## New to Higherspace + updating the wiki

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### New to Higherspace + updating the wiki

OK, so I'm quite interested in polyhedra, and while this seems like a wonderful place to continue my forays, I don't exactly have the mathematical expertise of a Coxeter or a Klitzing. I have a rudimentary grasp of higher dimensions, but my eyes glaze over at the sight of advanced mathematical writing. So I'm sorry if I start going on and on about something it turns out I have no comprehension of.

I was browsing the wiki, and on the http://hi.gher.space/wiki/Segmentotope page, the cells for K 4.141 (pentagon || pentagonal pyramid) aren't listed, when I'm pretty sure they consist of a pentagonal prism, 2 pentagonal pyramids, and 5 square pyramids. It seems unlikely that Klitzing would have overlooked the cell counts of his segmentotopes, so I'm assuming this is just a case of the wiki being slightly understaffed.
Similarly, K 4.51 (trigonal magnabicupolic ring) consists of 1 P6, 2 Q3, 3 P3, and 3 Y4. (How do you feel about Bonnie Stewart's notation? I've worked extensively with the Stewart toroids and the notation is quite efficient.)
May I update the wiki when I see things like this that aren't present, or do I completely misunderstand how polychora work? I'd like to get the bicupolic rings' pages a bit more complete, at least. (Don't worry, I won't put Stewart's notation on the wiki. You've worked hard on your own notation - I can't help but admire your elemental names of convex regular polytopes!)

Also, is there any software available that I could use to better interact with CRF polytopes? I'm not sure if I want to spring for Stella4D, and I heard that Polyview was something that only Quickfur and Keiji had access to at the moment. Is there anything else that I may not have heard of?

New Kid on the 4D analog of a Block
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### Re: New to Higherspace + updating the wiki

Welcome here.

A good deal of what happens here, such as the xo5ox&#xt for pentagonal antiprism, is my insight. Names like 'pap' are due to Bowers, but Klitzing keeps the master register. I maintain the general register for the xo5ox&#xt stuff.

Much of my stuff was due to sifting through and correcting Coxeter's book (Regular Polytopes).

Is Stewart the same as Ian Stewart?
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### Re: New to Higherspace + updating the wiki

Good thing I've just picked up a copy of Regular Polytopes from my university's library; that might help me get on the same page as you.
The notation with x's, o's, and numbers is part of your "lace" shorthand for Coxeter-Dynkin diagrams, right?

No, it's Bonnie M. Stewart, author of Adventures Among the Toroids, which is a masterful exploration of regular-faced toroidal polyhedra. Alex Doskey has laid out the notation here: http://polyhedra.doskey.com/Stewart02.html I'll abstain from using it here in the future.

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### Re: New to Higherspace + updating the wiki

The notation was devised in the 1980s. The lace-bit is more recent (about 2006 or so).

The notation is fairly robust. The crowd here have certainly pushed it to the limit. None the same it was meant to be 'computer input' from the start (long before i had a computer, and i was still using punched cards). It's not in Coxeter's books. I was so frustrated that Coxeter was more or less using a different notation for every problem, I sat down and wrote a unified one.

There are certainly a lot of people here who do different things. I usually prowl around the higher dimensions at the moment, or into something absurd like weights and measures. Reconsiling the cgs and si systems has created a new model of gravity, I am afraid. (The curvature in GRT derives specifically from matter tieing up space).
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wendy
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### Re: New to Higherspace + updating the wiki

welcome from my side as well!

The cited segmentochoron page of the hi.gher.space wiki is keiji's.

The official author's page on that topic is https://bendwavy.org/klitzing/explain/segmentochora.htm.
There lots of additional stuff can be found, like the interrelation of my notion of segmentotopes and Wendy Krieger's notion of lace prisms, my original paper on that topic from 2000 - which already provided all required cells each - and even several compilations of known higher dimensional analogues.

BTW, there also is an own page for the collection of most of the known CRFs as well, cf. https://bendwavy.org/klitzing/explain/johnson.htm#crf, including a downloadable spreadsheet, providing the respective cell counts each.

Your given cell lists for the mentioned 2 segmentochora were completely correct, as you might check too in the sources cited above.

