by **d023n** » Tue Nov 14, 2017 9:15 pm

I should add a bit more for clarity.

The n-simplex only captures the rotational structure of the (2n+1)-sphere and the (2n+2)-sphere indirectly, offering a useful way of organizing the types of rotation that can be carried out. The actual visualization of the rotational structure is somewhere between really difficult and impossible.

Nevertheless, I have tried categorizing the structures that emerge when rotations are performed. For example, in dimension 3, the 2-sphere can only be 1-rotated. When this happens, 3 structures of note that appear are (1) the polar point pair, (2) the equatorial circle/1-torus, and (3) the latitudinal circles/1-tori in between the equator and either polar point. The polar points can be thought of as remaining fixed while the equator contains the points that move the farthest/fastest.

Back in dimension 2, the 1-sphere can also only be 1-rotated, but there isn't a leftover axis to cause a rotation differential across the sphere. In fact, all odd-spheres embedded in even-spaces don't have any polar or latitudinal structures because of this fact. However, there can still exist rotation differentials across the odd-spheres, leading points to move different distances at different rates. This can't happen to the 1-sphere, but it can happen to all higher odd-spheres because they can do more than 1-rotate.

For example, in dimension 4, the 3-sphere can 2-rotate symmetrically (isoclinically), 2-rotate asymmetrically, or just 1-rotate; each producing different rotational structures. When the 3-sphere 2-rotates symmetrically (i.e. when it rotates the same amount in both planes of rotation), this can be seen as the equivalent to the 1-sphere rotation, because every point in the 3-sphere moves the same distance at the same rate, and there isn't really any special structure (in fact, any pair of orthogonal planes can be considered the planes of rotation). On the other hand, when the 3-sphere 2-rotates asymmetrically (which includes 1-rotation), there is a differential, and there are 2 specific orthogonal planes that are the designated planes of rotation. At this point, it becomes useful to think of the 3-sphere as a collection of 2-tori spanning between each circle/1-torus (degenerate 2-torus) in the 2 rotation planes. 2-tori have 2 radii. For the degenerate 2-torus, which I prefer to simply call a "degenerate" or "degen" (dee-jin), in one of the planes of rotation, one of its radii is 0; and, for the other degen, its opposite radius is 0; while the 2-tori in between will have various ratios of radii. Because of this, there will be a perfectly symmetrical 2-torus directly in the middle, which I call the "symmetor" (I like to refer to the rest of the 2-tori as "sheets of constant asymmitude," which sort of aligns with the idea of an equator and latitudes). If the 3-sphere has radius 1, then both radii of the symmetor will be 2^{-1/2} (about 0.7071). However, the symmetor will only be symmetrical with respect to its radii, not the 2 rates of the 2-rotation. If the 3-sphere is 1-rotating, then one degen will remain fixed, the other degen will contain points that move the farthest/fastest, and the asymmitudes in between will contain points moving intermediate distances at intermediate rates (e.g. points on the symmetor will move the square root of half the distance covered by the degen that is moving, at half of the rate, and strictly in a plane parallel to that degen). If the 3-sphere is 2-rotating asymmetrically, then there will actually be an asymmitude, which I like to call the "asymmetor," somewhere in between the symmetor and the slower degen where the component of translational motion parallel to one plane of rotation will be equal to the orthogonal component (i.e. if the 3-sphere rotates faster in the XY plane than it does in the ZW plane, then there is an asymmitude between the ZW degen and the symmetor where the ZW distance covered by points in a certain time is equal to the XY distance). As the 3-sphere rotates less and less asymmetrically, this behavior will be passed to asymmitudes closer and closer to the symmetor from the direction of the slower degen, until it finally passes to the symmetor itself, and the rotation becomes isoclinic.

