a,w,v,u)
(2+1)+1+1+1 - circle*circle*glome - a(x,y),b(a,z),c(b,w,v,u)
2+2+2 - a(x,y),b(z,w),c(v,u),d(a,b,c)
2+2+1+1 - a(x,y),b(z,w),c(a,b,v,u)
2+1+1+1+1 - circle*petaglome - a(x,y),b(a,z,w,v,u)

So, in 5D we have 12 possible toratopes, only 8 of which are "products" in the traditional sense. In 6D only 16 out of 33 are "products" - less than a half.

[Keiji]

There's a few shapes that I'm not sure about yet. For instance, is this an illegal shape? If it is, what kind of rule should we have to prevent it?
a(x,y), b(z,w), c(a,b)

I'm a bit confused about that one. I gathered from your post that:

a(x,y,z) is a point in a sphere because xyz is a realm.
a(x,y) is a point in a circle because xy is a plane.
a(x) is a point in a line because x is a line.

Okay, so what's c(a,b)? Well, as far as I can see, a and b are points, and two points makes a line. So c(a,b) should be a point in a line. So the shape that you wrote up there would contain all the points on all the lines intersecting the adjacent faces of a duocylinder. If that makes sense.

[Marek14]

Nope - according to his notation, a(x,y) is a vector pointing to a specific point on a circle, and so is b. c, then, is a point on a circle in a plane given by vectors a and b (already known by the time you compute it). Every point on the surface is given by unique combination of three angles.

By the way, I decided to call the "beast" a(x,y),b(z,w),c(a,b) "tiger", since we're talking toratopes here and "tora" is Japanese for "tiger" More specifically, this is circle/circle tiger. 5D has "sphere tiger" 3+2 and "torus tiger" (2+1)+2. 6D has "glome tiger" 4+2, "sphere*circle tiger" (3+1)+2, "circle^3 tiger" ((2+1)+1)+1)+2, "tiger tiger" (2+2)+2, "circle*sphere tiger" (2+1+1)+2, "sphere/sphere tiger" 3+3, "sphere/torus tiger" 3+(2+1), and "torus/torus" tiger (2+1)+(2+1).

Also, 5D has a "circtiger" 2+2+1, and 6D has "sphere circtiger" 3+2+1 and "torus circtiger" (2+1)+2+1, as well as "sphertiger" 2+2+1+1 and "leopard", or "circle/circle/circle leopard" 2+2+2.

Yep, I have nothing to do.

Now, I'll try to derive tiger's parametric equations:

Let's start with:

x = A * cos a (where a is angle and A is the radius)
y = A * sin a

z = B * cos b
w = B * sin b

Now to add terms for "c". c should be a circle with one basic direction in xy plane and other in zw plane.
So it should be
u = C * cos c
v = C * sin c
where u is some combination of x and y and v is some combination of z and w.

Maybe it would go like:
x = C * cos a * cos c
y = C * sin a * cos c
z = C * cos b * sin c
w = C * sin b * sin c

Now, if we add all of these together, parametric equations of tiger would be (correct me if I'm wrong):

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

[PWrong]

To be honest, at first I was almost certain that the "tiger" wouldn't exist. It's essentially the product of a duocylinder and a glome, and I didn't think you could multiply two 4D objects and get another 4D object. But now I don't know what to think . Maybe I should be less judgemental of shapes with ugly formulas. I got the same parametric equations for the tiger as you did. It's the sum of the equations for a duocylinder and a glome.

Before we get carried away though, we should try to prove the existance or non-existance of this creature. I think it should be possible to find out whether or not it obeys my first rule, that every point on the object is unique. If we can find two different sets of vectors a, b, c, with the same sum, then the shape will be illegal.

If we only count the glome, three 4D torii and this beast among "toratopes", each cross-section is either one 3D toratope or two toratopes which differ either in one of their dimensions or in one coordinate of their centers.

Actually, I count all the rotatopes as separate toratopes. I was thinking of toratopes as a combination of rotatopes, which are sums of n-balls, and ordinary torii, which are products of n-spheres (They don't have to be balls and spheres respectively, but it makes my classification easier to understand). So the toratopes include all rotatopes, and all torii, as well as combinations of these, like the torinder (circle*circle + solid line-segment).
I've found a formula for the number of toratopes in n dimensions, discounting tigers and other beast-like shapes. I'm quite proud of it.

As we know, any rotatope can be represented as a partition. Also, I think Wendy mentioned earlier that the number of these simple torii in n dimensions is 2^(n - 2) -1. I'll define the function T(n) = 2^(n-2) -1, since we'll need it later on. The "-1" isn't really neccessary, since we could just count the n-sphere as a kind of torus. But the final result looks much nicer this way.

Now, what I noticed is that you can make a toratope by taking a rotatope, and replacing one of the spheres with a torus of the same dimension. For instance, if you take a spherinder, (3+1), you can replace the sphere with a 3D torus to make a torinder. If we write the torus as a product of circles, 1*1 (in topological notation), then the torinder is "1*1+1".

Now, rotatopes in higher dimensions are made up of lots of n-spheres, any of which can be replaced by the available toratopes. Here's a complicated example. Consider the 10D rotatope, (4 + 3 +3). Here we have a 4D sphere, which can be left as it is, or replaced by 3 different torii giving 4 combinations. But we also have two 3D spheres, either of which can be replaced by a 3D torus. There are three combinations: two spheres, a sphere and a torus, or two torii. Now we have 4 combinations for the 4D shape, and 3 for the 3D shapes. This means the rotatope (4+3+3) corresponds to 4*3 = 12 unique toratopes, including the rotatope itself. Here's a few of them.
(2*1) + 3 + 3
(1*2) + (1*1) + 3
You don't even need the brackets:
1*1*1 + 1*1 + 1*1

Remember that multiplication isn't commutative, but addition is. To get the number of dimensions, just add up the numbers, but add one to each bit with multiplication (because 1*1 is a 2-sphere in the topologists sense, but a 3-sphere in the geometer's sense. I know it's confusing, but it seems to work better this way)

Now, I said I found a formula, and I have. It's a bit complicated, so you can skip my derivation if you want.