WRT the purpose of Bonnie Stewart, i.e. solely being concerned in compiling up specific finite hollow complexes of 3D cells in euclidean 3D space (or, to some extend quite similarily, when considering most of the convex 4D Wythoffian polytopes) his list of systematic cell names might serve useful. Even a bit more general, whenever one is dealing a lot with those most often encountered cells. - Sometimes I too use his nomenclature (cf. e.g. at https://bendwavy.org/klitzing/dimensions/flat.htm#3D - just scroll down a little). - But beyond those frequently being used cells that nomenclature becomes a bit ad hoc, if according acronyms would be provided at all. And for higher dimensions there is no such analogue - up to my knowledge. This is why I use those rather sparely. OTOH the acronyms of Jonathan Bowers (aka Polyhedron Dude) are in the same spirit, but derive quite systematically and, even more, are pronounceable.

--- rk
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### Re: New to Higherspace + updating the wiki

I don't mean to be boring you with problems you've solved decades ago, but this one seems pretty important.
I've started to understand how some of the simpler CRF polychora "fold up," like the Pyroursachoron and the bicupolic rings, and I accidentally figured out the Staurohedral Antiprism K4.15 yesterday. But how do we know if a polytope closes up? Or if it's convex or regular-faced? Do we deduce all of the angles by hand when we find a new way to pack some cells together? Is there software that can compute the coordinates? What would its input and output look like? I'd really like to be in the loop on this one.
Assuming we use regular-faced cells (even though they'll look distorted from being folded into 4D while being visualized in a lower dimension), the resultant polychoron should be regular-faced, but not necessarily convex. The tests would probably be for convexity (and maybe self-intersections), then. Am I anywhere close to correct on this?

Downloaded the Excel file, will use it for reference. Turns out I didn't invent the K4.15.
And I'll do my best to memorize the more commonly used Bowers acronyms. I had no idea they were systematic!

On a less polyhedral note, the wiki won't let me log in. How should I address this situation?

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### Re: New to Higherspace + updating the wiki

Hello New Kid. You got through the vetting, i see.

The lace-tower notation will always produce closed figures. It is not thus hard to verify convexity. I wrote a spreadsheet on Richard's website, that calculates the height from a given pair of layers. He modified it to something grand. If you have a polytope over three layers, and there is a polygon likewise, it suffices that if the height of layers AC = AB + BC, then the three layers are convex.

If i recall correctly, you have to PM the user Keiji, who is the administrator here and also manages the wikipedia. You need a separate log-on identity for the wiki. Not everyone has one.

As to Bower's acronyms, I don't know a lot of them either. Some of the more common ones. Bowers discovered all of the starry uniform polychora (4d polyhedra), for which one has a good deal of advantage in shortening something like "rhombotruncated icosadodecahedron" into rID or whatever. But he has thousands of examples, and has worked up to six dimensions. (gosh). I mainly use the lace-notation.

The cute thing about the lace-notation is that you can directly evaluate the number and shape of each of the surtopes (surface polytopes of lesser dimensions).
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### Re: New to Higherspace + updating the wiki

In four dimensions there is no block!

The ground plans for 4d cities is like the human body. You don't have level crossings for where the blood crosses the digestive system, and bits of muscle are not completely surrounded by blood vessels like bits of habitation are surrounded by motorcar streets (whence, blocks). But the thing is well understood, and we welcome you aboard.
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### Re: New to Higherspace + updating the wiki

The set of convex segmentochora once had been designed to be a well-suited didactical means in order to ease to grasp the 4th dimension.

The vertex set of segmentochora is defined to be on a 4D hypersphere as well as within a pair of 2 parallel hyperplanes. In fact this is equivalent to say that the vertex set happens to be on the surface of two distinct 3D spheres, which just are being shifted a bit in a perpendicular fourth direction. Thus you can visualize all these segmentochora by the two base polyhedra, which are being laced together in a similar manner as usual prisms, antiprism or cupola are get laced within 1 dimension less, just that the there lacing faces here become lacing cells. Because segmentochora are just monostratic, the depth of diving into the fourth dimension is as small as possible - and therefore quite easily manageable.

You even could project 4D space into 3D space from some point somewhere outside the to be considered polychoron. For segmentochora this amounts in two concentric spheres (the vertex layers of the bases) and the lacing cells then would be the ones which connect those two layers. In fact this is very similar to the dimensionally next step of Schlegel diagrams for polyhedra. Sure, the inner sphere would be scaled down a bit and the lacing cells get thereby a bit forshortened in a perspectivical sense (cf. the converging lines effect).

