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A New Solid of Constant Width

Ed Pegg

Ed Pegg, Wolfram Research

Posted 10 years ago

And here it is: A curve of constant width, such as the Reuleaux triangles, can be generated from any polygon. From any point there is another point at distance 1 with no points at a greater distance. In 3D, the problem is much harder. The sphere is a solid of constant width. If spherical curves are added to a tetrahedron, midpoints of opposing edges are at a distance greater than 1 apart. This flaw can be fixed with the special curve-smoothing techniques of the Meissner tetrahedra. Several curves of constant width can be the basis of a solid of revolution to produce a solid of constant width. According to the paper "On Curves and Surfaces of Constant Width" by Howard L. Resnikoff, that's pretty much it. His paper introduces a few new possible surfaces, such as this one. SphericalPlot3D[1 + 1/10 Cos[3 t] Sin[3 u] Sin[u]^2, {u, -Pi, Pi}, {t, -Pi, Pi}, Boxed -> False, SphericalRegion -> True, ViewAngle -> Pi/10, ImageSize -> {600, 600}, Mesh -> None, Axes -> False, PlotPoints -> 200] Is that really a solid of constant width? Let's make a function to find farthest points. FarthestPoints[pts_List, singlepoint_, num_] := SortBy[Complement[pts, Nearest[pts, singlepoint, Length[pts] - num ]], -MyEuclidean[singlepoint, #] &]; Then we generate many points on this surface, and find the point that is farthest away. resni = Flatten[Table[FromSphericalCoordinates[{(1 + 1/10 Cos[3 t] Sin[3 u] Sin[u]^2), u, t}], {u, .001, 3.14, .05}, {t, -3.14, 3.14, .1}], 1]; dists = Monitor[Table[EuclideanDistance[resni[[k]],FarthestPoints[resni,resni[[k]],1][[1]]],{k,1,Length[resni]}],k]; MinMax[dists] {1.99925, 2.07099} Not the desired result. It's close to being a solid of constant width, but not nearly close enough. Eleven months ago, I opened up the discussion Biggest Little Polyhedra, which led to a blog article on biggest little polyhedra, and eventually I improved results and made a Demonstration for biggest little polyhedra. I've had 1-3 computers steadily improving the results for nearly a year now. In short, you have $n$ points in 3 space, no two more than 1 apart. What is the greatest volume that can be bounded with the $n$ points with unit diameter? The classic solution, widely published, is to use the Thomson problem solutions, but they are far from optimal. For example, with Thomson, 121 points are needed to bound a volume of 1/2. It turns out that 84 points with diameter 1 will suffice. One feature of the biggest little polyhedra is that they have lots and lots of unit length diagonals. Can they be used to make solids of constant width? With most of my smallest solutions there are problems similar to the trouble the tetrahedron has. There may be Meissner-like methods for fixing the flaws, but I haven't explored that yet. During the long search, I came across the page Greater Self-Dual Solids. I especially liked the Self-Dual Icosioctahedron #4, and found that I could make a caltrop out of it, where every vertex was a unit distance from all corners of the opposing face. But it wasn't the optimal way to bound space with 28 points. Extending it to a solid of constant width caused the same sorts of flaws as the tetrahedron. I extended that self dual polyhedron to one on 76 points with tetrahedral symmetry. Points 1, 13, 25, 29, 41, and 53 are as follows: {{0.0833`, 0.0833`, 0.4930122817942774`}, ({C1, C1, C4}) {0.32530527130128584`, -0.20709494964790603`, 0.32530527130128584`}, ({C3, -C0, C3}) {0.28875291001058745`, 0.28875291001058745`, 0.28875291001058745`}, ({C2, C2, C2}) {-0.2142`, 0.40369721678726284`, -0.2142`}, ({-C6, C5, -C6}) {-0.07272969962634213`, 0.35355339059327373`, -0.35355339059327373`}, ({-C7, C8, -C8}) {0.07587339432355446`, 0.44185`, -0.23402687345687453`}} ({C9, C10, -C11}) The full vertices are as follows. There is something more elegant using the tetrahedral group, but I haven't gotten to that yet. cal76 ={{C1,C1,C4},{C1,-C1,-C4},{-C1,-C1,C4},{-C1,C1,-C4},{C4,C1,C1},{C4,-C1,-C1},{-C4,-C1,C1},{-C4,C1,-C1},{C1,C4,C1},{C1,-C4,-C1},{-C1,-C4,C1},{-C1,C4,-C1}, {C3,-C0,C3},{C3,C0,-C3},{-C3,C0,C3},{-C3,-C0,-C3},{C3,-C3,C0},{C3,C3,-C0},{-C3,C3,C0},{-C3,-C3,-C0},{C0,-C3,C3},{C0,C3,-C3},{-C0,C3,C3},{-C0,-C3,-C3}, {C2,C2,C2},{C2,-C2,-C2},{-C2,-C2,C2},{-C2,C2,-C2}, {-C6,C5,-C6},{-C6,C6,-C5},{-C5,C6,-C6},{-C6,-C5,C6},{-C6,-C6,C5},{-C5,-C6,C6},{C6,-C5,-C6},{C6,-C6,-C5},{C5,-C6,-C6},{C6,C5,C6},{C6,C6,C5},{C5,C6,C6}, {-C7,C8,-C8},{-C8,C8,-C7},{-C8,C7,-C8},{-C7,-C8,C8},{-C8,-C8,C7},{-C8,-C7,C8},{C7,-C8,-C8},{C8,-C8,-C7},{C8,-C7,-C8},{C7,C8,C8},{C8,C8,C7},{C8,C7,C8}, {C9,C10,-C11},{C9,C11,-C10},{-C10,C11,C9},{-C11,C10,C9},{-C11,-C9,-C10},{-C10,-C9,-C11},{C9,-C10,C11},{C9,-C11,C10},{-C10,-C11,-C9},{-C11,-C10,-C9},{-C11,C9,C10},{-C10,C9,C11},{-C9,-C10,-C11},{-C9,-C11,-C10},{C10,-C11,C9},{C11,-C10,C9},{C11,C9,-C10},{C10,C9,-C11},{-C9,C10,C11},{-C9,C11,C10},{C10,C11,-C9},{C11,C10,-C9},{C11,-C9,C10},{C10,-C9,C11}}; Here's what it looks like with all 150 unit diagonals added. With the Farthest Point function from earlier, random points at unit distance can be added. For each random point on a sphere, we move it closer or farther from the origin so that the farthest point is at distance 1. The polar opposite is also generated. randomUnitDistancePointPolar[pointset_] := Module[{k},With[{randomspherepoint = .54 coord[{RandomReal[{0,2Pi}],RandomReal[{-1,1}],1}]},{randomspherepoint(k/.Quiet[Solve[{MyEuclidean[k randomspherepoint, FarthestPoints[pointset,randomspherepoint,1][[1]]]==1,0<k<2}]][[1,1]]),(-randomspherepoint)(k/.Quiet[Solve[{MyEuclidean[k (-randomspherepoint), FarthestPoints[pointset,(-randomspherepoint),1][[1]]]==1,0<k<2}]][[1,1]])}]]; Using the 76 points, a lot more points can be added. is76scw = Table[randomUnitDistancePointPolar[cal76], {10000}]; The crucial check -- are any two polar opposites more than 1 apart? Select[is76scw, EuclideanDistance[#[[1]], #[[2]]] > 1 &] There are no flaws. So far as I know, this is the first solid of constant width that has been found with tetrahedral symmetry. To generate the image at the top, use Graphics3D[ConvexHullMesh[Join[cal76,Flatten[is76scw,1]]]["GraphicsComplex"], Boxed-> False, ViewAngle-> Pi/10, ImageSize->{600,600} ] And there you have it, a new solid of constant width. It's also the best known way to bound space with 76 points. Mathematica just needed to make a few trillion calculations, optimizations, and improvements over an eleven month period to find it.