To get the number of toratopes corresponding to a particular rotatope R, we first take the partition of the rotatope in vector form. Let k_n be the nth component, that is, the number of n-spheres in R. So in our example, we have 4 + 3 + 3 = 0*1 + 0*2 + 2*3 + 1*4 + 0*5 + ... + 0*10, so the partition is (0,0,1,2,0,0,..., 0). We therefore have k₃ = 2, and k₄ = 1.

Now we want the number of possible combinations available by replacing n-spheres with torii, for each n. Each n-sphere can be replaced by T(n) different torii, or not replaced at all. But the rotatope may have several of each n-sphere. Since order isn't important, it turns out the number of combinations reminds us of pascal's triangle, except it's a square, not a triangle. The formula is in fact:
(T(n) + k_n)!
T(n)! * k_n!

The total number of toratopes for this particular rotatope is the product of the numbers for each n. And finally, to get the total number of toratopes, we have to sum over all of the partitions. So here's my formula, as promised. (it's such a shame I can't use proper notation here )

Number of toratopes in n dimensions =

Code: Select all

        n      (T(i) + k(i))!
Sum ( Product (--------------))
k     i=1      T(i)! k(i)!


where T(n) = 2^(n-2) -1, and the sum is over all partitions of n.

I've worked this out up to n=8, and I'm writing a program to do it for higher dimensions. If you're interested, the numbers of toratopes for 1, 2, ... are 1, 2, 4, 9, 19, 43, 93, 207...

I looked up this sequence on Sloane's, but the closest sequence is this: A026776
Here, the first seven are ok, but the 8th term is 212.

[Marek14]

To be honest, at first I was almost certain that the "tiger" wouldn't exist. It's essentially the product of a duocylinder and a glome, and I didn't think you could multiply two 4D objects and get another 4D object. But now I don't know what to think . Maybe I should be less judgemental of shapes with ugly formulas. I got the same parametric equations for the tiger as you did. It's the sum of the equations for a duocylinder and a glome.

Before we get carried away though, we should try to prove the existance or non-existance of this creature. I think it should be possible to find out whether or not it obeys my first rule, that every point on the object is unique. If we can find two different sets of vectors a, b, c, with the same sum, then the shape will be illegal.

If we only count the glome, three 4D torii and this beast among "toratopes", each cross-section is either one 3D toratope or two toratopes which differ either in one of their dimensions or in one coordinate of their centers.

Actually, I count all the rotatopes as separate toratopes.

I decided not to exactly because the simple rules which occur on the cross-sections. I think I already managed to find rules that govern unshrinkable circles and higher unshrinkable surfaces (I posted in a new thread for that). For example, one result I got is that one should expect to have torus surface which can't be shrunk into a circle. The talk here was always just about spheres.

So I propose to only call "toratopes" those shapes which are smooth, i.e. which don't have any lower elements than their surfaces. Torinder doesn't belong here, since it has two 2D faces where its three separate faces meet.

However, there should be, indeed, a higher category combining rotatopes and toratopes and containing their various products. Perhaps "rotopes" might be a good name?

Now, how to prove that the tiger exists? I think the simplest way might be noticing that each of its four parametric equations arises - in some way - as parametric equation of torus in 3D. Not sure how to go about it yet, though.

However, I wouldn't banish tigers from the zoo just yet. You said that the parametric equations are sum of duocylinder and glome, but that is not exactly correct. If you only use the A and B formulas, you end with a 2D surface, since you only have two variables. I think that this is the duocylinder MARGIN - a surface that is topologically equivalent to torus, and which is also completely smooth. The connection between duocylinder and tiger suggests that there's something between toratopes and rotatopes I haven't quite grasped yet.

Ah, yes - the thing is that if we limit the prismatic product to SURFACES and ignore the inner volumes, then we get rotatopes which are smooth, but can have much lower dimension than the one they are embedded in. The simplest case here is product of two circles - the duocylinder margin. The tiger should be then gotten by "inflating" this margin much as you get torus by inflating the circle.

[PWrong]

So I propose to only call "toratopes" those shapes which are smooth, i.e. which don't have any lower elements than their surfaces. Torinder doesn't belong here, since it has two 2D faces where its three separate faces meet.

However, there should be, indeed, a higher category combining rotatopes and toratopes and containing their various products. Perhaps "rotopes" might be a good name?

Ok, in that case what I called a torus is a toratope, and what I called a toratope is a rotope. The formula above is actually a formula for the number of rotopes in n dimensions. What do you think of it, by the way?

Ah, yes - the thing is that if we limit the prismatic product to SURFACES and ignore the inner volumes, then we get rotatopes which are smooth, but can have much lower dimension than the one they are embedded in. The simplest case here is product of two circles - the duocylinder margin. The tiger should be then gotten by "inflating" this margin much as you get torus by inflating the circle.

I think I can see what you're getting at here. You spherate the surface of the duocylinder by a circle, but because the surface of the duocylinder is 2D, it's possible to do this without changing the dimension.

But the surface of the duocylinder is topologically the same as a torus, so does this mean duocylinder*circle is topologically equivalent to torus*circle = circle^3?

This is all reminding me of the linear algebra stuff I learned this year. Basis vectors, subspaces and stuff would be really useful here, if only this was actually linear.