Interesting convex polyhedra with 1:1:√2 triangles

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby mr_e_man » Mon Aug 31, 2020 11:20 pm

quickfur wrote:
mr_e_man wrote:[...]
It's interesting (but perhaps expected?) that the polynomials bulge outward rather smoothly, with smaller coefficients at the ends, and smallest at the highest-degree term.
[...]

This isn't always the case, but generally yes, that's expected. It's a similar phenomenon to binomial expansions.


Here's something related on MathOverflow. I wasn't searching, but I stumbled upon it anyway.
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby mr_e_man » Mon Aug 31, 2020 11:40 pm

quickfur wrote:Last night I ran into a case I could not solve. :(

[...]

Or perhaps it's time to reformulate the problem in a more sophisticated way than the current hamfisted approach of just coming up with a 13-variable polynomial system and handing over the whole thing to Singular to solve. :lol: :D (Either that, or upgrade my PC to have more RAM and hope for the best. :P)


Well, there are plenty of possible ways to describe a polyhedron; maybe some of them will have fewer variables:

Vertex coordinates. Distances between vertices (discussed earlier). Face normal vectors and plane distances from the origin. Dihedral angles, or their cosines or complex exponentials. Quaternions for face orientations.

The last two descriptions, unlike the first two, exploit the fact that the faces have fixed shapes.
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Tue Sep 01, 2020 12:19 am

It's true that some formulations are simpler than others, e.g., an icosahedron in general position/orientation will have a lot more variables than an icosahedron in pyritohedral orientation (where there is a lot of mirror symmetry w.r.t. coordinates axes, along which the variables to be solved are placed). This incidental complexity can be eliminated by identifying the essence of your system and formulating it such that the incidental complexity is eliminated.

However, for every polyhedron there's an irreducible "core" or inherent complexity that cannot be reduced any further. E.g., an icosahedron will inherently involve the golden ratio, a quadratic irrational; no matter how much you try to reduce that you will not be able to get away from that. (The only way you can get rid of the quadratic complexity altogether is if you assumed it as an axiom in the first place, e.g., by working in a coordinate system or formulation where (1+√5)/2 is built into the definition. But that's merely moving the quadratic complexity into the definition; it doesn't really get rid of it.)

In this case, I think you may be right that there's some incidental complexity that might possibly be eliminated. Orientation comes to mind: the way I set it up, the hexona complex is aligned to the vertical axis and the horizontal axis -- I chose this mainly to simplify most of the hexona vertex coordinates. However, this forces the luna and esp. the megacorona to be in a much less symmetry orientation, which in all likelihood introduces a lot of incidental complexity to the system. Probably I should try to see if the system might become more tractible if I orient it so that the megacorona is in maximally symmetric orientation (because that's the part with the majority of the polynomial constraints). Perhaps I might even be able to reduce the number of variables in the system.

Another, more radical way, is to break the problem into smaller pieces: the hexona complex, the luna, and the megacorona can all be treated as self-contained complexes, each with some number of degrees of freedom. You'd then derive, say, some trigonometric equation describing say the width of the hexona complex, and ditto for the luna and megacorona, with 1 or 2 degrees of freedom, then equate various parts where they interface (e.g., one of the megacorona's parameters could be a width that matches the width of the hexona complex, so you could equate the two trigonometric expressions). This should reduce the number of variables, which hopefully will bring it more within the realm of solvability. The trig equations can be easily translated into a polynomial system, hopefully simpler than the one I have right now, and afterwards I can solve for the various parameters (widths, angles, etc.), and plug it back into the trig equations to derive the coordinates for the points within each complex.

This does mean the solving won't be automated, though. :\ :sweatdrop:
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Tue Sep 01, 2020 12:29 am

I'm very tempted, however, to implement a ball-n-springs Monte Carlo style numerical solver, though. Have a set of points (with some designated as immovable, to have some control over the final orientation), add a bunch of constraints between them, and place the points in some initial guessed configuration. Then iterate over the initial guess by computing the amount of "spring force" between constrained points -- the springs would be set up, of course, such that the force will be 0 when the distance between two points is at the desired value; compressed springs (length is too short) causes repulsion, and stretched springs (length too long) causes attraction. At each iteration move the points in the direction of the total accumulated force. Repeat until the points stop moving (up to some given tolerance). If there is a solution, the resulting point positions should represent it (or at least one of the solutions, if multiple solutions are possible).

Disadvantages: if initial configuration is too far away from the solution, the simulation may fail to find it (gets stuck in a local minimum instead). Also, must be careful about termination conditions otherwise can get stuck in an infinite loop. And also may be vulnerable to numerical stability problems. And when there are multiple solutions, does not give information about how many (other) solutions there are, and may be tricky to coax the simulation to converge to a different solution (it may be very sensitive to initial conditions).

But the advantage is that it doesn't suffer from exploding algebraic complexity in large systems. :)
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby mr_e_man » Tue Sep 01, 2020 2:33 am

quickfur wrote:I tentatively named it hexonalunamegacorona: it consists of a hexona complex (4 Q-triangles + 2 equilaterals as a pseudo hexagonal pyramid), one side of which is attached to a modified luna that has Q-triangles instead of equilaterals. This complex has a hexagonal rim with 2 long edges (length √2), which may be closed up with a modified megacorona complex with 12 triangles, 2 of which are Q-triangles to mate with the 2 long edges of the modified luna. The resulting polyhedron has 1 square face, 8 Q-triangles, and 12 equilateral triangles, and has an overall shape like a bent american football or croissant-like shape, with only bilateral symmetry. Very interesting shape indeed, but I'm unable to solve the actual coordinates, so no pictures. :cry:


hexonalunamegacorona.png
hexonalunamegacorona.png (34.59 KiB) Viewed 1570 times
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Wed Sep 02, 2020 5:27 pm

So yesterday I threw together a simple ball-n-springs simulator, and with some tweaks, I could solve many of the polyhedra posted here numerically, and with fairly good accuracy too. And it runs very fast, compared to the polynomial solver.

However, I soon ran into a wrinkle: some of the polyhedra have dependencies between coordinates, e.g., <0, A, B> and <-A, 0, B>, which a simple ball-n-springs model does not express. In order to remedy this, I introduced the idea of a Bar, which behaves sort of like an equalizer between two coordinates of two different balls. It always applies force proportional to the average of two bound coordinates such that it tries to center them on the origin. It's a simplistic hack, but with some care, it does work; I could successfully solve for things like sphenohexocingulum and so forth, with good accuracy and very fast speed (compared to the hours it took to solve it algebraically :lol:). Here, however, I began to run into some of the limitations I mentioned about numerical solvers: in certain systems that depends too much on Bars, the solution converges very slowly -- sublinearly, probably even sublogarithmically. Generally, springs cause fast convergence, but Bars cause slow convergence (unless they're complemented by springs). So Bars are really only useful for maintaining the relationship between +X and -X (to prevent lopsided "solutions" -- local minima in which their absolute values no longer correspond).