POSTED BY: Ed Pegg

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Ed Pegg

Ed Pegg, Wolfram Research

Posted 10 years ago

The unit distance graph generated by the caltrop is interesting. The 150 long diagonals all have unit length, but it's hard to tell what's going on. Here's a clearer picture.

POSTED BY: Ed Pegg

Ed Pegg

Ed Pegg, Wolfram Research

Posted 10 years ago

Over at reddit, some excellent questions/ observations were made. God, for a second I thought it was a sphere. Is there a formal proof that it has constant width? For the first, what does happen when we compare this with a sphere? The caltrop-based surface of constant width will be done in yellow, and a sphere of radius .5 in blue. It's quite odd what happens. Graphics3D[{{Yellow,ConvexHullMesh[Join[cal76,Flatten[is76scw,1]]]["GraphicsComplex"]},{Blue, Sphere[{0,0,0}, .5]}},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}] For the second, how can this be verified? I have a decent method in my post -- generate a lot of points, and find the farthest point from each point. If the farthest point is always at distance one as the number of random surface points gets arbitrarily large, then the solution works. I applied that method to the Resnikoff surface. But not to my own surface. On my own surface, I did a weaker check. That was rather presumptuous of me. The program I used for picking points at distance 1 was flawed. It was -- get a random point, find the farthest point in a pointset, move the point to distance 1. But the farthest point could change in the process of moving. Here's more careful code. randomUnitDistancePointPolar[pointset_]:= Module[{k,randomspherepointPositive, randomspherepointNegative,farpointsPositive, farpointsNegative,multiplierPositive,multiplierNegative }, randomspherepointPositive= coord[{RandomReal[{0,2Pi}],RandomReal[{-1,1}],1}]; randomspherepointNegative = -randomspherepointPositive; farpointsPositive = FarthestPoints[pointset,randomspherepointPositive,4]; farpointsNegative = FarthestPoints[pointset,randomspherepointNegative,4]; multiplierPositive = Last[Sort[k/.Solve[{MyEuclidean[k randomspherepointPositive,#]==1}]]]&/@farpointsPositive; multiplierNegative = Last[Sort[k/.Solve[{MyEuclidean[k randomspherepointNegative,#]==1}]]]&/@farpointsNegative; {Sort[Table[{Max[Table[MyEuclidean[multiplierPositive[[k]] randomspherepointPositive,farpointsPositive[[j]]],{j,1,4}]], multiplierPositive[[k]] randomspherepointPositive},{k,1,4}]][[1,2]], Sort[Table[{Max[Table[MyEuclidean[multiplierNegative[[k]] randomspherepointNegative,farpointsNegative[[j]]],{j,1,4}]], multiplierNegative[[k]] randomspherepointNegative},{k,1,4}]][[1,2]]} ]; Here are the 76 points. cal76={{-0.4083582798988096`,-0.2742546402707427`,-0.09034731123170289`},{-9.958465404071062`^-9,0.14648426006691528`,0.4782019350351851`},{6.73274361415451`^-9,0.4020250213642393`,-0.2975073298522745`},{0.4083582831245314`,-0.2742546411604119`,-0.09034729395120787`},{-0.46005112774248885`,0.19669672299517796`,0.06479750202204874`},{-0.37646381006368895`,0.2828182459595357`,0.18117449242974917`},{-0.37646380664714646`,0.335125174376982`,0.02239368272554362`},{-0.08358732592409791`,-0.4267626938092579`,0.2557774801658645`},{-8.245298000993142`^-9,-0.3406411708449002`,0.37215447057356493`},{0.08358731417059766`,-0.42676269399136524`,0.25577748370302866`},{0.37646380301309024`,0.2828182451393534`,0.18117450836056592`},{0.3764638064296328`,0.33512517355679966`,0.0223936986563603`},{0.46005112542898596`,0.19669672199288823`,0.0647975214900296`},{-0.08358731053654152`,-0.19118072552497012`,-0.45934567478913835`},{1.0558800920201747`^-8,-0.05275227414316607`,-0.5017494940856435`},{0.0835873295581541`,-0.19118072570707753`,-0.45934567125197423`},{-0.13399475927338292`,0.3401518113710143`,-0.3479610215959084`},{3.935963916780783`^-9,0.478209035807638`,-0.16140272895373037`},{0.13399475688050796`,0.06673824586943174`,0.48200221421544004`},{0.3029245473152117`,-0.38342759987801794`,-0.12631198330894344`},{-0.4369193136085767`,-0.24537037448950066`,0.060246290844122634`},{-0.43691930813168356`,-0.1615196817990518`,-0.19428750195276642`},{-0.3029245428054219`,-0.3834275992180518`,-0.12631199612779967`},{-0.1339947771319936`,0.06673824616135929`,0.48200220854518433`},{-8.445753645264267`^-9,0.2886461632884318`,0.41402670839047334`},{0.1339947747391187`,0.3401518110790869`,-0.3479610159256526`},{0.4369193105245585`,-0.2453703754413943`,0.060246309333234584`},{0.4369193160014517`,-0.16151968275094553`,-0.1942874834636544`},{0.19857232806547456`,0.2414884119962342`,0.39691559128602183`},{-0.3014276658828522`,-0.39793238086123595`,0.07797943132815037`},{-0.19857234433563078`,0.24148841242885352`,0.39691558288303747`},{0.1985723403860086`,0.4301145723332891`,-0.17567040020422556`},{-0.4999999995937601`,-0.06907794446944336`,-0.022756256323355057`},{-0.1985723320150967`,0.4301145727659085`,-0.17567040860720995`},{-1.0624722918507781`^-8,-0.08736602396293898`,0.49765127053454294`},{9.81224354438672`^-9,0.2255219139911501`,-0.4521387790463301`},{0.30142766171604196`,-0.3979323815179409`,0.07797944408366307`},{0.30142766983247427`,-0.27367060390090675`,-0.29922461400696254`},{-0.3014276577664199`,-0.27367060324420184`,-0.29922462676247524`},{0.5000000004062395`,-0.06907794555876769`,-0.022756235164857957`},{-0.4779171238955934`,0.02608392988012896`,0.16331449187242988`},{-0.3660857189235138`,0.30324695712170446`,-0.17256680260947826`},{0.1469531522570092`,0.07747070721983303`,-0.47766079233150865`},{0.25878454841831394`,-0.10773258907155243`,-0.42092875729515894`},{0.47791711704144224`,0.026083928838915534`,0.16331451209644585`},{-0.3660857295010557`,0.1413058351635909`,0.31901515884135884`},{-0.25878454580394866`,-0.33682020241617083`,0.2744803855716309`},{-0.25878453084058534`,-0.10773258850775182`,-0.42092876824614284`},{-0.2191325810215751`,0.4546554290281564`,0.03203318211204315`},{-0.146953151409411`,-0.2215982971327089`,0.4301810525405597`},{-0.11183139061700911`,-0.35110079192819443`,-0.3463794825711219`},{0.11183139555301191`,-0.4882286748864234`,0.06988041180109335`},{0.11183140450982235`,-0.3511007921718357`,-0.3463794778387532`},{0.14695313272265692`,-0.22159829745286824`,0.43018105875917506`},{0.21913258065656127`,0.45465542855074365`,0.03203319138