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After much thought and exploration of the basic 4D geometric shapes, I have attempted visualizing them many times. Take the Duocylinder, for instance. At first it was a mountain hidden in the fog. I knew it had to be there, but I just couldn't see what it actually looked like. Then came the journey of learning its attributes and abilities: how it rolls, what the surface panels are, how they are arranged, etc. At some point, I had compiled a fairly decent list of all unique properties of the duocylinder. I think I gradually built up the image and feel of the shape, by overlaying concepts and visuals. That is, I am thinking of all abilities at once, and stacking several transparencies together, into one complete 4D shape.

Duocylinder

Thinking of the duocylinder, I make the first observation of how it can be created by lathing a 3D cylinder into a new, rounded 4D shape. It can be said that the 3D cylinder rolls along a line, perpendicular to its footprint. It has two flat sides to stand it on, and one round side to roll it on. Cutting it in half will make a circle, oval, or square. This lathing into 4D creates a new, higher expansion, adding on a new ability and structure, in addition to the cylinder's property. There is only one kind if bisecting rotation a cylinder can do to make a duocylinder: to rotate its flat circle ends around in a circular path. This motion traces, or carves out, a new rounded rolling side. This new rolling side is made by sacrificing the only flat sides that the 3D cylinder had. And, on top of that, the original rolling side of the cylinder is still intact, and fully functional. That means this new 4D duocylinder has two rolling sides, and zero flat ones. Make this concept a transparency.

Another observation I can make about the duocylinder is its 90° edge, that snakes its way around the surface. This sharp knifeblade edge is in one piece, and takes curves in the wildest ways. It joins the two individual rolling sides together, and is their only boundary. Now, it is impossible for this structure to exist, in 3D. There is an effective way to quiet the silent argument in your head about how this edge is possible. "If there are two linear rolling sides, then how can there be one separation?" Well, this happens to be one of the things that become possible, when given four dimensions to build with. Make this another transparency.

A third observation I can make about the duocylinder, is how it can roll. Possessing only two rolling sides means it has no flat sides to stand it on. It will always want to roll somewhere, like a 3D ball. Not only that, but it has two such sides. You could pick it up, rotate it 90°, and place it on the ground again. It will now be on its other rolling side, and could still roll away. Placed flat on our 3D plane, it has the footprint of a vertical circle, standing up on its edge. One of its rolling directions is perpendicular to the circle footprint, along our 2D ground. The other direction comes about if we were to physically roll the vertical circle on its edge, like a wheel, in a 90° direction to the first. Both rolling directions would cover a 2D plane of freedom, much like a 3D sphere. Make this concept another transparency.

A fourth observation I can make is the cross sections of the duocylinder. By starting at its footprint, which is right on the surface, we can begin to take thin slices, as we move deeper into the middle. What we see is the thin 2D circle begins to expand into a thin 3D cylinder, with very short height. As we get deeper into the duocylinder, the height of the slice gets taller and taller, eventually becoming a unit cylinder in the middle. This sequence is identical to the other slicing direction. I'm not entirely sure what an oblique duocylinder midsection would look like, but I'll bet that CalcPlot3D applet I use could make it. I haven't explored any open toratopes yet, but that could be the next frontier. Hmm, an oblique angle cut of a cyltrianglinder .... interesting .....

Now, take these four overlays:

1) Made from lathing 3D cylinder, turned flat sides into new rolling side, plus original, into 2 round sides only

2) A single, continuous 90° edge, joining the two rolling sides together

3) Rolls like a sphere, in two free directions along a 2D board

4) Mid sections are a 3D cylinder that collapses its height to a circle, then vanishes as we move away from center

Then, meditate on these, and put them together into one object. This is the real, actual duocylinder, in its four dimensional entirety. Overlaying these visuals and abilities can transform that hidden mountain in the fog into a clear, crisp image. You can then pick up the duocylinder, and place it in your hands. While you are holding it, you feel its strange two flat and rounded sides. You run your fingers along its single sharp edge, that joins the two round sides. It's also much heavier, as if it were made out of uranium. There's so much more of it than you initially realize, because it extends into a higher, extra direction of space. It will contain a lot more mass because of this. You then place it on the ground, and give it a nudge. You see it randomly roll one way, then change to another, as it follows the subtle contours and imperfections on the ground. And, now we pick it back up, dust it off, and place it in the showcase with the rest.

Duocylinder

Thinking of the duocylinder, I make the first observation of how it can be created by lathing a 3D cylinder into a new, rounded 4D shape. It can be said that the 3D cylinder rolls along a line, perpendicular to its footprint. It has two flat sides to stand it on, and one round side to roll it on. Cutting it in half will make a circle, oval, or square. This lathing into 4D creates a new, higher expansion, adding on a new ability and structure, in addition to the cylinder's property. There is only one kind if bisecting rotation a cylinder can do to make a duocylinder: to rotate its flat circle ends around in a circular path. This motion traces, or carves out, a new rounded rolling side. This new rolling side is made by sacrificing the only flat sides that the 3D cylinder had. And, on top of that, the original rolling side of the cylinder is still intact, and fully functional. That means this new 4D duocylinder has two rolling sides, and zero flat ones. Make this concept a transparency.

