Clifford, Flat or simply 4?

Discussion of shapes with curves and holes in various dimensions.

Clifford, Flat or simply 4?

Found in many locations besides youtube
Found only in this particular clip
Also found in many places when using "hypertorus" as the search term
http://www.lboro.ac.uk/departments/ma/g ... /ht800.gif
Found only in http://www.lboro.ac.uk/departments/ma/g ... torus.html

My question is
Which is which?
(4 torus, Clifford torus, Flat torus, others)

Also an attempt reconstruction of the 1st clip gives me this (Projected to 2D screen)
http://img191.imageshack.us/img191/6769/33013764.png (Made using powerpoint therefore lighting is wrong)
http://img51.imageshack.us/img51/1529/10675051.png
What torus is this?
Secret
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Re: Clifford, Flat or simply 4?

This makes me no sense.

A torus, or space where w^2+x^2 = c1, y^2+z^2=c2. Putting w,x -> r,þ, (þ = theta), and y,z = R,Þ, the torus is marked in coordinates of þ,Þ.

http://www.youtube.com/watch?v=kNva9WpQXvM 7 http://www.lboro.ac.uk/departments/ma/g ... /ht800.gif

Torus, as above, with Lissajous rotation as motif. Basically, this puts þ, Þ as wt (omega * time) and Wt (Omega * time), where w = 20* W. For any given þ,Þ, one projects onto r, R to get a Lissajous figure. This is how the sun would move, where there were 20 days in a year.
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wendy
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Re: Clifford, Flat or simply 4?

wendy wrote:http://www.youtube.com/watch?v=sdnzcEO- ... re=related.

This makes me no sense.

A torus, or space where w^2+x^2 = c1, y^2+z^2=c2. Putting w,x -> r,þ, (þ = theta), and y,z = R,Þ, the torus is marked in coordinates of þ,Þ.

http://www.youtube.com/watch?v=kNva9WpQXvM 7 http://www.lboro.ac.uk/departments/ma/g ... /ht800.gif

Torus, as above, with Lissajous rotation as motif. Basically, this puts þ, Þ as wt (omega * time) and Wt (Omega * time), where w = 20* W. For any given þ,Þ, one projects onto r, R to get a Lissajous figure. This is how the sun would move, where there were 20 days in a year.

Mind elaborate your statments using as little equations/maths as possible? as I haven't learnt 4D geometry yet (i.e. failed to understand the equations, Lissajous etc.)
Also it is said the 1st clip (the re=related one) is a 3D cross section of a (certin kind of) 4D torus passing through 3D space
Secret
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Re: Clifford, Flat or simply 4?

Some maths is needed, but i will make it little.

In four dimensions, ye have four axies, eg w,x,y,z. You can divide these into w,x and y,z, and set a rotation in each of these. Every point wxyz would then move as the point wx in the first 2-space, and yz in the second one.

When the rotations are equal, every point moves in a simple circle around the centre. From a point on a rotating sphere, you can't detect where the particular axies wx and yz cut through. Clifford rotations are related to 'clifford parallels', which are circles equidistant from another circle, but not in the same space as it. Clifford rotations are not mirror-identical. A rotation that makes w->x in 90°, can bring y to z or (differently) z to y by 90° also. Since in physics, the rotation of objects, like planets, tend to equalise the energies in the different modes of rotation, this is what a planet tends to acquire.

A simple rotation is when one of the 'axial' rotations is still, eg there is no change in y-z coordinates. As in three dimensions, there is an axis of N-2 dimensions that are perpendicular to it. Like three dimensions, one steers a wheel by slowing down the axis perpendicular to the wheel, by slowing down or leaning into the direction one wants to turn.

The general rotation in four dimensions is when the axies rotate at different rates, so that wx and yz are at speeds like 2:1 or 5:2. If you look at the projection in the 2-space like xy or wy or wz, the pattern formed by a point follows the well-known Lissajous figures.

In four dimensions, the 'torus' corresponds to those points that lie at the points wxyz, where wx and yz are circles in those planes. This is the same surface, that one gets if one rotates a cylinder of height x, and circle yz, is made to make a circle in wx. This represents a set of points that are equidistant from a given great circle, and if the great circle is an axis of rotation, the torus rotates inside itself. A rotation where one axis rotates 18 times faster than the other, would give a corkscrew, with 18 turns around one axis, and one turn around the other. This is what one sees in some of the pictures.
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wendy
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Re: Clifford, Flat or simply 4?