507527`},{0.25878453345495067`,-0.33682020297997145`,0.27448039652261474`},{0.3660857163091486`,0.14130583436601876`,0.3190151743330062`},{0.36608572688669044`,0.3032469563241323`,-0.17256678711783088`},{-0.4779171178890406`,0.11804366107396037`,-0.11583477230549717`},{-0.2191325855925641`,0.3846740382638743`,0.24446587938495315`},{-0.14695313187505868`,0.07747070753999229`,-0.47766079855012394`},{-0.11183139957381949`,-0.48822867464278213`,0.06988040706872475`},{0.21913257608557227`,0.3846740377864614`,0.2444658886579853`},{0.4779171230479952`,0.1180436600327469`,-0.11583475208148117`},{-0.407514327993345`,-0.16976437931749935`,0.24910041749277848`},{-5.161330127384547`^-9,0.43139871213507613`,0.2661466534980246`},{0.2897103239985093`,-0.048388805251119736`,0.41311661996952853`},{0.2897103406552224`,0.20662382688011618`,-0.3609894772467069`},{0.40751431708230673`,-0.16976438020532986`,0.2491004347375597`},{-0.4075143161517443`,0.011529357945156408`,-0.301227565200711`},{-0.28971034158578496`,-0.04838880461994275`,0.4131166077098581`},{-0.28971032492907184`,0.20662382751129316`,-0.3609894895063774`},{-0.11780398699190484`,-0.4682581594410395`,-0.15425759872920716`},{-3.4621769187396477`^-10,0.5051176070036564`,0.04236853897527871`},{0.11780399249945263`,-0.46825815969769297`,-0.15425759374409645`},{0.4075143289239075`,0.011529357057325836`,-0.30122754795592976`}}; Now we can generate 20000 random points and their polar opposites. This takes awhile to run. Then we combine them with the original 76 to get a checklist of 40076 points. For each point, we can find the distance to the farthest point. If there are any flaws, we can pick out the points that generated a distance over 1. Then we can get a minmax of the distances to see how bad the errors are, if any exist. is76scw=Monitor[Table[randomUnitDistancePointPolar[cal76],{jj,1,20000}],jj]; check76 = Join[cal76, Flatten[is76scw, 1]]; dists=Monitor[Table[EuclideanDistance[check76[[k]],FarthestPoints[check76,check76[[k]],1][[1]]],{k,1,Length[check76]}],k]; bad = check76[[Flatten[Position[Sign[Chop[dists - 1]], 1]]]]; MinMax[dists] {1., 1.00002} Looks like some pairs of points can be a micron to far apart. Out of the 40076 points generated and the 803022850 possible pairs, here are the 50 points that can generate a distance of 1.000002 with some other point in the "bad" list. bad = {{0.217299,0.453677,0.00289042},{0.21081,0.313122,0.327902},{-0.30425,-0.397356,0.0198628},{-0.305056,-0.396024,-0.00758598},{0.0769693,-0.491524,0.0703061},{0.499554,-0.0557945,0.003875},{0.0000341373,0.335519,0.371979},{-0.182439,-0.460237,0.0734066},{-0.15309,0.325789,-0.351062},{-0.421063,-0.266799,0.0628566},{0.316308,0.174691,0.346821},{0.262122,-0.118299,-0.414443},{-0.155536,0.323949,-0.351397},{-0.269515,-0.0325779,0.424797},{-0.0608613,0.167056,-0.467461},{0.324612,0.156324,-0.348921},{-0.332738,0.259147,-0.268588},{0.376774,0.333399,0.00612808},{0.247323,-0.0868552,-0.429567},{0.365602,0.0904816,-0.329314},{0.374647,0.0745269,-0.324088},{-0.0775596,0.150317,-0.470526},{0.16768,-0.243541,0.405892},{0.284112,-0.195228,-0.362151},{-0.334145,-0.0915662,0.361428},{0.295381,-0.276727,-0.301267},{-0.0000968207,0.504594,0.0277361},{0.127589,-0.204336,0.441693},{-0.280701,-0.283995,-0.306032},{0.465323,0.144349,-0.124355},{0.0174747,-0.355603,-0.350556},{-0.295778,-0.245216,-0.323275},{0.292672,-0.303975,0.27156},{-0.378837,-0.137164,0.29839},{0.21301,0.277084,-0.358325},{-0.0950728,0.132381,-0.473214},{-0.213059,0.329729,0.310047},{-0.352811,-0.0363017,-0.353029},{0.247033,0.436895,0.0305884},{-0.174493,-0.250587,0.397601},{-0.484321,0.0887515,-0.101515},{0.228601,0.410957,-0.176389},{-0.331244,0.164913,0.339026},{-0.430212,-0.218733,0.132927},{-0.473575,0.113459,0.114474},{-0.267608,-0.34923,0.243012},{-0.340969,-0.0984442,0.352577},{0.333913,0.0509263,0.368648},{0.485222,-0.0979485,-0.0753553},{-0.432159,-0.224615,0.117287}}; We can take a look at where these points are located in relation to the original 76 points. Graphics3D[{{Opacity[.6],ConvexHullMesh[cal76]["GraphicsComplex"]},Table[Line[{FarthestPoints[bad,bad[[n]],1][[1]],bad[[n]]}],{n,1,Length[bad]}]},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}] All of these bad point pairs connect an edge point to an edge point. For future checks, I can focus on those. There is plenty of play in the given caltrop, currently I'm optimizing for volume. How might I optimize for the edges working just slightly more nicely with each other? One method might be canonicalization. Here's some code adapted from George Hart's Canonical Polyhedra page. view[v_, f_] := Graphics3D[{Opacity[.7],Polygon /@ Map[v[[#1]] & , f, {2}]}, SphericalRegion-> True,Boxed -> False] face2edges[face_] := MapThread[Sort[{#1, #2}] & , {face, RotateLeft[face]}]; f2e[f_] := Union @@ face2edges /@ f closest[{P1_,P2_}]:=With[{L=P2-P1},P1-(L.P1 L)/L.L] recenter[v_, e_] := With[{centroid = Plus @@ (Chop[closest[#]]& /@ Map[v[[#1]] & , e, {2}]/Length[e])}, (#1 - centroid & ) /@ v] tangentify[v_, e_] := Module[{newV = v, t, c}, Scan[(t = closest[v[[#1]]]; c = 0.5(1 - Sqrt[t . t])t; newV[[#1[[1]]]] += c; newV[[#1[[2]]]] += c; ) & , e]; newV] unit[x_] := With[{mag2 = x . x}, If[mag2 != 0, x/Sqrt[mag2], x]] cross[{ax_, ay_, az_}, {bx_, by_, bz_}] := {aybz - azby, azbx - axbz, axby - aybx} approxNormal[face_] := unit[Plus @@ MapThread[unit[cross[#1 - #2, #2 - #3]] & , {face, RotateLeft[face, 1], RotateLeft[face, 2]}]] planarize[v_, f_] := Module[{newV = v, faceXYZ, n, centroid}, Scan[(faceXYZ = v[[#1]]; n = approxNormal[faceXYZ]; centroid = Plus @@ faceXYZ/Length[faceXYZ]; if[n . centroid < 0, n = -n]; Scan[newV[[#1]] += 0.2n . (centroid - v[[#1]])n & , #1]; ) & , f]; newV] canonicalize[v_, f_] := Module[{newV = N[v], e = f2e[f], oldV, maxChange}, Do[oldV = newV; newV = tangentify[newV, e]; newV = recenter[newV, e]; newV = planarize[newV, f]; maxChange = Max[Abs[oldV - newV]]; If[maxChange < 10.