Another observation I can make about the duocylinder is its 90° edge, that snakes its way around the surface. This sharp knifeblade edge is in one piece, and takes curves in the wildest ways. It joins the two individual rolling sides together, and is their only boundary. Now, it is impossible for this structure to exist, in 3D. There is an effective way to quiet the silent argument in your head about how this edge is possible. "If there are two linear rolling sides, then how can there be one separation?" Well, this happens to be one of the things that become possible, when given four dimensions to build with. Make this another transparency.

A third observation I can make about the duocylinder, is how it can roll. Possessing only two rolling sides means it has no flat sides to stand it on. It will always want to roll somewhere, like a 3D ball. Not only that, but it has two such sides. You could pick it up, rotate it 90°, and place it on the ground again. It will now be on its other rolling side, and could still roll away. Placed flat on our 3D plane, it has the footprint of a vertical circle, standing up on its edge. One of its rolling directions is perpendicular to the circle footprint, along our 2D ground. The other direction comes about if we were to physically roll the vertical circle on its edge, like a wheel, in a 90° direction to the first. Both rolling directions would cover a 2D plane of freedom, much like a 3D sphere. Make this concept another transparency.

A fourth observation I can make is the cross sections of the duocylinder. By starting at its footprint, which is right on the surface, we can begin to take thin slices, as we move deeper into the middle. What we see is the thin 2D circle begins to expand into a thin 3D cylinder, with very short height. As we get deeper into the duocylinder, the height of the slice gets taller and taller, eventually becoming a unit cylinder in the middle. This sequence is identical to the other slicing direction. I'm not entirely sure what an oblique duocylinder midsection would look like, but I'll bet that CalcPlot3D applet I use could make it. I haven't explored any open toratopes yet, but that could be the next frontier. Hmm, an oblique angle cut of a cyltrianglinder .... interesting .....

Now, take these four overlays:

1) Made from lathing 3D cylinder, turned flat sides into new rolling side, plus original, into 2 round sides only

2) A single, continuous 90° edge, joining the two rolling sides together

3) Rolls like a sphere, in two free directions along a 2D board

4) Mid sections are a 3D cylinder that collapses its height to a circle, then vanishes as we move away from center

Then, meditate on these, and put them together into one object. This is the real, actual duocylinder, in its four dimensional entirety. Overlaying these visuals and abilities can transform that hidden mountain in the fog into a clear, crisp image. You can then pick up the duocylinder, and place it in your hands. While you are holding it, you feel its strange two flat and rounded sides. You run your fingers along its single sharp edge, that joins the two round sides. It's also much heavier, as if it were made out of uranium. There's so much more of it than you initially realize, because it extends into a higher, extra direction of space. It will contain a lot more mass because of this. You then place it on the ground, and give it a nudge. You see it randomly roll one way, then change to another, as it follows the subtle contours and imperfections on the ground. And, now we pick it back up, dust it off, and place it in the showcase with the rest.

It is by will alone, I set my donuts in motion

- ICN5D
- Pentonian
**Posts:**1135**Joined:**Mon Jul 28, 2008 4:25 am**Location:**the Land of Flowers

The duocylinder is not so strange to understand if you take the right approach. I independently rediscovered it while studying duoprisms, and initially called it the "double torus", because, just like you described, it consists of two equal surfaces shaped like toruses, joined at a toroidal 2D interface that is equally deformed everywhere. Well, no need to repeat what I've already said on my webpage, so first read up on duoprisms, and then on prismic cylinders, which eventually lead to the duocylinder.

The simplest way (for me, anyway) to visualize it is in the following projection:

First, note that there's an hourglass-like column in the middle, then an outer "bloated cylinder". In between these two, circling around the hourglass column, is a flattened torus with a thin ellipsoidal cross-section. The hourglass column and the outer "bloated cylinder" are two halves of a single torus that's curving in and out of the 4th direction; the torus in between is the other torus. These two are actually exactly the same shape, seen from different angles, and they close up into what we call the duocylinder.

The curved 2D manifold where these two toruses touch each other, is the ridge of the duocylinder. If you take this manifold and inflate it, you get a tiger.

The simplest way (for me, anyway) to visualize it is in the following projection:

First, note that there's an hourglass-like column in the middle, then an outer "bloated cylinder". In between these two, circling around the hourglass column, is a flattened torus with a thin ellipsoidal cross-section. The hourglass column and the outer "bloated cylinder" are two halves of a single torus that's curving in and out of the 4th direction; the torus in between is the other torus. These two are actually exactly the same shape, seen from different angles, and they close up into what we call the duocylinder.

The curved 2D manifold where these two toruses touch each other, is the ridge of the duocylinder. If you take this manifold and inflate it, you get a tiger.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

Yeah, that's pretty much how I've been seeing it. At some point, I made the connection between the animated gif of a rotating tesseract and how a cylinder could do the same, when rotating into 4D. I saw the circle ends squeezing down through the middle, then expanding around the outside, as they flowed inside out, like a magnetic field. This motion would trace out the new rolling surface, while leaving the original one intact. I likened the rolling feel of a duocylinder to those weird gel-filled things you see for sale some places. You know, they roll inside out, and roll out of your hands? even if your holding it with a closed fist, it's in the shape of a torus, and filled with gel. A duocylinder would have this inside out rolling ability, when manifested in the 3D projection. It can roll on the outside, like a barrel, or inside out, like those gel-filled gag gifts.

