Visualizing 6D objects using interrelated 3D objects

[Teragon]

I'd like to share with you something that came to me yesterday evening and kept me from sleeping last night. It somehow feels like I've rediscovered something I've witnessed in another dimension.

In theory it works in any number of dimensions, but it works nearly as nice, if not nicer in 5D and 6D than in 4D and becomes a lot more complicated und confusing from 7D onwards. In 5D it's visually particulary clear, but conceptually it's not as beautiful as 6D. 6 is also a very interesting number of dimensions, so I'm going to explain it just refering to 6D.

The basic idea is that we can seperate hexaspace in two interrelated 3D sections.
One with the coordinates x, y and z. I'm gonna call this section XYZ. It's a function of u, v and w, as every combination of these coordinates gives a distinct section XYZ(u,v,w).
One with the coordinates u, v and w. I'm gonna call this section UVW. It's a function of x, y and z, as every combination of these coordinates gives a distinct section UVW(x,y,z).

If we consider an object in hexaspace its section in UVW will look like a 3D-object with its shape depending on x, y and z. We can visualize this relationship by placing the object's center in a point (x, y, z) in an outer coordinate system. We can then imagine it as an object that morphs its shape as it moves through XYZ. A bit more abstract, though more appropriate, we can imagine that every set (x, y, z) is associated with a shape in UVW.

The position of a dot inside the object is given by the coordinates u, v and w. If we fix a point in UVW and vary x, y and z, we keep our location inside the object, moving through XYZ. As the object looks different at any location and certainly it ends somewhere, there is a region, where the chosen point crosses the border of the object (it looks like the shape's border is moving, but it's actually the point moving in XYZ). We can draw the shape of this border in XYZ, which might look different for every combination of u, v and w - the location of the dot inside of the object in UVW.

To sum it up we have one object in UVW, which is given by the set of points that lie inside of the 6D-object for a given set (x, y, z) and another object XYZ, which is given by the set of points that lie inside of the 6D-object for a given set (u, v, w). The concept of an "inner" and an "outer" object is just a crutch for our imagination. In reality, and I think there lies the full beauty of the concept, both objects play equal roles and are exchangeable. The position of the dot inside the smaller object can as well be seen as the location of the larger object, with every point (u, v, w) having a different shape attributed to it. And this time we see why the shape changes with x,y and z which changes its shape with u, v and w and we've come full circle.

Here is an example. It's the carthesian product of a triangle prism and a sphere. The green cylinder is actually defined at every point inside the cuboid shape (and the black dot as well - it's in fact a volume), but is just drawn examplary at some points in order to show, how it relates to x, y and z. The cuboid shape marks the region where the black dot crosses the green cylinder, which is where the object ends in x-, y- and z-direction for a given set of (u, v, w). All the dots inside of the cuboid are also inside their cylinders. You can now mentally walk around in 6D by changing the position of one cylinder inside the cuboid (position in XYZ) and the position of the black dot (which is entangled with the cuboid) inside the cylinder (position in UVW) and test out the relations between the different coordinates. Don't worry, I'm planning to add some animations.





One more remark: It's a matter of taste, how to chose the subspaces. One could as well take XUW and YZV or any of the other 8 possible combinations. Every time it might look different. Appling rotations to the object would certainly help to comprehend relations between the two cross sections geometrically and vice versa.

[Klitzing]

In theory it works in any number of dimensions, but it works nearly as nice, if not nicer in 5D and 6D than in 4D and becomes a lot more complicated und confusing from 7D onwards. In 5D it's visually particulary clear, but conceptually it's not as beautiful as 6D. 6 is also a very interesting number of dimensions, so I'm going to explain it just refering to 6D.

The basic idea is that we can seperate hexaspace in two interrelated 3D sections.
One with the coordinates x, y and z. I'm gonna call this section XYZ. It's a function of u, v and w, as every combination of these coordinates gives a distinct section XYZ(u,v,w).
One with the coordinates u, v and w. I'm gonna call this section UVW. It's a function of x, y and z, as every combination of these coordinates gives a distinct section UVW(x,y,z).

If we consider an object in hexaspace its section in UVW will look like a 3D-object with its shape depending on x, y and z. We can visualize this relationship by placing the object's center in a point (x, y, z) in an outer coordinate system. We can then imagine it as an object that morphs its shape as it moves through XYZ. A bit more abstract, though more appropriate, we can imagine that every set (x, y, z) is associated with a shape in UVW.

Ain't that just a further shift of Wendy's lace tower (1D configuration space) and lace city (2D configuration space) ideas to a 3D configuration space?

