student91 wrote:[...]

Yes, I guess I'm rushing a bit, letting you do the rendering and doing the fun myself. You have to know I really appreciate your renderings, and am very reliable on them. For all discoveries I've done so far have I have heavilly used your renders (and klitzings site for the non-visual info I needed). Your sites are indispensable.

Thanks! I have to say, I also found Klitzing's incmats site extremely valuable for the lace towers it provides for the uniform polychora. It greatly simplified many of my recent CRF constructions, because having the CD symbols readily available means I only need to compute the height of the layer (and sometimes not even, if I'm copying a section of a 4D uniform intact -- the height is already encoded in the coordinates, just have to sort by the first coordinate (or 4th, if you align your construction that way)), and the rest can be automatically generated with wendy's apacs/epacs/apecs/etc. operators from known coordinates of 3D uniforms.

quickfur wrote:[...]

It's not just about CVP2 values being rearrangeable, though. The reason is that solving via pythagoras (which is implicitly what's happening in all of these constructions) requires a computation of the form X = √(edge_length^{2} - Y^{2}), where X is the coordinate being solved for, and Y is usually a difference between two vectors. The thing is, usually the value of Y will itself involve square roots (e.g., the golden ratio, or √2, or √3, etc.), and in theory the difference between the edge length squared and Y is, generally speaking, not a perfect square, so usually X will involve nested square roots.

However, what I found was that when the golden ratio is involved in constructions derived from the 120-cell family uniforms, somehow (edge_length^{2} - Y^{2}) keeps working out to be some kind of perfect square, so X becomes again something with only a single square root that can be rewritten in terms of combinations of the golden ratio. So that means something special is going on with constructions involving the golden ratio, probably due to its connection with the 120-cell family polychora, that causes these values to repeatedly work out to nicely-closed forms.

I think it is more related to phi itself than the 120-family. The fact that phi^2=phi+1 probably highly contributes to this. Furthermore, A*sqrt(5) + B might just always be a perfect square, I'll try to work this out tomorrow.

No, that's not true. The pentagon on the plane, for example, has nested square roots resulting from a combination of phi that

isn't a perfect square.

Also, not just phi but

any quadratic rational (i.e., numbers of the form x+y√z, for fixed z) will exhibit similar properties to phi, because numbers of this form are actually closed under field operations (+,-, and * should be obvious; division also applies because you just multiply a denominator x+y√z by its conjugate x-y√z to get rid of the reciprocal square root term and turn the denominator into a non-square root number, then it can be rewritten once again in the form x+y√z). Given any quadratic polynomial Ax^2 + Bx + C, a root R of this polynomial will (obviously) satisfy the relation AR^2 + BR + C = 0, or, in other words, AR^2 = -BR-C. Dividing through by R produces the relation AR = -B-C/R, or, in more familiar form, C/R = -B-AR. In the case of phi, A=1, B=-1, C=-1, so we have phi^2 = phi+1 and 1/phi = phi-1. But as you can see, other values of A, B, and C will produce numbers that also exhibit special relationships with their square and reciprocal. And, as I mentioned, since these quadratic rationals are closed under field operations, adding, subtracting, or multiplying them will all, again, produce numbers of the same form.

So you see, phi isn't

that special after all. Well, it still is, but not just because of its relations with its square and reciprocal.

The reason phi crops up more often than not is only because it is one of the simplest quadratic rationals, and because of its connection with the pentagon.

The special behaviour of phi that I've been observing, though, appears to be intimately tied to the 120-cell family polychora, in the sense that certain kinds of combinations of phi seem to somehow gravitate towards perfect squares, even though there are many situations where this doesn't happen (e.g., in the pentagon on the 2D plane). This makes me suspect that there might be some kind of underlying phi-based tessellation, or something like that, in 4D, that causes these numbers to "magically" work out in nice forms in many cases where it would normally be unexpected. IOW, the CRFs we've been finding so far are all just CRF fragments of this underlying tiling or grid or whatever it is, and that many more such fragments exist.

.[...]

Yep. So that means the best approach to find a CVP3 4D CRF may be to work topologically, rather than by direct construction as we have been doing so far. Once we have a topologically-closed cell complex that is likely to be CRF, we can then try to solve for the coordinates to see if it is CRF-able.

I've actually tried topological constructions before, but they are pretty hard because you have to be able to visualize 4D and perform topological assembly operations in your head. Drawing diagrams on paper quickly becomes a nasty tangle of incomprehensible lines that lead nowhere. It can be alleviated somewhat if you're a careful artist and use colored pens/pencils to indicate edges belonging to different cells, but that distracts you with the mechanics of 2D diagramming rather than letting you focus on the 4D construction at hand. Perhaps what is needed is some kind of interactive 4D-doodle program that lets you manipulate topological polyhedra with 3D rendering that helps prevent the clutter of edges that will quickly become unmanageable.

The doodle program sounds fun to play with. Maybe even people that are not interested in polytopes would like to play with it.

The topological constructions might just be more difficult that we would think. I can think of a lot of polytopes consisting of tetrahedra that work out topologically, yet they should be ,impossible according to mr and mrs Blind. I don't know how this would be extednded to CRF's, but I think not everything that works topologically will work "in real" as well.

Of course, there will be many more topological constructions that are possible to realize in a CRF way. So you can't just blindly enumerate topological constructions and hope to hit lots of CRFs, because chances are that the vast majority of them are not CRF-able.

The idea is to explore topological constructions that are

likely to be CRF-able, but without needing to actually check whether they are actually CRF at every step, and only verify that at the end.