It is by will alone, I set my donuts in motion

- ICN5D
- Pentonian
**Posts:**1135**Joined:**Mon Jul 28, 2008 4:25 am**Location:**the Land of Flowers

Sometimes in order to visualize a duocylinder I use time as the fourth dimension and visualize a single 3d shape at a time and change the shape in my head through time to produce the 4d shape I want to visualize. This is the method I used to figure out that it's possible for there to be two perpendicular rings with the same center that do not intersect in 4d. Sometimes I also visualize it by visualizing a cylinder and having the ends of the cylinder to be more faded than the middle with the more faded part representing the part that is further in the fourth dimension and imagining the ends to connect to each other. I also sometimes visualize it using the same kind of projection quickfur showed. I can visualize things in 4d but the things I can visualize are very limited. I can visualize Toratopes but I cannot visualize polytopes well enough to understand them through visualization.

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

- anderscolingustafson
- Tetronian
**Posts:**315**Joined:**Mon Mar 22, 2010 6:39 pm

ICN5D wrote:[...] At some point, I made the connection between the animated gif of a rotating tesseract and how a cylinder could do the same, when rotating into 4D. I saw the circle ends squeezing down through the middle, then expanding around the outside, as they flowed inside out, like a magnetic field. This motion would trace out the new rolling surface, while leaving the original one intact.

One interesting thing about this kind of animation, is that as the tesseract rotates along the (apparent) axis in the "squeezing/expanding" or "inside-out" manner, it can also rotate around this (apparent) axis, and you can have two simultaneous rotations, even with independent rates of rotation! This of course corresponds with the 4D double rotation, also something new that isn't possible in 3D and below. Here's another one of this same animation that I did on my website:

Now one thing that isn't immediately obvious about this double rotation, is that when the two rates of rotation become equal, it becomes a special kind of rotation called a Clifford rotation, in which there are infinite stationary planes. First, note that when the two rates of rotation are not equal, the vertices of the rotating tesseract will trace out spiralling curves in 4D, while points that lie on the (apparent) axis (which is actually the projection of a plane) will trace out circles, and points on the perpendicular plane bisecting the axis will also trace out circles. When the two rotation rates are equalized, however, all the spirals flatten out into circles, and it becomes impossible to differentiate between the "original" two planes of rotation! The resulting Clifford rotation, then, will have an infinite number of stationary planes, instead of just two. However, it still has a direction, in that it is not the same as a Clifford rotation in which the original (apparent) axis is rotated out-of-plane! And even better yet, the Clifford rotation remains chiral. So here you have this incredible rigid motion that's both like an infinite number of simultaneous rotations, and yet it still has direction and orientation (chirality).

The circles traced out by this incredible compound rotation is, in fact, none other than the Hopf fibration itself. Now imagine if instead of the two toroidal manifolds of the duocylinder, you mark out more than 2 points on the rotating circum-3-sphere, equally spaced, so that they trace out n circles. Inflate these circles into toruses until they touch each other. The result is none other than one of Jonathan Bowers' polytwisters. Now take the boundary of these toruses and delete their interior. Inflate it in the same manner as the tiger. The result is a kind of "poly-tiger" which has n toroidal holes.

I likened the rolling feel of a duocylinder to those weird gel-filled things you see for sale some places. You know, they roll inside out, and roll out of your hands? even if your holding it with a closed fist, it's in the shape of a torus, and filled with gel. A duocylinder would have this inside out rolling ability, when manifested in the 3D projection. It can roll on the outside, like a barrel, or inside out, like those gel-filled gag gifts.

Yes, though keep in mind that this inside-out effect is really just an artifact of the projection from 4D into 3D. If viewed from the side in 4D, it would appear exactly the same as a "normal" 3D rotation. This effect is analogous to 3D rotations as seen from 2D:

Of course, our brain spontaneously interprets this animation in a 3D way, but suppose for a moment that we suppress the 3D interpretation of it. Look at it from a purely 2D point of view. The rotating square then becomes no longer a rotating square spinning in 3D; it becomes a crazy morphing quadrilateral in 2D that's performing impossible inside-out contortions. But actually, it's a perfectly normal square rotating in 3D. In the same way, when we 3D beings look at a 4D rotation, it appears to be some kind of incredible inside-out transformation. But actually, when seen from 4D, it's a perfectly normal rotation, no different from any other rotation.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

I no longer think in 3D anymore. Especially after coming to terms with a few particular 9, 15, and 16D toratopes

It is by will alone, I set my donuts in motion

- ICN5D
- Pentonian
**Posts:**1135**Joined:**Mon Jul 28, 2008 4:25 am**Location:**the Land of Flowers

That's what I thought I did, too.

But actually, every now and then I catch myself applying 3D logic to 4D constructions, and have to stop and remind myself that the 3D projection model I'm working with is not actually a 3D object. This tripped me up when I was first trying to construct a CRF with two triangular hebesphenorotunda cells... unconscious 3D-centric thinking led me to think the construction was impossible, until student91 pointed out that it actually should be possible. Once I realized my mistake, I got past the blockade in the construction and discovered the J92 rhombochoron.