• In lace towers you just consider mD = 1D + nD, i.e. an axial stack of nD perpendicular sections (n=m-1).
• In lace cities you consider mD = 2D + nD, i.e. a city map like display of nD perpendicular sections (n=m-2).
• And yours now considers mD = 3D + nD, i.e. a 3D positioning of nD perpendicular sections (n=m-3).

For curved figures the applied grid could be anything, even so not too dense (too many tiny pics) and not too sparse (one still ought get the feeling on what happens inbetween). For polytopes OTOH one might reduce that arbitrariness of the grid to just vertex positions, as there only do occur conceptual changes. E.g. in lace towers usually vertex layers are used.

Thus, for instance we have x3o3o5o = o3o5o || x3o5o || o3o5x || f3o5o || o3x5o || f3o5o || o3o5x || x3o5o || o3o5o. (Representation of 600-cell within axially-ixosahedral subsymmetry.) The same figure also could be given within o5o2o5o subsymmetry as the following lace city:

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                 o5o           o5o                 
                        o5x                       
                                                   
            x5o                     x5o           
     o5o                                   o5o     
                        f5o                       
                 o5f           o5f                 
                                                   
     o5x                                   o5x     
            f5o                     f5o           
                                                   
o5o                     x5x                     o5o
                                                   
            o5f                     o5f           
     x5o                                   x5o     
                                                   
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                   
                        x5o                       
                 o5o           o5o                 

Even so lace towers and lace cities are well suited for display within the paper plane / screen plane, yours would provide some problems in such representations. The additional 3rd coord. would have to be shown somewhere obliquely foreshortened. This then provides some ambiguity of Interpretation of display. OTOH, when using real 3D displays, such like physical models (3D printers), interactive VRMLs, or at least continuously spinning GIFs, this ambiguity might get overcome and thus add a further level indeed.

--- rk

[wendy]

Lace cities precedes lace towers by decades, rather like the nodes precede the branches by decades.

The two usual representations of polytopes, is either to project the whole onto a plane, so each vertex, edge, etc is shewn differently, or to project perpendicular to a hedrix of symmetry. That is, we project onto a minimum of points. There are excellent projections of the twelftychoron and the octagonny that one can see the various elements in projections. I tend to prefer this form.

If you take the usual vertex-first listings of the fifhundchoron (pt, I, D, fI, ID, fI, D, I, pt), it is less informative than a projection orthogonal to a polygon, which Richard's lace-city of the this figure shows. One can see more clearly how things fit together, and can see how the pentagons vary in the various loops. Richard's xoo3ooo3oxoAoox&x is actually the central core of the 2_21, less the outer three vertices, something I wrote down when I was trying to decipher Coxeter's cryptic 'regular polytopes'.

Of course, a lace city or tower, is simply an expansion of a lace compound, laid out in a suggestive pattern. The {3,3,5} is x5x2o5o + f5o2x5o + o5fo2o5x + x5o2o5f + o5x2f5o + o5o2x5x. But we project this variously down the 'x' axis (ie nodes 1,2) or 'y' axis (3,4). But {3,4,3} produces different versions orthogonally on the triangular axis.

The lace city, sufficiently large enough, is easily committed to memory, and i used such when answering some question for John Conway.

You can of course, do projections down the 3-axis, or down a 4-axis. But there is a sufficiently more potent device here than lace cities for this: one can use stations of a lattice, and shells (spheres radiating from a point). This is how i derive many of the lace towers and lace cities for the gosset figures.

[Klitzing]

Richard's xoo3ooo3oxoAoox&x is actually the central core of the 2_21, less the outer three vertices, something I wrote down when I was trying to decipher Coxeter's cryptic 'regular polytopes'.

Here Wendy is refering to  xoo-3-ooo-3-oxo *b3-oox-&x  when considered as 3-dim jak ("tedjak" for short), cf. the lace city of jak being:

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o           where:
    E       E = o3o3x *b3o (hex)
X       o   O = o3o3o *b3x
    O       X = x3o3o *b3o
o           o = o3o3o *b3o (point)
         
|   |   +   o3o3o *b3o3o (point)
|   +----   x3o3o *b3o3o (hin)
+--------   o3o3o *b3o3x (tac)

The lower part of that display, by the way, shows how a "city" indeed has "lanes" (orientational directions) of orthogonally placed "towers".

--- rk

[Teragon]

It certainly is related to the idea of lace towers. Where in the forum was it discussed? Unfourtunatly I'm not familiar with the notation you used to describe polytopes.

At least it seems that no attemps have been made to really visualize them in the forum. And this is just what I'm most interested in. Boosting the multidimensional imagination. This form of depiction might be easier to learn from than just a 4D cross-section, because all the information is there at any point in time and maby this promotes the ability to keep in mind the structure of the whole object at once. We'll see.