Anyway, after getting it to solve bimegalunamegacingulum (aka tetrafolium), I finally felt confident enough to attempt hexonalunamegacorona... only to run into the most feared problem in ball-n-spring simulators: convergence to the wrong solution among multiple local minima. I'm still not 100% sure this is an inherent problem -- in the process I did discover a mistake in one of the constraints, so it's possible the problem is caused by an error in another constraint -- but it appears to be converging on numerous non-convex solutions, and is very hard to control which solution it converges to. It's highly sensitive to initial conditions and also spring strength: having too high of a spring strength causes convex/concave oscillations in the early iterations that throws off the solution path, but having too low a spring strength slows down convergence significantly. Once the right spring strength has been found, it seems to be highly sensitive to the initial ball positions; a small shift can dramatically change the solution path and lead to very different solutions (all of which, so far, appear to have concavities in one place or another, and it's very hard to control where it will go). There are also many initial conditions that cause the system to diverge altogether.

So right now, I'm stuck trying to find the right initial conditions that will yield the desired solution. And also wondering if I should review the constraints yet again to make sure I didn't screw up the formulation of the system! :sweatdrop: There's also the ever-present threat that perhaps this high sensitivity to initial conditions is a sign that there's no convex solution after all. :(
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Wed Sep 02, 2020 7:21 pm

I hang my head in shame. I had indeed screwed up my constraints: there was a completely wrong constraint connecting the wrong vertices, which screwed up the entire structure, as well as a wrong variable range that caused an eversion of one of the megacorona's vertices. :oops: :oops: :oops:

But on a positive note: after fixing those mistakes, I found that the extreme sensitivity to initial conditions seems to have gone away, and the Monte Carlo simulator was able to converge on the solution with default settings (no spring strength tweaks and so forth). Meaning to say, I have a numerical solution!!! :D :XD: :nod: :evil: :lol: 8) And so, here it is, the beast itself:

14) Hexonalunamegacorona:

Image

I've already described the structure, but now you can see for yourself its american-football-like shape. It's not quite, though; it's more bent, almost like a croissant shape. And is somewhat flat, with two sides stretched out. And looks like some kind of weird modification of a regular icosahedron.

And in retrospect, the megacorona looks quite asymmetrical (except for the same global bilateral symmetry), so reformulating the system to center on it probably would not have significantly simplified it.

Anyway, I'll post more pictures later. Currently attempting to solve the system algebraically again, on the off-chance that fixing the mistakes in the original system may have removed some of its inherent complexity, and brought the solution within reach in a reasonable amount of time. The absence of high sensitivity to initial conditions implies fewer local minima, which might mean that the polynomials involved would also be lower-degree. So I hope, anyway.

Edit: Singular is still running (and has been for a long time), but I'm noticing that the memory usage pattern seems to be less than before. There have been some plateaus in total memory usage, as opposed to a steadily increasing usage before, which could mean Singular is finding some sanely-sized polynomials in the Gröbner basis, as opposed to being lost in a sea of virtually unbounded growth. Promising signs!
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Wed Sep 02, 2020 8:00 pm

Ahhh, in my excitement, I totally forgot to actually post the coordinates. :lol: :oops: :roll: So here they are:

Code: Select all
<0, 0, -A>
<±1, ±B, 0>
<±C, 0, D>
<±1, E, F>
<±G, H, J>
<0, N, P>
<0, Q, R>

where
   A = 1.175639098977154
   B = 1.271956252768225
   C = 2.542066641385793
   D = 0.064480730214184
   E = -1.373214507591875
   F = 1.997435046711174
   G = 1.601963552734183
   H = 0.518186256475521
   J = 1.751990489091918
   N = 0.117016787163775
   P = 2.880164160386993
   Q = 1.713081764310372
   R = 1.674935307128810

These coordinates are accurate to approx 1e-14 or 1e-15, or thereabouts (close to hardware precision of 64-bit floating-point).

Another interesting fact: the dihedral angle at the two triangles on the front right, that look almost coplanar, is 176.744255°, which has an angle defect of only 3.26°. Not as extreme as the < 1° angle defect found in dihexonacingulum, but getting pretty close. This was the part I said I wasn't completely sure was convex, because the angle defect is small enough to be ambiguous in the model I made from my son's coarse magnetic tiles. But happily, it turned out to be convex after all.

More interesting facts: the cluster of 5 triangles at the top is almost a Johnson pentagonal pyramid; the chord (the value of G) is 1.6019, which is within less than 1% of the Golden Ratio. Just a tiny bit more, and it could have been diminished to yield a polyhedron with a regular pentagon. ;) But it's just a teeny bit off, so we can't.
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Wed Sep 02, 2020 9:39 pm

So, Singular still failed to find a solution. :\ :glare: :angry:

But on another interesting note: thanks to my new Monte Carlo numerical solver, I can now experiment with various parameters without getting stuck in polynomial hell, and I just discovered that hexonolunamegacorona can be constructed without √2 edges in the luna and megacorona! Here's the alternate version of hexonolunamegacorona, in which the luna and megacorona are all unit-edged, and only the hexona complex has Q-triangles:

Image

This version is even flatter, but somewhat less elongated on the sides. It has 16 equilateral triangles, 4 Q-triangles, and 1 square.

The coordinates are the same as the first variant, since they are topologically identical, except that the values of the variables are as follows:
Code: Select all
   A = 1.053149866281406
   B = 1.375091036677738
   C = 2.363067250593586
   D = 0.501170940107053
   E = -0.591255502286166
   F = 1.840000504082833
   G = 1.688032667597141
   H = 1.286330022693727
   J = 1.875828491817867
   N = 0.821260074170541
   P = 2.842397504411134
   Q = 2.281612061321615
   R = 1.475879274154433
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Thu Sep 03, 2020 12:48 am

Found another one:

15) Spicamegacorona:

Image

This one is a larger version of spicamesocorona, with a 16-triangle corona rounding out the other side of the spike of 6 Q-triangles. The dihedral angle between the Q-triangles is 178.286826°, only about 1.7° away from becoming coplanar and merging into squares. Arguably a Johnson near miss? :-)

Due to the high symmetry, the system was relatively simple and was solvable with Singular. The polynomials were not pretty, though. :D They are 48th degree in x^2 and 24th degree in x, with gigantic coefficients. Anyway, here are the coordinates:

Code: Select all
<0, 2/√3, 0>
<±1, -1/√3, 0>
<0, -2*A/√3, B>
<±A, A/√3, B>
<0, 2*C/√3, D>
<±C, -C/√3, D>
<0, -2*E/√3, F>
<±E, E/√3, F>
<0, 0, G>
   
where
   A = 1.66913403130821
   B = 1.08512709967512
   C = 1.44363896309135
   D = 1.93328200061527
   E = 1.41405551968898
   F = 3.06331986173855
   G = 4.21827843948763