Wendy clearly is right in her statement that my set of segmentotopes and her set of lace prisms has a very big set theoretical intersection. But segmentotopes always will have unit edges only (by definition), whereas lace prisms well could be applied to various edge sizes as well. OTOH lace prisms (and, more generally, lace towers, i.e. hulls of multistratic stacks of lace prisms) are bound to use polytopes for bases (or base parallel sections - aka vertex layers) which get Wythoff constructed (as those layers are described by Coxeter-Dynkin diagrams), whereas segmentotopes by definition also are free to use various diminishings or gyrations therefrom as well.

If you get your mind to wrap around to read the Coxeter-Dynkin diagrams (like x4o3o being a cube), then this provides a very effective means to write down lace prisms by putting the 2 vertex layers together: say you have a cube atop a cuboctahedron - which I would have written simply as "cube || co", Wendy would read here x4o3o || o4x3o instead, which could then be folded into xo4ox3oo&#x, where each node position just gets crowded by one member of each layer and the trailing bit is just the notion that we have additionally (&) lacing edges (#) of the edge size x (as well). That symbol then allows to read of the required cells directly: the bases just are the two layers (i.e. the first and second representants of all node positions respectively) and the lacings here are obtained similarily as the facets get derived from the usual (one-layered) Coxeter-Dynkin diagrams, i.e. by omitting any single node position. Thus in our example we get lacing cells being xo4ox ..&#x (aka square antiprisms) and .. ox3oo&#x (tetrahedra in the sense of triangular pyramids). The further case xo .. oo&#x here can be neglected as it happens to become subdimensional (just triangles).

--- rk
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### Re: New to Higherspace + updating the wiki

And I thank you, Dr. Klitzing and your collaborators, for creating such an effective didactic.

So I've figured out the crown jewel D4.10 in order to learn more of the intricacies of lace notation. It's a good thing its lace contains those notes.
Each row represents the points (vertices) of a point group, layered "on top of" each other in 4D. f seems to represent an active mirror that separates points by the Golden Ratio instead of by unit length 1, so I assume F separates points by phi^2?
Connections (edges) between different layers' vertices are represented by those laces being adjacent in the diagram. Sometimes the connections aren't "linear," hence the close-up of the lace tower that really fleshes out the nonlinear connectivity.
D4.10 is built up from two "sides," from the antipodal tetrahedra. The lace tower has central symmetry because each of its ends is one of those tetrahedra, being built upon and approaching the center. The lace tells us where the vertices are, and it's "left as an exercise to the reader" to find the edges. So not all of the cells are laid out in the lace; just enough for the vertices to be defined. For example, the pentagonal antiprisms don't appear in the lace because their pentagons' vertices are defined in the J91s (one pentagon from either side), so there's no need to express them in the lace. I didn't actually find the squippys, but I assume they're mere interstitials like the paps.
I'm not sure why I'm telling you this; you were involved in its discovery. It's like explaining radiation to Madame Curie.

In other news, I thought I found a new crown jewel, but it turned out to have some edges with only 2 cells. I bit off more than I could chew, I know, but at least I caught the mistake before I came running to the forum with a bloated incidence matrix and fevered descriptions of sphenocoronae. Anyway, back to the drawing board...

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### Re: New to Higherspace + updating the wiki

The short-chords of the polygons occur often enough for them to get special symbols.

The shortchord is the third side of a triangle, formed by two edges of a regular polygon. They are

$$x = 1$$ for the triangle

$$q = \sqrt{2}$$ for the square.

$$f = \frac 12 + \frac 12 \sqrt{5}$$ for the pentagon and $$v = \frac 12\sqrt{5} - \frac 12$$ for the pentagram

$$h = \sqrt{3}$$ for the hexagon

$$u = 2$$ for the horogon.

The golden rectangle can be written as f2x or x2v. You can have a golden prism of f2x2v, from the centre, all edges correspond to the short-chord of a polygon above, and the face-diagonals are their complement. The other prism with this effect is x2x2q.

Other letters are used in Richard's project, but they are considered local, rather than global.

The cube || icosahedron, has an implementation as a section of xo5oo3of&#xf, where the top layer is trimmed back to a cube, ie x5o3o -> f4o3o. It has a pyritohedral representation of 3ox * ox2%x &xt , but this notation is not very stable. (It's a decorated conway-thurston notation, which i have been bangang at).

For $$\LaTeX$$, install greasemonkey and visit: https://greasyfork.org/en/users/188714-wendy-krieger You want the hi.gher.space one.
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wendy
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### Re: New to Higherspace + updating the wiki

"D4.10" is what is found on my website here. There even providing all its cells too.
And "cube || icosahedron" is being found here.

--- rk
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