The Clifford torus and flat torus are the same thing. We call it the duocylinder or 22 here. It's topologically the same as a regular torus, but flattened out into 4D (hence "flat torus").

I'm not sure what the first video is. Certainly not a flat torus. It could be a tiger, but it's probably a 3-torus, which is actually a 4D shape. We call it ((21)1), or "ditorus", for some reason. Whatever it is, it's represented by a 3D cross section with the 4th axis represented by time.

The second one seems to be a (31) or toraspherinder, which is a completely different shape. See how it looks like a regular torus but each smaller circle actually turns out to be a sphere? I wonder if the guy who made the video knew this. I'm pretty sure Wendy is wrong about this one.

[http://www.youtube.com/watch?v=kNva9WpQXvM]The third video [url] is titled "4D hypersphere" but it's actually a flat torus/Clifford torus/duocylinder.

Unfortunately I can't tell you what your images are, I can't see enough information.

PWrong
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Re: Clifford, Flat or simply 4?

PWrong wrote:The Clifford torus and flat torus are the same thing. We call it the duocylinder or 22 here. It's topologically the same as a regular torus, but flattened out into 4D (hence "flat torus").

So that means a flattened 2-torus (clifford torus) is the duocylinder? Or are they two different shapes?

PWrong wrote:I'm not sure what the first video is. Certainly not a flat torus. It could be a tiger, but it's probably a 3-torus, which is actually a 4D shape. We call it ((21)1), or "ditorus", for some reason. Whatever it is, it's represented by a 3D cross section with the 4th axis represented by time.

how does the Tiger look like? as I can't find it elsewhere on the internet except this forum and the related wiki. Btw so this (3-torus) is what many people referred as the hypertorus?

PWrong wrote:The second one seems to be a (31) or toraspherinder, which is a completely different shape. See how it looks like a regular torus but each smaller circle actually turns out to be a sphere? I wonder if the guy who made the video knew this. I'm pretty sure Wendy is wrong about this one.

A toraspherinder, according to the wiki here is a shape formed when you bend the spherinder and stick their ends together. If this is really the case, this will clear my confusion on which is the 3-torus (http://www.youtube.com/watch?v=i5MRtkKVZ5c VS first video)

PWrong wrote:[http://www.youtube.com/watch?v=kNva9WpQXvM]The third video [url] is titled "4D hypersphere" but it's actually a flat torus/Clifford torus/duocylinder.

Ok

PWrong wrote:Unfortunately I can't tell you what your images are, I can't see enough information.

See if these description helps:

In the picture you'll see three 2-tori (big one that "covers" two narrow ones) between the two narrow tori is an empty space. The image is produced by revolving a catesian product of two circles around a circle placed in 4D (might be wrong with the understanding of the catesian product and the attempt reconstruction of the object from the 3D slices from the 1st video)
Secret
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Re: Clifford, Flat or simply 4?

Trying to get sense out of 4D, by using 3D names gets horibly complex. It's in part why i developed the polygloss.

A rhombus in 2D is a pretty well defined idea. It is variously, a parallelopied of equal angles, or a covering (tegum) of unequal lines at right angles.

In three dimensions, just as the square gives the cube (prism) or gives the octahedron (tegum), so does the two definitions of the rhombus in two dimensions give different three dimensional things. The "rhombohedron" is a paralleloped of three lines. In PG it is 'triangular anti-tegum', a prism-like construction. The faces of the rhombic dodecahedron etc, give rise to a tegum (a covering of figures at right-angles).

In four dimensions, the thing gets utterly confusing, since words like 'plane' (which among other things, divide space). The words confuse as much as the maths. Still.

A 'tiger' is apparently the "spherated bi-glomlatric prism". 'Spherate' is the result of replacing points by a sphere centred thereon. A torus in 2d is a "spherated circle". bi- means two (independently). glomo- is 'sphere-shaped' latr is 1D, -ix (ic) is cloth. a glomolatron is a sphere-shaped 1D patch (and its interior), ie a 2D disk, while a glomolatrix is a 1D fabric bent into sphere-shaped. prism is a form of cartesian product. So, eg

"spherated glomolatrix" is the result when ye run a sphere around the surface of a 1d sphere - ie a torus or anchor-ring or donought.

