Getting started?

[oddron]

I'm starting to learn to visualize shapes in 4D, and I have all sorts of questions.

Which 4D shapes are easiest to visualize? Right now, the only shapes I can begin to see are the tesseract and the pentachoron. I can imagine the tesseract as a cube within a cube, and I can pick out the 8 cubic cells (the big cube, the small cube, and 6 squashed cubes). I can imagine the pentachoron as a point within a tetrahedron, and I can pick out the 5 tetrahedral cells. On the other hand, I have no idea how to rotate either of these shapes.

What kinds of things should I try to visualize? In addition to mentally rotating a shape, some ideas that come to mind are finding cross-sections of a shape, figuring out how (or if) two shapes can fit together, and seeing how a shape can be unfolded.

How would I handle round shapes? While the tesseract is a collection of cubes that fit together in a particular way, a 4D sphere doesn't seem to have any constituent parts. Should I save round shapes for later?

[wendy]

Welcome.

I started at a different point. Have a look at my page http://www.os2fan2.com/gloss/, particularly the Hyperspace thread. There are some useful ideas there to think about when you start making things. Things that 'divide' are relative to solid space, so the blade of a 4D knife is 2-dimensional.

A good amount of time can be wasted if you use common words for things. I make a lot of note about this in my glossary. For example, a 2D space does not divide 4-space, while a plane carries the image of dividing space. So i have a new word for 2d space which does not carry the meanings of plane.

Rotation is hard, because it is stranger than it is in 3D. If you are not adverse to reading þ as th, then http://www.os2fan2.com/glossn/ is derived from the same source, but more current.

I started off with the starry night, and cows and things, and eventually replaced cardboard cutouts with the real thing when i calculated what they should look like.

[Klitzing]

Which 4D shapes are easiest to visualize? Right now, the only shapes I can begin to see are the tesseract and the pentachoron. I can imagine the tesseract as a cube within a cube, and I can pick out the 8 cubic cells (the big cube, the small cube, and 6 squashed cubes). I can imagine the pentachoron as a point within a tetrahedron, and I can pick out the 5 tetrahedral cells.

The natural extension of that Ansatz is what once was described as Convex Segmentochora, cf. esp. the original paper. With restriction to non-diminished or non-gyrated bases, this happens to be the same as Wendy's lace prism idea (when that one being restricted to unit edges only). In fact you just use 2 polyhedra (or even subdimensional things) and stack them within the 4th direction. Thus you'll have the 2 bases of that "prism". Further you will have to lace these together by connecting "close" vertices by edges. These edges then will span further cells.

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!

1.) The easiest ones clearly are the prisms themselves. Take any polyhedron, take the unit prisms for any of its faces, pile them above those, and finally close the top by a mirror Image of the bottom polyhedron. - Indeed, it is the mirror Image. Consider for that purpose e.g. the prism of the snub cube: for the bases you'll need an enantiomorphic pair! - These prisms then are the direct extrapolation of the tesseract, you already managed.

2.) The next step then is the generalization of the pentachoron, i.e. the general pyramids. For that purpose take any orbiform polyhedron (i.e. vertices on a sphere, edges all unity), which additionally has a circumradius strictly lesser than unity. Then you can oppose that by a point as other base, and you'll be done most easily.

3.) But even other shapes (general lace prisms) then can be accessed quite easily. Say, take a truncated cube as one base and a great rhombicuboctahedron as the other. Then it is pretty straight forward, to lace the octagons each by octahedral prisms, to lace the triangles to the hexagons each by triangular cupolae, and to lace the octagon-octagon edges to the squares each by digonal cupolae (i.e. by triangular prisms based on a latteral square).

