Visualizing 3-hyperplanes

[Klitzing]

Hmmm, that mail of yours, Philip, bewilders me, cause first you show that |O always commutes into O| and likewise >O into O>, but then you procede giving examples where it is allowed and where not.

Either you've got some before set choice how to make one shape becoming an other one, and thus now have fix assignments of symbols to shapes. But setting up a strict definition how operations should apply thereafter only. You then uses that definition to show the commutativity to hold, but then recognizes that some of your shapes won't allow that. - This one smells like that you have applied your operations somewhere not according to your now chosen definition!

Or I misunderstood you in having provided that | and O always commute, resp. that > and O always commute. - In that case you should be more precise. Not providing just examples where it applies and where not, rather you should get down to general preconditions instead.

--- rk

[quickfur]

I acknowledge that there are other ways to spin things. Diagonally, what have you. But, those methods and motions are currently outside of this notation's ability, for now. Perhaps I will incorporate it at some point. They are all still real and valid ways to spin something. But, rotating something by 45 degrees cannot be described in the current notation.
[...]

Actually, there's nothing about rotating 45° in what I pointed out, I merely mentioned 45° as a way of showing congruence between, say, a square and a diamond. But 45° rotation is not necessary at all. Consider this: start with a line segment, and taper it on both sides (i.e., make a bipyramid out of it). What do you get? A diamond. Extrude it, and what do you get? A square. Disregarding the fact that they are congruent to each other under a 45° rotation, these two shapes will behave differently under subsequent operations; for example, spinning around the vertical axis turns the diamond into a bicone (two cones joined end-to-end), but turns the square into a cylinder (its dual!).

Tapering the square on both sides yields an octahedron, but tapering the diamond on both sides also yields an octahedron -- in a different orientation. These must be regarded as distinct shapes, since they will behave differently under subsequent operations, even though they are congruent, geometrically-speaking.

Note that I did not use any rotation to produce these different series of sometimes-congruent shapes; all I did was to begin from a line segment and to taper it on both sides or extrude it, which is included in your notation, if I read it correctly.

[ac2000]

This diagram represents two intersecting hyperplanes. The green and red parallelopipeds represent sections of two hyperplanes in some orientation. It just so happens that they intersect at 90°, but the additional degree of freedom in 4D means that even then, they still can vary in lateral orientation. The yellow square represents a square section of their 2D intersection (note that hyperplanes in 4D, in general, will intersect in a 2D plane).

Hopefully, the analogy with the previous diagram of intersecting planes in 3D is clear.

Thank you, quickfur, that you're so patient with me and provide this additional example and illustration to make things clear.
I now have understood that the other example with intersecting planes (the purple & brown one) was to show the fallibility of one's tendency to interpret everything in 3D. That makes perfect sense to me now in theory.
However, when I look at this illustration of intersecting hyperplanes, I just can't help it: I still see them as two intersecting 3D parallelopipeds, although I know that's the wrong way to look at them, because they're meant to be hyperplanes. Obviously I look at them in a different way then you do. I'm pretty sure that you rather see their hyperplane nature.
Where do you look at exactly, when you want to see these as 4D hyperplanes?
I mean, do you focus your attention rather to the outline/perimeter and try to ignore the shading of the sides, so that the brain doesn't interpret them in a 3D way? Or do you focus your attention more on the yellow intersection?

[ac2000]

And perhaps now, we're prepared to understand how things can rotate "around" a plane in 4D. Take the middle cube, for example. Imagine a horizontal plane that bisects it. This particular animation is a bit tricky to work with because I didn't put in any frame of reference in the images, but hopefully it's clear that the cross-section of this horizontal plane with the cube is actually a static square of constant area. Well, except for the one frame where the cube is inverting itself inside-out -- but actually, even in that frame, the actual intersection is still the same square, because the corners of the cube at that point are actually far out along the W axis (the frame where the cube appears as a flat square is when it is exactly 90° to our 3D space). This horizontal plane, therefore, represents the stationary plane of the rotation: points that lie on this plane don't move as the cube rotates. Furthermore, notice how the top/bottom faces of the cube expand and contract as it rotates: the expansion simply means it's moving towards the 4D viewpoint, and the contraction means it's moving away from the 4D viewpoint (remember, this is perspective projection: nearer things appear bigger, and farther things appear smaller). When the top face crosses the horizontal plane in the projection image (but not in 4D space!), it is nearest to the 4D viewpoint, that is, it is displaced by -k units away from the center of the cube, and appears at its largest. Then as it shrinks again, it crosses the horizontal a second time, this time in the smallest appearance -- now it's +k units away from the center of the cube. If you thus trace its motion, you'll see that it's actually moving around the horizontal plane! Remember that the apparent crossing of the horizontal plane does not actually intersect it, because the horizontal plane lies in the 3D space where the stationary center of the cube is, that is, 0 units along the 4th axis, whereas at the apparent crossings, the cube's square face is displaced ±k units along the 4th axis.

Thanks again, quickfur, for spending so much effort and time to exemplify everything so thoroughly. I wish I could say that I'd "grokked" how these projection images work, but I haven't yet.
The first part of your explanations are quite clear I hope, but I'm still struggling with the above paragraph.

Did I understand this right that the assumed horizontal "bisecting" plane is identical with one plane of our 3D space (e.g. the xz plane, when y runs upwards)?
And then the cube rotates around the xz-plane in such a way that it protrudes into ana/kata space alternatingly?

I don't understand, where I, as the observer, would be located, when watching this animation. I guess, I would see it somewhere from a 4D viewpoint, but where? Which coordinates could that be?

Where do you look at, when watching this animation? Do you look at the whole outline or follow one of the the vertices/planes/edges with your eyes as they're moving? Or is it best to focus on the assumed horizontal "bisecting" plane?

I tried to sum up the differences between the rotational behaviours to understenad them better, but I'm not sure if they're correct:

in 2D:
it's possible to rotate around a 0D-point
it's possible to rotate in a 2D-plane

in 3D:
it's possible to rotate around a 1D-axis
it's possible to rotate in a 2D-plane

in 4D:
it's possible to rotate around a 2D-plane
it's possible to rotate in a hyperplane (which would be similar to rotating somehow in a 3D solid??)

[ac2000]

You're welcome! I felt the same way, when I saw the notation. It seemed impossibly complicated. Then, I don't know, it sort of "clicked" in my head, and I went crazy with it! I started with linear ops, since it was probably the easiest way to generalize into 4D. Then, I took a 5 year hiatus away from this forum, for no particular reason. During that time, I began to create a notation around linear ops, to better understand it. Then, finding more and more ways to create the same shape, I kept adding to the notation, refining it, and I came back to this forum. I think I wanted to finally understand the tiger! And, I did. In the process, I adapted a new kind of operation, the cartesian product, to the existing system. That's why my notation looks horrendously complex. It combines four notations, that I use in four algorithms, to create and define shapes.
[...]

I have spent better part of 6 years developing an algorithm to connect shapes in this manner. I call it the " Perspective Product", and it works on every shape of every dimension.

That sounds pretty cool and I think these 6 years have been well spent. Thank you for sharing all of this here. Although, I have to admit, I only understand part of your achievement, but that's not because you didn't explain it well, it's just because I'm a bit too stupid for things like this.

Just to go beyond 4 and 5D, I want to show you a frightening 6D shape, as created and expressed in my notation. I understand that this is not very elementary! But, I want to show you that 6D isn't that impossible to see. Since you picked up the notation better than you expected, ac2000, let's push the limits a little bit.

I really appreciate your confidence in my mental abilities, but that's all too much for my little brain right now. Maybe in a few years or so I will have developed a little more geometrically inclined brain tissue, so I can follow all this a bit better ...

Here is the Conindric Trianglinder: it is a cartesian product of two 3D shapes. These will multiply together, and make a 6D shape. You can think of it as an entire cone branching off of every point, on and within a triangle prism. This is pretty much what cartesian products do. At each corner of this 6D triangle-prism structure, there is an entire cone. Along each edge, there is a cone-prism ( coninder ). Spread out along each 2-D panel, is a cone-diprismic space, if we remove the edges and corners. Connecting all of the curved surfaces together will create a single torus, the rolling surface of this crazy thing.

To be honest, I can't really imagine this strange thing. From your description it sounds a bit like a cactus that somehow got out of control . But I'm sure it's an object of intricate beauty, if one knows how to visualize it adequately. The name itself "Conindric Trianglinder" sounds pretty cool too. Did you name it that way on the basis of the parts it is build from? Because it's the only "Conindric Trianglinder" that is on google. Does that mean, this object didn't exist before and you discovered it, or is it a known object, and other people name it differently?

[quickfur]

This diagram represents two intersecting hyperplanes. The green and red parallelopipeds represent sections of two hyperplanes in some orientation. It just so happens that they intersect at 90°, but the additional degree of freedom in 4D means that even then, they still can vary in lateral orientation. The yellow square represents a square section of their 2D intersection (note that hyperplanes in 4D, in general, will intersect in a 2D plane).