Here are the solution polynomials (not for the faint of heart :lol:):
Code: Select all
        3675*A^24 + 32760*A^23 - 1056332*A^22 + 7183576*A^21 - 13089294*A^20 - 61182104*A^19 + 318010804*A^18 - 191037832*A^17 - 1774150661*A^16 + 3855738848*A^15 + 2180728160*A^14 - 16254051536*A^13 + 12643322672*A^12 + 25447074176*A^11 - 50778610016*A^10 + 2867230784*A^9 + 68010537632*A^8 - 48488595200*A^7 - 31189214464*A^6 + 41324772608*A^5 + 2728022016*A^4 - 11672278016*A^3 + 169823744*A^2 + 762872832*A + 42189568        [1.669 < A < 1.670]
        64596985700625*B^48 + 52991347700793600*B^46 + 290251605548162304*B^44 + 5700872430769683456*B^42 + 111650086740888686592*B^40 - 1722098880857707315200*B^38 - 7935957816985852968960*B^36 + 106941263707923041746944*B^34 + 158051897295228144451584*B^32 - 3082911414758987626708992*B^30 + 963796839409308414246912*B^28 + 46013238972601491970326528*B^26 - 78009024775824485948325888*B^24 - 308790370171299192977227776*B^22 + 1034171644059969249293107200*B^20 + 83612748633466029981302784*B^18 - 4427457087669198409967861760*B^16 + 6694599718327193240202117120*B^14 - 1258795372872244486612189184*B^12 - 5330676628579351675164164096*B^10 + 4022753829256876529274585088*B^8 + 698470556708024606036852736*B^6 - 1464103568766113139199574016*B^4 + 176106786027079988661977088*B^2 + 113821133301270803115933696 [1.085 < B < 1.086]
        2304*C^24 - 15360*C^23 - 53760*C^22 + 547840*C^21 + 190976*C^20 - 8180992*C^19 + 5738752*C^18 + 67835648*C^17 - 84268064*C^16 - 348072512*C^15 + 552502880*C^14 + 1164150400*C^13 - 2130954544*C^12 - 2608597168*C^11 + 5179907104*C^10 + 3991372576*C^9 - 7997770111*C^8 - 4262815712*C^7 + 7562399648*C^6 + 3235688192*C^5 - 3954359936*C^4 - 1647962112*C^3 + 863594496*C^2 + 403914752*C + 6688768      [1.4 < C < 1.5]
        31381059609*D^48 + 1245944292624*D^46 + 11042000497152*D^44 - 98228255254272*D^42 - 1438448004495936*D^40 + 3875147504775936*D^38 + 75800564545817088*D^36 - 141917991316678656*D^34 - 2190571145960569344*D^32 + 4836428210890027008*D^30 + 35388420407958159360*D^28 - 106348476148384333824*D^26 - 283094521586045632512*D^24 + 1295518034807944445952*D^22 + 459627252692985446400*D^20 - 8222169542467203366912*D^18 + 10760099870768181608448*D^16 + 7190261192114999132160*D^14 - 29493400252457492152320*D^12 + 29587326637230620934144*D^10 - 15612816594986933944320*D^8 + 4625403017985005715456*D^6 - 589994511256065671168*D^4 - 4453456399585771520*D^2 + 27101862113050624 [1.93 < D < 1.94]
        12544*E^24 - 136192*E^23 - 610560*E^22 + 18168064*E^21 - 54991008*E^20 - 385377472*E^19 + 1538432624*E^18 + 1084177904*E^17 - 10529345855*E^16 + 6913627392*E^15 + 30813874976*E^14 - 49457062016*E^13 - 37794056256*E^12 + 128854356992*E^11 - 3745782272*E^10 - 172928882688*E^9 + 58910426624*E^8 + 122131873792*E^7 - 56266399744*E^6 - 39808106496*E^5 + 16729489408*E^4 + 3719561216*E^3 + 250216448*E^2 + 6815744*E + 65536   [1.414 < E < 1.415]
        930196594089*F^48 + 149287061549304*F^46 + 13577424641963676*F^44 + 1740244727457819144*F^42 + 23800686541694303382*F^40 - 824096555806235959224*F^38 - 3514273171796951981604*F^36 + 86975954316277133452344*F^34 + 284012027488322597765961*F^32 - 2319739040887870816069632*F^30 - 19220133914515638774525312*F^28 - 115806229837034691960953856*F^26 + 931579439603819594958056448*F^24 + 7975483563544583430066241536*F^22 - 27928012940988302573810122752*F^20 - 148940159052187558852570644480*F^18 + 486504624042825337081965182976*F^16 + 726102725031837674294728458240*F^14 - 3388038448665333081603833856000*F^12 + 2972687096502795689624082579456*F^10 - 838675605205949203929374916608*F^8 + 79124838301236348018862587904*F^6 + 1604376357471873910574153728*F^4 - 747034542388463141375705088*F^2 + 50325224100889303443308544   [3.06 < F < 3.07]
        3486784401*G^48 - 297538935552*G^46 + 9917964518400*G^44 - 152736653583360*G^42 + 536892479262720*G^40 + 25531044662476800*G^38 - 719087750230966272*G^36 + 12224152593824219136*G^34 - 161174213511578910720*G^32 + 1805197980052802764800*G^30 - 17099966297782005792768*G^28 + 130006768588968645623808*G^26 - 819765159867408256598016*G^24 + 4149198867038566210338816*G^22 - 6875228111439614699372544*G^20 - 106379709716088954342604800*G^18 + 719362852825154764419366912*G^16 - 376480280347722180187914240*G^14 - 6138928102362484373786198016*G^12 + 3282431656459654165369454592*G^10 + 20564981141251910783519948800*G^8 + 21730223530164520021024309248*G^6 + 5924669700000883774857412608*G^4 - 1008683313641507608709824512*G^2 + 30983446494107742247059456      [4.2 < G < 4.3]
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Thu Sep 03, 2020 5:02 am

quickfur wrote:[...]
15) Spicamegacorona:
[...] The dihedral angle between the Q-triangles is 178.286826°, only about 1.7° away from becoming coplanar and merging into squares. Arguably a Johnson near miss? :-)[...]

Whoa, in fact, this one is listed as in Orchid Palms as near-miss #4. :o_o: :nod: :D
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Thu Sep 03, 2020 4:43 pm

And yet another related one:

16) Snub dispica:

Image

Basically two spica complexes in anti orientation joined together by a band of triangles. Can be considered a kind of Siamese twin of two copies of (15).

The dihedral angle between the Q-triangles' long edges is about 173°, and the larger dihedral angle between the triangles in the equatorial band is about 172°. So pretty close to the coplanar boundary.