"bi-glomolaton prism" is a prism product of a two cylinders, eg a 'duocylinder'.

bi-glomolatrix prism is the prism-product of two circle-perimeters, a 2d sheet folded in 4d (a 'torus')

spherated bi-glomolatrix prism is a solid made by making the aforementioned 2d sheet into a 4d object.

"clifford torii" don't really exist. The clifford parallels equidistant from a given great circle, do form a "biglomolatric prism", or 'torus", but it's not something that stands out. It's the four-dimensional equal of 'lattitude' on a 3d sphere.

A 'torocylinder' is the comb of a circle and a sphere, or a 'spherated circle' in four dimensions.

"hyper" means 'above'. We're hyperspace to two dimensions. In six dimensions, hyperspace is 7D, not 4D. A 'four dimensional hypersphere' is actually a 5D thing, invoked to solve problems in 4D.
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wendy
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Re: Clifford, Flat or simply 4?

So that means a flattened 2-torus (clifford torus) is the duocylinder?

Yes. It's embedded in 4 dimensions, but it's actually a 2-dimensional surface, like a torus or a klein bottle.

how does the Tiger look like? as I can't find it elsewhere on the internet except this forum and the related wiki.

We invented it here on the forum. As far as I know noone else is aware of its existence. I can't really say what it looks like, and I'm no good at programming these videos and pictures. It's the result of taking a duocylinder, and replacing each point with a circle, in such a way that the whole thing is still embedded in 4 dimensions (this is called "spherating" by a circle). It's code is (22), and its implicit equation is
(sqrt(x^2 + y^2) - r1)^2 + (sqrt(z^2 + w^2) - r2)^2 = r3^2

Btw so this (3-torus) is what many people referred as the hypertorus?

Yes. It's subtly different from the tiger.

torus = circle spherated by circle
3-torus = (circle spherated by circle) spherated by circle
tiger = (circle x circle) spherated by circle

A toraspherinder, according to the wiki here is a shape formed when you bend the spherinder and stick their ends together.

That's right.

The image is produced by revolving a catesian product of two circles around a circle placed in 4D

If I don't take that literally it could mean either a tiger or a 3-torus. If I do, I'm still not sure what shape it would make. Sorry.

might be wrong with the understanding of the catesian product

You may have heard that a torus is the cartesian product of two circles. This is only true if you position the circles just right, in which case it's equivalent to spheration. If you put the circles in orthogonal planes, you get a duocylinder. Generally when we say the cartesian product AxB we assume the special case when A and B are in othogonal hyperplanes.

PWrong
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Re: Clifford, Flat or simply 4?

PWrong wrote:We invented it here on the forum. As far as I know noone else is aware of its existence. I can't really say what it looks like, and I'm no good at programming these videos and pictures. It's the result of taking a duocylinder, and replacing each point with a circle, in such a way that the whole thing is still embedded in 4 dimensions (this is called "spherating" by a circle). It's code is (22), and its implicit equation is
(sqrt(x^2 + y^2) - r1)^2 + (sqrt(z^2 + w^2) - r2)^2 = r3^2

hmm I'll try to 'make' one later...

PWrong wrote:Yes. It's subtly different from the tiger.

torus = circle spherated by circle
3-torus = (circle spherated by circle) spherated by circle
tiger = (circle x circle) spherated by circle

Does the operation "spheration also invented here? As I also cannot find this operation elsewhere
Also: http://img248.imageshack.us/img248/9541/tesmx.jpg

PWrong wrote:If I don't take that literally it could mean either a tiger or a 3-torus. If I do, I'm still not sure what shape it would make. Sorry.

Doesn't matter, I'll try to 'make' a better one later (Now I know which is the 3 torus, I won't mixed it up with the toraspherinder anymore)

PWrong wrote:You may have heard that a torus is the cartesian product of two circles. This is only true if you position the circles just right, in which case it's equivalent to spheration. If you put the circles in orthogonal planes, you get a duocylinder. Generally when we say the cartesian product AxB we assume the special case when A and B are in othogonal hyperplanes.