4.) 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: just attach 6 triangular prisms (in the sense of digonal cupolae) onto the sides of the cube, while not breaking the pyrithohedral symmetry (those neighbouring digons all are mutually perpendicular). Then attach onto all 12 un-matched squares a square pyramid. These will match by one triangle each to a neighbouring triangular base of the prisms as well. Further introduce 8 tetrahedra by their tips opposing the cube at its vertices. Then again all their latteral faces become matched to some so far un-matched triangles of the square pyramids. And then, you guess what? The remaining un-matched faces all are triangles, infact one from every square pyramid and one from every tetrahedron, giving space for the final icosahedron!

5.) Now, having managed these monostratic things, you could take the next step. Just stack some lace prisms (or segmentochora) on top of each other in the sense of an external blend, i.e. erase the connecting face and join the 2 bodies into one. Alternatively these can be viewed as the first step of lace towers as well (the bistratic ones). 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". - Well, we'll start with one of those first: it attaches square pyramids on top of each of the 6 squares of the cuboctahedron, and onto the formers 8 triangles attach octahedra. then the triangles of those lacing octahedra next to the already matched base can connect to the sides of the pyramids. But doing so, the other lacing triangles of these trigonal antiprisms (what these lacing octahedras are used as) will mutually connect pairwise as well. Thus only the opposing bases of these antriprisms remain, and give space for the other base polyhedron, the final octahedron. - Now, when adjoining 2 such figures at their cuboctahedra, right that cell (each) will be erased. The outer 2 base octahedra clearly remain. And so do the 2x8 lacing trigonal antiprisms (lacing octahedra). And the 2x6 square pyramids will connect, as they are co-realmic, into further octahedra. Thus we have in total 2+16+6=24 octahedra. - What we just have constructed here is - give me a wow - nothing but the famous 24-cell!

--- rk

[oddron]

Thank you. This is very helpful information.

Things that 'divide' are relative to solid space, so the blade of a 4D knife is 2-dimensional.

This makes sense. In 3D, if I try to cut an object with a thin pin, it won't work. All I can do is puncture the object. With many pin-holes, I can perforate an object to weaken it, but I can't directly cut it. In 4D, if I try to cut an object with a 1D edge of a polygon, I will only create a bunch of 2D holes. While this may weaken the object, it won't cut it. I would need to use a face of a polyhedron as a blade.

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 = v₀ + M * r_P, where v_P is the velocity vector at point P, r_P is the position vector for point P, v₀ is the velocity at the origin, and M is a skew-symmetric matrix. In n dimensions, vector v₀ represents translation, with n degrees of freedom, and matrix M represents rotation, with _nC₂ degrees of freedom. In 4D, this gives me 4 dof for translation and 6 dof for rotation.

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?

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?

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.

[wendy]

Welcome aboard.

The 24choron has two analogs in three dimensions, but these two analogs in turn have two further analogs in 4D.

1. In 4D There is x3o3o3x, which is the 4D Simplex-prism-circuit. It corresponds to a lace-tower xxo3ooo3oxx&#xt.
2. In 3D there is the cuboctahedron, being x3o3x or o3x4o
3. In 4D, the 24choron, x3o4o3o, being variously o3x3o4o and o3m3o4o.
4. In 3D, the rhombic dodecahedron m3o3m = o3m4o. 3&4 are double-cubes, ie a cube with pyramids on each face
5. In 4D the m3o3o3m, a polytope bounded by twenty obtuse rhombohedra. the short diagonals go from the vertex of a 5-ch to those of its dual.

There is an analog between 1,2 between 2,3, between 3,4 and between 4,5.

The polytope o3x3o3o is the 4D analog of the x2x3o, this class of figure cumulates in 9D as a tiling, is called the gosset series.

The segmentotope Richard talks of (ike || cube), is xo3 * xx2o% *#t, if i am not mistaken.

[Klitzing]

Thank you. This is very helpful information.

[...]

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 = v₀ + M * r_P, where v_P is the velocity vector at point P, r_P is the position vector for point P, v₀ is the velocity at the origin, and M is a skew-symmetric matrix. In n dimensions, vector v₀ represents translation, with n degrees of freedom, and matrix M represents rotation, with _nC₂ 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.

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.

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.

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