Coordinates:
Code: Select all
<0, 0, ±A>
<0, 2*B/√3, C>
<±B, -B/√3, C>
<0, -2*B/√3, -C>
<±B, B/√3, -C>
<0, -2*D/√3, E>
<±D, D/√3, E>
<0, 2*D/√3, -E>
<±D, -D/√3, -E>

where
   A = 2.68284390955696
   B = 1.4116292331122
   C = 1.5239347050472
   D = 1.5273472128174
   E = 0.471596752709668

Due to the higher degree of symmetry, the solution polynomials are quite nice, compared to the ones we've been seeing. They're only degree 8 in x^2 and degree 4 in x:
Code: Select all
      3*A^8 - 14*A^6 - 93*A^4 + 216*A^2 + 432         [2.6 < A < 2.7]
      4*B^4 - 28*B^3 + 7*B^2 + 80*B - 64              [1.4 < B < 1.5]
      81*C^8 + 1836*C^6 - 2268*C^4 - 5760*C^2 + 256   [1 < C < 2]
      D^4 - 11*D^2 + 8*D + 8                          [1 < D < 2]
      81*E^8 + 270*E^6 - 63*E^4 - 288*E^2 + 64        [0.4 < E < 0.5]
quickfur
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Thu Sep 03, 2020 9:32 pm

More pretty Q-polyhedra:

17) Trigonal hebesphenoscindocupola (aka Stott-expanded spicamesocorona):

Image

A kind of modification of the elongated trigonal cupola, augmented with penti clusters (2 Q-triangles + 3 triangles), slightly deformed in order to accomodate the width of the augments. Very reminiscient of the triangular hebesphenorotunda (J92), hence the parallel naming scheme. It can also be understood as the Stott-expansion of spicamesocorona (3).

Coordinates:
Code: Select all
<0, 2/√3, H>
<±1, -1/√3, H>
<±A, (A+2)/√3, B>
<±1, -(2*A+1)/√3, B>
<±(A+1), (A-1)/√3, B>
<0, 2*C/√3, D>
<±C, -C/√3, D>
<±1, ±√3, 0>
<±2, 0, 0>

where   
   A = 1.39634279970237
   B = 1.94693344694474
   C = 2.67898396031137
   D = 1.07082324799101
   H = 3.13027717594284

Solution polynomials:
   16*A^8 - 96*A^7 + 24*A^6 + 488*A^5 - 167*A^4 - 760*A^3 + 96*A^2 + 320*A + 64 [1.3 < A < 1.4]
   729*B^16 + 4860*B^14 - 38394*B^12 - 126468*B^10 + 552969*B^8 + 1024608*B^6 - 2931840*B^4 - 755712*B^2 + 2166784      [1.94 < B < 1.95]
   15*C^8 - 100*C^7 + 26*C^6 + 964*C^5 - 1393*C^4 - 1784*C^3 + 2592*C^2 + 1728*C - 576  [2.67 < C < 2.68]
   54675*D^16 + 1529280*D^14 + 7454592*D^12 - 13879296*D^10 - 90771456*D^8 + 132775936*D^6 + 213909504*D^4 - 452984832*D^2 + 201326592  [1.07 < D < 1.08]
   2187*H^16 - 58320*H^14 + 396576*H^12 - 518400*H^10 + 12061440*H^8 - 49987584*H^6 - 484638720*H^4 - 253755392*H^2 + 616562688 [3.1 < H < 3.2]


18) Stott-expanded trihexona:

Image

A progression from (17) by adding another cupolaiec cap on the bottom, and merging the penti clusters into hexona complexes. Basically a Stott-expansion of (6).

Coordinates:
Code: Select all
<0, 2/√3, ±H>
<±1, -1/√3, ±H>
<±A, (A+2)/√3, ±1>
<±1, -(2*A+1)/√3, ±1>
<±(A+1), (A-1)/√3, ±1>
<0, 2*B/√3, 0>
<±B, -B/√3, 0>
   
where
   A = 1.40846620713632
   B = 2.57725498617837
   H = 2.16402917681362
   
Polynomials:
   4*A^4 + 4*A^3 + A^2 - 12*A - 12      [1.4 < A < 1.5]
   4*B^4 - 8*B^3 - 17*B^2 + 18*B + 27   [2.5 < B < 2.6]
   3*H^4 - 6*H^3 - 17*H^2 + 16*H + 40   [2.1 < H < 2.2]
quickfur
Pentonian
 
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Thu Sep 03, 2020 11:03 pm

The shape of the variant of hexonalunamegacorona with unit edges in the luna and megacorona made me wonder if there's room to fit another hexona complex in the place of the luna. Turns out, it's indeed possible:

19) Dihexonamegacorona:

Image

Beautiful thing, isn't it? Very reminiscient of the regular icosahedron. It has 4 degree-6 vertices, 2 degree-4 vertices, and 8 degree-5 vertices. The dihedral angle at the long edges is about 172.12°, just a little over 7° away from being coplanar. Betcha this is another near-miss Johnson. ;) :D

Coordinates:
Code: Select all
<±1, 0, 0>
<±A, 0, B>
<0, ±C, D>
<±1, ±E, F>
<±G, 0, H>
<0, ±1, J>

where
   A = 2.12796336485761
   B = 1.65157459641976
   C = 1.59171679265575
   D = 0.682962408905282
   E = 1.46687373106241
   F = 2.41050812753967
   G = 1.66378199003398
   H = 3.59696289988692
   J = 4.07844964906969