But actually, every now and then I catch myself applying 3D logic to 4D constructions, and have to stop and remind myself that the 3D projection model I'm working with is not actually a 3D object. This tripped me up when I was first trying to construct a CRF with two triangular hebesphenorotunda cells... unconscious 3D-centric thinking led me to think the construction was impossible, until student91 pointed out that it actually should be possible. Once I realized my mistake, I got past the blockade in the construction and discovered the J92 rhombochoron.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

Yeah, tell me about it. Every time I explore some other +5D shape, I still find out something new about 4D. It's still a strange, mysterious place that we have to explore mentally. Every voyage sharpens the view, and sometimes, there are radical breakthroughs in understanding, that completely change everything! Even the projection of the tesseract, as simple as it is, still has a lot of hidden meaning in it's logic, and is tough for normal 3D thinking to make sense of it. Take the double ring feature, for example! I believe conquering that shape, and knowing exactly why it's put together that way, is the best introduction into 4D concepts. Or any above four concept. The fifth dimension is no more strange than 4D, it's merely a matter adding another " mysterious extra direction", that extends into a new up and down. Compressing 4D flat again, means 5D is the familiar above and below that thin sheet, by perceptual analogy.

In my toratope explorations, I have learned a fantastic concept with reducing dimensions. Especially when you apply this new understanding to how much more expansive five dimensions actually is, compared to 3D! What I learned is the cross-over patterns we see with reducing a shape by a certain number of dimensions. We have to do this, in order to actually see it, in our 3D viewing plane. What we see, is roughly identical to a 1D cut of a 3D shape. This would be a thin needle puncturing through a 3D solid. I say roughly identical, as in the positions of the 5->3 cut shapes are in this 1D cut pattern. With any 1D cut, we always get points along a line. Since the cutting needle is so thin, we have an entire 2D plane to move in, and trace out the rest of the shape. This causes the points to dance around in complex symmetrical patterns, as a terribly reduced representation of the whole 3D shape. Same goes with a 5D shape, reduced to 3D. By a direct analogy, our 3D cutting plane is as thin as a needle, being given a 2D array to move in, and trace out the missing 5D structure. That was a mind awakener right there Holy crap, man, when I finally saw how that worked, it really WAS expanding my mind! Or, think of it as looking at a painting through a thin hollow tube, we see a rough estimate of 1D vision, and we have to move in 2 independent directions to find more colors.

But then, there's 6D. Which is a three dimensional loss, to get to 3D. Our natural 3D viewplane is actually thinner than a needle, in 6D. This is the realm of where you start getting larger arrays of cut shapes. Especially in those 2x2x2, or 2x1x4, or 4x1x2 arrays. What I learned about why those arrays come about, what's making them show up as four things by 2 things, was incredible. By now, I'd have to say that it was the toratopes that really expanded my understanding into incomprehensible high dimensions. For years, I had played with all others: tapertopes, rotatopes, orthotopes, simplices, etc. Just the real basic ones, not anything CRF-like. Then came time to explore the toratopes. They ended up being highly complex, yet highly predictable with their midsections, and merge sequences. It is this predictability that makes very high-D comprehension still attainable.

I think the best example was coming to terms with the midsection cut lattice of nine dimensional triotorus tiger : (((II)I)((II)I)((II)I)) . It looks horrendous at first, but once I show what's happening here, it won't be so bad . Ultimately, its only non-empty cut shows up as (((I))((I))((I))), which means a 4x4x4 cubic array of 64 spheres. Now, firstly, it's important to point out the notation for a torus: ((II)I) , and as we can see in triotorus tiger, there are three torii in it. This is key. Now, consider what happens when we make a 1D cut of a torus: ((I)), we get 4 points along a line. A needle puncturing through the side of an innertube will hit rubber four times. We reduced the torus by 2 dimensions. Now, consider the cubic array again: (((I))((I))((I))) , there are three torii that we have cut 2 times each, in order to get a 9D shape down to 3D. Each leg of the 4x4x4 array, corresponds to the 1D cut of a torus. When we finally reduce triotorus tiger to 3D, we get a line puncturing an innertube to the power of three! Hidden in plain sight, within the 4x4x4 array of 64 spheres, are three hollow torii, arranged orthogonally, multiplied together. The true entire 9D shape is the single surface of all three combined. When we cut it down 6 dimensions, we get four points along a line, to the power of three, making 64 spheres along a 4x4x4 cubic array. Each leg of the array contains an entire separate 2D cut array of its native torus. This gives us a full 6D cut array to move in and trace out the rest of the missing 9D structure. Our 3D line of sight is so incredibly thin; in fact, it's three times thinner than a needle!

In my toratope explorations, I have learned a fantastic concept with reducing dimensions. Especially when you apply this new understanding to how much more expansive five dimensions actually is, compared to 3D! What I learned is the cross-over patterns we see with reducing a shape by a certain number of dimensions. We have to do this, in order to actually see it, in our 3D viewing plane. What we see, is roughly identical to a 1D cut of a 3D shape. This would be a thin needle puncturing through a 3D solid. I say roughly identical, as in the positions of the 5->3 cut shapes are in this 1D cut pattern. With any 1D cut, we always get points along a line. Since the cutting needle is so thin, we have an entire 2D plane to move in, and trace out the rest of the shape. This causes the points to dance around in complex symmetrical patterns, as a terribly reduced representation of the whole 3D shape. Same goes with a 5D shape, reduced to 3D. By a direct analogy, our 3D cutting plane is as thin as a needle, being given a 2D array to move in, and trace out the missing 5D structure. That was a mind awakener right there Holy crap, man, when I finally saw how that worked, it really WAS expanding my mind! Or, think of it as looking at a painting through a thin hollow tube, we see a rough estimate of 1D vision, and we have to move in 2 independent directions to find more colors.