Hopefully, the analogy with the previous diagram of intersecting planes in 3D is clear.

Thank you, quickfur, that you're so patient with me and provide this additional example and illustration to make things clear.
I now have understood that the other example with intersecting planes (the purple & brown one) was to show the fallibility of one's tendency to interpret everything in 3D. That makes perfect sense to me now in theory.
However, when I look at this illustration of intersecting hyperplanes, I just can't help it: I still see them as two intersecting 3D parallelopipeds, although I know that's the wrong way to look at them, because they're meant to be hyperplanes. Obviously I look at them in a different way then you do. I'm pretty sure that you rather see their hyperplane nature.
Where do you look at exactly, when you want to see these as 4D hyperplanes?
I mean, do you focus your attention rather to the outline/perimeter and try to ignore the shading of the sides, so that the brain doesn't interpret them in a 3D way? Or do you focus your attention more on the yellow intersection?

In interpreting a 3D diagram that's meant to convey a 4D scene, first you have to interpret it in 3D. The image on the computer screen is only 2D, you see, so if you ignore the 3D aspect of the model, you won't be able to get to 4D. So first, you have to see it as two parallelopipeds intersecting in 3D.

But don't stop there; once you have a good mental picture of what the diagram looks like in 3D, you must then layer an additional interpretation on top of it. So now you reinterpret the parallelopipeds as cubes being seen from an angle in 4D -- just as a square, when seen from an angle in 3D, may appear as various parallelograms, so cubes in 4D may appear as various parallelopipeds, depending on your viewpoint. The direction of the distortion gives you some idea as to how the cube is oriented in 4D space, just like when you see a parallelogram in the 3D case, you unconsciously interpret that as a square being seen at an angle. This part is probably the hardest step, since once you get this, everything else will fall into place. The point is that since our brain isn't wired to interpret 4D automatically, we have to make conscious effort to reinterpret the image in a 4D way -- staring at the image is not going to make 4D suddenly pop into your head. So the key is really to look at one of the parallelopipeds and interpret it as a cube in 4D, seen from some oblique angle.

Once you get that, then you can extrapolate the hyperplane from that cube -- just as the diagram of the 3D case doesn't actually show you any 2D planes (that's impossible since 2D planes are infinitely wide and long!), so here we aren't actually showing you an entire 3D hyperplane. We're showing you a cube-shaped cutout of it. And just as the square-shaped cutout in the 3D case serves as a good basis for us to extrapolate the rest of the plane, so here the cube gives us an idea of the orientation and slanting angles of the hyperplane, and so we may extrapolate the rest of the hyperplane by imagining similarly-oriented cubes tiled next to each other in 3 directions, to span the entirety of the hyperplane thus represented.

Of course, it's useful to still just regard the cube as being representative of the entire hyperplane, since if you fill it out, it would just fill up the entire volume of the screen (or 3D space of the diagram), which kinda makes it hard to understand what's going on. So now we're really looking at two cubes in 4D intersecting each other. From the different way in which they're slanted, we can extrapolate a particular orientation of the hyperplanes they represent. Once we understand that, then it's easy to see that these are two intersecting hyperplanes (represented as their respective representative cubes intersecting).

The yellow square, then, represents the intersection of these two representative cubes, which means that, suitably extrapolated, it represents the 2D plane that forms the intersection between two 3D hyperplanes.

Now, I admit that this diagram isn't particularly good, because I just threw it together to give an idea of the kind of 3D diagrams I have in mind, and I didn't actually sit down to think it through as to which hyperplanes exactly it represents. A more helpful diagram should probably include thin lines to indicate where the coordinate axes are, so that you have some frame of reference to compare each hyperplane's orientation with, and so get a better idea of what's actually going on. This diagram also uses oblique projection (well, it's supposed to represent an oblique projection, but I didn't actually calculate it from an actual 4D model, I just threw in a bunch of shears into the 3D matrix, so if it turns out to be inconsistent, it's my bad). Arguably, perspective projection would be better, because it would let you see which part of the hyperplane (its representative cube) is farther away from the 4D viewpoint, based on the relative sizes of the representative cube's faces, which would make it easier to tell what's the orientation. This particular diagram isn't actually very good at conveying orientation because of the oblique projection -- it's just the easiest thing I could make in povray, so again, my bad.

One of these days, when I have a bit more free time, I should do an image with actual 4D->3D projections, so that the diagrams will actually be more realistic than this. In any case, that doesn't change the basic principles behind interpreting such diagrams -- first you have to get the 3D-ness of it, and then use that 3D mental model as the stepping stone into 4D. Trying to short-cut the 3D step will 99% of the time only lead to confusion.

[quickfur]

And perhaps now, we're prepared to understand how things can rotate "around" a plane in 4D. Take the middle cube, for example. Imagine a horizontal plane that bisects it. This particular animation is a bit tricky to work with because I didn't put in any frame of reference in the images, but hopefully it's clear that the cross-section of this horizontal plane with the cube is actually a static square of constant area. Well, except for the one frame where the cube is inverting itself inside-out -- but actually, even in that frame, the actual intersection is still the same square, because the corners of the cube at that point are actually far out along the W axis (the frame where the cube appears as a flat square is when it is exactly 90° to our 3D space). This horizontal plane, therefore, represents the stationary plane of the rotation: points that lie on this plane don't move as the cube rotates. Furthermore, notice how the top/bottom faces of the cube expand and contract as it rotates: the expansion simply means it's moving towards the 4D viewpoint, and the contraction means it's moving away from the 4D viewpoint (remember, this is perspective projection: nearer things appear bigger, and farther things appear smaller). When the top face crosses the horizontal plane in the projection image (but not in 4D space!), it is nearest to the 4D viewpoint, that is, it is displaced by -k units away from the center of the cube, and appears at its largest. Then as it shrinks again, it crosses the horizontal a second time, this time in the smallest appearance -- now it's +k units away from the center of the cube. If you thus trace its motion, you'll see that it's actually moving around the horizontal plane! Remember that the apparent crossing of the horizontal plane does not actually intersect it, because the horizontal plane lies in the 3D space where the stationary center of the cube is, that is, 0 units along the 4th axis, whereas at the apparent crossings, the cube's square face is displaced ±k units along the 4th axis.

Thanks again, quickfur, for spending so much effort and time to exemplify everything so thoroughly. I wish I could say that I'd "grokked" how these projection images work, but I haven't yet.
The first part of your explanations are quite clear I hope, but I'm still struggling with the above paragraph.

Did I understand this right that the assumed horizontal "bisecting" plane is identical with one plane of our 3D space (e.g. the xz plane, when y runs upwards)?
And then the cube rotates around the xz-plane in such a way that it protrudes into ana/kata space alternatingly?

Yes! That's exactly it.

I don't understand, where I, as the observer, would be located, when watching this animation. I guess, I would see it somewhere from a 4D viewpoint, but where? Which coordinates could that be?

It would just be some distance away in the ana direction. So when the square face of the cube is protruding in the ana direction, it would appear bigger, and when it protrudes in the kata direction, it would appear smaller. In terms of coordinates, if we say the first coordinate is ana/kata, it would be something like (5,0,0,0), where the 5 can really be any arbitrary number depending on the relative size of the cube. (In the models I used to create this animation, the cube has edge length 2, with coordinates (0,±1,±1,±1), and the 4D viewpoint is actually at (5,0,0,0)). Note that if you weren't displaced in the ana direction, you'd be standing in the center of the cube at (0,0,0). In other words, there is zero 3D displacement, only 4D displacement.

And, of course, the cube itself is located at (0,0,0,0), and the 4D viewer is, obviously, pointing its eyes in the (-1,0,0,0) direction.

Where do you look at, when watching this animation? Do you look at the whole outline or follow one of the the vertices/planes/edges with your eyes as they're moving? Or is it best to focus on the assumed horizontal "bisecting" plane?

The best place to focus is the center of the cube. For example, look at this square:



Where do your eyes instinctively focus on? Probably the blank area in the middle of the square, even though there's "nothing" there. A 2Der, OTOH, would spontaneously focus their attention to the edges of the square, because that's what's immediately visible to them -- they can't see "inside" a square. Unless the edges were made transparent, of course.

Which is the analogous situation here: I deliberately made the faces of the rotating cube transparent so that you can see through it, and most importantly, look at the cubical volume inside it -- because that's where a 4Der's attention would spontaneously focus. The faces, edges, and vertices are really just peripheral landmarks to help you to perceive the shape of the piece of 3D space enclosed inside the cube. It's important not to get overly distracted about what's going on with the faces and edges, but to see the volume inside the cube and how it (apparently) changes over time. Once you see this cube-shaped volume, then seeing the 4D rotation becomes easier: just as the 3D rotation causes a square to appear to "squish" into trapezoids, so the 4D rotation here is causing the cube-shaped volume to appear to "squish" into various "squished" shapes, but actually, the volume doesn't change at all throughout the rotation; it's just that you're seeing it from a 4D angle, so the squishing effect is actually perspective distortion that tells you how the volume is oriented relative to your 4D viewpoint: the sharper the angle, the more "squished" it appears.