Polynomials:
   893025*A^56 - 31026240*A^55 + 419597496*A^54 - 2160635472*A^53 - 8217455076*A^52 + 172454091120*A^51 - 768982902552*A^50 - 1336526431344*A^49 + 26480256205158*A^48 - 79505363714256*A^47 - 208420333298136*A^46 + 1966468921212784*A^45 - 2744380025386884*A^44 - 17553192010232432*A^43 + 72635603191619480*A^42 + 18837443224155056*A^41 - 698314078576746111*A^40 + 1124866556563619792*A^39 + 3348481462664023904*A^38 - 12964495821377515872*A^37 - 2326342915055662816*A^36 + 78145254282088356416*A^35 - 85092646824878117696*A^34 - 284647421890442084096*A^33 + 692662687031338508672*A^32 + 489943578353251507200*A^31 - 3132215021431489656832*A^30 + 1023318269986488774656*A^29 + 9458665992010428681216*A^28 - 10528832816826345017344*A^27 - 18852786923103266033664*A^26 + 40832727266813217685504*A^25 + 18428008951400213843968*A^24 - 104384037345109528412160*A^23 + 24277425299882136305664*A^22 + 193584328884521527148544*A^21 - 147705748275208165261312*A^20 - 266934380160825692258304*A^19 + 353611221687298204631040*A^18 + 274355384795418402488320*A^17 - 571068784604196749967360*A^16 - 213251620592762748928000*A^15 + 683214529123810474459136*A^14 + 141271671508113064722432*A^13 - 615102546934305919074304*A^12 - 107327649423249282957312*A^11 + 403869869505113887866880*A^10 + 93113264851534443708416*A^9 - 175888191501391221489664*A^8 - 60176193958372628758528*A^7 + 39878588549551882240000*A^6 + 19469019601194845208576*A^5 - 1554055188276272168960*A^4 - 1145951621233537712128*A^3 + 169256532995651665920*A^2 + 33074435663408922624*A - 5471873547255152640   [2.1 < A < 2.2]
   797493650625*B^112 + 23812111549800*B^110 - 630988864801524*B^108 - 9027028731885480*B^106 + 245702089326644478*B^104 + 103267377324712728*B^102 - 32412419572342887252*B^100 + 149253162674857338024*B^98 + 1759096828667127854637*B^96 - 15635967215463665329248*B^94 - 28286781409600443513360*B^92 + 683231050144144161853792*B^90 - 956393361664917363725720*B^88 - 15178479822078034737387552*B^86 + 46179417908855184292088240*B^84 + 217200514721286121756904224*B^82 - 823647899737985911062247022*B^80 - 3972553508321257876737848272*B^78 + 16005703959033566450679350184*B^76 + 71689730660861967259148515408*B^74 - 396864740694663077252190296268*B^72 - 471159966947985180623659776432*B^70 + 5884281950445339307371016597928*B^68 - 1931752331439212149578152378832*B^66 - 61964232777005625999508851532654*B^64 + 66856490149793477793780533539616*B^62 + 768849212726592954643905940583856*B^60 - 2551692220551568751784362356960288*B^58 - 2456141559887782361571585022336408*B^56 + 33967526913922326551453751346853216*B^54 - 83604620291221488429296888078382608*B^52 + 7699748133433789994440497109391776*B^50 + 496874028647498287979704282960340141*B^48 - 1518871141167684402433753910422533720*B^46 + 2229789462502053651015688219375479724*B^44 - 873896042199994972066173927262957032*B^42 - 3495266767965995251041545314174770946*B^40 + 8700851724716533801079613273106805080*B^38 - 9928660624374408563756069427091567092*B^36 + 4191664746756725779175143744950419560*B^34 + 5630569944375668157818410921767745601*B^32 - 12662735529690249323569968891402543616*B^30 + 12511103080929474482056756019786293248*B^28 - 6994983693038870480248417659333902336*B^26 + 1272136050357564752619410384603840512*B^24 + 1392857622203362777771926515284443136*B^22 - 1358112851840058776152898648133337088*B^20 + 510784544329606830447800855363059712*B^18 - 4972368435729540054832279275438080*B^16 - 84326077828528276327746516329431040*B^14 + 36772305053799124183146568163000320*B^12 - 4178368005878322185496595712180224*B^10 - 1982482222515286651379147627036672*B^8 + 872360280191861869757319729905664*B^6 - 123189647319200242298273094696960*B^4 + 456565947350209890692471193600*B^2 + 1079600980061354218741982822400   [1.65 < B < 1.66]
   15854469120*C^56 - 160054640640*C^55 - 981215477760*C^54 + 12755574718464*C^53 + 24670679072768*C^52 - 469460463910912*C^51 - 268643552460800*C^50 + 10613040592584704*C^49 - 964331806302208*C^48 - 165211534223523840*C^47 + 80039042618675200*C^46 + 1879819807473707008*C^45 - 1419012122769733376*C^44 - 16177261366654920960*C^43 + 15695791592513490496*C^42 + 107298457608549570400*C^41 - 126028558657319086815*C^40 - 552283413626264460640*C^39 + 779729087747639935200*C^38 + 2191722259891411198080*C^37 - 3841613960201591349888*C^36 - 6511009570909503049728*C^35 + 15415001162253466777600*C^34 + 13182727260812660563968*C^33 - 51290036266059100237824*C^32 - 11068873526057518235648*C^31 + 143728378637558259122176*C^30 - 35690588150736961404928*C^29 - 343586860770477530415104*C^28 + 185564631514881682571264*C^27 + 706146922782627175858176*C^26 - 469552380648662310584320*C^25 - 1247796044852570127073280*C^24 + 809530187592611238248448*C^23 + 1879713676264483648962560*C^22 - 1011160119547469135609856*C^21 - 2378335153932611333652480*C^20 + 911807100096068668358656*C^19 + 2484166276882992237379584*C^18 - 556492180605482691461120*C^17 - 2108043169089897652813824*C^16 + 170919549641105004822528*C^15 + 1434962460618972857368576*C^14 + 54640345008196957503488*C^13 - 775383437895021057015808*C^12 - 103336473521826843066368*C^11 + 328348598695365126914048*C^10 + 67542143188138014539776*C^9 - 106640876032519353401344*C^8 - 27045097547078595248128*C^7 + 25685424790778423541760*C^6 + 6884418563180653969408*C^5 - 4392522846161030086656*C^4 - 1019182610072452726784*C^3 + 497485629237854470144*C^2 + 71481133285624512512*C - 26517194605957480448   [1.59 < C < 1.60]
   251364191077033574400*D^112 + 14501604873881832652800*D^110 - 48429784224292310876160*D^108 - 4183095054921895262552064*D^106 + 17559884793157491072434176*D^104 + 494286172743617650253365248*D^102 - 3398011284828409567077466112*D^100 - 24066389148344369585229660160*D^98 + 279403408507686938703707504640*D^96 + 107594358545903975320705302528*D^94 - 10052099530462352961069102661632*D^92 + 24304044158617959598197111783424*D^90 + 177636839080711776058059230789632*D^88 - 863838014025608635014911798419456*D^86 - 1354511504474613446597902740405760*D^84 + 14889756152039510688606281534348160*D^82 - 5149129678713200080335250181339711*D^80 - 159051110452606626788312839746108984*D^78 + 238781608998172473633788913424746540*D^76 + 1139082279402503437228330303460618296*D^74 - 2924676997835464397338485432900908130*D^72 - 5519312270285879216453655831462694088*D^70 + 22385665863131512398565520860071004940*D^68 + 16400745749602920866581333056572061384*D^66 - 123131424449197410905411220849869270771*D^64 - 11178152580962546151719558515030992096*D^62 + 513106604375934529558509062858417219184*D^60 - 168824835202632069595657109347434578720*D^58 - 1658344047033334143015378899745247248152*D^56 + 1049166928013391109642291296411874399840*D^54 + 4189278856179012071649653826986504333616*D^52 - 3642622827823272105930968710244919551328*D^50 - 8235714742592972181871885026445551387374*D^48 + 8806019902577269696017997935117451985776*D^46 + 12383438846468987105223347298181401924968*D^44 - 15585721064708451801071419818067843432304*D^42 - 13743289850767616588786648108028001089036*D^40 + 20320751740612101902210156629357425068048*D^38 + 10488878268972299998608891781349823451880*D^36 - 19209669636213644029204503836845827926544*D^34 - 4608869603132757520867667133396729849646*D^32 + 12724505235544830080600650837069979270816*D^30 + 278698503958438072115142939442816834608*D^28 - 5583343137864702547609777046718040372896*D^26 + 860920087143830526638740440913848473320*D^24 + 1486651577177598921928795249094464007648*D^22 - 458733875125445237584233025305409315472*D^20 - 207237388093611056896478930105669029472*D^18 + 105762022700585783125992918922262337837*D^16 + 7801435553922015883901917902639728904*D^14 - 11848555397219704829231536960799293492*D^12 + 1436459031371053906413550954885357816*D^10 + 515903162590611489278668174719538334*D^8 - 188452642379540278897228723463718792*D^6 + 25342496268675398910912253134535020*D^4 - 1597929636450694341983225941789816*D^2 + 39417601043777067773010945292321   [0.68 < D < 0.69]
   15854469120*E^56 - 323129180160*E^55 + 1138753536000*E^54 + 21886540972032*E^53 - 220804053729280*E^52 + 228229926354944*E^51 + 6192053552414720*E^50 - 28144712164966400*E^49 - 39684149735563264*E^48 + 570058849029308416*E^47 - 720689896887103488*E^46 - 5326036679062198272*E^45 + 16409321351468322048*E^44 + 21853253526343715072*E^43 - 154942231174986667712*E^42 + 41181566523067759840*E^41 + 867609991030547718577*E^40 - 1065064516582337574544*E^39 - 3034891356886821378384*E^38 + 6696856605857617414560*E^37 + 6127214818882307084560*E^36 - 24279699016073317333056*E^35 - 5220719357161667392576*E^34 + 54863268322320089768832*E^33 + 11720298405126710324576*E^32 - 73624531201344864491776*E^31 - 169368402287996567020288*E^30 + 78084272176767094754816*E^29 + 1005605037164518070814976*E^28 - 353784993853763538152448*E^27 - 3578453293699217874263040*E^26 + 1866438725214038989260800*E^25 + 8896556977512066952339712*E^24 - 6088741729994262722617344*E^23 - 16261975624919497993814016*E^22 + 13684616764388423948894208*E^21 + 21697657466836670280237056*E^20 - 23294730222037108222590976*E^19 - 19248107283361740391186432*E^18 + 32732737703899108076421120*E^17 + 6637533897768499683524608*E^16 - 41324757061865621440954368*E^15 + 9182742859776628994605056*E^14 + 48406166440332399300575232*E^13 - 15718134632685993315008512*E^12 - 49325467657755980027920384*E^11 + 8350172193198624661831680*E^10 + 38783966232794634303569920*E^9 + 3178874240684026629193728*E^8 - 20428425012245666126102528*E^7 - 7204777822622216435531776*E^6 + 5614259423917831442399232*E^5 + 4023023046115022814052352*E^4 - 46958779987113122201600*E^3 - 771939372532249927876608*E^2 - 266082609933026137735168*E - 29758921646535782432768   [1.46 < E < 1.47]
   251364191077033574400*F^112 + 52216493298337972224000*F^110 + 4702698554155898807255040*F^108 + 239800764994006509623967744*F^106 + 7515452334328598160895639552*F^104 + 143388950877075577911581343744*F^102 + 1375116808101335450342903513088*F^100 - 3636479237491155370918574817280*F^98 - 281397735310511250111678054924288*F^96 - 3172284891866591199768162275950592*F^94 - 3639685975287214533761519652438016*F^92 + 278534928380225336434110744220925952*F^90 + 2842473432228172287635410843095384064*F^88 + 1907402495259351300135234931743842304*F^86 - 180006822776744866868375895290388444672*F^84 - 1349992435897158405178194692730041556608*F^82 + 1445568859678837116435360004817903888737*F^80 + 77527810550615296068549521035862568952000*F^78 + 351206048604857587273475139515646742409440*F^76 - 1726566676378409462829035558299274730236800*F^74 - 21782598762959588103881719517682681559904832*F^72 - 12476788444196171342482424949840287016467968*F^70 + 696628435483841534316498920272246776326913536*F^68 + 1898467592371016265374708478670939154489124864*F^66 - 14365665017380253081699971150335449364044503552*F^64 - 59481152235443542964624182934270003037290389504*F^62 + 181064854308685768822977273847153690701528981504*F^60 + 1147467873572499275096049977466364486634753458176*F^58 - 354344602714443728776144753126964724331264884736*F^56 - 18971503419094063920781703969554616123546147618816*F^54 - 31901517515730507264849829548857001267941591613440*F^52 + 306694824430139408344328027720207681066615573053440*F^50 + 426609472161673193960553105758004165648658819252224*F^48 - 2998657034546476652637350544759705925740740592271360*F^46 - 2968322530218619877954598264769460621808343554457600*F^44 - 940047090940085035236955833446877474615927119020032*F^42 + 129645673085828510297781908159853561092769709987725312*F^40 - 39703439112333397321748911315021834562919922306908160*F^38 - 1852948787150961611714721796109095517185406197961850880*F^36 + 5231388709248253239760421946999838759061914264797184000*F^34 - 6810795187626436815489587313794179808052320485228150784*F^32 + 16327986329228745855504956505799581198700554675996852224*F^30 + 8554225730004326124003550105589730996299724609869053952*F^28 - 482208955467302137586170200459421525149806647781978275840*F^26 + 2288753803477778399348141359144158534635183202224262086656*F^24 - 5620704333085410615075643903771756171140866114793418784768*F^22 + 8649988839607671226100568470064761153832917003412776681472*F^20 - 9023681012423783686842236028409473438467784256246303948800*F^18 + 6770897232176192549443598589822231165282358389193731735552*F^16 - 3889509050602682276890750355268417548650265578991574319104*F^14 + 1778707700167616025872261104289279077236682146657595293696*F^12 - 629429622890767227698199850596216272700044150388119568384*F^10 + 160129113390947395098583916916416909223262000714283483136*F^8 - 26706285398795834203485759362010776450539353174712516608*F^6 + 2203612563388463175564422980537259878528730313969893376*F^4 - 83141208528725280799348803878162865730107682376908800*F^2 + 1158772266535146661009936216474655410584142479360000   [2.4 < F < 2.5]
