Exploring Hyperspace with the Geometric Product

Higher-dimensional geometry (previously "Polyshapes").

Exploring Hyperspace with the Geometric Product

Postby tbriggs » Fri Sep 08, 2006 7:35 pm

This message is presented as a PDF link (7 pages) at http://www.bayarea.net/~kins/thomas_briggs . In this article I show how geometric products can prevent confusion regarding the dimensionality of a figure and open some areas for research. Included are the duocylinder, duoprisms, various wormholes, prism cylinders, and the hypersphere, with illustrations. I also show that topology texts may obscure higher dimensional geometry by not distinguishing between topological and geometric morphologies. Updated 11/15/06.
Last edited by tbriggs on Wed Nov 15, 2006 9:25 pm, edited 1 time in total.
-Tom Briggs
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Postby wendy » Sun Sep 10, 2006 7:55 am

There are prehaps a dozen known geometric properties... :S

You are obviously talking of the prism-product...
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Postby tbriggs » Sun Sep 10, 2006 9:33 pm

Thank you for your reply. Besides the prism-product, this article looks at solid tori in general (including the closed wormhole), open wormholes, and the 3-sphere.
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Postby PWrong » Mon Sep 11, 2006 4:12 am

I think you missed a few of the solid torii in 4D. I'm not sure what your closed wormholes are, and how they relate to torii.
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Postby tbriggs » Mon Sep 11, 2006 8:02 pm

Only solid tori that are geometric products (except for those composing the 3-sphere) were included in my link. These are the duoprisms, the duosphere, prism cylinders, and closed wormholes which I will try to describe here: Luminet used the label "closed wormhole" for the product of a circle and a sphere. This should qualify as a torus- it has a genus of one. I also generalize the solid torus catagory (maybe too much) to include the product of three circles, which has a genus of three. Among the non-product solid tori not included in this article are those composing the boundary of the 24-cell.
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Postby wendy » Tue Sep 12, 2006 8:37 am

The torus product is a subclass of the comb product. Unlike the comb product, the torus product is not communitive (but is associative), ie

(a ## b ) ## c = a ## ( b ## c)

but a ## b <> b ## a

The surface is topologically identical, but the interior is not. You can use torus product with any set of shapes, eg dodecahedron - dodecahedron torus.

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Postby PWrong » Tue Sep 12, 2006 3:57 pm

The torus product isn't associative, because A#(B#C) doesn't exist.
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Postby wendy » Wed Sep 13, 2006 6:42 am

The torus product can be applied over any number of different figures. The dimension of the resulting surface is the sum of surfaces of the elements.

One can, in 4d, form six different products from three different polygons, where different polygons form different roles.

Consider a sheet of paper. You can make it into first a cylinder, and then a torus. In four dimensions, this torus can be extended into a kind of torus-cylinder, which can then be bent into a torus.

The first instance of a non-ring torus of three bases happens in seven dimensions, since the surface becomes 2d, and the 2+2+2 = 6, and 6+1 = 7. None the same, they do exist.
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Postby thigle » Thu Sep 14, 2006 5:16 am

wendy wrote
One can, in 4d, form six different products from three different polygons, where different polygons form different roles.

from what polygons you get what products ?
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Postby wendy » Thu Sep 14, 2006 6:54 am

If you start off with {5} and {6}. This starts as a 5*6 rectangle. One can roll it up into a cylinder, either 5 around or 6 around. One can then connect the ends to get a tyre that has a tread 5 squares long, or six. Different ways give different tread-lengths.

One starts with a {5}{6}{7} block in 4d. One rolls this into a prism * cylinder prism. The cylinder here is say 5 around.

The rectangle gets rolled into a prism, to give a torus 5 around, 6 tread-length, by a column of 7.

One rolls the height of 7 in, to ge a new torus-shaped tyre of length 7.

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Postby PWrong » Thu Sep 14, 2006 7:21 am

Consider a sheet of paper. You can make it into first a cylinder, and then a torus. In four dimensions, this torus can be extended into a kind of torus-cylinder, which can then be bent into a torus.

This could either be a 4D or 5D shape. I suppose what you're describing could be called 2#(2#2). This is equivalent to (2#2)#2, and we use the latter notation because it's not ambiguous.

Finding A#B requires that we look at the tangent space of A, and put a B in there. The problem is that B is embedded in R<sup>k</sup>, which is not the same as a kD tangent space. R<sup>k</sup> has an origin, k axes, and a standard basis. The tangent space has an origin, but it doesn't have a standard basis, or any axes to speak of. So there is no unique way to put B into the tangent space unless B is a sphere.