If you can somehow internalize this, then that's when you begin to really see 4D.

I tried to sum up the differences between the rotational behaviours to understenad them better, but I'm not sure if they're correct:

in 2D:
it's possible to rotate around a 0D-point
it's possible to rotate in a 2D-plane

in 3D:
it's possible to rotate around a 1D-axis
it's possible to rotate in a 2D-plane

in 4D:
it's possible to rotate around a 2D-plane
it's possible to rotate in a hyperplane (which would be similar to rotating somehow in a 3D solid??)

No, in 4D, it's possible to rotate in a 2D-plane. It's still only a 2D plane. Remember what I said about rotation being an inherently 2D phenomenon? That holds in every dimension, which is why that approach to rotation is so much easier to understand. Of course, in 4D (and above) you also have the monkey wrench of Clifford-style double rotations, but even those beasts ultimately decompose into two plane rotations, so the key is really to first grasp plane rotations, which always rotate in a 2D plane, and once you've mastered that you can put them together to make more complicated things.

[ICN5D]

Whew, man! This defining of the spin is an ugly process! I'm feeling a little worn out, now. You must understand, my brain has cycles of interest that wax and wane, like the moon. I've spent a lot of energy lately, trying to solidly define it for you guys.

But, that's a good thing. I'm glad you guys are challenging me. Without skepticism, there would be no progress.

I think the problem may be a language barrier. I have no formal higher math education, and you guys probably do. I don't really have a way to express the different spins in a current, widely taught mathematical context. I self-taught all I know, making my perspective on the descriptions a little biased, and potentially missing a crucial element.

But, I still believe that these three:

*** If the spin follows an extrude, |O , we treat this spin as the " Prismic Spin", where the last constructed axis is in motion.

*** If the spin follows a taper, >O , we treat this spin as the " Triangular Spin", where the second-last constructed axis is in motion.

*** If the spin is paired with sequence of spins, OOOO we treat this spin as the "Spherical Spin", where the moving axis is ambiguous. However, this does not hold up when we add a spin op to the cylinder form of ||O. If there is another extrude ( other than the "starting line"), the rule changes into a special case.

.... should be the ones we are focusing on.


- The other two spins ( toric and trianglindric ) are special cases, and modify the rule because they are uniquely different.

- In the methods of the two special cases, the single moving axis is somewhere it normally isn't, according to the above three. But, just like quickfur is pointing out, there are many ways to spin a shape into N+1. These two special cases are elaborations and descriptions of some of those alternative ways.



-The spin operator that I use is rotating around a bisecting n-plane.

-This n-plane has n-1 dimensions to the shape being rotated.

-The surface elements that lie on the axes of this bisecting n-plane remain stationary during the spin, and rotate in place. We then add the spin operator to the end of the construction sequence of the shapes present.

- This leaves only one axis left over, which is the rotating axis ( moving axis ).

- The surface elements on this rotating axis trace out the path of a circle, into N+1, and therefore become joined into a torus. This will reduce the number of elements, on this axis, from 2 to 1, by joining them together into the new rolling surface. We then add the torus operator (O) to the end of the construction sequence of the single shape present.


Richard, as for the break in the commutative rule, I also am not sure how to explain it, other than a formality in the patterns. That doesn't help, but it opens up the topic for discussion:

* I feel it may be related to how the cyltrianglinder |>|O has a single spin at the end of a non-commuting tapertope, the triangle prism |>|. But, that doesn't hold up with the cylindrone ||>O, where the spin commutes to all three. It's the only one that does that in 4-D.

* Or, maybe it's related to how the cartesian product with a circle can be interchanged with linear sequence operators. In my notation, the brackets [N] mean cartesian product with shape N. The cyltrianglinder can be |>|O and |>[|O] and |O[|>]. But not |O|>, which would be the cylindrone, interestingly enough. Actually, in using the spin op, the cyltrianglinder is the only non-commuting, and the cylindrone is the most commuting. Both are related by the interchanging of the operator sequence, in a cartesian product context. Like I said, perhaps a fluke in the patterns!



ac2000, you're not stupid, just being exposed to fresh things. All of these crazy symbols, motions, hyperplanes, and such are Mandarin Chinese, but you're starting to get it. It's been crazy off and on with me, with surges of interest, then loss of interest. But, now is the most I've ever been on here. I'm just now starting to really explain my thought process to everyone. It's just as foreign to me to try to teach it

-- Philip

[ICN5D]

To be honest, I can't really imagine this strange thing. From your description it sounds a bit like a cactus that somehow got out of control . But I'm sure it's an object of intricate beauty, if one knows how to visualize it adequately. The name itself "Conindric Trianglinder" sounds pretty cool too. Did you name it that way on the basis of the parts it is build from? Because it's the only "Conindric Trianglinder" that is on google. Does that mean, this object didn't exist before and you discovered it, or is it a known object, and other people name it differently?

Well, it sort of IS a cactus that got out of control! And, it does have an intricate beauty. I can't say that I really discovered it, but maybe illuminated it from the dark depths of the 6th dimension. It has always existed mathematically, I am probably the first to detail it. I'm probably the first to try naming it, too. I derived it from the standard names on the HDDB Wiki, and combined them together. The name Conindric Trianglinder is really describing the cone-prism |O>| times the triangle |>, as in cartesian products. Another way to represent it is by |O>|[|>]. It's a cone-prism sticking off of every point in a triangle. Or, a triangle sticking off of every point in a cone-prism. Just like saying that a 3-D cylinder |O|, or can of beans, is a circle sticking off of every point along a line. The triangle, in this case, is a 2-D plane cut into a triangle, that has an entire 4D cone-prism sticking off of it in the same manner. Just one of the ridiculously high number of basic 6D shapes.

[Klitzing]

But, I still believe that these three:

*** If the spin follows an extrude, |O , we treat this spin as the " Prismic Spin", where the last constructed axis is in motion.

Yes, that one is an unambiguous, concise definition. (Whether that is what was intended or not, then would be a totally different question.)

Thus you'll have any kind symbol |...|O to be nothing but |... x |O, the cartesian product of that former part shape |... and the disk |O. E.g. that in your post later on discussed exceptional shape |>|O truely becomes the cyltrianglinder: at the one hand you'd extrude the triangle |> say eaually to either side and then spin around that former plane which then became its midplane between the bases (of that triangular prism). Then the triangle becomes kind a torus with triangular cross-section (instead of the minor radius), whereas the former 3 lacing squares would all spin into cylinders. Thus the sides of the cross-secting triangles of the torus then become the lacing lines at the surface of the cylinders. - On the other hand that shape truely is the cross product of a disk and a triangle, infact the limit of a 3,n-duoprism, n running to infinity, while keeping the circumradius of that n-gon fixed.

*** If the spin follows a taper, >O , we treat this spin as the " Triangular Spin", where the second-last constructed axis is in motion.

Being applied to |...|>O indeed works. And this surely is what you had in mind. But it does not apply to |...>>O, so!
Thus neglect for a moment the cases |...O>...>O, where the 2nd ellipsis uses any amount of > signs only. Thus you could restrict to |...|>...>O (2nd ellipsis again only >'s). Then your intend most probably would have been to use the mid-hyperplane of |...| between the 2 bases |... of the last extrusion. Then you'd generate the span of that hyperplane and the tips (points) of all following taperings. This then indeed generates a n-1 flat. Thus the spin becomes well-defined as being rotating around that fixed space.

Therefore the remaining case could be addressed, if we might result in the general commution of > and O. Infact, then we would have shifted the remainder to the following definition of |...O...O (2nd ellipsis using only O's).

But note, as a first step that commutation has to be checked somehow for all |...>O / |...O> pairs (where the ellipsis does not contain any further O) generally. But further it still cannot be proved so far, because any such |...O>...>O still lacks an own definition so far. (So, if you want to, you could reverse the process and define |...O>...>O just as being |...>...>OO. - Provided that that then would what you'd decide to do with all that shapes you had derived before going into that more explicite definition process...)

- Thus still open for discussion!

*** If the spin is paired with sequence of spins, OOOO we treat this spin as the "Spherical Spin", where the moving axis is ambiguous. However, this does not hold up when we add a spin op to the cylinder form of ||O. If there is another extrude ( other than the "starting line"), the rule changes into a special case.

That one I do not understand, sorry.

Actually, in using the spin op, the cyltrianglinder is [...] non-commuting

Got it. | and O not generally do commute: |>|O is the cyltrianglinder (as I outlined above). But |>O| obviously is the cone-prism, i.e. the extrusion of |>O, where that one (as being outlined in that already settled part of the |...>O definition) indeed is nothing but the requested for cone.

(That | and > do not commute aready can be seen from those symbols which do not contain any (so far ambiguous) O operand. - Thus your comparision of |>|O and |O|> was not an allowed one.)

--- rk

[ICN5D]

I should probably also note how the linear operators follow the XYZWVUTSRQP axis flow, where each operator is done into N+1 along the respective axis in the sequence. So, |>| means to | along X, > along Y, and |along Z .