But then, there's 6D. Which is a three dimensional loss, to get to 3D. Our natural 3D viewplane is actually thinner than a needle, in 6D. This is the realm of where you start getting larger arrays of cut shapes. Especially in those 2x2x2, or 2x1x4, or 4x1x2 arrays. What I learned about why those arrays come about, what's making them show up as four things by 2 things, was incredible. By now, I'd have to say that it was the toratopes that really expanded my understanding into incomprehensible high dimensions. For years, I had played with all others: tapertopes, rotatopes, orthotopes, simplices, etc. Just the real basic ones, not anything CRF-like. Then came time to explore the toratopes. They ended up being highly complex, yet highly predictable with their midsections, and merge sequences. It is this predictability that makes very high-D comprehension still attainable.

I think the best example was coming to terms with the midsection cut lattice of nine dimensional triotorus tiger : (((II)I)((II)I)((II)I)) . It looks horrendous at first, but once I show what's happening here, it won't be so bad . Ultimately, its only non-empty cut shows up as (((I))((I))((I))), which means a 4x4x4 cubic array of 64 spheres. Now, firstly, it's important to point out the notation for a torus: ((II)I) , and as we can see in triotorus tiger, there are three torii in it. This is key. Now, consider what happens when we make a 1D cut of a torus: ((I)), we get 4 points along a line. A needle puncturing through the side of an innertube will hit rubber four times. We reduced the torus by 2 dimensions. Now, consider the cubic array again: (((I))((I))((I))) , there are three torii that we have cut 2 times each, in order to get a 9D shape down to 3D. Each leg of the 4x4x4 array, corresponds to the 1D cut of a torus. When we finally reduce triotorus tiger to 3D, we get a line puncturing an innertube to the power of three! Hidden in plain sight, within the 4x4x4 array of 64 spheres, are three hollow torii, arranged orthogonally, multiplied together. The true entire 9D shape is the single surface of all three combined. When we cut it down 6 dimensions, we get four points along a line, to the power of three, making 64 spheres along a 4x4x4 cubic array. Each leg of the array contains an entire separate 2D cut array of its native torus. This gives us a full 6D cut array to move in and trace out the rest of the missing 9D structure. Our 3D line of sight is so incredibly thin; in fact, it's three times thinner than a needle!

It is by will alone, I set my donuts in motion

- ICN5D
- Pentonian
**Posts:**1135**Joined:**Mon Jul 28, 2008 4:25 am**Location:**the Land of Flowers

ICN5D wrote:[...] The fifth dimension is no more strange than 4D, it's merely a matter adding another " mysterious extra direction", that extends into a new up and down. Compressing 4D flat again, means 5D is the familiar above and below that thin sheet, by perceptual analogy.

Actually, 5D is markedly different from 4D. Or, should I say, 4D has some strange properties that make it stand apart from the other dimensions. For example, there are some math theorems that have been proven for 5D and above, and 3D and below, but not in 4D, because the currently-known proof breaks down in 4D. Another property is the chirality of the 4D rotation group. 4D is the first dimension to have double rotations; 5D and above all have them (and n-dimensions in general have n/2-multiple rotations). However, it's only in 4D that these double rotations are chiral. In 5D, you can just flip it upside-down to change the sense of the rotation, but in 4D, these chiral pairs cannot be converted to each other except via a reflection (i.e. mirror image). 4D also has the strange appearance of the regular 24-cell, that simply doesn't have any direct analogues in any other dimension. From what I can tell, this is somehow related to the fact that 1+1+1+1 = 2^2 = 2+2 = 2*2. Not to mention that 4D is also the place where the alternated 4-cube is equal to the 4-cross (i.e. 16-cell), and the 4-cross tiles space. (The only other n-cross that tiles n-space is in 2D, where it is identical to the 2-cube, aka square.) And 4D is the last dimension to have convex pentagonal polytopes. And this last point may seem trivial, but recently with the CRF project we've been finding that something is going on between 4D and the Golden Ratio, that is making a lot of coordinates involving the latter "magically" working out in many combinations that one would normally not expect it to work out. I still haven't quite figured this one out yet, but I'm pretty sure there's something going on here.

In my toratope explorations, I have learned a fantastic concept with reducing dimensions. Especially when you apply this new understanding to how much more expansive five dimensions actually is, compared to 3D!

Someone said once that going from 3D to 4D feels more like 3D to infinity-D instead of just adding 1, because that additional dimension actually adds an infinitude of parallel hyperplanes to the current space. Same thing for going from 4D to 5D.

What I learned is the cross-over patterns we see with reducing a shape by a certain number of dimensions. We have to do this, in order to actually see it, in our 3D viewing plane.

This is what I've been doing for 4D visualization, because it conveniently coincides with how a native 4D being with eyes analogous to ours would see their own world.

But my plan for 5D is to see it not from 3D, but from 4D. I hope to hone my 4D visualization skills enough to be able to be comfortable enough in 4D, that I can see 5D objects as 4D projections. Seeing 5D with 3D projections is, like trying to understand a cube while looking at a point-projection of it, that is, almost impossible. As you said, it's just a needle prick in the object. I'm hoping to be able to see 5D in its full 4D splendor... though I don't know how far I'll succeed.

[...]

But then, there's 6D.

You should talk to Wendy, who regularly works with up to 8D or 9D or somewhere thereabouts. I think she far surpasses most of us here in visualizing higher-dimensional shapes.

