show/hide visualization coordinates
a : (1.0389580132421052, 0.09402784013684218, 0.015152841747368417)
b : (0.43569592824210523, -0.3253019521631578, -0.6632530727526316)
c : (0.8109043326421054, 0.5875645542368422, -0.8241427527526316)
d : (1.3143977566421055, 1.049154465436842, -0.09378073975263163)
e : (0.6602656748421053, 0.7955755469368422, 0.6188259486473684)
f : (0.24569292474210536, -0.11390084666315786, 0.5874274175473684)
g : (0.7396414119421054, -0.8532152503631578, 0.12979309774736836)
h : (0.06092830754210532, -1.2372121199631578, -0.49622452565263164)
i : (-0.5309719050578947, -0.4337849208631579, -0.4312117677526316)
j : (-0.16460255845789468, 0.4739622886368422, -0.6357587260526316)
k : (0.3396207961421053, 1.2319077139368422, -0.22189074905263162)
l : (0.8635827568421054, 1.759458901136842, 0.4468051593473684)
m : (-0.2919561070578947, 0.6897559281368422, 0.3323427591473683)
n : (-0.7521661298578947, -0.16999845506315786, 0.5537599934473685)
o : (-0.2007526525578947, -0.9962087263631578, 0.43835622064736834)
p : (-0.8890888332578947, -1.3386972690631578, -0.20109562255263164)
q : (-1.4554380755578946, -0.5265139856631579, -0.06130678345263163)
r : (-1.0683466838578948, 0.3821268362368422, -0.2177250957526316)
s : (-1.1563649571578947, -1.0687005486631578, 0.7239263972473684)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['i', 'q', 'p'], ['h', 'p', 'o'], ['s', 'q', 'p'], ['d', 'a', 'c'], ['d', 'a', 'e'], ['d', 'e', 'l'], ['r', 'n', 'q'], ['d', 'k', 'l'], ['o', 'n', 'f'], ['l', 'k', 'e'], ['m', 'n', 'f'], ['r', 'i', 'q'], ['s', 'n', 'o'], ['d', 'k', 'c'], ['b', 'j', 'i'], ['b', 'g', 'a'], ['r', 'j', 'm'], ['r', 'm', 'n'], ['s', 'p', 'o'], ['b', 'a', 'c'], ['r', 'i', 'j'], ['b', 'j', 'c'], ['g', 'h', 'o'], ['o', 'g', 'f'], ['b', 'g', 'h'], ['i', 'h', 'p'], ['m', 'e', 'f'], ['s', 'n', 'q'], ['a', 'e', 'f'], ['b', 'i', 'h'], ['k', 'j', 'c'], ['m', 'k', 'e'], ['g', 'a', 'f'], ['k', 'j', 'm']]
Coordinate Data:
d : [4785822717 / 2500000000, 8060781277 / 5000000000, 4444066411 / 10000000000]
c : [3527089157 / 2500000000, 5752831721 / 5000000000, -2859553719 / 10000000000]
n : [-1522347997 / 10000000000, 3930033349 / 10000000000, 10919473743 / 10000000000]
s : [-556433627 / 1000000000, -5056987587 / 10000000000, 12621137781 / 10000000000]
q : [-4277533727 / 5000000000, 364878043 / 10000000000, 2384402987 / 5000000000]
p : [-2891575031 / 10000000000, -7756954791 / 10000000000, 3370917583 / 10000000000]
i : [689594251 / 10000000000, 1292168691 / 10000000000, 1069756131 / 10000000000]
m : [3079752231 / 10000000000, 12527577181 / 10000000000, 43526507 / 50000000]
a : [8194446717 / 5000000000, 6570296301 / 10000000000, 2766701113 / 5000000000]
b : [1294534073 / 1250000000, 1188499189 / 5000000000, -1250656919 / 10000000000]
e : [252039401 / 200000000, 13585773369 / 10000000000, 2314026659 / 2000000000]
r : [-4684153537 / 10000000000, 4725643131 / 5000000000, 3204622851 / 10000000000]
j : [4353287717 / 10000000000, 5184820393 / 5000000000, -243928363 / 2500000000]
k : [9395521263 / 10000000000, 17949095039 / 10000000000, 1581483159 / 5000000000]
l : [1463514087 / 1000000000, 23224606911 / 10000000000, 4924962701 / 5000000000]
h : [6608596377 / 10000000000, -67421033 / 100000000, 52453569 / 1250000000]
f : [8456242549 / 10000000000, 4491009433 / 10000000000, 703509249 / 625000000]
g : [13395727421 / 10000000000, -725533651 / 2500000000, 3339902393 / 5000000000]
o : [498973347 / 1250000000, -1083017341 / 2500000000, 1953087203 / 2000000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 19
|E| = 51
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 500001059754905084378194083657382924383781541022517411020889 / 1000003133711064324280200072028382864485100000000000000000000
Collision distance in [35355321 / 50000000, 70710643 / 100000000] ~ [0.70711, 0.70711]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [1901 / 10000, 1903 / 10000] ~ [0.1901, 0.1903]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 518352873245453971057928742581461 / 10000000000000000000000000000000000000000
rho in [22767 / 100000000, 1423 / 6250000] ~ [0.00023, 0.00023]
sigma_min ^ 2 / (16 * E ^ .5) in [3613801 / 11426285488, 3621409 / 11426285472] ~ [0.00032, 0.00032]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [8910561 / 100000000, 4484433 / 50000000] ~ [0.08911, 0.08969]
LHS DEN := 8 * |E| ^ .5 in [357071421 / 6250000, 714142843 / 12500000] ~ [57.13143, 57.13143]
LHS := (LHS NUM) / (LHS DEN) in [8910561 / 5713142744, 1494811 / 952190456] ~ [0.00156, 0.00157]
CD / |V| ^ .5 in [70710642 / 435889895, 70710643 / 435889894] ~ [0.16222, 0.16222]
Success: LHS < CD / |V| ^ .5
Success: existence proven