But it does not apply to |...>>O, so!

Sure it does! Rotating a tetrahedron around a plane, in a certain way, can create the dicone : |>>O = |>O> = |O>>. By addressing the "second-last constructed axis" as the one in motion, which ends up being the y-axis for |>>, the two triangle-faces that lie skewered by the y-axis join into a torus, the triangle-torus surcell of a dicone. Then, the base-triangle will rotate stationary, and the x-axis triangle as well, to make two cones. Since the vertex is 0-D, its rotation will still make a vertex, giving us the cone + cone + triangle-torus surface of a dicone. But, of course, it's possible that I'm wrong about this one. This notation is in a perpetual experimental phase, where I'm always finding the nuances and patterns in it.

*** If the spin is paired with sequence of spins, OOOO we treat this spin as the "Spherical Spin", where the moving axis is ambiguous. However, this does not hold up when we add a spin op to the cylinder form of ||O. If there is another extrude ( other than the "starting line"), the rule changes into a special case.

That one I do not understand, sorry.

In the case of an n-sphere rotation around a bisecting (N-1)-plane, there are no well defined surcell pairs to place on the axes. Only orthogonal, tapertope, and cylindrical shapes have this ability. The surface of an n-sphere will evenly occupy all axes, it has circular symmetry. This means the moving axis can be any axis, the end result will always be an (N+1)-sphere. It is not necessary to identify which axis should be the one in motion, when an n-sphere rotates into N+1 around a bisecting (N-1)-plane. Since n-spheres have circular symmetry on every axis, the addition of another axis, with circular symmetry, will always make an (N+1)-sphere.

|O - circle, 2-sphere
|OO - sphere, 3-sphere
|OOO - glome, 4-sphere
|OOOO - pentasphere, 5-sphere
|OOOOO - hexasphere, 6-sphere
etc.

Then the triangle becomes kind a torus with triangular cross-section (instead of the minor radius), whereas the former 3 lacing squares would all spin into cylinders. Thus the sides of the cross-secting triangles of the torus then become the lacing lines at the surface of the cylinders.

I noticed how you said " lacing squares", between two triangles for |>|. That's a nice way to put it, I've always call them "connecting" squares.



I also want to elaborate on another effect with commuting operators. Before, I mentioned how adding the spin O to the end of ||O will make the ||OO. This sequence is also a |OO|, spherinder. Adding the spin to the end of |O| will make |O|O, duocylinder. If using XYZW as the context of orientation, the two OO paired together, after ||, are different than a |O, where |O spins like a prism. This means OO will spin according to the special case, the Toric spin, " O_m". The toric spin catagory means the moving axis lies on one of the toratope surcell main circles ( major radius ). Only shapes that have curved toratope surcells can have a toric spin option. However, some shapes can ONLY be done by the toric spin, like the duocylinder. This is because the duocylinder has no flat sides, only toratope sides.

Which brings me to my next point: The rotation of a duocylinder |O|O and a spherinder |OO| into 5D will BOTH make the cylspherinder |OO|O. This rotation is done to a 4-D shape, which means it will rotate around a bisecting 3-plane. The 3-plane has 3 axes that remain stationary, leaving one of them left over: the moving axis. Lathing a duocylinder will unavoidably place the moving axis on one of the main circles of its 2 toratope surcells. When a toratope rotates its main circle into N+1, the major radius turns into an (N+1)-sphere. The other toratope surcell then has its minor-radius cross section undergo a spin, and its main circle is unaffected.


|O| + O == |O|O, (circle,circle)-prism, duocylinder

||O + O_m == ||OO_m == |OO|, (sphere,line)-prism, spherinder


|O|O + O == |O|OO = |OO|O , (sphere,circle)-prism, cylspherinder

|OO| + O == |OO|O , (sphere,circle)-prism, cylspherinder

||OO + O == ||OOO = |OO|O , (sphere,circle)-prism, cylspherinder

But, we can also do a toric spin to the |OO|, or ||OO. Using the sequence ||OO, we have ||OOO_m, which makes the glominder |OOO|. It might sound more complicated now, with the addition of the other types of spins. But it's only because of the additional ways to spin. They must be addressed and defined separately, in order for the system to work. This is ultimately what you are after, these precise definitions The type of spin defines which is the moving axis relative to the surcells.



In the computation tables I use, the math looks like this:

shape N == [ X-axis-pair , Y-axis-pair , Z-axis-pair, W-axis-pair ] or,
shape N == [ XY-plane-torus , Z-axis-pair, W-axis-pair ] or,
shape N == [ XYZ-plane-toratope , W-axis-pair ], etc


|O|O == [ |Ozw(O)xy , |Oxy(O)zw ] : 2 perpendicular circle torii |O(O) on surface, bound orthogonally

|OO| == [ |w(OO)xyz , |OOxyz--2 ] : a line-toraspherinder |(OO), or (line,glomohedrix)-prism, the curved toratope that laces the 2 sphere |OO endcaps together

|OO|O == [ |Owv(OO)xyz , |OOxyz(O)wv ] : a torisphere bound orthogonally to a spheritorus ( toraspherinder ((III)I) bound to toracubinder ((II)II) )


In the tables, for the spin, Ox means axis x is the moving axis


|O|O ------V------> Ox == |OO|O
-----------------------------------------------------------
|Ozw(O)xy -------> Ox == [ |Ozw(OO)xyv ]

|Oxy(O)zw -------> Ox == [ |OOxyv(O)zw ]


|OO| ------V------> Ow == |OO|O
------------------------------------------------------------
|w(OO)xyz -------> Ow == [ |Owv(OO)xyz ]

|OOxyz--2 --------> (O)w == [ |OOxyz(O)wv ]




-Philip

[ICN5D]

Accordingly you'd get (with at least one ">" symbol)
2D:
"|>" = triangle = "ooo&#x" = "ox&#x" = x3o"
3D:
"||>" = squippy (Bowers acronym ["OBSA" = official Bowers style acronym] for 'square pyramid') = "oxx&#x" = "ox4oo&#x"
"|>>" = tet (OBSA for 'tetrahedron') = "oooo&#x" = "oox&#x" = "ox3oo&#x" = "x3o3o"
"|>|" = trip (OBSA for 'triangle prism') = "xxx&#x" = "ox xx&#x" (line atop square) = "xx3oo&#x" = "x x3o"
4D:
"|||>" = cubpy (OBSA for 'cube pyramid') = "oxx oxx&#x" = "oxx4ooo&#x" = "ox4oo3oo&#x"
"||>|" = squippyp (OBSA for 'square pyramid prism') = "oxx xxx&#x" = "xx ox4oo&#x" (line atop cube)
"||>>" = squippypy (OBSA for 'square pyramid pyramid') = "oox oox&#x" = "oox4ooo&#x" = "xo ox4oo&#x" (line atop perp square)
"|>||" = tisdip (OBSA for 'triangle,square-duoprism') "xxx4ooo&#x" = "ox xx4oo&#x" (square atop cube) = "xx xx3oo&#x" = "x3o x4o" = "x3o x x"
"|>|>" = trippy (OBSA for 'triangle prism pyramid') = "oxxx&#x" = "oox oxx&#x" = "oxx3ooo&#x" = "ox ox3oo&#x"
"|>>|" = tepe (OBSA for 'tetrahedron prism') = "xxxx&#x" = "xxx oox&#x" = "xx xo ox&#x" (square atop ortho square) = "xx ox3oo&#x" (line atop axial trip) = "xx3oo3oo&#x" = "x x3o3o"
"|>>>" = pen (OBSA for 'pentachoron') = "ooooo&#x" = "ooox&#x" = "oox3ooo&#x" = "ox3oo3oo&#x" = "x3o3o3o"

Wow, that's actually really cool! I just started picking it up, and I noticed that it does a lot of the same stuff as my notation. I see that the perspectives are mostly there. That's actually what my "Perspective Product" is, a computation that derives the lacing elements between two shapes. In addition, I have done them for the curved ones as well. I see that you, too have found the "square atop ortho square" relationship with the tetrahedron prism. That's really interesting right there. Perhaps I can provide another expansion to this notation that includes rotopes.

Here's a few of what I found ( some may already be in your post ):

||> - line atop triangle
|>> - line atop ortho line

|>O| - line atop cylinder
|||> - line atop tetrahedron
|>|> - line atop sq pyramid
||>| - square atop triangle prism
||>> - line atop tetrahedron
|>|O - circle atop cylinder
|>O> - line atop perp circle

|>O|O - circle atop duocylinder
|O>>> - triangle atop ortho & perp circle +3D
|>O>O - line atop perp sphere
|>>|O - circle atop cyltrianglinder, cylinder atop ortho cylinder
||>|O - circle atop tesserinder, cylinder atop cyltrianglinder
|>||O - cylinder atop cubinder
|O>[|>] - triangle atop cyltrianglinder, cone atop coninder
|>|O> - circle atop cylindrone


--Philip

[quickfur]

[...]

But it does not apply to |...>>O, so!