^(-8), Break[]], {i, 80}]; Print["Solved within ", maxChange]; newV] I think I'll go back to my original set of vertices: verts = {{0.0833`,0.0833`,0.4930122817942774`},{0.0833`,-0.0833`,-0.4930122817942774`},{-0.0833`,-0.0833`,0.4930122817942774`},{-0.0833`,0.0833`,-0.4930122817942774`},{0.4930122817942774`,0.0833`,0.0833`},{0.4930122817942774`,-0.0833`,-0.0833`},{-0.4930122817942774`,-0.0833`,0.0833`},{-0.4930122817942774`,0.0833`,-0.0833`},{0.0833`,0.4930122817942774`,0.0833`},{0.0833`,-0.4930122817942774`,-0.0833`},{-0.0833`,-0.4930122817942774`,0.0833`},{-0.0833`,0.4930122817942774`,-0.0833`},{0.32530527130128584`,-0.20709494964790603`,0.32530527130128584`},{0.32530527130128584`,0.20709494964790603`,-0.32530527130128584`},{-0.32530527130128584`,0.20709494964790603`,0.32530527130128584`},{-0.32530527130128584`,-0.20709494964790603`,-0.32530527130128584`},{0.32530527130128584`,-0.32530527130128584`,0.20709494964790603`},{0.32530527130128584`,0.32530527130128584`,-0.20709494964790603`},{-0.32530527130128584`,0.32530527130128584`,0.20709494964790603`},{-0.32530527130128584`,-0.32530527130128584`,-0.20709494964790603`},{0.20709494964790603`,-0.32530527130128584`,0.32530527130128584`},{0.20709494964790603`,0.32530527130128584`,-0.32530527130128584`},{-0.20709494964790603`,0.32530527130128584`,0.32530527130128584`},{-0.20709494964790603`,-0.32530527130128584`,-0.32530527130128584`},{0.28875291001058745`,0.28875291001058745`,0.28875291001058745`},{0.28875291001058745`,-0.28875291001058745`,-0.28875291001058745`},{-0.28875291001058745`,-0.28875291001058745`,0.28875291001058745`},{-0.28875291001058745`,0.28875291001058745`,-0.28875291001058745`},{-0.2142`,0.40369721678726284`,-0.2142`},{-0.2142`,0.2142`,-0.40369721678726284`},{-0.40369721678726284`,0.2142`,-0.2142`},{-0.2142`,-0.40369721678726284`,0.2142`},{-0.2142`,-0.2142`,0.40369721678726284`},{-0.40369721678726284`,-0.2142`,0.2142`},{0.2142`,-0.40369721678726284`,-0.2142`},{0.2142`,-0.2142`,-0.40369721678726284`},{0.40369721678726284`,-0.2142`,-0.2142`},{0.2142`,0.40369721678726284`,0.2142`},{0.2142`,0.2142`,0.40369721678726284`},{0.40369721678726284`,0.2142`,0.2142`},{-0.07272969962634213`,0.35355339059327373`,-0.35355339059327373`},{-0.35355339059327373`,0.35355339059327373`,-0.07272969962634213`},{-0.35355339059327373`,0.07272969962634213`,-0.35355339059327373`},{-0.07272969962634213`,-0.35355339059327373`,0.35355339059327373`},{-0.35355339059327373`,-0.35355339059327373`,0.07272969962634213`},{-0.35355339059327373`,-0.07272969962634213`,0.35355339059327373`},{0.07272969962634213`,-0.35355339059327373`,-0.35355339059327373`},{0.35355339059327373`,-0.35355339059327373`,-0.07272969962634213`},{0.35355339059327373`,-0.07272969962634213`,-0.35355339059327373`},{0.07272969962634213`,0.35355339059327373`,0.35355339059327373`},{0.35355339059327373`,0.35355339059327373`,0.07272969962634213`},{0.35355339059327373`,0.07272969962634213`,0.35355339059327373`},{0.07587339432355446`,0.44185`,-0.23402687345687453`},{0.07587339432355446`,0.23402687345687453`,-0.44185`},{-0.44185`,0.23402687345687453`,0.07587339432355446`},{-0.23402687345687453`,0.44185`,0.07587339432355446`},{-0.23402687345687453`,-0.07587339432355446`,-0.44185`},{-0.44185`,-0.07587339432355446`,-0.23402687345687453`},{0.07587339432355446`,-0.44185`,0.23402687345687453`},{0.07587339432355446`,-0.23402687345687453`,0.44185`},{-0.44185`,-0.23402687345687453`,-0.07587339432355446`},{-0.23402687345687453`,-0.44185`,-0.07587339432355446`},{-0.23402687345687453`,0.07587339432355446`,0.44185`},{-0.44185`,0.07587339432355446`,0.23402687345687453`},{-0.07587339432355446`,-0.44185`,-0.23402687345687453`},{-0.07587339432355446`,-0.23402687345687453`,-0.44185`},{0.44185`,-0.23402687345687453`,0.07587339432355446`},{0.23402687345687453`,-0.44185`,0.07587339432355446`},{0.23402687345687453`,0.07587339432355446`,-0.44185`},{0.44185`,0.07587339432355446`,-0.23402687345687453`},{-0.07587339432355446`,0.44185`,0.23402687345687453`},{-0.07587339432355446`,0.23402687345687453`,0.44185`},{0.44185`,0.23402687345687453`,-0.07587339432355446`},{0.23402687345687453`,0.44185`,-0.07587339432355446`},{0.23402687345687453`,-0.07587339432355446`,0.44185`},{0.44185`,-0.07587339432355446`,0.23402687345687453`}}; Because I had a list of faces for those points. faces = { {13,21,17},{14,22,18},{15,23,19},{16,24,20}, {28,29,41,30},{12,53,41,29},{22,54,41,53},{4,30,41,54},{28,31,42,29},{8,55,42,31},{19,56,42,55},{12,29,42,56},{28,30,43,31},{4,57,43,30},{16,58,43,57},{8,31,43,58},{27,32,44,33},{11,59,44,32},{21,60,44,59},{3,33,44,60},{27,34,45,32},{7,61,45,34},{20,62,45,61},{11,32,45,62},{27,33,46,34},{3,63,46,33},{15,64,46,63},{7,34,46,64},{26,35,47,36},{10,65,47,35},{24,66,47,65},{2,36,47,66},{26,37,48,35},{6,67,48,37},{17,68,48,67},{10,35,48,68},{26,36,49,37},{2,69,49,36},{14,70,49,69},{6,37,49,70},{25,38,50,39},{9,71,50,38},{23,72,50,71},{1,39,50,72},{25,40,51,38},{5,73,51,40},{18,74,51,73},{9,38,51,74},{25,39,52,40},{1,75,52,39},{13,76,52,75},{5,40,52,76},{11,10,68,59},{17,21,59,68},{10,11,62,65},{20,24,65,62},{12,9,74,53},{18,22,53,74},{9,12,56,71},{19,23,71,56},{8,7,64,55},{15,19,55,64},{7,8,58,61},{16,20,61,58},{6,5,76,67},{13,17,67,76},{5,6,70,73},{14,18,73,70},{4,2,66,57},{24,16,57,66},{3,1,72,63},{23,15,63,72},{2,4,54,69},{22,14,69,54},{1,3,60,75},{21,13,75,60}}; With the vertices and faces, we can canonicalize the placement of the vertices. canon76 = canonicalize[verts, faces]; Let's have a look. Doesn't look too much different. Unfortunately, a lot of unit diagonals are gone. Graphics3D[{Polygon[canonical76[[#]]] & /@ faces}, Boxed -> False, ViewAngle -> Pi/10, ImageSize -> {600, 600}] Why is that nice? Since this is a self-dual polyhedron, we can combine it with the dual so that all edges intersect and are perpendicular to each other. Graphics3D[{Polygon[canonical76[[#]]]&/@faces, Polygon[-canonical76[[#]]]&/@faces},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}] That doesn't happen in my unit diagonal and volume-optimized version, many diagonals are skew and don't intersect. Graphics3D[{Polygon[verts[[#]]]&/@faces, Polygon[-canonical76[[#]]]&/@faces},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}] In short, for a formal proof that this is a solid of constant width, it fails. The currently given solution is off by an amount that wouldn't be detectable if it were 3D printed, but it's still not perfect. However, it's very likely fixable by giving up the volume optimization, and instead going for a canonicalization biased optimization, while still maintaining all 150 unit diagonals. That's very doable, I just need to figure out how to do it.