[...] For years, I had played with all others: tapertopes, rotatopes, orthotopes, simplices, etc. Just the real basic ones, not anything CRF-like.

Maybe you should try out something fun, like the 120-cell. I think it was while trying to "see" the 120-cell in my mind's eye, that I suddenly came upon the realization of how 4D surfaces are 3D. That was something I'd known in theory before, but could never quite grok. When I "saw" the 120-cell in my mind's eye for the first time, it was a major "wow" moment where I glimpsed, for the first time, what 4D "looks like" from a native's POV.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

Those grokking moments are precious, aren't they? It always brings me back for more!

It is by will alone, I set my donuts in motion

- ICN5D
- Pentonian
**Posts:**1135**Joined:**Mon Jul 28, 2008 4:25 am**Location:**the Land of Flowers

Very beautiful description of the duocylinder, ICN5D! Here is an animation of the 3D image of its edge - the Clifford torus with color coding the distance. I've learned to see the duocylinder (I find the term "duocircle" more appropriate) as an association of circles. If it's rotated, the composite circles in the image just morph as normal circles rotating into the depth-dimension, making up the behaviour of the whole object, as you can recognize in the animation.

What is deep in our world is superficial in higher dimensions.

- Teragon
- Trionian
**Posts:**136**Joined:**Wed Jul 29, 2015 1:12 pm

Yeah, the duocylinder isn't really cylindrical, even though it does make cylindrical cross-sections and projections. It's basically the Cartesian product of two circles (well, discs), and shares a lot in common with the duoprisms (Cartesian product of two polygons). In fact, it's the limiting shape of the n,n-duoprisms as n approaches infinity.

There's also the interesting intermediate series of shapes produced by the Cartesian product of a disc with a polygon, which contains n cylinders as surface elements (where n is the degree of the polygon). Perhaps one might justify the duocylinder's name by considering it as also the limiting shape of these disc-polygon Cartesian products as the degree of the polygon n approaches infinity.

There's also the interesting intermediate series of shapes produced by the Cartesian product of a disc with a polygon, which contains n cylinders as surface elements (where n is the degree of the polygon). Perhaps one might justify the duocylinder's name by considering it as also the limiting shape of these disc-polygon Cartesian products as the degree of the polygon n approaches infinity.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

quickfur wrote:Perhaps one might justify the duocylinder's name by considering it as also the limiting shape of these disc-polygon Cartesian products as the degree of the polygon n approaches infinity.

One might, but it's somehow far-fetched, because the height of these cylinders you obtain if m goes to infinity goes to zero if n goes to infinity. So actually you are creating a totally new object of infinitesimally small cylinders. Names should reflect properties of objects, helping people to understand. I find the name "duocylinder" misleading, creating an idea that stands in the way of understanding the object properly.

How would you call a cartesian product of two cylinders? If I want to be systematic in naming I'd probably call it a duocylinder...

What is deep in our world is superficial in higher dimensions.

- Teragon
- Trionian
**Posts:**136**Joined:**Wed Jul 29, 2015 1:12 pm

Oh don't get me wrong, I agree that "duocylinder" isn't really a particularly good name. I "rediscovered" it by accident when considering the duoprism series, and realized that the limiting shape would have two bounding surface manifolds that are both the same shape, a circular torus (but not of the kind embeddable in 3D; this one has its larger circle lying in an orthogonal plane to the circular cross sections). So I dubbed it the "double torus". I never made the connection with the duocylinder until one day, after having derived equations describing the double torus, I suddenly realized that it was identical with the equations for the duocylinder. Prior to that I had no idea how to visualize a duocylinder.

But that's how names go when the objects are first discovered. Quite often you realize the need for a name, but at the time you may not have discovered its true place within the larger context of similar shapes just yet, and so your choice of name is usually not very consistent in hindsight. Much of the existing terminology in published mathematical literature describing higher dimensions suffer from this effect, having been coined ad hoc as different people considered different directions of generalization from the same 3D concept. As a result you have the messy situation where "face", for example, may refer to only a 2D element in an n-dimensional polytope, or to the (n-1)-dimensional bounding elements, or worse, to any j-dimensional element where 0≤j≤n.

And need I mention the horrible ways in which the "hyper-" prefix has been abused to qualify just about any extension you can imagine of a 3D or 2D concept, with no regard for whether it even makes any sense. For example, a 4D cube is often referred to as a "hypercube" but in the context of 5D space, calling a 4D cube "hyper" is ridiculous, since it is but a flat surface patch. Neither does it make sense to arbitrarily designate anything higher than 3D as "hyper", since that's a completely arbitrary decision that doesn't give you much information and doesn't accurately reflect the generality of the mathematics behind it. So a 5D cube and a 17D cube are both "hyper", but that conveys nothing about the relative difference in dimension between the two objects.

And here in this forum, in the CRF polytopes section, we discovered and named quite a few polytopes that we thought were unique, only to later discover that they are actually part of a larger pattern of so-called EKF polytopes, and that our names for them were pretty arbitrary and don't really describe their exact relation with each other within the larger scheme of things. But those ad hoc names have somewhat stuck, partly because of frequent use prior to discovering the larger pattern, and partly because of our first mental impressions of them that were made before the larger pattern became clear.

I wouldn't be surprised if the duocylinder was actually already known under a different name before the denizens of this forum came upon the scene and rediscovered it.