Similarly, the universe is not the same as R<sup>3</sup>, because the universe has no origin and no standard basis.
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Postby wendy » Thu Sep 14, 2006 7:34 am

Somehow, something is missing. You can fold a torus from a sheet of paper in the manner described. There is no origion on the torus, because the join is perfectly relative.

Likewise, you can connect R3 into a tripple-modulus sheet, and connect these into three circles in four dimensions. Modulus can be defined in terms of reals.

I fail to see how you get 5d from folding a sheet, laid out in the xyz space, firstly into a circle in wx, then in a circle in wy, and then in a circle in wz. It gives something that is scattered over wxyz, which us 4d when i last checked.

Also, it should be remembered, that the torus, like any comb product, or indeed, any cartesian product, has a grain, and therefore a basis for setting up standard coordinates.

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Postby PWrong » Thu Sep 14, 2006 8:28 am

Somehow, something is missing. You can fold a torus from a sheet of paper in the manner described. There is no origion on the torus, because the join is perfectly relative.

I never said there was an origin on the torus. There is an origin of the torus (the centre point, which isn't actually on the surface), but that's not the point. Pick a point on a surface and consider the tangent plane at that point. The tangent plane has an origin (the point on the surface), but it has no x and y axis.

Finding A#B requires that we look at the tangent space of A, and put a B in there.

I should be talking about the normal space here, which is just the space perpendicular to the tangent space. A torus has a normal vector and a tangent plane, while a curve in 3D has a normal plane and a tangent vector. Also note that we often have to add a dimension or two. The argument still holds, but I should reword it a bit.

I fail to see how you get 5d from folding a sheet, laid out in the xyz space, firstly into a circle in wx, then in a circle in wy, and then in a circle in wz. It gives something that is scattered over wxyz, which us 4d when i last checked.

You could have the torus in xyz, extend it into w, and then curve the line into a circle in wv. This would be (2#2)x2 or (21)2. I assumed you were talking about the 4D shape.

Also, it should be remembered, that the torus, like any comb product, or indeed, any cartesian product, has a grain, and therefore a basis for setting up standard coordinates.

What's this about a grain? How can it give you a unique x and y axis on a normal plane?
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Postby wendy » Thu Sep 21, 2006 8:12 am

A cylinder is a prism-product of a line and a circle. We see that there is a grain, in that you can determine height from circumference on the surface.

When you bend a cylinder in 3d, to make a torus, it does not become 4d, but the axis of bending includes one of the axis of the base. This is the axis of pondering (or dimension-loss).

Likewise, you can make in 4d, a prism the height a line, the base a torus. You can then bend this in the hedrix of wx or wy or wz, to make a figure that is the product of three circles: viz a tri-circular torus.

There is also some nonsense that this is different to the spheration of the margin of the bi-circular prism. This is in fact also a torus, since one multiplies three circles, in prism product, and then fold the result downwards. The comb product of a tri-circle is a chorix (3fabric) in six dimensions, and this folds down onto four dimensions in an unknown number of ways.

But i can not see how two different figures can arise from the tri-circular torus. They are all topologically 3hc+3ch, so they ought be deformable into each other.

Note that a simple torus product, eg the dodecagon-dodecahedral torus, is topologically hc, while it gives a dodecahedron-dodecagon torus of genus ch. These are indeed different!

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Postby PWrong » Thu Sep 21, 2006 9:58 am

When you're replying, it's helpful if you quote me so that I know exactly what you're replying too. It's especially hard when we're almost using different languages. I don't even know how much of my notations and arguments you understand.

A cylinder is a prism-product of a line and a circle. We see that there is a grain, in that you can determine height from circumference on the surface.

When you bend a cylinder in 3d, to make a torus, it does not become 4d, but the axis of bending includes one of the axis of the base. This is the axis of pondering (or dimension-loss).

You seem to be talking about a line drawn on a torus :?. I asked for a unique pair of axes drawn on the normal plane to a duocylinder.

There is also some nonsense that this is different to the spheration of the margin of the bi-circular prism. This is in fact also a torus, since one multiplies three circles, in prism product, and then fold the result downwards. The comb product of a tri-circle is a chorix (3fabric) in six dimensions, and this folds down onto four dimensions in an unknown number of ways.

This isn't a proof. You've given a vague construction for two objects, and you claim they both give the same object. We do indeed multiply three circles, but we use two different multiplications. (2x2)#2 is not (2#2)#2.

But i can not see how two different figures can arise from the tri-circular torus. They are all topologically 3hc+3ch, so they ought be deformable into each other.

Deformable doesn't mean the same. I don't know what your h,c notation means. Could you define it in terms of one of our notations, or using equations?

You've seen the equations for the 3-torus and the tiger, and I've explained why this proves they are different objects. If you can prove otherwise using real mathematics, do so. Otherwise, why are you making vague arguments about why they are the same?