Over at reddit, some excellent questions/ observations were made.

God, for a second I thought it was a sphere.
Is there a formal proof that it has constant width?

For the first, what does happen when we compare this with a sphere? The caltrop-based surface of constant width will be done in yellow, and a sphere of radius .5 in blue. It's quite odd what happens.

Graphics3D[{{Yellow,ConvexHullMesh[Join[cal76,Flatten[is76scw,1]]]["GraphicsComplex"]},{Blue, Sphere[{0,0,0}, .5]}},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}]

caltrop76 and sphere

For the second, how can this be verified? I have a decent method in my post -- generate a lot of points, and find the farthest point from each point. If the farthest point is always at distance one as the number of random surface points gets arbitrarily large, then the solution works. I applied that method to the Resnikoff surface. But not to my own surface. On my own surface, I did a weaker check. That was rather presumptuous of me. The program I used for picking points at distance 1 was flawed. It was -- get a random point, find the farthest point in a pointset, move the point to distance 1. But the farthest point could change in the process of moving. Here's more careful code.

randomUnitDistancePointPolar[pointset_]:=
Module[{k,randomspherepointPositive, randomspherepointNegative,farpointsPositive, farpointsNegative,multiplierPositive,multiplierNegative },
randomspherepointPositive= coord[{RandomReal[{0,2Pi}],RandomReal[{-1,1}],1}];
randomspherepointNegative = -randomspherepointPositive;
farpointsPositive =  FarthestPoints[pointset,randomspherepointPositive,4];
farpointsNegative =  FarthestPoints[pointset,randomspherepointNegative,4];
multiplierPositive = Last[Sort[k/.Solve[{MyEuclidean[k  randomspherepointPositive,#]==1}]]]&/@farpointsPositive;
multiplierNegative = Last[Sort[k/.Solve[{MyEuclidean[k  randomspherepointNegative,#]==1}]]]&/@farpointsNegative; 
{Sort[Table[{Max[Table[MyEuclidean[multiplierPositive[[k]]  randomspherepointPositive,farpointsPositive[[j]]],{j,1,4}]], multiplierPositive[[k]]  randomspherepointPositive},{k,1,4}]][[1,2]], 
Sort[Table[{Max[Table[MyEuclidean[multiplierNegative[[k]]  randomspherepointNegative,farpointsNegative[[j]]],{j,1,4}]], multiplierNegative[[k]]  randomspherepointNegative},{k,1,4}]][[1,2]]}
];

Here are the 76 points.