But that's how names go when the objects are first discovered. Quite often you realize the need for a name, but at the time you may not have discovered its true place within the larger context of similar shapes just yet, and so your choice of name is usually not very consistent in hindsight. Much of the existing terminology in published mathematical literature describing higher dimensions suffer from this effect, having been coined ad hoc as different people considered different directions of generalization from the same 3D concept. As a result you have the messy situation where "face", for example, may refer to only a 2D element in an n-dimensional polytope, or to the (n-1)-dimensional bounding elements, or worse, to any j-dimensional element where 0≤j≤n.

And need I mention the horrible ways in which the "hyper-" prefix has been abused to qualify just about any extension you can imagine of a 3D or 2D concept, with no regard for whether it even makes any sense. For example, a 4D cube is often referred to as a "hypercube" but in the context of 5D space, calling a 4D cube "hyper" is ridiculous, since it is but a flat surface patch. Neither does it make sense to arbitrarily designate anything higher than 3D as "hyper", since that's a completely arbitrary decision that doesn't give you much information and doesn't accurately reflect the generality of the mathematics behind it. So a 5D cube and a 17D cube are both "hyper", but that conveys nothing about the relative difference in dimension between the two objects.

And here in this forum, in the CRF polytopes section, we discovered and named quite a few polytopes that we thought were unique, only to later discover that they are actually part of a larger pattern of so-called EKF polytopes, and that our names for them were pretty arbitrary and don't really describe their exact relation with each other within the larger scheme of things. But those ad hoc names have somewhat stuck, partly because of frequent use prior to discovering the larger pattern, and partly because of our first mental impressions of them that were made before the larger pattern became clear.

I wouldn't be surprised if the duocylinder was actually already known under a different name before the denizens of this forum came upon the scene and rediscovered it.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

But on another note, the cartesian product of two cylinders sounds like a very interesting shape! The nice thing about the Cartesian product is that it is associative and commutative, and that allows us to break down the product to "see" different aspects of it. A cylinder is, of course, nothing but the Cartesian product of a disc and a line segment, or loosely speaking, a circle and a line, so the Cartesian product of two cylinders would be (circle * line) * (circle * line) = (circle * circle) * (line * line) = duocylinder * square. So we can see it as a 6D object produced by extruding a duocylinder twice in two orthogonal directions.

Alternatively, one can also factor it as (circle * line * line) * circle = (circle * square) * circle, and circle * square is just one of those products between a circle (disc) and a polygon, so its surface has 4 cylindrical surface elements and a square toroid. In fact, there's also an ad hoc name for it: cubinder (portamenteau of "cube" and "cylinder"). Admittedly another not-very-consistent name, but one that's been around for a while and probably stuck for good. So the Cartesian product of two cylinders can also be understood as the product of a cubinder with a circle, or, to use a different viewpoint, what you get if you embed a cubinder in 6D and take the trace of a cubinder as you drag it around a circular path outside the 4D subspace it resides in.

Alternatively, one can also factor it as (circle * line * line) * circle = (circle * square) * circle, and circle * square is just one of those products between a circle (disc) and a polygon, so its surface has 4 cylindrical surface elements and a square toroid. In fact, there's also an ad hoc name for it: cubinder (portamenteau of "cube" and "cylinder"). Admittedly another not-very-consistent name, but one that's been around for a while and probably stuck for good. So the Cartesian product of two cylinders can also be understood as the product of a cubinder with a circle, or, to use a different viewpoint, what you get if you embed a cubinder in 6D and take the trace of a cubinder as you drag it around a circular path outside the 4D subspace it resides in.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

Nice visual, Teragon! It's Interesting to read my old thoughts here. Later on, I realized the single edge of the duocylinder is a 2D sheet, that's shaped like a torus. Which is strange to think about: the duocylinder has two tori as rolling surfaces, bound together by an additional torus, that's stretched into 4D. So, two tori on different coordinate 3-planes can touch their entire surface together, in the shape of a 4D-stretched torus. Which is, whoa.

You can also think of the cylinder*cylinder prism as a bisecting rotated tesserinder (cube*circle), and a double bisecting rotation of a tesseract (into 5D, then 6D). In other words, if I = extrude, and O = bisecting rotate, then IIO = IOI (the bisecting spin of square = extrude of circle). It would then follow that IIOIIO = IOIOII = IIIOIO = IOIIOI = IIOIOI = IIOOII = IIIOOI = IIIIOO =/= IOOIII . The sphere*cube prism IOOIII is the only non-commutative.

You can also think of the cylinder*cylinder prism as a bisecting rotated tesserinder (cube*circle), and a double bisecting rotation of a tesseract (into 5D, then 6D). In other words, if I = extrude, and O = bisecting rotate, then IIO = IOI (the bisecting spin of square = extrude of circle). It would then follow that IIOIIO = IOIOII = IIIOIO = IOIIOI = IIOIOI = IIOOII = IIIOOI = IIIIOO =/= IOOIII . The sphere*cube prism IOOIII is the only non-commutative.

It is by will alone, I set my donuts in motion

- ICN5D
- Pentonian
**Posts:**1135**Joined:**Mon Jul 28, 2008 4:25 am**Location:**the Land of Flowers

Well, to be precise, the two tori of the duocylinder do not lie on any coordinate 3-planes, because they are not flat. Rather, they are each 3-manifolds that are wrapped around in a circle. A kind of cylindrical 3-space, if you will.