Note that a simple torus product, eg the dodecagon-dodecahedral torus, is topologically hc, while it gives a dodecahedron-dodecagon torus of genus ch. These are indeed different!

They're not simple, and they're not different. They're both equally non-existent. I've said plenty of times that the torus product is only defined for a sphere on the right hand side.
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Postby wendy » Fri Sep 22, 2006 7:40 am

One understands that grain is not a unique axis, but a unique direction. Wood has grain: it does not mean that every peice of wood has a set axis, but that it is useful to lay the axis to follow a given direction.

Likewise, when one takes products, like the torus-product, the nature of the product produces a local direction: that is at any point P, one can run parallel to either of the bases. This does not mean that there is an origin, but rather that if you set P as an origin, then the axies follow.

The difference between (2x2)#2 and (2#2)#2 is to me very vague. If you take a torus, for which the axle through the tyre is in the x axis, and the other two axis are y and z, the difference here is that the two shapes arise depending on whether the additional circle formes in wx or wy hedrices. If you use wx, then you do indeed get a tiger.

Deformable doesn't mean the same. I don't know what your h,c notation means. Could you define it in terms of one of our notations, or using equations?


The nature of the torus product, like any other product, is that the result can be deformed without destroying the product. Even the cartesian product can be implemented on oblique coordinates.

So it's not a matter of if the figure is deformable, it's more a matter of if the deformation will destroy the product. It does not in this case.

In higher dimensions, the nature of holes is legion. It is possible to resolve holes into an exterior and interior component. These holes can be closed by adding a fabric of 2d (h), 3d (c), 4d (t) &c. Since every hole has an exterior and interior component, one has hc holes (in 4d), or ht holes in 5d.

The genus of a figure is then the irreducable holes: for the tiger, it is 3hc + 3ch : ie a dozen holes.

Note that a simple torus product, eg the dodecagon-dodecahedral torus, is topologically hc, while it gives a dodecahedron-dodecagon torus of genus ch. These are indeed different!


They're not simple, and they're not different. They're both equally non-existent. I've said plenty of times that the torus product is only defined for a sphere on the right hand side.


The torus product is indeed defined for polytopes, as well as spheres. One can place on the torus in 3d, a set of 30 squares, that unfold to a 5*6 rectangle. Your claim that these figures don't exist is because your definition of the torus product is incomplete.
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Postby PWrong » Fri Sep 22, 2006 7:48 am

The torus product is indeed defined for polytopes, as well as spheres.

Define it then.
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Postby wendy » Sat Sep 23, 2006 8:18 am

The torus product is the comb product, or cartesian product of the surfaces, with directions of pondering.

Consider the cartesian product of two figures X, Y. The dimensions of these solids are x+1, y+1 dimensions. The cartesian product of the surfaces gives in x+y+2 dimensions, a marginix of x+y dimensions. One can also place a surface of x+y+1 dimensions to cover this marginix. These are the cartesian-products of the interior of X and the surface of Y, and of the interior of Y, and the surface of X. The cartesian product of the interiors of X, Y complete the picture.

Since the marginix now divides the surface into two, we now will call one the inside, and one the outside. Now let the outside be holed with an infinity, and stretched out to fall in a plane (x+y+1 dimensions). The result is that the partition remains, and there are two holes formed in the process. One hole corresponds to the surface of X, the other corresponds to the surface of Y.

Since this topological transformation can be made to preserve straight lines, it follows that one can make X, Y into polytopes.

In the example of a dodecahedron-decagon torus, we have the interiors of these polytopes as c, h. The cartesian product of these produce a five-dimensional solid, made of 120 pentagonal prisms.

One set of terons of this prism product consists of 10 decahedron-line prisms, the other consists of 12 pentagon-decagon prisms.

If we set the hole to be in the in a dodecahedron-line prism, then unfolding the figure to four dimensions, will produce a set of 12 pentagon-decagon prisms, joined into the shape of a dodecahedron, by their decagonal faces. The exterior will be then the 120 pentagonal prisms, the line of 10, runs through the hole. We can pass a linear string through this, and make a linkage. The nature of this hole is ch, since to block the hole, one must restore one of the dodecahedrons (ie a chorid or 3d solid)

The second orientation is to make the outside fall in the pentagon-decagon prism. The resulting shape is then a ring of 10 distorted dodecahedron-line prisms, fashioned into a circle. A linear rope can not be made to link with this, one needs something bigger. The nature of this hole is hc. To prevent any non-vanishing sphere forming on the outside, one needs to restore a further decagon (hedrid = 2d solid), and to prevent non-vanishing circles on the inside, a dodecahedron (ie a chorid). Therefore the hole is hc.

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