Any pic to illustrate the catesian product of two circles? (Not the projection of the resulted shape, only the two circles in perspective projection)

Wendy wrote:A rhombus in 2D is a pretty well defined idea. It is variously, a parallelopied of equal angles, or a covering (tegum) of unequal lines at right angles.

In three dimensions, just as the square gives the cube (prism) or gives the octahedron (tegum), so does the two definitions of the rhombus in two dimensions give different three dimensional things. The "rhombohedron" is a paralleloped of three lines. In PG it is 'triangular anti-tegum', a prism-like construction. The faces of the rhombic dodecahedron etc, give rise to a tegum (a covering of figures at right-angles).

Would be better if you can illustrate these with a pic

Wendy wrote:A 'tiger' is apparently the "spherated bi-glomlatric prism". 'Spherate' is the result of replacing points by a sphere centred thereon. A torus in 2d is a "spherated circle". bi- means two (independently). glomo- is 'sphere-shaped' latr is 1D, -ix (ic) is cloth. a glomolatron is a sphere-shaped 1D patch (and its interior), ie a 2D disk, while a glomolatrix is a 1D fabric bent into sphere-shaped. prism is a form of cartesian product. So, eg

"spherated glomolatrix" is the result when ye run a sphere around the surface of a 1d sphere - ie a torus or anchor-ring or donought.

"bi-glomolaton prism" is a prism product of a two cylinders, eg a 'duocylinder'.

bi-glomolatrix prism is the prism-product of two circle-perimeters, a 2d sheet folded in 4d (a 'torus')

spherated bi-glomolatrix prism is a solid made by making the aforementioned 2d sheet into a 4d object.

"clifford torii" don't really exist. The clifford parallels equidistant from a given great circle, do form a "biglomolatric prism", or 'torus", but it's not something that stands out. It's the four-dimensional equal of 'lattitude' on a 3d sphere.

A 'torocylinder' is the comb of a circle and a sphere, or a 'spherated circle' in four dimensions.

Need some time to digest the concepts, and to build a richer interpretation of 4D, see if i can 'fold' this thing later
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Re: Clifford, Flat or simply 4?

Does the operation "spheration also invented here? As I also cannot find this operation elsewhere
Also: http://img248.imageshack.us/img248/9541/tesmx.jpg

We invented it (and many other concepts), although technically it's a special case of the cartesian product. The second picture is correct.

Spheration is a difficult thing to describe precisely. Do you know about normal vectors and normal planes? You fill up as many dimensions as you can in the normal plane, and introduce new axes to take the rest of the sphere.

Suppose you have 4D surface embedded in 6D space, and you spherate that by a glome (3-sphere living in 4 dimensions). You have two (6 - 4) dimensions given by the normal plane, so you have to make 2 more. The resulting shape is then a 7D surface embedded in 8D space.

Any pic to illustrate the catesian product of two circles? (Not the projection of the resulted shape, only the two circles in perspective projection)

It's the same as a duocylinder. You'd draw a circle in the x-y plane, another circle in the z-w plane

PWrong
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Re: Clifford, Flat or simply 4?

I know it's a challenge to learn 4d geometry but it would be easier to understand what Wendy is talking about
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Re: Clifford, Flat or simply 4?

Understanding the higher dimensions is pretty difficult when the words give you mixed messages.

Cell, in usual parlance, is a solid element of something, the notion being a tiling or foam. For example, people who play board games call the squares or hexagons 'cells'. This is because the cells and the whole of playing space (the plane here), are two-dimensional. In essence, one is playing on a regular 2d foam. Games for zero players, like Conway's "Game of Life" are called 'cellular automata'.

When one looks at quickfur's renders, one sees lots of bubbly things which might be thought of as a foam of cells. They are in effect projections of the surface, and the cells cease to be cells in the same manner that cells on a square lattice cease to be cells on the surface of a cube.

Grasping four dimensions, is more than three dimensions. With the right terminology, things are much easier to hold onto.

Any view of four dimensions, is essentially a picture, a plan or a projection. In a picture, there's a top and bottom, so things would fall to the bottom. A plan is a view of the ground from above, so things don't fall in it. A projection is a distorted image of something, usually a plan, such that one can see the innards.