Could also be interesting for future 5D computer games to have two worlds side by side, each showing two dimensions plus the height. When walking inside one of the worlds, the other one morphs.

In 6D there are 15 possible single rotations, 45 possible double rotations (if I counted correctly) and 15 possible triple rotations.

One nice thing about these 3D fields of shapes is that 6 out of the 15 rotations can be easily visualized. The other 9 involve morphing shapes and are thus much harder to program, but one might be able to get a feeling for that with the aid of the version rotated by 90°.

Here is a rotation in the vw-plane:



Everything is keeping its place, just the smaller objects are rotating in situ. And with them of course the dots that are indicating where the cuboid-shaped cross-section is currently taken.

A rotation in the xy-plane takes the cylinders with it, while the cylinders are keeping their orientation (the coordinates u, v and w are not involved in the rotation).



Now, for 9 of the 45 double rotations, XYZ and UVW do not merge and thus involve only combinations of those two types of rotations.



Adding another rotation in the zu-plane would look no different if this was really a "Triangle Spherinder", because the cross-section in zw is a circle, but as it turned out this is rather the carthesian product of a dicone and a cuboid (a "Dicone Tesserinder" with unequal side lengths)*.

If we consider a rotation in zw, the edge, where the cylinders have their highest point in w would move to the upper/lower side of the cube (+-z) with the maxium height of the cylinders decreasing. At the same time, the edge where the cylinders have their lowest point would be move in the opposite direction. With the two edges at the top and bottom side of the cube, two more edges would come about, decreasing the height of the cuboid by a factor of sqrt(2) and making the heights of the cylinders grow as they move from +-z towards +-w. After 45° the height of the cylinders would be equal everywhere in the cuboid. So what is shown in the animations is not the simplest representation of the object. Still it's a very simple object given that most of it is only a hypercube, where coordinates don't depend on each other.

This is in fact the simplest type of a double and triple rotation. For some triple rotations, all three coordinates of one cross-section merge with coordinates of the other cross-section.

What I can do at the moment is walking around in UVW. If we are moving up and down in w with constant speed during the double rotation we get this:



As the dots are crossing the borders of their cylinders, their center points are also leaving the cross-section in XYZ, which is just defined by where in XYZ one is still inside the object for a given position in UVW. The height of the cuboid is changing at a constant rate with a constant speed in w, reflecting the shape of the sqare. The egdes express themselves as where the height of the cuboid reaches zero and where its change reverses.

If the cylinders are aligned conveniently one can directly see a projection of the square in zw as shown in red. It looks distorted because the cylinders are drawn shorter than they really are. This makes it also easier to see how a rotation in the zw-plane would look like. As you can see this cross-section is the same for every x, y, u and v, but at the same time, u and v (the radius of the cylinder) are changing with y (left-right). One can also recognize that the cross-section in the xyu-plane is not really the cross-section of a dicone, because it's rounded at the top*.



*Due to a mistake it's actually two half periods of a positive and negative sine-function rotated in the uv-plane. Maby I will change that so that it's not needlessly complicated.

Edit: Made some adjustments, because it's a different shape than I thought initially.

[Polyhedron Dude]

This reminds me of the way I rendered poke sections of polytera, except for those it is a plane of 3-D sections. This 3-D grid of sections would work for polypeta, the only issue would be the sheer complexity of the sections and the fact most of the sections would be buried in the center, I would likely need a 21x21x21 grid to show the detail. I would need virtual reality and a way to "swim" through the sections to get to the center to pull that off.

[Prashantkrishnan]

This process is the same as expressing a shape as a Cartesian product of two shapes.

If we consider an object in hexaspace its section in UVW will look like a 3D-object with its shape depending on x, y and z. We can visualize this relationship by placing the object's center in a point (x, y, z) in an outer coordinate system. We can then imagine it as an object that morphs its shape as it moves through XYZ. A bit more abstract, though more appropriate, we can imagine that every set (x, y, z) is associated with a shape in UVW.

The position of a dot inside the object is given by the coordinates u, v and w. If we fix a point in UVW and vary x, y and z, we keep our location inside the object, moving through XYZ. As the object looks different at any location and certainly it ends somewhere, there is a region, where the chosen point crosses the border of the object (it looks like the shape's border is moving, but it's actually the point moving in XYZ). We can draw the shape of this border in XYZ, which might look different for every combination of u, v and w - the location of the dot inside of the object in UVW.

This is an interpretation of the fact that every object is a Cartesian product of a point and itself. If the shape does not change when you change the location, it means that the shape you obtain is a Cartesian factor of the object in hexaspace.