Sure it does! Rotating a tetrahedron around a plane, in a certain way, can create the dicone : |>>O = |>O> = |O>>. By addressing the "second-last constructed axis" as the one in motion, which ends up being the y-axis for |>>, the two triangle-faces that lie skewered by the y-axis join into a torus, the triangle-torus surcell of a dicone. Then, the base-triangle will rotate stationary, and the x-axis triangle as well, to make two cones. Since the vertex is 0-D, its rotation will still make a vertex, giving us the cone + cone + triangle-torus surface of a dicone. But, of course, it's possible that I'm wrong about this one. This notation is in a perpetual experimental phase, where I'm always finding the nuances and patterns in it.

I can vouch for the tetrahedron producing a dicone under a particular rotation, namely, in the plane spanned by one of its edges and a vector perpendicular to the hyperplane the tetrahedron lies in. This rotates the edge into a circle (disk), while keeping the opposite edge stationary, thus it traces out a shape with two cones joined at their bases at an angle, with their tips connected by the stationary edge, i.e., a dicone.

This particular construction requires a very specific rotation though, and I think that's the issue here. How do you define the O operator such that it always selects the "most ideal" rotation among the many possibilities? It is very non-obvious to me, for example, why that particular rotation would be chosen for the tetrahedron, as opposed to many other possibilities that are arguably more "intuitive" or more "symmetrical". I mean, if I didn't know beforehand that I'm supposed to get a dicone out of a tetrahedron, I wouldn't know to pick that particular rotation out of the other more obvious ones that would come to mind first. If you give me some arbitrary shape X to rotate, what is the procedure by which I can decide what the rotation plane should be, without consulting a list of explicit selections? Either such a procedure exists, and consistently selects a single rotation out of many, or we should admit the fact that multiple possible rotations exist, and focus our attention on how to go about specifying the particular rotation we have in mind.

[ICN5D]

This particular construction requires a very specific rotation though, and I think that's the issue here. How do you define the O operator such that it always selects the "most ideal" rotation among the many possibilities? It is very non-obvious to me, for example, why that particular rotation would be chosen for the tetrahedron, as opposed to many other possibilities that are arguably more "intuitive" or more "symmetrical". I mean, if I didn't know beforehand that I'm supposed to get a dicone out of a tetrahedron, I wouldn't know to pick that particular rotation out of the other more obvious ones that would come to mind first. If you give me some arbitrary shape X to rotate, what is the procedure by which I can decide what the rotation plane should be, without consulting a list of explicit selections? Either such a procedure exists, and consistently selects a single rotation out of many, or we should admit the fact that multiple possible rotations exist, and focus our attention on how to go about specifying the particular rotation we have in mind.

This is exactly my dilemma! You nailed it right on the head. Perhaps there is no singular way, and we really do have to detail them all. That sounds more daunting than I want to get into, honestly . However, can we agree on the Prismic and Spherical cases as being blindly followable?

Another way to select out of all possibilities could be to abide by the commutative rule, where accordingly, we can't get anything BUT a dicone out of a tetrahedron, by establishing how |>>O = |>O> = |O>> . By doing so, we can narrow down the spin type. Hopefully. What do you think about that? I had a tough enough time defining it myself over the years, so it only stands to reason that it's a legitimately difficult process. I think that's what I'm really aiming for, preserving the commutative property. I like how it works and I try to find the specific spin that does this.

I apologize in advance if I accidentally hijacked the thread First, we were talking about visualizing. Then it got into my notation. Now, we're talking about the spin operator. But, maybe you feel it's worth being here, I don't know.

[Klitzing]

But it does not apply to |...>>O, so!

Sure it does! Rotating a tetrahedron around a plane, in a certain way, can create the dicone : |>>O = |>O> = |O>>. By addressing the "second-last constructed axis" as the one in motion, ...

Let's see:

• |O>> is immediate. |O is the disc, |O> then is the cone, and |O>> the dicone
  (btw, one should be careful with this name: it is not a disc with a tapering at both opposite sides, rather, as your symbol shows, a tapering into 2 perpendicular directions. I.e. that dicone is not an element of 3D, rather of 4D.)
• |>O> too is immediate according to your "second-last" rule: |> is the triangle, then |>O will be the cone, because the triangle spins around the span out of the midhyperplane of | (second-last, thus a point) plus the just added tip, i.e. around the midline of the triangle. Finally |>O> then becomes, by tapering, the dicone again.
• But if you'd apply your "second-last" rule to |>>O, you would have to apply it around the span out of the line defined by the tapering of the first added operation, i.e. the triangles diagonal, and the tip added by the second tapering. Thus this 2D fixed plane (of the requested 4D spin) would be defined by the midpoint of | plus the 2 tips. So again the 2 tips stay fixed, but that | becomes a disc. In fact, you are right, that is the 4D dicone again.

Thus, my doubting with respect to |...>>O has been come around - at least for the case where that ellipsis '...' is the empty symbol sequence.

Okay, at least this second-last rule then selects a definite hyperplane to spin around. So we'll settle this such.
And we'll get therefrom by definition that

• |...>>O spins around the span of the span of the midhyperplane A of |... (however that one would look like), and both tips of either tapering.
• |...>O> spins around the span of the span of the same midhyperplane A of |... and the tip of the first tapering. Then that spun figure becomes tapered again. - Looks like those would come out the same, then.
• |...O>> first spins around the same midhyperplane A of |... . Then that spun figure becomes tapered twice. - Thus it lookes like that too becomes the same.

So in conclusion we get that symbols > and O can be indeed commuted freely, both in cases |...|>O and |...>>O.
Remains just the investigation of |...O>>O .

--- rk

[Klitzing]

*** If the spin is paired with sequence of spins, OOOO we treat this spin as the "Spherical Spin", where the moving axis is ambiguous. However, this does not hold up when we add a spin op to the cylinder form of ||O. If there is another extrude ( other than the "starting line"), the rule changes into a special case.

That one I do not understand, sorry.

In the case of an n-sphere rotation around a bisecting (N-1)-plane, there are no well defined surcell pairs to place on the axes. Only orthogonal, tapertope, and cylindrical shapes have this ability. The surface of an n-sphere will evenly occupy all axes, it has circular symmetry. This means the moving axis can be any axis, the end result will always be an (N+1)-sphere. It is not necessary to identify which axis should be the one in motion, when an n-sphere rotates into N+1 around a bisecting (N-1)-plane. Since n-spheres have circular symmetry on every axis, the addition of another axis, with circular symmetry, will always make an (N+1)-sphere.

|O - circle, 2-sphere
|OO - sphere, 3-sphere
|OOO - glome, 4-sphere
|OOOO - pentasphere, 5-sphere
|OOOOO - hexasphere, 6-sphere
etc.

Well, that is the easier part. That one is quite clear, for sure.
But what I asked to get a definition for, was rather the case |...OO (where the ellipsis could be anything).
At least I could not map your earlier comment to that general case.

--- rk

[ICN5D]

Remains just the investigation of |...O>>O .

This one, my friend, was the MOST difficult for me to define! I battled for better part of 5 years, going back and forth with it. However, upon categorizing the spins into Prismic, Triangular, Spherical, I had formal definitions to apply. Once again, in wishing to maintain the commutative property, I finally, and only in the last year, satisfied it.

The formal syntax can be written |O_a>_b , which is an n-sphone. N-sphone means "any number of spin + taper operators" after the axiom line. Because the cone contains an n-sphere, along with tapertope symmetry, it can have a triangular spin applied. The end result is always a sphone: |>OO = |O>O = |OO> . According to the "second-last" application in the triangular spin, the N-sphere base has the moving axis. In the same case as a spherical spin, the base becomes an (N+1)-sphere. Note how the cone can also have the Toric spin O_(m), which ends up making the same thing!

But what I asked to get a definition for, was rather the case |...OO (where the ellipsis could be anything).

Okay, I see what you mean. I answer this on the footnote ## towards the bottom. I need to clarify some things a little better, first. I think a clearer way to put it may be ||_aO_b>_c. This is because of the first | is like an axiom, where all shapes really start with a 0-D point, in the linear construction format. Omission of this first | can change the outcome. Spinning a 0-D point will create nothing new, so it's sort of like multiplying by zero. Same with tapering a 0-D point, there is no hypervolume to shrink while moving into N+1, and so becomes a line.