   41860546875*G^56 - 1478806875000*G^55 + 21705332062500*G^54 - 152170554825000*G^53 + 167694055976250*G^52 + 6193577943699000*G^51 - 54205045550587900*G^50 + 171889752039234920*G^49 + 319036031041153071*G^48 - 4973458091498870496*G^47 + 16949483079141637296*G^46 + 394327912683075168*G^45 - 199062014257516660488*G^44 + 658546434940493932384*G^43 - 296823728819162574832*G^42 - 4186918758703978176736*G^41 + 13487326696044268976486*G^40 - 9642491987823636320528*G^39 - 50977920905213271299720*G^38 + 173799558203125234879568*G^37 - 166531981294208466626660*G^36 - 396163956400422456670448*G^35 + 1567166365807435149236216*G^34 - 1760837584696232204657232*G^33 - 2159987748317329439603706*G^32 + 10210854417899332407145632*G^31 - 12184929955436545098603440*G^30 - 8568990431507830579835680*G^29 + 48815708165842395650729016*G^28 - 58584920320409677473709856*G^27 - 25995747414057149765932848*G^26 + 173047596910003036955794336*G^25 - 198060753897903668421332001*G^24 - 61949487388562481657416504*G^23 + 445622316269068706002207620*G^22 - 477607892607694905920972904*G^21 - 101717859318384648742047558*G^20 + 828682214417456400788916856*G^19 - 828950471915177074156260060*G^18 - 125864183668972559773482456*G^17 + 1088518329545823754517593419*G^16 - 944097470958923036490288640*G^15 - 129422724015270041190790944*G^14 + 872397102221272513197001216*G^13 - 619675393158442909136645312*G^12 - 37765667551671249655795712*G^11 + 329392208657798559772977664*G^10 - 210217533340223279473676288*G^9 + 33671799655463564298912256*G^8 + 27878314491288115792969728*G^7 - 19944505573465418936082432*G^6 + 6087065335384779165138944*G^5 - 998554853505647795683328*G^4 + 78019354266479382495232*G^3 - 518029967188637384704*G^2 - 225395424769613496320*G + 150545125327175680   [1.663 < G < 1.664]
   1752305384674072265625*H^112 + 37460783630566406250000*H^110 - 3432933108506759765625000*H^108 - 32504852405222967656250000*H^106 + 4867598118224317457123437500*H^104 + 94489385240568898033769250000*H^102 - 792178164461795202361350035000*H^100 - 29693162464566117860504784582800*H^98 - 261290566242404085197571480774634*H^96 + 74688413649659612185612910572912*H^94 + 85792553391906203051407582025118472*H^92 + 276941586956850896786468809316676432*H^90 - 17433799204566012308554299069368291076*H^88 + 277919758464879483990342184764086697840*H^86 + 3885542272654782452694503881323967064568*H^84 - 80204040152719735541732295113241260834224*H^82 - 236561606040380441069865755018502661325735*H^80 + 5646318570551906801060496661091449727730240*H^78 - 44933890547775534946743176171189109286387872*H^76 + 311530209904564747432880005000006322654270848*H^74 + 1355254948886787194009409273200775992612782528*H^72 - 38680133798672159286068815099192087540416094720*H^70 + 1533691510357246030480206179894145027027671750144*H^68 - 5242171520138488314974169988146663812084226975744*H^66 - 218369156948803276218130703179360044360749327837696*H^64 + 1653850062051156267466840097679575297564339174203392*H^62 - 961512726387096342358628116981752290802634730381312*H^60 - 74860813902850405789168477708238803403393775832236032*H^58 + 1494315687519168129168792950543643546986111996394192896*H^56 - 7834402799596360324458849191939347329392145028377608192*H^54 - 29689660055011205395016410542493592110309792422842728448*H^52 + 254564675353220498942913535723784068878433578906180648960*H^50 + 301292948333593042969370036470135983387291049658527842304*H^48 - 2176893703014167974762613474810400505790017638872907776000*H^46 - 4705507149420859735005502438714860148906325006696086241280*H^44 - 65473106325033026989571827699064189252584247557597213753344*H^42 + 463895398675373150517922961024441700000890413779491133325312*H^40 - 152870024616354789201975429429689819864428810085393911250944*H^38 - 1919765217555811634861160249584009965522002117982162975195136*H^36 - 3848475305369154692295503693930308959178714251830616781750272*H^34 + 4809757110251452138977127991367209111289202603450724637474816*H^32 + 122264416125872378621668861549764813419200621629015933770006528*H^30 - 348390109080128704999250488203612016146005885448506519666556928*H^28 - 245252858946115411336168490573209164046499754735307421506338816*H^26 + 2348841226732575496687103734324772946001415801061798795842420736*H^24 - 2889420437725065690492752007380257613183903517779023860248608768*H^22 - 2291964438929787939124243870172897210879299334546076007101628416*H^20 + 9038820927178595757096183544759257831192560168822970181488738304*H^18 - 8403244634452957192462342540170600924735006345827216196947148800*H^16 + 1339969351366941933734358828761514853747148719429683525623742464*H^14 + 3416640028151542668979806039209025489894133282238214293570977792*H^12 - 3150515564526449226111314720555929848126938181106720260510187520*H^10 + 1370652681329314177890109765864046796652694260269441240916819968*H^8 - 364522793107787815693655634977115534266011351022084459625185280*H^6 + 60553213277957780459332979104821686849901893084180949604761600*H^4 - 4877159692737540730914920815519104517497943561935323450572800*H^2 + 91931849205356964613290185901550908900040214385304535040000   [3.5 < H < 3.6]
   J^112 + 112*J^110 + 3632*J^108 + 3776*J^106 - 799776*J^104 + 45460224*J^102 + 1152417536*J^100 - 23795248128*J^98 - 606942311168*J^96 + 7713111056384*J^94 + 154247174356992*J^92 - 3080343415750656*J^90 - 49835446869393408*J^88 + 482735100599730176*J^86 + 8983780146523406336*J^84 - 25210478620314697728*J^82 - 447826804548983193600*J^80 + 6284722963358347689984*J^78 + 51561116637341700587520*J^76 - 466897971835313760239616*J^74 - 8798874658615781211766784*J^72 - 72366747501185961010659328*J^70 - 72589772917122645933162496*J^68 + 4176949562314364934443499520*J^66 + 17696232633429291363771351040*J^64 + 249108354104212411814848233472*J^62 + 6203864640322774732464193536000*J^60 + 37783165568256272458036671414272*J^58 - 259203415697511395142370792046592*J^56 - 5470880605204995372809399763468288*J^54 - 26048075884094463197020975725019136*J^52 - 37382968340742743126836763703640064*J^50 + 562326063173451353854883792198565888*J^48 + 17839536566462120986089448498316443648*J^46 + 138331063190622059877020894964918255616*J^44 - 39764164128805216533892156071547502592*J^42 - 7073778116653612179769608093778451628032*J^40 - 42572145573015968921254087016502836330496*J^38 + 23512431644522093244375710049327976022016*J^36 + 1332360515816926641344408905865849282232320*J^34 + 4783160820296218511240530955808734246338560*J^32 - 5432281670652800982000119034660221167337472*J^30 - 81446865590701165002516323253925428821229568*J^28 - 218794778868605728443639422884392627779665920*J^26 - 61049420482997384100483236243329622021767168*J^24 + 1269696162099168003111674142912940917731098624*J^22 + 3829111839340506708774398398109316679066451968*J^20 + 2822143047261468694184285229238001638846758912*J^18 - 7483659403550239765682948729556968326593249280*J^16 - 10887665026046834673136833549572865850448281600*J^14 + 10762200187191968249451854751298464746295001088*J^12 + 10740769559618368454048540525332890676577173504*J^10 - 14935820165649164841686404007167493264180772864*J^8 + 5327118371723425915457827789649228489268658176*J^6 - 276519614175969829717587307877851936176734208*J^4 - 154173729378026005322643244201246992277438464*J^2 + 14507274551409785469248394362462044384395264   [4.0 < J < 4.1]

