## Algebraic coordinates for 3D crown jewels

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Algebraic coordinates for 3D crown jewels

So, recently I've been wrangling with the 3D crown jewels, particularly with the disphenocingulum (J90). Been trying to find defining polynomials that would yield theoretically infinite-precision coordinates. It became pretty clear early on that doing this by hand would be infeasible, and after several feeble attempts I decided that the only way to proceed was to ask the computer to do it for me. So I wrote a utility, currently still rather imperfect, that uses a modified Buchberger algorithm to compute a Gröbner basis for the system of polynomial equations that describe the coordinates of J90.

I'm happy to report that it was a success, and in spite of the current limitations of the program, it was able to discover defining polynomials for J90's coordinates, which I've posted on my website on the J90 page. As far as I can tell, no one has posted these polynomials online anywhere before; the only sources of J90's coordinates that I have found online so far appear to be low-precision (only accurate up to 6-7 digits, and apparently acquired by numerical methods, so it was unclear how one might obtain more digits of precision). Given the polynomials I've found, along with the root intervals that I've posted alongside, one could in theory extract coordinates of arbitrary precision using a suitable polynomial root-finding algorithm (e.g., Newton's method, which should be widely available in numerical libraries, yet simple enough to implement oneself, or even compute by hand, if necessary).

What I wanted to discuss here, however, is the polynomials themselves. As it turns out, J90's coordinates require solving 24th degree polynomials (with terms of even power) and 12th degree polynomials (with terms of all powers), and these polynomials have huge coefficients (up to 10 digits!). Furthermore, they are irreducible (at least according to Wolfram Alpha), unlike the case with the snub disphenoid (J84), where solving the associated polynomial system yields a quartic that can be factored into a linear term and a cubic.

A slightly less extreme case is the snub square antiprism (J85), whose coordinates involve solving 12th degree polynomials with terms of even degree, and 6th degree polynomials with terms of all powers. These are also irreducible polynomials.

So far, I have not been able to obtain defining polynomials for J88 and J89. Given that they involve polynomial systems with ≥7 unknowns, I'm expecting that they will be at least as complex as the polynomials for J90, in all likelihood a lot more complex.

Anyway, what I'm trying to get at, is that all of this is reminding me of our discussions of polytope complexity in various guises like CVP, minimum polynomial degree, etc.. While this isn't proof, per se, the huge coefficients of J90 and the irreducibility of the 12th degree polynomials is leading me to think that it's probably unlikely that there will be any higher-dimensional CRFs that contain J90's as cells, besides the trivial prisms (and higher-dimensional prism products). I just can't envision any way closure could happen in a CRF way unless the other cells can somehow line up exactly along the peculiar angles of J90, besides J90 itself. But J90 itself has coordinates that are so specific that I can't envision how it could be used to bridge the gap with other J90's in a CRF way, other than direct closer in the J90 prism. When the coordinates and angles involve 12th degree polynomials, it just seems really far-fetched that edge lengths / coordinates could resolve in any other way than the extremely specific way said coordinates were obtained in the first place.

Furthermore, as far as CVP is concerned, I wonder if it makes sense to claim that J85 has CVP 6 and J90 has CVP 12. I think the original argument was that the CVP would be reducible to the maximum of its prime factors, but given the irreducibility of these polynomials and the fact that they are higher than the 5th polynomials that are known to have roots that cannot be expressed in terms of radicals, I'm doubting that their inherent complexity can be rationalized into mere cubics. If anything, their large coefficients that AFAIK are virtually "randomly" distributed makes it unlikely that they fall into the special reducible cases that are products of lower polynomials.

Thoughts?
quickfur
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### Re: Algebraic coordinates for 3D crown jewels

Brief update: due to a current limitation in my utility, I've not been able to solve the J88 system just yet. But I did manage to squeeze out a defining polynomial for one of its variables. It turns out to be a 34th-degree polynomial(!) with terms of even degree (so a 17th-degree polynomial if we take a square root afterwards -- and since 17 is prime, this surely puts J88's CVP at 17 (!)), and with absolutely humongous coefficients: many 12-digits coefficients, and one 13-digit coefficient.

I'm currently working on fixing my utility so that it can extract defining polynomials for the other variables as well (it's a 7-variable system, so highly-nontrivial). But the fact that we're dealing with a 17th degree polynomial (or 34th degree in its original form) surely means that it's probably impossible for any polytope containing J88 to close up in a CRF way except if the other polytope elements are perpendicular to it (i.e., prisms and higher-dimensional prism products)?

Now, there remains the possibility that the 34th (or 17th) degree polynomial can be factored into a product of polynomial factors of composite degree -- the coefficients are so large that the polynomial exceeds Wolfram Alpha's online input size limit, so currently I don't know whether it's factorable -- but I consider this quite unlikely. If it turns out to be irreducible, and I consider the chances of that being very high, then we have on our hands an example of a polytope with CVP > 3, in fact, CVP = 17.
quickfur
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### Re: Algebraic coordinates for 3D crown jewels

Haha... my ignorance is showing. As it turns out, one A. V. Timofeenko back in 2008 has already derived coordinates and their associated defining polynomials for all the 3D crown jewels[*], and the polynomial for the sphenomegacorona (J88) is an irreducible 16th degree polynomial, not 17th degree.

The 17th degree thing may have been either a partial result, or an incorrect partial result, as I have since discovered that there is an inconsistency in my input data that's causing the algorithm to fail. I've yet to fix this problem. The coordinates I obtained for J90 are consistent with Timofeenko's results, though he did find less scary presentations for J90's coordinates (but still with 12th-degree polynomials).

(Note[*]: linked article is in Russian.)
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