show/hide visualization coordinates
a : (-0.5076699410684211, -0.22586189353157893, 0.4288308376315789)
b : (-0.15687355986842105, -0.8941609394315788, -0.22715565926842107)
c : (-0.0054258010684210545, -1.027837256631579, 0.752228952731579)
d : (0.08265739933157895, -0.15537768713157896, 1.232911435531579)
e : (-0.5268405242684211, 0.6000271508684211, 0.9923375146315789)
f : (-0.6567925774684211, 0.6665935602684211, 0.003054250531578928)
g : (-0.7737692706684212, -0.16916016963157887, -0.533445661868421)
h : (-0.30295484536842104, -0.741752481431579, -1.204617637968421)
i : (0.447520809031579, -0.27870974873157894, -0.733048215868421)
j : (0.679029370531579, -0.550707238931579, 0.20098666103157897)
k : (0.870335294531579, -0.7669427978315788, 1.158402049631579)
l : (0.9380827193315789, 0.1771848713684211, 0.8358602217315789)
m : (0.37187801563157896, 0.7892481327684211, 1.3879388369315788)
n : (0.25822714973157895, 0.7630434618684211, 0.39476372873157894)
o : (0.14544640313157897, 0.6643709736684211, -0.593944642468421)
p : (-0.7715231056684211, 0.7211244945684211, -0.9888445764684209)
q : (-1.0120269399684212, -0.10256130063157898, -1.502361485768421)
r : (-0.05875465636842109, 0.1955418142684211, -1.453305536668421)
s : (0.9794540605315789, 0.3359370542684211, -0.1505910727684211)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['r', 'q', 'p'], ['d', 'a', 'c'], ['o', 'p', 'f'], ['d', 'a', 'e'], ['r', 'i', 'o'], ['d', 'k', 'l'], ['l', 's', 'n'], ['o', 'n', 'f'], ['s', 'n', 'o'], ['j', 's', 'i'], ['r', 'p', 'o'], ['d', 'k', 'c'], ['b', 'j', 'i'], ['b', 'g', 'a'], ['m', 'e', 'n'], ['s', 'i', 'o'], ['b', 'a', 'c'], ['l', 'k', 'j'], ['h', 'g', 'q'], ['e', 'n', 'f'], ['d', 'e', 'm'], ['r', 'h', 'i'], ['d', 'm', 'l'], ['b', 'j', 'c'], ['l', 's', 'j'], ['b', 'g', 'h'], ['g', 'p', 'f'], ['a', 'e', 'f'], ['b', 'i', 'h'], ['k', 'j', 'c'], ['g', 'q', 'p'], ['r', 'h', 'q'], ['l', 'm', 'n'], ['g', 'a', 'f']]
Coordinate Data:
d : [3093597 / 7812500, 1611114519 / 2500000000, -651642177 / 1000000000]
c : [1210159041 / 2500000000, 15169053771 / 10000000000, -854798471 / 5000000000]
n : [68878333 / 312500000, -1369876707 / 5000000000, 932527649 / 5000000000]
s : [-1252040613 / 2500000000, 765655331 / 5000000000, 7318603313 / 10000000000]
q : [14906647553 / 10000000000, 5916294211 / 10000000000, 20836307443 / 10000000000]
p : [1250160921 / 1000000000, -2320563741 / 10000000000, 314022767 / 200000000]
i : [311170063 / 10000000000, 1919444673 / 2500000000, 1642896843 / 1250000000]
m : [1067597997 / 10000000000, -3001800123 / 10000000000, -1008336973 / 1250000000]
a : [2465769391 / 2500000000, 357465007 / 500000000, 1524384209 / 10000000000]
b : [794389219 / 1250000000, 13832290599 / 10000000000, 4042124589 / 5000000000]
r : [5373924717 / 10000000000, 1467631531 / 5000000000, 1271609247 / 625000000]
e : [2513695849 / 2500000000, -34674697 / 312500000, -4110682561 / 10000000000]
j : [-62622361 / 312500000, 5198876797 / 5000000000, 152113039 / 400000000]
k : [-489621849 / 1250000000, 12560109183 / 10000000000, -5771327911 / 10000000000]
l : [-57430613 / 125000000, 3118832491 / 10000000000, -19889919 / 78125000]
h : [7815926607 / 10000000000, 12308206019 / 10000000000, 3571773793 / 2000000000]
f : [1419287991 / 1250000000, -887627199 / 5000000000, 18069219 / 31250000]
g : [626203543 / 500000000, 6582282901 / 10000000000, 2786787301 / 2500000000]
o : [1665957061 / 5000000000, -438257133 / 2500000000, 1175213901 / 1000000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 19
|E| = 51
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 624974376489388945523755023218261647679575492303597106689 / 1044308925233737554944197248400030219784000000000000000000
Collision distance in [38680013 / 50000000, 77360027 / 100000000] ~ [0.7736, 0.7736]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [1111 / 2500, 2223 / 5000] ~ [0.4444, 0.4446]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 208733081998634540291 / 125000000000000000000000000000000000000
rho in [129223243 / 100000000000000000, 1129223243 / 100000000000000000] ~ [0.0, 0.0]
sigma_min ^ 2 / (16 * E ^ .5) in [1234321 / 714142843, 1647243 / 952190456] ~ [0.00173, 0.00173]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-1249 / 6250000, 10073 / 50000000] ~ [-0.0002, 0.0002]
LHS DEN := 8 * |E| ^ .5 in [357071421 / 6250000, 714142843 / 12500000] ~ [57.13143, 57.13143]
LHS := (LHS NUM) / (LHS DEN) in [-1249 / 357071421, 1439 / 408081624] ~ [-0.0, 0.0]
CD / |V| ^ .5 in [77360026 / 435889895, 77360027 / 435889894] ~ [0.17748, 0.17748]
Success: LHS < CD / |V| ^ .5
Success: existence proven