oddron wrote:Thank you. This is very helpful information.

[...]

wendy wrote:Rotation is hard, because it is stranger than it is in 3D.

The strange thing about trying to visualize 4D is that I have to use mathematical "eyes" to figure out what I ought to see. I know that an instantaneous rigid body motion can be described with a velocity field according to the equation v

_{P} = v

_{0} + M * r

_{P}, where v

_{P} is the velocity vector at point P, r

_{P} is the position vector for point P, v

_{0} is the velocity at the origin, and M is a skew-symmetric matrix. In n dimensions, vector v

_{0} represents translation, with n degrees of freedom, and matrix M represents rotation, with

_{n}C

_{2} degrees of freedom. In 4D, this gives me 4 dof for translation and 6 dof for rotation.

Sure. But the thing Wendy was having in mind is that a 4D rotation still acts within a 2D subspace. Thus the fixed space here no longer is a point (rotation within 2D) or a line (axis: rotation within 3D), but becomes itself a 2D subspace. I.e. a rotation here runs AROUND a full 2D subspace. - And then, this opens a new degree of complication, you can have SIMULTANUOUSLY a further, unrelated rotation within this orthogonal subspace as well, then just rotating around the former action subspace! E.g. consider the general 4D rotation matrix (in according orientation)

- Code: Select all
`cos(a) -sin(a) 0 0`

sin(a) cos(a) 0 0

0 0 cos(b) -sin(b)

0 0 sin(b) cos(b)

This is what is termed as a Clifford (double) rotation.

And, just as 2D polygons resp. 3D prisms, antiprism, pyramids, or cupolae might have a rotational symmetry, 4D polychora might have double rotational symmetries - in this context also being called swirl symmetries. - The easiest examples here are the duoprisms for sure. Those then have the Dynkin diagrams (o)-N-o-2-(o)-M-o, where N,M are some integers larger than 2 (or even rationals larger than 2). (The cases with N or M equating to 2 become degenerate, i.e. subdimensional only.) - Clearly of Special interest are those, where the 2 independent rotations become commensurate, say M=N; or, more general, N=n/a, M=n/b; etc.

Klitzing wrote:The neat thing about this Ansatz is that you always can "visualize" them by means of projection, like Schlegel diagrams: you can have one base scaled down somehow, positioned then concentrically to the other one! ... The easiest ones clearly are the prisms themselves. ... The next step then is the generalization of the pentachoron, i.e. the general pyramids.

So I either have a vertex in the middle of a polyhedron, of I have a shrunken mirror image in the middle of a polyhedron. Either way, I just have to connect the outside to the inside, and I have my Ansatz projection of a polychoron. Now there are two things in particular that I am curious about. What would the projection look like if I rotate the polychoron? What would a 3D cross-section look like?

The special feature of orthogonality of prisms or pyramids (both: axis to base) makes it quite easy to derive continuous section by (d-1)D subspaces parallel to the base with height coordinate as parameter. For prisms this is completely anoying, you just have nothing (outside), then, when entering, the base. But beyond this section figure will never change, until you reaches the opposite base. And beyond again nothing (outside). - For pyramids it is only a tiny bit better: Again nothing, then base polyhedron, but now this base polyhedron continuously shrinks concentricly to a point, and beyond: again nothing.

The other special orientation here would be having the axis either completely immersed within your cutting (d-1)D subspace, or at least being parallel to. Then a corresponding cutting movie would just provide pics series, which are either prisms or pyramids in turn, and their respective bases then would be just the corresponding (d-2)D sections of the (d-1)D bases.

But instead of doing sectionings you also could consider projectioning. You already did this in a static way when having the top base somehow forshortened concentrically within the bottom base. But consider an ordinary 3D cone or pyramid. The similar projection here would be from exactly somewhere atop, right on the axis - or similarily, from below. But you well could leave that axial position. Then the projection Image of your tip no longer would be concentrically aligned with the bottom base, but will be ANYWHERE else (in this subspace). The same hold when going up one dimension: any lace prism / segmetotope might be projected this way as well, getting the 2 (possibly subdimensional) bases no longer aligned concentrically, but rather aligned somehow shifted to each other - while their rotational orientation thereby always is kept fixed.

Klitzing wrote:You even might try to consider a cube as one base and an icosahedron as the other. - This ain't work, you might think? - But, in fact, this outstanding segmentochoron does exist.

I can almost, but not quite, visualize the 3D projection that you have described. What is the name of this segmentochoron?

Well, my own naming sheme here generally was "[top-base polytop] atop [bottom-base polytope]", where the (parallely) "atop" being symbolized by "||".

Jonathan Bowers made up some acronymic referencing of polytopes in general, which reduces the often rather longuish names to the most dominant consonants only, and then filling in again some vowels, in order to get some easy pronounceable "words". For sure these are standardized ("OBSA" = official Bowers style acronyms), so that such a shortname is unique always. Thus a tetrahedron just becomes a "tet", a truncated tetrahedron a "tut", a cube remains "cube" (already rather short), a truncated cube is a "tic", a dodecahedron becomes a "doe", an icosahedron becomes "ike", etc. In the sequel of that segmentochoron research he also provided sylabels for that atop alignment (-a-), atop gyroalignment (-ag-), atop inverted alignment (-al- for alternate), etc.

Thus you might refer to

that figure either as "cube atop ike", as "cube || ike", as "cubaike". Or you just refer to it by the numbering of that

article: "K4.21".

A true lace prism description is not possible, at least not with restriction to Coxeter groups only, because the axial symmetry here would be a pyrithohedral one. - But you still might write os3os4xo&#x, i.e. something being laced from o.3o.4x. (top cube) and .s3.s4.o (bottom ike). - But this alternated faceting of snubbing semiation here not only applies in the sense of afore to be 3D-ly applied only, and thereafter to be stacked to become 4D, it even can be considered the other way round, i.e. applying some alternation onto some 4D (then already stacked) figure! - Only this comutativity of processes (providing the same results, for sure) is what allows to use that snubbed lace prismatic description here as well.

Klitzing wrote:Now, having managed these monostratic things, you could take the next step. ... Thus you might consider a quite easy one here: take the mirror-join of 2 "octahedron atop cuboctahedron" at the larger base, i.e. the stack "octahedron atop (pseudo) cuboctahedron atop octahedron" ... nothing but the famous 24-cell!

This one just seems so strange. The 24-cell has no analog in 3D, so I suppose it serves as a "gateway" to 4D visualization.

Indeed:

---rk