* == 0-D point
*O == *
*> == |
*| == |
*>> == |> , which resembles Wendy's notation quite a bit
*(O) == (O) , hollow circle, or point-torus, point extruded along path of circle

Using the mnemonic " STEMP " that stands for Spin, Taper, Extrude, Manifold, Product, we have:
---------------------------------------------------------------------------------------------------------------------

• mO - rotate shape M around N-1 plane into N+1
• m> -taper shape M into N+1
• m| - extrude shape M into N+1
• m(q_a) - extrude shape M along surface of shape Q into N+a
• m[q] - cartesian product with shape M and Q, the (m,q)-prism

* Where "a" is number of operators in parentheses



General Shape Families:
-----------------------------

• |_a = N-Cube
• |>_a = N-Simplex
• |O_a = N-Sphere
• |O_a>_b = N-Sphone
• ||_a>_b = N-Pyramid
• |>_a|_b = N-Pyramid
• |>_a|_b>_c = N-Pyramid
• ||_aO_b = N-Cylinder
• ||_aO_b>_c = N-Cylindrone ( the Polycylindramids, as I've previously called them)
• |...[|O] = N-Cylinder
• |...[|>] = N-Trianglinder

* Where "a,b,c" is ≥ 1



Within the spin O, are 5 sub-types, which determine the 1 moving axis during rotation.
If we allow " α " to stand for an axis, using the format example xyzwO, α is axis V, α-1 is axis W, α-2 is axis Z, α-3 is axis Y, etc...
-----------------------------------------------------------------------------------------------------------------------------------------------

• O_| - Prismic Spin : α-1 in motion

• O_> - Triangular Spin : α-2 in motion

• O_s - Spherical Spin : any axis can be in motion

• O_(m) - Toric Spin : axis in motion is a on toratope surcell's main circle

• O_[p] - Product Spin : α-2 in motion, spin like triangle

## In the case of |...OO, the spin is part of the last constructed axis, if using XYZW as a reference frame of orientation. A spin adds an axis with circular symmetry. Therefore, the Prismic Spin, where α-1 is in motion, will have the same effect as the Toric Spin. This will place the moving axis on the main circle of the shape's toratope surcell. This makes the flat endcaps rotate stationary, turning ||_aO_bO_(m) into |O_(b+1)|_a . Also, ||O_aO == |O_a|O , where a ≥ 1 .

(btw, one should be careful with this name: it is not a disc with a tapering at both opposite sides, rather, as your symbol shows, a tapering into 2 perpendicular directions. I.e. that dicone is not an element of 3D, rather of 4D.)

You are correct. Only one letter sets them apart: Bicone vs Dicone. If we allow the operator " U " to be the "flip", then |>U will make the bicone. But, I haven't really developed on the Flip operator. The bicone seems like it can also be described as a triangle torus attached at one side, |>U = |>(O) =/= |>O>


-- Philip

[ICN5D]

I want to elaborate on the toratope notation as well:

I use this starting 0-D point to describe toratopes, where " (O) " means hollow circle, or " point-torus", or to "extrude a point along the path of a circle". The whole symbol, parentheses and spin, represent one transforming motion into N+1. A hollow circle is truly 1-dimensional, and is perfectly symbolized by the single (O) symbol. The formal definition, N(M), means "extrude shape N along the path of a line that has been modified by an M operator". This is a verbose way of stating that the parentheses represent a defaulting extrusion, that was modified by some operator, M. The symbols in the parentheses are from a shape ( most likely an n-sphere ), where the first extrusion is removed:

" (O) " is the glomolatrix, hollow form of |O
" (OO) " is the glomohedrix, hollow form of |OO
" (OOO) " is the glomochorix, hollow form of |OOO
" (O)(O) " is the "torahedrix" ( my word for it), hollow version of |O(O), where we can then embed a circle into this 2-manifold surface to make the ditorus |O(O)(O)

This system also allows for other shapes with holes:
" (>) " is the hollow triangle, the "triangulatrix"
" (|) " is the hollow square, the " tetralatrix "
" (O>) " is the hollow 3-D cone, where we embed into the 2-manifold surface of a cone only, the "conihedrix"
" (O|) " is the hollow 3-D cylinder, the "cylindrihedrix"
etc..

|O>(O>) would make the " Duoconterix ", a 5-D shape that has a cone embedded into the 2-D surface manifold of another cone, hollow inside. It's a cone atop cylconinder, laced only by a coninder-torus. I like to call these "complex manifold toratopes", or CMT's, not to be confused with Country Music Televison


Therefore, |O(O) is the circle-torus, |OO(O) is the spheritorus ( toracubinder ), |O(OO) is the torisphere ( toraspherinder ). This method seems more intuitive to me, where the subshape ( minor radius shape ) is more clearly defined separately from the hollow manifold ( major radius shape ). This is my linear step by step way of creating every toratope on the HDDB. Marek's notation reflects the actual equation, but can be a little ambiguous with some, in my opinion. According to Marek's language, N is the meat, M is the bones.

After adapting cartesian products into the notation, I was finally able to describe the tiger as |O[(O)(O)], where the brackets [N] represent the cartesian product with shape N. So, [(O)(O)] means the cartesian product of two hollow circles, which makes the strange wireframe bone structure that the circle embedded into. Embedding a circle into the surface manifold of a hollow shape is "spheration" as Wendy calls it. [(O)(O)] is the duocylinder margin, and adding the circle |O to the beginning is "inflating" this margin to make the tiger. A great definition of "spheration".


-- Philip

[Klitzing]

Using the mnemonic " STEMP " that stands for Spin, Taper, Extrude, Manifold, Product, we have:
---------------------------------------------------------------------------------------------------------------------

• mO - rotate shape M around N-1 plane into N+1
• m> -taper shape M into N+1
• m| - extrude shape M into N+1
• m(q_a) - extrude shape M along surface of shape Q into N+a
• m[q] - cartesian product with shape M and Q, the (m,q)-prism

* Where "a" is number of operators in parentheses

Thus, besides of |,>,O you like to use further operators too. According to your above mnemonic definitions, those then would both act kind as a cartesian product (or direct sum), just as for duoprisms. |...[|...] then truely is just the duoprism of both ellipis parts. But |...(|...) reuses from the second one the hollow surface only. - For polytopes this most probably is not used too heavily, but you probably would have all sorts of tori shapes in mind here.

General Shape Families:
-----------------------------

• |_a = N-Cube
• |>_a = N-Simplex
• |O_a = N-Sphere
• |O_a>_b = N-Sphone
• ||_a>_b = N-Pyramid
• |>_a|_b = N-Pyramid
• |>_a|_b>_c = N-Pyramid
• ||_aO_b = N-Cylinder
• ||_aO_b>_c = N-Cylindrone ( the Polycylindramids, as I've previously called them)
• |...[|O] = N-Cylinder
• |...[|>] = N-Trianglinder

* Where "a,b,c" is ≥ 1

What?
"|" and ">" definitely do not commute at all. So ||_a>_b surely is not the same as |>_a|_b. And both surely would be different from |>_a|_b>_c (even if the index numbers would be set up independently so that the dimensions would align - at least for none of them becoming zero, i.e. the trivial case).

Thus by "N-Pyramid" you most probably seem to mean some largely more general class, than mere pyramids with base N. (Note that I equate here tapering and pyramidisation!) Moreover the equal sign would be wrong. (Btw. in any of those cases.) As the left is just a single representant, but the right is a class. It rather should be an element-of sign. - But still, you ought to define what you mean by N-Pyramid. As it is surely not, what I just mentioned. (And therefore a different name would be appreciated...)

--- rk

[ac2000]

No, in 4D, it's possible to rotate in a 2D-plane. It's still only a 2D plane. Remember what I said about rotation being an inherently 2D phenomenon? That holds in every dimension, which is why that approach to rotation is so much easier to understand. Of course, in 4D (and above) you also have the monkey wrench of Clifford-style double rotations, but even those beasts ultimately decompose into two plane rotations, so the key is really to first grasp plane rotations, which always rotate in a 2D plane, and once you've mastered that you can put them together to make more complicated things.

Thank you, quickfur, for patiently answering all my additional questions.
I'm sure the information will help me to make some progress in developing the necessary mindset to ultimately be able to visualize at least some basic 4D geometry.

[Klitzing]

No, in 4D, it's possible to rotate in a 2D-plane. It's still only a 2D plane. Remember what I said about rotation being an inherently 2D phenomenon? That holds in every dimension, which is why that approach to rotation is so much easier to understand. Of course, in 4D (and above) you also have the monkey wrench of Clifford-style double rotations, but even those beasts ultimately decompose into two plane rotations, so the key is really to first grasp plane rotations, which always rotate in a 2D plane, and once you've mastered that you can put them together to make more complicated things.

Thank you, quickfur, for patiently answering all my additional questions.
I'm sure the information will help me to make some progress in developing the necessary mindset to ultimately be able to visualize at least some basic 4D geometry.

Just to provide you some feeling on that mentioned Clifford stuff:

Also in 3D the mere rotation is a 2D phenomenon (as quickfur already said). Thus there you will have an 1D orthogonal space (the axis) which remains pointwise fixed. Now you can do within that orthogonal subspace something different meanwhile the rotation is running in the other subspace. Okay, in a 1D subspace we have not much choice. But still we can have some translation. That is, you could consider some new transformation, which acts like a rotation in the 2D subspace, and like a translation within the orthogonal 1D subspace. Yes, you'd know, that is an helical transformation, kind a skrewing action.

The same holds true for Clifford- or double rotations within a 4D space: Again a mere rotation just acts within a 2D subspace. The orthogonal subspace, here being 4 - 2 = 2 dimensional as well, remains pointwise fixed. So you can consider to apply to that subspace an different action meanwhile. In case that one is a rotation too, and that both orthogonal rotations just act around that specific point, which is the common intersection of those 2 orthogonal subspaces, then this total action is what Clifford once had considered.

