show/hide visualization coordinates
a : (0.7984779656157895, -0.5152283628842106, 0.06490141869473698)
b : (1.3438543680157895, 0.32103412521578945, 0.008131037994736967)
c : (0.6485749102157895, 0.22492775001578946, 0.720417188294737)
d : (-0.09528070918421044, -0.4060029958842105, 0.49993947679473694)
e : (0.003461165715789527, -1.0021476271842105, -0.2968434541052631)
f : (0.8003006590157895, -0.8432424642842105, -0.879766800905263)
g : (0.6523224552157896, 0.12690785341578942, -0.6876308828052631)
h : (0.4319842732157895, 0.7246086916157894, 0.08320609999473694)
i : (1.1796543864157896, 1.0693877940157894, 0.6507720083947369)
j : (0.23986340801578954, 1.0964265679157894, 0.991422868094737)
k : (-0.19307116738421043, 0.23604707361578944, 1.260336990394737)
l : (-1.0030432622842105, -0.08418492618421058, 0.769011166994737)
m : (-0.8876801957842104, -0.6055065494842105, -0.07651547540526299)
n : (-0.7652822272842104, -1.1194629137842105, -0.925552128005263)
o : (-0.07615119108421048, -0.41713069788421053, -1.103946773305263)
p : (-0.30258784598421046, 0.30432879621578945, -0.4495382887052631)
q : (-0.5023351601842105, 0.7162666447157895, 0.43951807449473695)
r : (-1.2427508876842104, 0.3287445261157894, -0.10964361420526308)
s : (-1.0303109445842105, -0.15577328528421053, -0.9582189127052632)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['e', 'a', 'd'], ['k', 'c', 'j'], ['g', 'f', 'o'], ['m', 'n', 's'], ['q', 'r', 'p'], ['l', 'r', 'm'], ['e', 'a', 'f'], ['m', 'r', 's'], ['i', 'c', 'j'], ['q', 'h', 'p'], ['e', 'm', 'n'], ['g', 'h', 'p'], ['g', 'o', 'p'], ['i', 'h', 'b'], ['a', 'c', 'b'], ['i', 'h', 'j'], ['g', 'a', 'b'], ['q', 'k', 'j'], ['g', 'h', 'b'], ['e', 'f', 'o'], ['k', 'c', 'd'], ['g', 'a', 'f'], ['l', 'r', 'q'], ['l', 'k', 'd'], ['i', 'c', 'b'], ['m', 'e', 'd'], ['a', 'c', 'd'], ['n', 'o', 's'], ['q', 'k', 'l'], ['r', 's', 'p'], ['q', 'h', 'j'], ['e', 'n', 'o'], ['l', 'm', 'd'], ['p', 'o', 's']]
Coordinate Data:
r : [4306895663 / 2500000000, 194564591 / 1250000000, 5967544661 / 10000000000]
s : [15103183221 / 10000000000, 3200847421 / 5000000000, 7226648823 / 5000000000]
h : [480231043 / 10000000000, -2402124927 / 10000000000, 4039047519 / 10000000000]
p : [1565190447 / 2000000000, 1800674027 / 10000000000, 4683245703 / 5000000000]
g : [-1723150777 / 10000000000, 714976691 / 2000000000, 11747417347 / 10000000000]
f : [-640586563 / 2000000000, 1659548329 / 1250000000, 854298533 / 625000000]
n : [778306003 / 625000000, 16038591127 / 10000000000, 14126629799 / 10000000000]
q : [9823425377 / 10000000000, -1159352229 / 5000000000, 237963887 / 5000000000]
j : [480287939 / 2000000000, -612030369 / 1000000000, -2521560081 / 5000000000]
i : [-6996470089 / 10000000000, -5849915951 / 10000000000, -327322313 / 2000000000]
m : [13676875733 / 10000000000, 2724756871 / 2500000000, 5636263273 / 10000000000]
e : [2382731059 / 5000000000, 14865438261 / 10000000000, 391977153 / 500000000]
l : [7415253199 / 5000000000, 5685811251 / 10000000000, -2819003151 / 10000000000]
k : [6730785449 / 10000000000, 2483491253 / 10000000000, -1546452277 / 2000000000]
o : [2780792843 / 5000000000, 1126908621 / 1250000000, 3977644063 / 2500000000]
d : [5752880867 / 10000000000, 2225997987 / 2500000000, -128286249 / 10000000000]
b : [-1727693981 / 2000000000, 1633620737 / 10000000000, 4789798139 / 10000000000]
c : [-1685675327 / 10000000000, 2594684489 / 10000000000, -583265841 / 2500000000]
a : [-3184705881 / 10000000000, 4998122809 / 5000000000, 1055523583 / 2500000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 19
|E| = 51
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 2802451588044741985615675324651587943671597401854474186201 / 6791064625616243914168515928775604460556250000000000000000
Collision distance in [80299 / 125000, 64239201 / 100000000] ~ [0.64239, 0.64239]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [3181 / 10000, 3183 / 10000] ~ [0.3181, 0.3183]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 128584940356397802517038553105297 / 10000000000000000000000000000000000000000
rho in [11339 / 100000000, 567 / 5000000] ~ [0.00011, 0.00011]
sigma_min ^ 2 / (16 * E ^ .5) in [10118761 / 11426285488, 3377163 / 3808761824] ~ [0.00089, 0.00089]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [2084821 / 100000000, 2126431 / 100000000] ~ [0.02085, 0.02126]
LHS DEN := 8 * |E| ^ .5 in [357071421 / 6250000, 714142843 / 12500000] ~ [57.13143, 57.13143]
LHS := (LHS NUM) / (LHS DEN) in [2084821 / 5713142744, 2126431 / 5713142736] ~ [0.00036, 0.00037]
CD / |V| ^ .5 in [12847840 / 87177979, 64239201 / 435889894] ~ [0.14737, 0.14737]
Success: LHS < CD / |V| ^ .5
Success: existence proven