Yeah, this is another one of those with hilariously high-degree polynomials with ludicrously huge coefficients. :roll:
quickfur
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Re: Interesting convex polyhedra with 1:1:√2 triangles

Postby quickfur » Sun Sep 06, 2020 2:40 pm

20) Q-snub tetraluna (AKA dihexona Q-snub gyrobifastigium):

Image

Construction: start with a hexona complex, and attach squares around its rim, then insert triangles between the squares at the tips of the Q-triangles (the ends of the long edges). This produces a kind of skew octagonal rim. Take two copies of this and join them together in anti orientation. As it turns out, the octagonal rim is not altogether symmetric, so the two halves will not actually join up perfectly; to remedy this, split the non-axial squares into pairs of Q-triangles to confer an additional degree of freedom to deform the octagonal rim into a symmetric figure so that the two halves can fit together perfectly.

That's the long description. :lol: The short description is to take 4 luna, in gyrobifastigial symmetry, and snub them with a skew tetragonal band of Q-triangle pairs, hence the name Q-snub tetraluna. You can also imagine it as snubbing a gyrobifastigium with Q-triangles by Stott-expanding pairs of squares and triangles apart, then cap the top and bottom with hexona complexes.

Coordinates:
Code: Select all
<0, 0, ±H>
<±1, ±A, -B>
<±C, 0, -D>
<±A, ±1, B>
<0, ±C, D>
<±E, ±1, -F>
<±1, ±E, F>

where
   A = 1.40297281027372
   B = 2.1194012647404
   C = 2.22045936618307
   D = 1.38309546608831
   E = 2.32703421094668
   F = -0.345673402923284
   H = 3.1351115057209

Solution polynomials:
Code: Select all
       7*A^20 + 84*A^19 - 1306*A^18 - 4652*A^17 + 54161*A^16 + 75688*A^15 - 861180*A^14 - 695672*A^13 + 6936473*A^12 + 4195204*A^11 - 31356346*A^10 - 15507036*A^9 + 82911311*A^8 + 31860608*A^7 - 128179664*A^6 - 32735488*A^5 + 107116576*A^4 + 13012992*A^3 - 34131712*A^2 - 132096*A - 4868352     [1.4 < A < 1.5]
       784*B^40 + 328608*B^38 + 18607544*B^36 + 367200792*B^34 + 2219666673*B^32 - 7641536468*B^30 - 95822087518*B^28 - 104503811276*B^26 + 892085120759*B^24 + 2406616247520*B^22 + 848036524*B^20 - 5549875880128*B^18 - 4589267635081*B^16 - 659541678628*B^14 + 719074129162*B^12 + 1140976050228*B^10 - 72229793055*B^8 - 8583107144*B^6 - 71920266064*B^4 + 4701318528*B^2 + 1991301376  [2.1 < B < 2.2]
       C^20 - 16*C^19 - 252*C^18 + 4848*C^17 - 6898*C^16 - 228768*C^15 + 1019732*C^14 + 3295856*C^13 - 28522471*C^12 + 10602192*C^11 + 318722304*C^10 - 673728000*C^9 - 1130146816*C^8 + 5474779136*C^7 - 3780657152*C^6 - 10516299776*C^5 + 22906667008*C^4 - 20854079488*C^3 + 17205035008*C^2 - 15233712128*C + 6375342080  [2.2 < C < 2.3]
       16*D^40 + 9504*D^38 + 2126648*D^36 + 235826648*D^34 + 14965158017*D^32 + 583533971076*D^30 + 14332437803986*D^28 + 219638857906828*D^26 + 2026933532144871*D^24 + 10982423691850704*D^22 + 38990392378999628*D^20 + 110052965683008464*D^18 + 112453973304738535*D^16 - 164867149502618604*D^14 - 254332257266272390*D^12 - 1500023687172449204*D^10 - 3250370064984084687*D^8 + 904894065556475256*D^6 + 73794197619354544*D^4 - 19605251192451712*D^2 + 369226985478400       [1.3 < D < 1.4]
       E^20 - 16*E^19 + 32*E^18 + 744*E^17 - 4196*E^16 - 6120*E^15 + 93736*E^14 - 116792*E^13 - 689034*E^12 + 1667192*E^11 + 1906744*E^10 - 7627688*E^9 - 1008308*E^8 + 14483848*E^7 - 1581096*E^6 - 10614568*E^5 - 332911*E^4 + 844792*E^3 + 470824*E^2 + 174624*E + 15760    [2.3 < E < 2.4]
       16*F^40 + 1120*F^38 + 32920*F^36 + 532600*F^34 + 5224033*F^32 + 32127000*F^30 + 122228208*F^28 + 261756528*F^26 + 192010020*F^24 - 354978664*F^22 - 705282440*F^20 + 115211344*F^18 + 799935430*F^16 - 110844504*F^14 - 526295552*F^12 + 212115600*F^10 + 122255396*F^8 - 112307064*F^6 + 32462432*F^4 - 4069000*F^2 + 185761   [-0.35 < F < -0.34]
       16*H^40 + 416*H^38 - 1320*H^36 - 87416*H^34 - 102383*H^32 + 6027160*H^30 - 6245800*H^28 - 178960640*H^26 + 2262433756*H^24 + 1788322808*H^22 - 94059776224*H^20 + 211291937024*H^18 + 859666619422*H^16 - 6204640938520*H^14 + 4406457085240*H^12 + 8762038298560*H^10 + 2541954617164*H^8 + 383918621608*H^6 + 122308904792*H^4 + 7190956408*H^2 + 7458361     [3.1 < H < 3.2]

Curiously enough, these polynomials seem to start with small coefficients that gradually explode into huge numbers, only to slightly shrink toward the end. A magnitude distribution somewhat more biased towards the end than the usual "barrel shaped" curve.
quickfur
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