Algebraic coordinates for 3D crown jewels

[quickfur]

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]

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]

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.)

[mr_e_man]

Quickfur, could you run your programs to find algebraic representations of other numbers related to these polyhedra? such as the squared distances between vertices, or the cosines of dihedral angles? Maybe these numbers have simpler polynomials; or maybe the coordinates can be expressed more easily in terms of these.

[quickfur]

Perhaps there are, but I doubt it. If there were a simple relationship hidden somewhere in there, it would have caused the relevant polynomials to be simplified. The process of finding the Gröbner basis -- the prerequisite to deriving the polynomials for these coordinates -- basically involves mutually reducing the starting polynomials recursively, in a scheme that generalizes Gaussian elimination, and effectively considers all combinations of polynomials and their mutual reductions of each other. If there were any hidden connections between them, they would have turned up during the Grobner basis computation, and would have acted as reducing factors to simplify the resulting polynomials.

The fact that in spite of these reductions the degree of the resulting polynomials still turn out so high indicates that the likelihood of a simple algebraic connection between vertices is low.

[mr_e_man]

http://hi.gher.space/wiki/Sphenocorona

I didn't know that the sphenocorona is constructible! You might want to include that on your page. I had the impression that it was CVP 3 or 4, but it's actually 2.

(Is there any such thing as CVP 4, considering solvability of quartics by complex cube roots? I guess that depends on the definition of CVP.)

[quickfur]

I'm pretty sure the sphenocorona is CVP 4 (whatever that means). The polynomials involved are degree 8 in x^2 and degree 4 in x. There used to be this idea that we only consider prime factors in the polynomial degree, but these days I'm starting to think that's not entirely an accurate measure of the true complexity of a polytope. But anyway, I'm doubtful of the concept of CVP these days, so this may not really be saying very much. But there definitely seems to be something about the polynomial degree that seems to somehow have some kind of correlation with complexity, even though I still don't know how to define it in an adequate and useful way.

[mr_e_man]

I must revise my definition again; and also not call it CVP, since it's not quite what you're after.

The reduced degree, of an algebraic number α, is the smallest number n, such that α is contained in some chain of fields F>G>...>ℚ, where each field extension has degree n or less. (By default, these are sub-fields of the complex numbers. We may modify the definition to require them to be real, but I don't know whether that makes a difference.)

For example, the number α = √(2 + √3) has minimal polynomial x⁴ - 4x² + 1, so it has degree 4; but it has reduced degree 2, because it's expressible using square roots (degree 2 field extensions). Any rational number has reduced degree 1. Due to the quartic formula, any number with degree 4 has reduced degree 3 or less. The reduced degree cannot exceed the ordinary degree.

I define the reduced degree of a polyhedron (which I previously called CVP) as the maximum of the reduced degrees of its (squared?) chord lengths. So the sphenocorona, or anything constructible, has reduced degree 2 or 1.

This differs from my linked definition, in that the fields are no longer required to be contained in K, which is generated by the chords.

[quickfur]

Do you have the minimal polynomials for the chords of the sphenocorona? I'm curious, if they are indeed only quadratic, then that might be a strong indication that it might be possible to construct a CRF with sphenocorona cells!

[mr_e_man]

No, the minimal polynomials are at least quartic. Using the variables here, A itself (or 2A) is a chord, and it involves √(213 - 57√6), which cannot be expressed using just a single square root.

"Constructible", or reduced degree 2, means it may be expressed using any number of square roots. An algebraic number with degree 2 uses a single square root.

[mr_e_man]

Okay, here are the polynomials for all squared distances between vertices of a sphenocorona with edge length 1:

x - 1; (for an edge)

x - 2; (for the diagonal of a square)

2x² - 4x - 1; (x≈2.224744)

225x⁴ - 3756x³ + 16774x² - 25580x + 9025; (x≈2.492783)

15x⁴ - 96x³ + 116x² + 96x - 36; (x≈2.578855)

15x⁴ - 84x³ + 62x² + 180x - 81; (x≈2.705453)

225x⁴ - 3576x³ + 18136x² - 30944x + 8464; (x≈2.908572)

225x⁴ - 4476x³ + 30214x² - 78844x + 61345; (x≈3.908572)

These are not simpler than the coordinate polynomials.