So far that might look kind of arbitrary. But there is one specific type of polytopal figures within 4D, which just bow to that special Clifford symmetry! Consider some n-gon, centered at the origin within the x-/y-plane. Consider then some m-gon, centered at the origin within the z-/w-plane. Then build the direct sum (cartesian product) of those polygons, and you will get the n,m-duoprism, a figure being bounded by m cells, being n-gonal prisms, plus n cells, being m-gonal prisms. Btw. the tesseract here is just the special case, where n=m=4. And obviously, that 4D polytope has an isometry which applies any n-fold rotation within the one direction (2D subspace) and simultanuously any m-fold rotation within the other!

--- rk

[Klitzing]

In that mail, I elaborated on those symbols which only use the | (extrusion) and > (tapering) signs. Moreover I related those to unit edged polytopes. This was done for all possible combination up to 4D.

But it just occured to me now, that the circumradii of the unit edged D-dimensional hypercubes obviously are given by R = sqrt(D)/2. Esp. the unit tesseract has unit circumradius too, and all higher dimensional hypercubes would have a circumradius value, which is larger than one.

So, what is the issue here?

It is just that e.g. the general sequence |>, ||>, |||>, etc. very fast comes to an end!
This is because of Pythagoras: for that tapering we derive (unit lacings subsumed) H² = 1 - R² = (4 - D)/4.
In more detail:

• The height of an all unit edged |> (triangle) clearly is H = sqrt(3)/2 = 0.866.
• The height of an all unit edged ||> (square pyramid) is H = 1/sqrt(2) = 0.707.
• The height of an all unit edged |||> (cube pyramid) is H = 1/2 = 0.5.
• But already an all unit edged ||||> becomes degenerate as it would have a zero height!
• And beyond an |...|> (the ellipsis using |-signs only) would not even be possible any more - when subject to all unit edges only!

--- rk

[ICN5D]

You are correct, the so called n-pyramids are better named as the tapertopes. There would be infinite combinations as they don't commute. I wanted to throw out a few of the endless examples

As for the degenerate tesseract pyramid, that's interesting. I didn't know about that with using unit edged n-cubes. I guess, in trying to keep the system intact, and having at least some way to symbolize the ||||>, the length of tapering along 5D, the height, would be greater than the edge length of ||||. There is no way to taper a tesseract along the same length as one of its edges? Surely there must be some way! Why does it have to stop in 5D? I cant visualize that one.

[Klitzing]

It is not really the height that is restricted to unity, it is that the lacing edges are. And as soon as the circumradius of the base exceeds unity, the lacing edges can no longer connect the bases vertices to the tip. - Or taken in a different way: the larger the base, the nearer the tip will come to the base, when using a constant lacing edge lengths. - This even is the same in 2D (isoceles triangle) or 3D, quite ready for Investigation. - And when the base becomes too large, there will be no way to connect those any more.

It just happens, that the hypercube of unit edges, centered at the origin, has some vertex at (1/2, 1/2, 1/2, ...). Accordingly its circumradius R is the distance from the origin to that vertex, i.e. R² = (1/2)² + (1/2)² + (1/2)² + ... = D/4. Accordingly R = sqrt(D)/2 might get larger than unity. And this happens when D exceeds 4.

--- rk

[ICN5D]

Wow, that's a very interesting, and unanticipated phenomenon! I tried visualizing it in my head, and it seems like the unit edge tesseract pyramid ( the hemhexadecateron as I've called it ) would have the vertex at the origin, with absolute no height. There are simply too many lacings from the 16 vertices to the final top vertex for it to have height. Does this hold true for the cubindrone |||O> as well? Does the 5th dimension degenerate the height many other shapes?

This effect looks very familiar to another shape, the cylconinder |O>|O. Through linear and cartesian product means, the surface elements can be derived as a circle atop duocylinder, laced by a cylinder torus and cone torus. The cone torus is the important feature, where one of the circle-torus surcells of the duocylinder-base lace only to the 1-manifold edge of the vertex-circle. The minor radius collapses by tapering means, and the major remains the same. This means, the vertex-circle is in the same plane, XY, as the surcell, having the appearance of no height above the base. The other surcell lies in the ZW plane, and will lace according to extrusion means, only the major radius collapses. If the circle was truly above the duocyl, then both torii surcells would lace to the entire circle, making two cylinder-torii. It shows the same property as if the cylconinder has no height in addition to being 5-D. Quite something, and worth an investigation!

[ICN5D]

Thus, besides of |,>,O you like to use further operators too. According to your above mnemonic definitions, those then would both act kind as a cartesian product (or direct sum), just as for duoprisms. |...[|...] then truely is just the duoprism of both ellipis parts. But |...(|...) reuses from the second one the hollow surface only. - For polytopes this most probably is not used too heavily, but you probably would have all sorts of tori shapes in mind here.

Yes, the m(q_a) operator would be used for any rotope, to describe one or more of its curved toratope surface elements. In addition to the whole toratopes themselves. And, a whole bunch of other strange shapes with a hole, that can be described by the notation definition, such as the |O>(O>). This is very much a product, where one factor is a hollow shape.

The same holds true for Clifford- or double rotations within a 4D space: Again a mere rotation just acts within a 2D subspace. The orthogonal subspace, here being 4 - 2 = 2 dimensional as well, remains pointwise fixed. So you can consider to apply to that subspace an different action meanwhile. In case that one is a rotation too, and that both orthogonal rotations just act around that specific point, which is the common intersection of those 2 orthogonal subspaces, then this total action is what Clifford once had considered.

So, does this mean that rotating a 4D shape into 5D, around a 2-plane will make two moving axes? Leaving two stationary? I see the 3-plane rotation, and leaving one axis left over for the moving, but if a 2-plane is used, it looks like two axes are swirling around, into 5D.

--Philip

[quickfur]

You are correct, the so called n-pyramids are better named as the tapertopes. There would be infinite combinations as they don't commute. I wanted to throw out a few of the endless examples

As for the degenerate tesseract pyramid, that's interesting. I didn't know about that with using unit edged n-cubes. I guess, in trying to keep the system intact, and having at least some way to symbolize the ||||>, the length of tapering along 5D, the height, would be greater than the edge length of ||||. There is no way to taper a tesseract along the same length as one of its edges? Surely there must be some way! Why does it have to stop in 5D? I cant visualize that one.

This is a well-known fact. The circumradius of an n-cube (i.e., the radius of the n-sphere that contains its vertices) is proportional to sqrt(n). If we fix edge length at 1, then the circumradius is sqrt(n)/2. So in 4D, the circumradius is equal to the edge length, which is a happy coincidence that makes the 24-cell a regular polytope. However, this same fact also means that in 5D, you can taper a tesseract with equal edge length, but it will be completely flat! You will only get a degenerate pyramid of height 0. And since in 5D, the circumradius of the 5-cube is sqrt(5)/2 = 1.11803..., that means any pyramid of the 5-cube cannot possibly have edge length 1, since that would require its height to be the square root of a negative number.

This is just one of the counterintuitive things that happen in higher dimensions, and is related to the fact that as n grows larger, the volume of the n-cube begins to concentrate around its vertices, so that for very large n, the volume of an n-sphere inscribed by the n-cube becomes very small relative to the volume of the n-cube. In fact, the ratio of the n-sphere's volume to the ratio of the n-cube's volume approaches 0 as n increases, meaning that in very high dimensions, almost all of the n-cube's volume is concentrated around its vertices (rather than its center!) so that you can fit an arbitrarily large number of n-spheres inside an n-cube by letting n grow.

If you keep the facets of the n-cube at a fixed distance from its center, then as n grows its corners will spread farther and farther away, and the ratio of its circumradius to its inradius grows without bound. Meaning to say that in sufficiently large n, it takes k times the amount of effort to get from its center to its corners than it takes to get from its center to the centroid of one of its facets, where k is proportional to sqrt(n). If you like, you can imagine this counterintuitive situation as the n-cube becoming more and more "star-like" in higher dimensions, such that its corners are the peak of very long "spikes" protruding from a tiny core that contains almost none of its total volume.

Of course, this extreme effect really only becomes obvious when n is very large, say n=100, but even at n=5, you begin to see some of this effect, one of which is the impossibility of a tesseract pyramid with unit edges.

There are also many other counterintuitive things that happen when you go up the dimensions. One example is that while the volume of the n-cube is concentrated around its corners, the volume of the n-sphere is, on the contrary, concentrated in a narrow band around its equator, such that for sufficiently large n, almost all of the n-sphere's volume is contained in a tiny band within epsilon units of the hyperplane that bisects it, where epsilon is a number that shrinks to 0 as n grows without bound. This strange effect is almost impossible to intuitively visualize, since it's so contrary to our 3D-centric experience; perhaps one way to think of it is that the n-sphere is like a cast-iron disk embedded inside a hollow plastic ball: almost all of the mass is concentrated in the disk, and the rest of the ball contributes almost nothing more. (This analogy isn't strictly accurate, of course, since an n-sphere is homogenous in all directions, whereas an iron disk embedded in a plastic ball is non-homogenous in its density. But it gives you some sort of crude idea of what happens in very high dimensions.)