But a projection that looses one dimension is going to be something solid. You have to imagine that you're looking at the solid and its innards, without surface or crossing any bit of it. This is how you see a three-dimensional thing captured to two dimensions. For the greatest part, we can't see the innards of three dimensions like we can see the innards of two dimensions. So people make the bits translucent (so you can see through it), or give an exploded diagram (so you can see part A goes onto part B etc). When you look at quickfur's renders, you have to imagine that the solid thing you build up, is only a picture of something, and you then have to imagine that it has depth perpendicular to the three-dimensional space.

You next have to understand that some ideas are not hooked to specific dimensions, but are related to solid space. One can hide behind a 2d panel in three dimensions, but in four dimensions, hiding behind a 2d panel is like hiding behind a 1d pole in 3d. You need a three-dimensional panel to hide behind. The words that suggest division of space (plane, wall, face, facing) are thus hooked to N-1 space, while the words that suggest 2d (hedron, hedrix) are hooked to a fixed dimension.

Because we're inventing a lot of new words for a lot of new ideas, we can use linguistic concepts to make life easier. Back in the days at school, one might be taught ladin and greek roots, and construct words like tele.phone = far+i see, and tele.vision = far + image etc. Then there is the idea of regular endings, like a verb /to own/ gives a noun to make the object (ownings), and a noun for the subject (owner). There are regular endings in the PG, for various ideas like this.

One has more room to distinguish between ar.round and sur.round /Sur/ means 'on', so a surface is a face (a dividing thing), that is on (ie contains) something. A surface is divided into surtopes: vertices, edges, surhedra, surchora, or counting down from N, faces, margins, ... The surface is measured in the space where the thing is solid. So the 'surface' of a hexagon is the bit of the hexagon that separates it from the hedrix it falls in.

The around-space is the space that is not part of the plane, but is said to 'ring' it. One sees for example, that one dances around the maypole, and the earth spins around its axis. The distinction is faint in 3d, but is useful to persue higher.

A poly.hedr.on is read as many.2d.patches. The patch image is merged into an unbounded cloth hedr.ix. Stems in /a/ and /o/ refer to patches, while those in /i/ refer to unbounded cloths. It then comes a matter of inventing words for 1d, (latr), 3d (chor), 4d (ter) &c. There are other ideas that have specific dimensions like /solid/ (you can have a solid area or a solid part of a line), gives rise to sol.id, and then -id becomes a solid or definite thing in that dimension. A hexagon is always a hedr.id because it is always a 2d solid. The opposit of solid is neblu.ous or cloudy. Something that is approximately 2d, but might have thickness in other spaces, is "hedr.ous".

You can add prefixes to give these things certian shape. A glom(o).id is a globe-shaped thing, a circle, a sphere, etc. A glomo.hedr.on is a globe-shaped thing bounded by a 2d shape, ie a 3d sphere. A glomo.hedr.ix is the 2d cloth that's bent into glomid shape - a sphere-surface.

While there is a cartesian product, there are many different things one can do to modify or multiply shapes. Truncation is about cutting off the corners, usually vertices. When this is used of higher dimensions, it's described as a bevel. A bevelled cube has long hexagon sides where the edges were.

Spheration is about giving any non-solid element a glomic arrounding (eg circular or spheric section). The Atomium is a spheration of a body-centred cube. The eight vertices and the centre become spheres (since 3d is 'around' 0d) and the edges are turned into thinner cylinders (because the arounding of an edge gives a 2d space, which holds a circle). The sizes of the added aroundings do not have to match.

When we look at eg the 'tiger', we see it's a 'spherated bi-glomohedrix prism'. The glomo.latr.ix is a sphere-shaped 1d cloth, ie something that might cover a 2d sphere (circle). A prism is a cartesian product. So this corresponds to a 2d fabric (hedrix) that lives in 4d. It's the 2d element in a bi-cylinder. Spherating it makes it solid (ie a 4d thing here). It's hard to imagine directly, i fess, but not so hard as something like a bi-curcular tegum.

In any case, start with a circle in 3d. When you spherate its perimeter, you get a torus, anchor ring, or doughnut. You replace the line by a bent cylinder. To get something that works like the 'tiger' you now have to spherate the surface of the 3d torus in four dimensions. This means that you have to imagine spaces on either side of 3d space, and replace each point of the surface with a circle, that is orthogonal to the hedrix. You get in essence, a circle, standing on its edge, so that all you see is a 1d section. The centre is the old 3d torus, the inner and outer points are the new surface in 4d.
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