cal76={{-0.4083582798988096`,-0.2742546402707427`,-0.09034731123170289`},{-9.958465404071062`*^-9,0.14648426006691528`,0.4782019350351851`},{6.73274361415451`*^-9,0.4020250213642393`,-0.2975073298522745`},{0.4083582831245314`,-0.2742546411604119`,-0.09034729395120787`},{-0.46005112774248885`,0.19669672299517796`,0.06479750202204874`},{-0.37646381006368895`,0.2828182459595357`,0.18117449242974917`},{-0.37646380664714646`,0.335125174376982`,0.02239368272554362`},{-0.08358732592409791`,-0.4267626938092579`,0.2557774801658645`},{-8.245298000993142`*^-9,-0.3406411708449002`,0.37215447057356493`},{0.08358731417059766`,-0.42676269399136524`,0.25577748370302866`},{0.37646380301309024`,0.2828182451393534`,0.18117450836056592`},{0.3764638064296328`,0.33512517355679966`,0.0223936986563603`},{0.46005112542898596`,0.19669672199288823`,0.0647975214900296`},{-0.08358731053654152`,-0.19118072552497012`,-0.45934567478913835`},{1.0558800920201747`*^-8,-0.05275227414316607`,-0.5017494940856435`},{0.0835873295581541`,-0.19118072570707753`,-0.45934567125197423`},{-0.13399475927338292`,0.3401518113710143`,-0.3479610215959084`},{3.935963916780783`*^-9,0.478209035807638`,-0.16140272895373037`},{0.13399475688050796`,0.06673824586943174`,0.48200221421544004`},{0.3029245473152117`,-0.38342759987801794`,-0.12631198330894344`},{-0.4369193136085767`,-0.24537037448950066`,0.060246290844122634`},{-0.43691930813168356`,-0.1615196817990518`,-0.19428750195276642`},{-0.3029245428054219`,-0.3834275992180518`,-0.12631199612779967`},{-0.1339947771319936`,0.06673824616135929`,0.48200220854518433`},{-8.445753645264267`*^-9,0.2886461632884318`,0.41402670839047334`},{0.1339947747391187`,0.3401518110790869`,-0.3479610159256526`},{0.4369193105245585`,-0.2453703754413943`,0.060246309333234584`},{0.4369193160014517`,-0.16151968275094553`,-0.1942874834636544`},{0.19857232806547456`,0.2414884119962342`,0.39691559128602183`},{-0.3014276658828522`,-0.39793238086123595`,0.07797943132815037`},{-0.19857234433563078`,0.24148841242885352`,0.39691558288303747`},{0.1985723403860086`,0.4301145723332891`,-0.17567040020422556`},{-0.4999999995937601`,-0.06907794446944336`,-0.022756256323355057`},{-0.1985723320150967`,0.4301145727659085`,-0.17567040860720995`},{-1.0624722918507781`*^-8,-0.08736602396293898`,0.49765127053454294`},{9.81224354438672`*^-9,0.2255219139911501`,-0.4521387790463301`},{0.30142766171604196`,-0.3979323815179409`,0.07797944408366307`},{0.30142766983247427`,-0.27367060390090675`,-0.29922461400696254`},{-0.3014276577664199`,-0.27367060324420184`,-0.29922462676247524`},{0.5000000004062395`,-0.06907794555876769`,-0.022756235164857957`},{-0.4779171238955934`,0.02608392988012896`,0.16331449187242988`},{-0.3660857189235138`,0.30324695712170446`,-0.17256680260947826`},{0.1469531522570092`,0.07747070721983303`,-0.47766079233150865`},{0.25878454841831394`,-0.10773258907155243`,-0.42092875729515894`},{0.47791711704144224`,0.026083928838915534`,0.16331451209644585`},{-0.3660857295010557`,0.1413058351635909`,0.31901515884135884`},{-0.25878454580394866`,-0.33682020241617083`,0.2744803855716309`},{-0.25878453084058534`,-0.10773258850775182`,-0.42092876824614284`},{-0.2191325810215751`,0.4546554290281564`,0.03203318211204315`},{-0.146953151409411`,-0.2215982971327089`,0.4301810525405597`},{-0.11183139061700911`,-0.35110079192819443`,-0.3463794825711219`},{0.11183139555301191`,-0.4882286748864234`,0.06988041180109335`},{0.11183140450982235`,-0.3511007921718357`,-0.3463794778387532`},{0.14695313272265692`,-0.22159829745286824`,0.43018105875917506`},{0.21913258065656127`,0.45465542855074365`,0.03203319138507527`},{0.25878453345495067`,-0.33682020297997145`,0.27448039652261474`},{0.3660857163091486`,0.14130583436601876`,0.3190151743330062`},{0.36608572688669044`,0.3032469563241323`,-0.17256678711783088`},{-0.4779171178890406`,0.11804366107396037`,-0.11583477230549717`},{-0.2191325855925641`,0.3846740382638743`,0.24446587938495315`},{-0.14695313187505868`,0.07747070753999229`,-0.47766079855012394`},{-0.11183139957381949`,-0.48822867464278213`,0.06988040706872475`},{0.21913257608557227`,0.3846740377864614`,0.2444658886579853`},{0.4779171230479952`,0.1180436600327469`,-0.11583475208148117`},{-0.407514327993345`,-0.16976437931749935`,0.24910041749277848`},{-5.161330127384547`*^-9,0.43139871213507613`,0.2661466534980246`},{0.2897103239985093`,-0.048388805251119736`,0.41311661996952853`},{0.2897103406552224`,0.20662382688011618`,-0.3609894772467069`},{0.40751431708230673`,-0.16976438020532986`,0.2491004347375597`},{-0.4075143161517443`,0.011529357945156408`,-0.301227565200711`},{-0.28971034158578496`,-0.04838880461994275`,0.4131166077098581`},{-0.28971032492907184`,0.20662382751129316`,-0.3609894895063774`},{-0.11780398699190484`,-0.4682581594410395`,-0.15425759872920716`},{-3.4621769187396477`*^-10,0.5051176070036564`,0.04236853897527871`},{0.11780399249945263`,-0.46825815969769297`,-0.15425759374409645`},{0.4075143289239075`,0.011529357057325836`,-0.30122754795592976`}};

Now we can generate 20000 random points and their polar opposites. This takes awhile to run. Then we combine them with the original 76 to get a checklist of 40076 points. For each point, we can find the distance to the farthest point. If there are any flaws, we can pick out the points that generated a distance over 1. Then we can get a minmax of the distances to see how bad the errors are, if any exist.

is76scw=Monitor[Table[randomUnitDistancePointPolar[cal76],{jj,1,20000}],jj];  
check76 = Join[cal76, Flatten[is76scw, 1]];
dists=Monitor[Table[EuclideanDistance[check76[[k]],FarthestPoints[check76,check76[[k]],1][[1]]],{k,1,Length[check76]}],k];  
bad = check76[[Flatten[Position[Sign[Chop[dists - 1]], 1]]]];
MinMax[dists]  

{1., 1.00002}

Looks like some pairs of points can be a micron to far apart. Out of the 40076 points generated and the 803022850 possible pairs, here are the 50 points that can generate a distance of 1.000002 with some other point in the "bad" list.