Other counterintuitive things in higher dimensions include:
- The ending of the uniform antiprisms after 3D;
- The unexpected appearance of the regular 24-cell in 4D, with no direct analogues in any other dimension;
- The ending of the pentagonal regular/uniform polytopes after 4D, which includes the ending of all regular star polytopes after 4D;
- The ending of the bifurcated CD diagram uniform polytopes (i.e., the Gosset polytopes) at 8D;
- The ending of many classes of CRF segmentochora with n-simplex symmetry at 8D;
- The ending of many classes of CRF segmentochora with n-cube symmetry at 19D, after which there are only 3 classes. (Why 19D? Because the equation that determines it is a quadratic polynomial in √2, which crosses 0 at an irrational root between 19 and 20. On the surface, though, it seems really strange for the number 19 to appear seemingly out of nowhere.)

I'm sure there are many others out there waiting to be discovered. Higher dimensions are a very strange place once you leave the familiar harbor of 2D and 3D and venture into the deep waters.

[quickfur]

[...]The same holds true for Clifford- or double rotations within a 4D space: Again a mere rotation just acts within a 2D subspace. The orthogonal subspace, here being 4 - 2 = 2 dimensional as well, remains pointwise fixed. So you can consider to apply to that subspace an different action meanwhile. In case that one is a rotation too, and that both orthogonal rotations just act around that specific point, which is the common intersection of those 2 orthogonal subspaces, then this total action is what Clifford once had considered.

So, does this mean that rotating a 4D shape into 5D, around a 2-plane will make two moving axes? Leaving two stationary? I see the 3-plane rotation, and leaving one axis left over for the moving, but if a 2-plane is used, it looks like two axes are swirling around, into 5D.

--Philip

You can't rotate around a 2-plane in 5D. A 5D plane rotation rotates around a 3-plane.

Like I keep saying, the concept of rotational axis or rotating around something is hard to generalize nicely to higher dimensions (or lower dimension, for that matter -- 2D rotations have no axes!). It is better to regard rotations as always happening in a 2D plane. The orthogonal subspace of this plane (i.e., the space spanned by the remaining (n-2) axes that aren't involved in the rotation) then forms the stationary hyperplane that you rotate "around". It's much easier to get it right if you start from rotating in a 2D plane, instead of around some fixed-dimensional plane, which inevitably becomes wrong as soon as you change the number of dimensions.

(Not to mention the existence of 5D double rotations that have a stationary linear axis, but the rotation doesn't happen "around" the axis the same way it does in 3D, but there are two simultaneous rotations both sharing that axis, and with two independent rates of rotation. And rotating the axis changes the rotation because these two simultaneous rotations have orientation w.r.t. the "axis". Try wrapping your brain around that one. )

[ICN5D]

And since in 5D, the circumradius of the 5-cube is sqrt(5)/2 = 1.11803..., that means any pyramid of the 5-cube cannot possibly have edge length 1, since that would require its height to be the square root of a negative number.

Hmm, that's interesting. Has anyone ever suggested an "imaginary height" for a true unit 5D-pyramid, non-degenerate? That would be a dimension within a dimension. Wow, I don't know, that's pretty crazy. With a complex conjugate volume? Just brainstorming here.....

Meaning to say that in sufficiently large n, it takes k times the amount of effort to get from its center to its corners than it takes to get from its center to the centroid of one of its facets,

Does this resemble hyperbolic space in a ways? Perhaps that's why the universe looks like it's accelerating while expanding. All of those extra Calabi-Yau extensions.

If you like, you can imagine this counterintuitive situation as the n-cube becoming more and more "star-like" in higher dimensions, such that its corners are the peak of very long "spikes" protruding from a tiny core that contains almost none of its total volume.

I used to see them as more sphere-like, but you're right, it would be probably more star-like. I would see all of the corners smoothing out into a sphere-like shape. The corners would actually be protruding, and smoothing out into a cone as more and more axes (edges) were added to each one. I guess the over all outer envelope could be sphere-like, with lots of space in between.

One example is that while the volume of the n-cube is concentrated around its corners, the volume of the n-sphere is, on the contrary, concentrated in a narrow band around its equator, such that for sufficiently large n, almost all of the n-sphere's volume is contained in a tiny band within epsilon units of the hyperplane that bisects it, where epsilon is a number that shrinks to 0 as n grows without bound. This strange effect is almost impossible to intuitively visualize, since it's so contrary to our 3D-centric experience; perhaps one way to think of it is that the n-sphere is like a cast-iron disk embedded inside a hollow plastic ball: almost all of the mass is concentrated in the disk, and the rest of the ball contributes almost nothing more.

You know, you're right. It does have that effect. That's very strange, I never would have thought about that before. Well, it's not that difficult to see. Each n-sphere sort of bulges out from the poles and center, it's in the nature of the n-sphere. So, if we keep multiplying this bulging effect, over and over again, soon we get this flattened thing, where its essence is further away from the poles and center. Really, once again, this is an awesome website, where I have seen some of the most amazing things in my mind. Weird mathematical effects, crazy shapes, counter-intuitive phenomena, it's expanding my visual capacity really far.

- The ending of many classes of CRF segmentochora with n-cube symmetry at 19D, after which there are only 3 classes. (Why 19D? Because the equation that determines it is a quadratic polynomial in √2, which crosses 0 at an irrational root between 19 and 20. On the surface, though, it seems really strange for the number 19 to appear seemingly out of nowhere.)

Yep, very strange indeed. I have no idea what any of those are, but maybe it has something with 19 being a prime number? I haven't computed a 19D shape yet, probably only up to 10. Only a few of them. I'll have to check those out.

This is a well-known fact. The circumradius of an n-cube (i.e., the radius of the n-sphere that contains its vertices) is proportional to sqrt(n). If we fix edge length at 1, then the circumradius is sqrt(n)/2. So in 4D, the circumradius is equal to the edge length, which is a happy coincidence that makes the 24-cell a regular polytope. However, this same fact also means that in 5D, you can taper a tesseract with equal edge length, but it will be completely flat!

Well, you're definitely right about that one. And all along it's been staring me in the face. The cylconinder, |O>|O or (II)'(II) , has no height, and I didn't notice this strange effect before. I mean, I saw that one of the lacing elements was a cone-torus, but I never once guessed why the circle would have been inside or in the same plane as the duocylinder. It's a very beautiful representation of this effect, with curved surfaces instead. Two torii bound orthogonally lace to a circle at origin, laced by a cone torus and cylinder torus ( torinder ). The cone torus is on the same plane as the circle, the cylinder torus is orthogonal. But, I'm only repeating myself now! So, where does that leave us with 6-D tapertopes? Like the penteract pyramid? Are they impossible, or is the unit height degenerate indefinitely? Surely we can still shrink something to a point indefinitely, right? Where is the concentration of volume in one of these?

(Not to mention the existence of 5D double rotations that have a stationary linear axis, but the rotation doesn't happen "around" the axis the same way it does in 3D, but there are two simultaneous rotations both sharing that axis, and with two independent rates of rotation. And rotating the axis changes the rotation because these two simultaneous rotations have orientation w.r.t. the "axis". Try wrapping your brain around that one. )

My god, I don't know where to begin.

But, I don't know, maybe I do? Lately, I've been playing around with the rotations of a cylconinder into 6-D, around a bisecting 4-plane. This is in light of finally seeing the rotations of a duocylinder. Ultimately, there are five bisecting 4-planes centered at origin, that bisect the n-1 elements. Rotation around one creates a horrendous-looking shape, with no elegant beautiful symmetry. It's a self intersecting toratope of all elements. Another two are much nicer and identical, the last two are unique, leaving 3 unique bisecting rotations. Two of them make a cylspherinder lacing to a sphere, the other a cylspherinder to circle. The cylspherinder to circle has an innertube+spherical toratope lacing. Of the two cylspherinder to sphere lacings, there is either an innertube+spherical toratope lacing, or two spherical. But, perhaps this isn't what you're talking about. Just a neat little effect. Between the surface elements of the three shapes, the spin commutes in and out of the manifold operator.

Or, it could be a rotation around a 3-plane, leaving two axes available for motion. A so-called N-2 plane rotation?



--Philip

[ICN5D]

Please keep those little facts about high-D geometry coming, quickfur. You wouldn't believe how hungry I am for little tidbits of generic phenomena.



You know, another degenerate pyramid in 5-D is the square pyramid diprism. It's a square atop tesseract, where the square is in the same plane as the tesseract-base. This means it has no height! Very cool, it all makes sense now. Even another is cube atop tesseract, the triangle triprism. It's mathematically destined by unbreakable rules to become this way.


I want to compile a list of all degenerate tapertopes/rotopes in 5D, I'm very interested in this.