show/hide visualization coordinates
a : (0.4820856278600001, 0.76641708556, 0.1922281573066667)
b : (0.98525434536, -0.09093159054, 0.08371791410666668)
c : (0.26718752876000007, 0.15331920096, -0.5679887648933334)
d : (-0.35386015633999995, 0.90299220826, -0.3393182770933333)
e : (-0.29586774463999993, 0.15056819086, 0.31680307180666667)
f : (0.4900617146600001, 0.09420688036000002, 0.9325453619066667)
g : (1.4683461264600002, -0.11135860523999996, 0.9590493124066667)
h : (0.8120840685600001, -0.8313053118399999, 0.7332244943066667)
i : (0.20144118126000005, -0.69793481324, -0.04736987409333332)
j : (-0.42053906463999996, -0.53032316134, -0.8122534743933333)
k : (-0.47734756693999997, 0.39348283786, -1.1908763098933335)
l : (-1.1978234954399998, 1.00601531576, -0.8657327022933333)
m : (-1.0119301208399998, 0.15120314776000002, -0.38123312579333335)
n : (-0.7870832126399998, -0.6939843521400001, 0.10363973650666669)
o : (-0.16200923143999996, -0.6623670330399999, 0.8835644801066667)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['a', 'b', 'f'], ['a', 'd', 'e'], ['o', 'i', 'n'], ['o', 'e', 'n'], ['j', 'k', 'm'], ['l', 'd', 'k'], ['j', 'n', 'm'], ['o', 'h', 'i'], ['b', 'f', 'g'], ['b', 'h', 'i'], ['j', 'c', 'i'], ['b', 'h', 'g'], ['o', 'f', 'e'], ['h', 'f', 'g'], ['c', 'd', 'k'], ['a', 'b', 'c'], ['a', 'f', 'e'], ['a', 'c', 'd'], ['b', 'c', 'i'], ['o', 'h', 'f'], ['d', 'l', 'm'], ['e', 'n', 'm'], ['j', 'i', 'n'], ['l', 'k', 'm'], ['d', 'e', 'm'], ['j', 'c', 'k']]
Coordinate Data:
e : [492361121 / 625000000, 2420042517 / 10000000000, 321772757 / 2500000000]
o : [1634798201 / 2500000000, 2637348689 / 2500000000, -876104611 / 2000000000]
m : [7519200849 / 5000000000, 603423237 / 2500000000, 2066863251 / 2500000000]
h : [-800435049 / 2500000000, 1529847193 / 1250000000, -2877123197 / 10000000000]
g : [-390574431 / 400000000, 2519655239 / 5000000000, -2567685689 / 5000000000]
a : [98244211 / 10000000000, -373844643 / 1000000000, 2532840173 / 10000000000]
l : [4224333861 / 2500000000, -1533607183 / 2500000000, 13112448769 / 10000000000]
d : [8457702053 / 10000000000, -5104197657 / 10000000000, 7848304517 / 10000000000]
c : [1123612601 / 5000000000, 37383319 / 156250000, 2027001879 / 2000000000]
n : [1598741577 / 1250000000, 10865567947 / 10000000000, 3418724381 / 10000000000]
b : [-1233360741 / 2500000000, 4835040331 / 10000000000, 723588521 / 2000000000]
j : [71285087 / 78125000, 9228956039 / 10000000000, 1257765649 / 1000000000]
k : [9692576159 / 10000000000, -9103953 / 10000000000, 3272776969 / 2000000000]
i : [2904688677 / 10000000000, 5452536279 / 5000000000, 4928820487 / 10000000000]
f : [18483343 / 10000000000, 1491827811 / 5000000000, -4870331873 / 10000000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 15
|E| = 39
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 41666662651011583408287416980757864691239022777512121118107 / 83333338850931487878078267771008799795550000000000000000000
Collision distance in [4419417 / 6250000, 70710673 / 100000000] ~ [0.70711, 0.70711]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [887 / 2000, 4437 / 10000] ~ [0.4435, 0.4437]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 4142148930760529127878229227 / 5000000000000000000000000000000000000000
rho in [910181183 / 1000000000000000, 920181183 / 1000000000000000] ~ [0.0, 0.0]
sigma_min ^ 2 / (16 * E ^ .5) in [786769 / 399679872, 19686969 / 9991996784] ~ [0.00197, 0.00197]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-9751 / 100000000, 30367 / 100000000] ~ [-0.0001, 0.0003]
LHS DEN := 8 * |E| ^ .5 in [624499799 / 12500000, 3122499 / 62500] ~ [49.95998, 49.95998]
LHS := (LHS NUM) / (LHS DEN) in [-1393 / 713714056, 30367 / 4995998392] ~ [-0.0, 1e-05]
CD / |V| ^ .5 in [23570224 / 129099445, 70710673 / 387298334] ~ [0.18257, 0.18257]
Success: LHS < CD / |V| ^ .5
Success: existence proven