bad = {{0.217299,0.453677,0.00289042},{0.21081,0.313122,0.327902},{-0.30425,-0.397356,0.0198628},{-0.305056,-0.396024,-0.00758598},{0.0769693,-0.491524,0.0703061},{0.499554,-0.0557945,0.003875},{0.0000341373,0.335519,0.371979},{-0.182439,-0.460237,0.0734066},{-0.15309,0.325789,-0.351062},{-0.421063,-0.266799,0.0628566},{0.316308,0.174691,0.346821},{0.262122,-0.118299,-0.414443},{-0.155536,0.323949,-0.351397},{-0.269515,-0.0325779,0.424797},{-0.0608613,0.167056,-0.467461},{0.324612,0.156324,-0.348921},{-0.332738,0.259147,-0.268588},{0.376774,0.333399,0.00612808},{0.247323,-0.0868552,-0.429567},{0.365602,0.0904816,-0.329314},{0.374647,0.0745269,-0.324088},{-0.0775596,0.150317,-0.470526},{0.16768,-0.243541,0.405892},{0.284112,-0.195228,-0.362151},{-0.334145,-0.0915662,0.361428},{0.295381,-0.276727,-0.301267},{-0.0000968207,0.504594,0.0277361},{0.127589,-0.204336,0.441693},{-0.280701,-0.283995,-0.306032},{0.465323,0.144349,-0.124355},{0.0174747,-0.355603,-0.350556},{-0.295778,-0.245216,-0.323275},{0.292672,-0.303975,0.27156},{-0.378837,-0.137164,0.29839},{0.21301,0.277084,-0.358325},{-0.0950728,0.132381,-0.473214},{-0.213059,0.329729,0.310047},{-0.352811,-0.0363017,-0.353029},{0.247033,0.436895,0.0305884},{-0.174493,-0.250587,0.397601},{-0.484321,0.0887515,-0.101515},{0.228601,0.410957,-0.176389},{-0.331244,0.164913,0.339026},{-0.430212,-0.218733,0.132927},{-0.473575,0.113459,0.114474},{-0.267608,-0.34923,0.243012},{-0.340969,-0.0984442,0.352577},{0.333913,0.0509263,0.368648},{0.485222,-0.0979485,-0.0753553},{-0.432159,-0.224615,0.117287}};

We can take a look at where these points are located in relation to the original 76 points.

Graphics3D[{{Opacity[.6],ConvexHullMesh[cal76]["GraphicsComplex"]},Table[Line[{FarthestPoints[bad,bad[[n]],1][[1]],bad[[n]]}],{n,1,Length[bad]}]},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}]

bad points in the 76-caltrop

All of these bad point pairs connect an edge point to an edge point. For future checks, I can focus on those. There is plenty of play in the given caltrop, currently I'm optimizing for volume. How might I optimize for the edges working just slightly more nicely with each other? One method might be canonicalization. Here's some code adapted from George Hart's Canonical Polyhedra page.

view[v_, f_] := 
Graphics3D[{Opacity[.7],Polygon /@ Map[v[[#1]] & , f, {2}]}, SphericalRegion-> True,Boxed -> False] 
face2edges[face_] := MapThread[Sort[{#1, #2}] & , {face, RotateLeft[face]}]; 
f2e[f_] := Union @@ face2edges /@ f
closest[{P1_,P2_}]:=With[{L=P2-P1},P1-(L.P1 L)/L.L]
recenter[v_, e_] := 
  With[{centroid = Plus @@ (Chop[closest[#]]& /@ Map[v[[#1]] & , e, {2}]/Length[e])}, 
   (#1 - centroid & ) /@ v]
tangentify[v_, e_] := 
  Module[{newV = v, t, c}, Scan[(t = closest[v[[#1]]]; 
        c = 0.5*(1 - Sqrt[t . t])*t; newV[[#1[[1]]]] += c; 
        newV[[#1[[2]]]] += c; ) & , e]; newV]
unit[x_] := With[{mag2 = x . x}, If[mag2 != 0, x/Sqrt[mag2], x]]
cross[{ax_, ay_, az_}, {bx_, by_, bz_}] := 
  {ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx}
approxNormal[face_] := 
  unit[Plus @@ MapThread[unit[cross[#1 - #2, #2 - #3]] & , 
     {face, RotateLeft[face, 1], RotateLeft[face, 2]}]]
planarize[v_, f_] := 
  Module[{newV = v, faceXYZ, n, centroid}, 
   Scan[(faceXYZ = v[[#1]]; n = approxNormal[faceXYZ]; 
        centroid = Plus @@ faceXYZ/Length[faceXYZ]; 
        if[n . centroid < 0, n = -n]; 
        Scan[newV[[#1]] += 0.2*n . (centroid - v[[#1]])*n & , #1]; ) & , f]; 
    newV]
canonicalize[v_, f_] := 
  Module[{newV = N[v], e = f2e[f], oldV, maxChange}, 
   Do[oldV = newV; newV = tangentify[newV, e]; newV = recenter[newV, e]; 
      newV = planarize[newV, f]; maxChange = Max[Abs[oldV - newV]]; 
      If[maxChange < 10.^(-8), Break[]], {i, 80}]; 
    Print["Solved within ", maxChange]; newV]

I think I'll go back to my original set of vertices:

verts = {{0.0833`,0.0833`,0.4930122817942774`},{0.0833`,-0.0833`,-0.4930122817942774`},{-0.0833`,-0.0833`,0.4930122817942774`},{-0.0833`,0.0833`,-0.4930122817942774`},{0.4930122817942774`,0.0833`,0.0833`},{0.4930122817942774`,-0.0833`,-0.0833`},{-0.4930122817942774`,-0.0833`,0.0833`},{-0.4930122817942774`,0.0833`,-0.0833`},{0.0833`,0.4930122817942774`,0.0833`},{0.0833`,-0.4930122817942774`,-0.0833`},{-0.0833`,-0.4930122817942774`,0.0833`},{-0.0833`,0.4930122817942774`,-0.0833`},{0.32530527130128584`,-0.20709494964790603`,0.32530527130128584`},{0.32530527130128584`,0.20709494964790603`,-0.32530527130128584`},{-0.32530527130128584`,0.20709494964790603`,0.32530527130128584`},{-0.32530527130128584`,-0.20709494964790603`,-0.32530527130128584`},{0.32530527130128584`,-0.32530527130128584`,0.20709494964790603`},{0.32530527130128584`,0.32530527130128584`,-0.20709494964790603`},{-0.32530527130128584`,0.32530527130128584`,0.20709494964790603`},{-0.32530527130128584`,-0.32530527130128584`,-0.20709494964790603`},{0.20709494964790603`,-0.32530527130128584`,0.32530527130128584`},{0.20709494964790603`,0.32530527130128584`,-0.32530527130128584`},{-0.20709494964790603`,0.32530527130128584`,0.32530527130128584`},{-0.20709494964790603`,-0.32530527130128584`,-0.32530527130128584`},{0.28875291001058745`,0.28875291001058745`,0.28875291001058745`},{0.28875291001058745`,-0.28875291001058745`,-0.28875291001058745`},{-0.28875291001058745`,-0.28875291001058745`,0.28875291001058745`},{-0.28875291001058745`,0.28875291001058745`,-0.28875291001058745`},{-0.2142`,0.40369721678726284`,-0.2142`},{-0.2142`,0.2142`,-0.40369721678726284`},{-0.40369721678726284`,0.2142`,-0.2142`},{-0.2142`,-0.40369721678726284`,0.2142`},{-0.2142`,-0.2142`,0.40369721678726284`},{-0.40369721678726284`,-0.2142`,0.2142`},{0.2142`,-0.40369721678726284`,-0.2142`},{0.2142`,-0.2142`,-0.40369721678726284`},{0.40369721678726284`,-0.2142`,-0.2142`},{0.2142`,0.40369721678726284`,0.2142`},{0.2142`,0.2142`,0.40369721678726284`},{0.40369721678726284`,0.2142`,0.2142`},{-0.07272969962634213`,0.35355339059327373`,-0.35355339059327373`},{-0.35355339059327373`,0.35355339059327373`,-0.07272969962634213`},{-0.35355339059327373`,0.07272969962634213`,-0.35355339059327373`},{-0.07272969962634213`,-0.35355339059327373`,0.35355339059327373`},{-0.35355339059327373`,-0.35355339059327373`,0.07272969962634213`},{-0.35355339059327373`,-0.07272969962634213`,0.35355339059327373`},{0.07272969962634213`,-0.35355339059327373`,-0.35355339059327373`},{0.35355339059327373`,-0.35355339059327373`,-0.07272969962634213`},{0.35355339059327373`,-0.07272969962634213`,-0.35355339059327373`},{0.07272969962634213`,0.35355339059327373`,0.35355339059327373`},{0.35355339059327373`,0.35355339059327373`,0.07272969962634213`},{0.35355339059327373`,0.07272969962634213`,0.35355339059327373`},{0.07587339432355446`,0.44185`,-0.23402687345687453`},{0.07587339432355446`,0.23402687345687453`,-0.44185`},{-0.44185`,0.23402687345687453`,0.07587339432355446`},{-0.23402687345687453`,0.44185`,0.07587339432355446`},{-0.23402687345687453`,-0.07587339432355446`,-0.44185`},{-0.44185`,-0.07587339432355446`,-0.23402687345687453`},{0.07587339432355446`,-0.44185`,0.23402687345687453`},{0.07587339432355446`,-0.23402687345687453`,0.44185`},{-0.44185`,-0.23402687345687453`,-0.07587339432355446`},{-0.23402687345687453`,-0.44185`,-0.07587339432355446`},{-0.23402687345687453`,0.07587339432355446`,0.44185`},{-0.44185`,0.07587339432355446`,0.23402687345687453`},{-0.07587339432355446`,-0.44185`,-0.23402687345687453`},{-0.07587339432355446`,-0.23402687345687453`,-0.44185`},{0.44185`,-0.23402687345687453`,0.07587339432355446`},{0.23402687345687453`,-0.44185`,0.07587339432355446`},{0.23402687345687453`,0.07587339432355446`,-0.44185`},{0.44185`,0.07587339432355446`,-0.23402687345687453`},{-0.07587339432355446`,0.44185`,0.23402687345687453`},{-0.07587339432355446`,0.23402687345687453`,0.44185`},{0.44185`,0.23402687345687453`,-0.07587339432355446`},{0.23402687345687453`,0.44185`,-0.07587339432355446`},{0.23402687345687453`,-0.07587339432355446`,0.44185`},{0.44185`,-0.07587339432355446`,0.23402687345687453`}};

Because I had a list of faces for those points.

faces = {
{13,21,17},{14,22,18},{15,23,19},{16,24,20},

{28,29,41,30},{12,53,41,29},{22,54,41,53},{4,30,41,54},{28,31,42,29},{8,55,42,31},{19,56,42,55},{12,29,42,56},{28,30,43,31},{4,57,43,30},{16,58,43,57},{8,31,43,58},{27,32,44,33},{11,59,44,32},{21,60,44,59},{3,33,44,60},{27,34,45,32},{7,61,45,34},{20,62,45,61},{11,32,45,62},{27,33,46,34},{3,63,46,33},{15,64,46,63},{7,34,46,64},{26,35,47,36},{10,65,47,35},{24,66,47,65},{2,36,47,66},{26,37,48,35},{6,67,48,37},{17,68,48,67},{10,35,48,68},{26,36,49,37},{2,69,49,36},{14,70,49,69},{6,37,49,70},{25,38,50,39},{9,71,50,38},{23,72,50,71},{1,39,50,72},{25,40,51,38},{5,73,51,40},{18,74,51,73},{9,38,51,74},{25,39,52,40},{1,75,52,39},{13,76,52,75},{5,40,52,76},{11,10,68,59},{17,21,59,68},{10,11,62,65},{20,24,65,62},{12,9,74,53},{18,22,53,74},{9,12,56,71},{19,23,71,56},{8,7,64,55},{15,19,55,64},{7,8,58,61},{16,20,61,58},{6,5,76,67},{13,17,67,76},{5,6,70,73},{14,18,73,70},{4,2,66,57},{24,16,57,66},{3,1,72,63},{23,15,63,72},{2,4,54,69},{22,14,69,54},{1,3,60,75},{21,13,75,60}};

With the vertices and faces, we can canonicalize the placement of the vertices.

canon76 = canonicalize[verts, faces];

Let's have a look. Doesn't look too much different. Unfortunately, a lot of unit diagonals are gone.

Graphics3D[{Polygon[canonical76[[#]]] & /@ faces}, Boxed -> False, ViewAngle -> Pi/10, ImageSize -> {600, 600}]

canonical caltrop 76

Why is that nice? Since this is a self-dual polyhedron, we can combine it with the dual so that all edges intersect and are perpendicular to each other.

Graphics3D[{Polygon[canonical76[[#]]]&/@faces, Polygon[-canonical76[[#]]]&/@faces},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}]

canonical caltrop and dual

That doesn't happen in my unit diagonal and volume-optimized version, many diagonals are skew and don't intersect.

Graphics3D[{Polygon[verts[[#]]]&/@faces, Polygon[-canonical76[[#]]]&/@faces},Boxed->False,ViewAngle->Pi/10,ImageSize->{600,600}]

unit diagonal optimized and dual

In short, for a formal proof that this is a solid of constant width, it fails. The currently given solution is off by an amount that wouldn't be detectable if it were 3D printed, but it's still not perfect. However, it's very likely fixable by giving up the volume optimization, and instead going for a canonicalization biased optimization, while still maintaining all 150 unit diagonals. That's very doable, I just need to figure out how to do it.

POSTED BY: Ed Pegg

EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 10 years ago

- another post of yours has been selected for the Staff Picks group, congratulations ! We are happy to see you at the tops of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

POSTED BY: EDITORIAL BOARD

Ed Pegg

Ed Pegg, Wolfram Research

Posted 10 years ago

In the above, I mentioned that I could bound a volume of 1/2 with 96 points, with no two points greater than 1 apart. The mysterious oleg567 managed to bound 1/2 with 86 points. That's much, much better than my best solution.

POSTED BY: Ed Pegg

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