All of these numbers are contained in the field generated by √(213 - 57√6), or equivalently by √(538 + 18√6), or more simply by √(13 + 3√6). The relations between these are

√(538 + 18√6) = √(213 - 57√6) (6 + √6)/3,

√(13 + 3√6) = √(213 - 57√6) (3 + 2√6)/15,

√(213 - 57√6) = √(13 + 3√6) (-3 + 2√6).

[quickfur]

Hmm. Is there by any chance a "simple" (low-degree polynomial) relationship with the chords of the augmented triangular/hexagonal prisms? I've been suspecting for a while that there's some kind of connection there, but have not had the time to investigate further.

[mr_e_man]

Indeed, their chordsquare fields are just ℚ(√6), which is a sub-field of the sphenocorona's ℚ(√(13+3√6)). In the triaugmented triangular prism, the chordsquare between a pyramid's tip and an opposite vertex of the prism is (3+√6)/2, and the chord between two pyramids' tips is (1+√6)/2.

You may ask why I want to square everything. It's because squaring vectors is equivalent to taking dot products:

||v||² = v∙v,

2u∙v = ||u+v||² - ||u||² - ||v||²,

and the dot product behaves nicely and can be used for many geometric constructions. It's also because the chordsquare field is generally much smaller than the chord field; for example, the truncated octahedron's chordsquares are integers, but its chords include √2,√3,√5,√7, making a 2⁴=16 dimensional field.

[quickfur]

BTW, I just realized my original post is quite outdated by now; since the time I wrote it, I've solved the polynomial systems for all the Johnson solids, including the crown jewels (a few with the help of Timofeenko's paper). The coordinates are all posted on my website's Johnson solids pages.

If you look at the Johnson solids pages, particularly those of the crown jewels, you'll see that things like J88 or J90 have an inherent complexity to them. The coordinates themselves come from roots of polynomials of a certain minimal degree, and even though the coordinates in and of themselves are somewhat artificial, being based on some arbitrarily-chosen coordinate reference frame, nonetheless they do serve as indicators of the inherent complexity -- because the chords of the polyhedra are algebraic combinations of the coordinate values; and the coordinates were chosen such that for the most part they maximize the symmetry of the polyhedron w.r.t. the coordinate axes. So for example, the coordinate values of J88 involve roots of irreducible 16th degree polynomials: so no matter what, the chords are not going to be significantly simpler -- since if there were such a relationship between the different coordinate values, it would have been possible to factor out some polynomial factor (e.g., the polynomial that defines the chord in terms of the roots) and divide through the defining polynomials, thus reducing their degrees, which would contradict the minimality/irreducibility of those 16th degree polynomials.

So even though we can't pinpoint exactly how the exact degree of the defining polynomials correspond to polyhedral complexity, there's definitely some kind of correlation there, where a higher-degree polynomials implies some kind of inherent complexity to the polyhedron.

[mr_e_man]

I forgot to post an update here.

The reduced degree, of an algebraic number α, is the smallest number n, such that α is contained in some chain of fields F>G>...>ℚ, where each field extension has degree n or less. (By default, these are sub-fields of the complex numbers. We may modify the definition to require them to be real, but I don't know whether that makes a difference.)

It does make a difference. Apparently there is a real number with complex-reduced-degree 5, but real-reduced-degree at least 10.
https://math.stackexchange.com/question ... intics-can

Also, the reduced degree of α can be determined by looking at the Galois closure of ℚ(α) (that is, the splitting field of the minimal polynomial of α); let's call this field G. We don't need to consider infinitely many fields. If α is contained in a chain F₁⊇F₂⊇...⊇ℚ where each extension has degree n or less, then it's also contained in the chain (G∩F₁)⊇(G∩F₂)⊇...⊇(G∩ℚ) of sub-fields of G, and these extensions still have degree n or less. Also, if the original chain of fields is real, then of course the intersection with G is still real. Thus, the (real or complex) reduced degree of α will appear in the lattice of sub-fields of G.

You wanted to find a way to express complicated algebraic numbers in terms of simpler algebraic numbers. (More discussion of that here.) This is one way to do it. For example, if α has degree n², and you want to write α=μ+ν where μ and ν have degree n, then α must be contained in the chain of fields ℚ(μ,ν)⊇ℚ(μ)⊇ℚ where the two extensions have degree n. (If the Galois closure shows that α in fact has reduced degree greater than n, then the decomposition α=μ+ν is not possible.)

Magma has functions dealing with this, e.g. computing sub-fields: https://magma.maths.usyd.edu.au/magma/handbook/text/423
(EDIT: It looks like the page has moved to .../text/425 . But I can't keep updating this.)