So, to recap the previous paragraph, when the 3-sphere 2-rotates asymmetrically, the 4 structures of note that appear are (1) a pair of circles/degenerate-2-tori, "degens," (2) an "asymmitudinal" span of 2-tori whose radii go from (0, r) to (r, 0) where r is the radius of the 3-sphere, and the square root of the sum of the squares of both radii is r, (3) a symmetrical 2-torus, the "symmetor," whose radii are both the square root of r/2, and (4) a particular asymmitude, the "asymmetor," between the slower degen and the symmetor whose components of translational motion are equal. When the 3-sphere 1-rotates, the asymmetor is the fixed degen. When the 3-sphere 2-rotates symmetrically (or doesn't rotate), there is no distinction between any of these structures because there is no absolute/specified pair of othogonal planes of rotation.

Now, to recap everything, having a leftover axis creates a rotation differential across the even-sphere embedded in odd-space, leading to a polar point pair, an equator, and latitudes; whereas having more than one plane of rotation creates a different rotation differential across the odd-sphere embedded in even-space, leading to degens, a symmetor, and asymmitudes.

What happens when you mix these phenomena? Well, you can finally do just that in dimension 5 with the 4-sphere! It has 2 planes of rotation and a leftover axis. This means that, while the 4-sphere's equator is essentially the 3-sphere, every latitude is a smaller 3-sphere, too. However, if the 4-sphere were only to 1-rotate, instead of a single degen remaining fixed, a degen in every 3-sphere latitude remains fixed. A "degen bubble" (2-sphere) remains fixed during a 1-rotation. Additionally, since every latitude contains a symmetor, the 4-sphere as a whole can be thought of as having a "symmetor bubble" where the radii of latitudinal symmetors span from 0 at the poles to the square root of r/2 at the equator. Only during a isoclinic rotation does the 4-sphere degenerate to simply having a polar point pair, a 3-sphere equator, and 3-sphere latitudes, with the other fine structures becoming indistinct.

In dimension 6, the 5-sphere no longer has polar points or an equator or latitudes. It does, however, have the option to 3-rotate. This means that there are now 3 degens and a 3-torus symmetor. There are also 3 2-tori that are partially degenerate 3-tori, which I call "2-degens" (they are technically also partial symmetors). On the unit 5-sphere, these 2-degens have 2 radii of 2^{-1/2} and one of 0, whereas the symmetor has 3 radii of 3^{-1/2} (about 0.57735). This is actually where the simplex idea came to me, because I realized that keeping track of the structure was beginning to get complicated. A triangle is useful for this rotational structure, because a degen is any of the 3 vertices, an asymmitude (where one radius is 0) is any point along the 3 edges, a 2-degen is any of the 3 edge midpoints, an asymmitude (with no 0 radii) is any point inside of the triangle, and the symmetor is the center. Also, connecting the vertices to the midpoints of the opposite edges partitions the inside of the triangle, showing the various ranges of asymmetry.

Dimension 7 retains the same basic rotational structure, but with the added concept of latitudes, which simply smears/shrinks dimension 6's 5-sphere "up" and "down" to point poles and so creates "bubbled versions" of the various degen, symmetor, and asymmitude structures.

Dimension 8 can finally use the tetrahedron to represent the basic rotational structure, showing that there are now 4 degens, 6 2-degens, 4 3-degens, and a symmetor.

Dimension 9 "bubbles" dimension 8's structures.

Dimension 10 is where the simplex method finally becomes impossible to properly visualize with dimension 4's pentachoron.

So, to recap one last time, dimension 2 rotation produces no unique structures, dimension 3 rotation produces poles/equator/latitudes because of a leftover axis, dimension 4 rotation abandons dimension 3's structures but produces degens/symmetor/asymmitudes because of an additional plane of rotation, and dimension 5 is the first to showcase all 6 structures because of a leftover axis and an additional plane of rotation. Generally though, only odd dimensions possess a leftover axis and, thus, poles/equator/latitudes.

I wish I could explain this in a more organized fashion, but I've simply never seen the degen/symmetor/asymmitude ideas or anything like them anywhere else.

Please, let me know if I need to better explain myself.

d023n.