Now, I'm not 100% sure, but I think the cylindrical 3-space of each torus is the same curved space that the curved square torus of the cubinder (cartesian product of circle and square) lies in. If this is correct, then it means that the duocylinder is a kind of 4D crind, formed by the intersection of two cubinders.

Now, I'm not 100% sure, but I think the cylindrical 3-space of each torus is the same curved space that the curved square torus of the cubinder (cartesian product of circle and square) lies in. If this is correct, then it means that the duocylinder is a kind of 4D crind, formed by the intersection of two cubinders.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

Thanks, ICN5D! I find those properties fascinating too.

Not sure what you mean exactly. The cylindrical 3-spaces you are talking about are the carthesian products disc*circle and circle*disk, right? If you compare both of them to the square torus (full square*circle), the points of one of them will form a subset of it while the other one will just be enclosed by it. Both form a subset of the cubinder. Which intersetion do you mean?

A duocylinder/duodisc can also be thought as a bisecting rotated cylinder, with the height of the cylinder rotating into the new dimension. The edges of the cylinder then connect to form the Clifford torus and it's also very clear from this point of view that the duocylinder has cylinders as cross sections in the main directions.

quickfur wrote:Now, I'm not 100% sure, but I think the cylindrical 3-space of each torus is the same curved space that the curved square torus of the cubinder (cartesian product of circle and square) lies in. If this is correct, then it means that the duocylinder is a kind of 4D crind, formed by the intersection of two cubinders.

Not sure what you mean exactly. The cylindrical 3-spaces you are talking about are the carthesian products disc*circle and circle*disk, right? If you compare both of them to the square torus (full square*circle), the points of one of them will form a subset of it while the other one will just be enclosed by it. Both form a subset of the cubinder. Which intersetion do you mean?

A duocylinder/duodisc can also be thought as a bisecting rotated cylinder, with the height of the cylinder rotating into the new dimension. The edges of the cylinder then connect to form the Clifford torus and it's also very clear from this point of view that the duocylinder has cylinders as cross sections in the main directions.

What is deep in our world is superficial in higher dimensions.

- Teragon
- Trionian
**Posts:**136**Joined:**Wed Jul 29, 2015 1:12 pm

I meant that placing two cubinders of the same dimensions at the origin, but with two different orientations, and taking their intersection, could yield a duocylinder? But as I said, I'm not 100% sure about this.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

I thought about it and my brain tells me that is case, if they are intersected in a way that their disk-like cross-sections lie perpendicular to each other.

We can see the cubinder as an extruded cylinder or as a square-prism changing its height like a circle with w: z = 2*sqrt(1-w²)

Imagining both "on top" of each other, moving through w the cross-section has always the shape of a cylinder with the height given by the height of the cube, which behaves like a circle as a function of w.

What we get is exactly the carthesian product of to discs.

Because the elements of a carthesian product are perpendicular and therefore independent of each other, it's even simpler to imagine (disk*square) and (square*disk), the disk of each cutting out a disk from each other's square, so that (disk*disk) remains.

We can see the cubinder as an extruded cylinder or as a square-prism changing its height like a circle with w: z = 2*sqrt(1-w²)

Imagining both "on top" of each other, moving through w the cross-section has always the shape of a cylinder with the height given by the height of the cube, which behaves like a circle as a function of w.

What we get is exactly the carthesian product of to discs.

Because the elements of a carthesian product are perpendicular and therefore independent of each other, it's even simpler to imagine (disk*square) and (square*disk), the disk of each cutting out a disk from each other's square, so that (disk*disk) remains.

What is deep in our world is superficial in higher dimensions.

- Teragon
- Trionian
**Posts:**136**Joined:**Wed Jul 29, 2015 1:12 pm

It's amazing how many roads lead to the duocylinder.

Yet another one: A cyltorus is a torus with a cylindrical cross section, where the surface normal of the cylinder's base face is pointing towards the center of the torus. It has two edges - an inner one and an outer one. Both are flat tori with one common radius and one radius deviating from one another in size. One of those tori may be a Clifford torus with identical radii. There are three faces - one surrounded by the inner flat torus, one surrounded by the outer one, both identical to one of the faces of the duocylinder and another face lying in between the two flat tori, something like a Clifford torus with a thickness in the plane of one of its loops. Two cyltori can be wrapped around each other which their inner faces enclosing a duocylinder. We can then put a squaretiger (a tiger made form a square instead of a disk) around them and the composite object will be a duocylinder with a duocylinder-shaped cavity inside.

Yet another one: A cyltorus is a torus with a cylindrical cross section, where the surface normal of the cylinder's base face is pointing towards the center of the torus. It has two edges - an inner one and an outer one. Both are flat tori with one common radius and one radius deviating from one another in size. One of those tori may be a Clifford torus with identical radii. There are three faces - one surrounded by the inner flat torus, one surrounded by the outer one, both identical to one of the faces of the duocylinder and another face lying in between the two flat tori, something like a Clifford torus with a thickness in the plane of one of its loops. Two cyltori can be wrapped around each other which their inner faces enclosing a duocylinder. We can then put a squaretiger (a tiger made form a square instead of a disk) around them and the composite object will be a duocylinder with a duocylinder-shaped cavity inside.

What is deep in our world is superficial in higher dimensions.

- Teragon
- Trionian
**Posts:**136**Joined:**Wed Jul 29, 2015 1:12 pm

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