show/hide visualization coordinates
a : (0.6678235123947368, 0.8546416074210527, 0.20546898452631573)
b : (0.6728991179947369, 0.4446125670210527, -0.7065893683736842)
c : (-0.1102137597052632, 0.9901040339210526, -0.40797044597368426)
d : (-0.2630548094052632, 0.7018134412210527, 0.5372957342263158)
e : (0.3822023659947368, -0.06065349707894735, 0.4894681826263157)
f : (1.2531287457947369, 0.06656434282105261, 0.014807273126315745)
g : (1.468576479894737, 0.9576626163210525, -0.3845999591736843)
h : (0.5947852712947369, -0.5204254900789473, -0.45638991697368425)
i : (-0.22914024270526318, 0.020958467721052698, -0.6238717434736842)
j : (-1.0278220409052632, 0.6178114260210527, -0.5472300812736843)
k : (-0.8830852382052632, 1.3588300448210529, 0.10847005802631571)
l : (-1.2300432027052632, 0.47476814862105265, 0.4216069660263157)
m : (-0.5801901069052632, -0.23253278107894737, 0.6998448855263157)
n : (0.04774958749473679, -0.9997083311789474, 0.5689507072263158)
o : (1.017526124994737, -0.8272630329789474, 0.39633566012631577)
p : (0.5157460588947368, -1.4767483355789475, -0.17496603807368427)
q : (-0.3075240149052632, -0.9414974131789473, -0.3639973924736843)
r : (-1.0591210597052632, -0.30964625817894736, -0.17461418197368428)
s : (-0.9302427896052632, -1.1192915565789474, 0.3979806763263158)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['h', 'p', 'o'], ['d', 'a', 'c'], ['b', 'g', 'f'], ['d', 'a', 'e'], ['d', 'k', 'l'], ['n', 'p', 'q'], ['b', 'h', 'f'], ['o', 'h', 'f'], ['r', 'i', 'q'], ['h', 'p', 'q'], ['d', 'k', 'c'], ['n', 'p', 'o'], ['b', 'g', 'a'], ['i', 'h', 'q'], ['m', 'e', 'n'], ['s', 'r', 'm'], ['j', 'i', 'c'], ['e', 'n', 'o'], ['s', 'm', 'n'], ['b', 'a', 'c'], ['b', 'i', 'c'], ['l', 'k', 'j'], ['r', 'i', 'j'], ['d', 'e', 'm'], ['f', 'e', 'o'], ['d', 'm', 'l'], ['r', 'j', 'l'], ['s', 'n', 'q'], ['a', 'e', 'f'], ['b', 'i', 'h'], ['k', 'j', 'c'], ['l', 'r', 'm'], ['s', 'r', 'q'], ['g', 'a', 'f']]
Coordinate Data:
d : [3304272377 / 10000000000, 3017661843 / 2500000000, 1073838579 / 1250000000]
c : [2416341437 / 5000000000, 14953553299 / 10000000000, -86195317 / 1000000000]
n : [3206158173 / 5000000000, -309035647 / 625000000, 4453629181 / 5000000000]
s : [-134704297 / 400000000, -3070201303 / 5000000000, 7197558053 / 10000000000]
q : [1429790161 / 5000000000, -1090615293 / 2500000000, -84444527 / 2000000000]
p : [554614053 / 500000000, -2428742599 / 2500000000, 1468090909 / 10000000000]
i : [910854511 / 2500000000, 5262097637 / 10000000000, -604193229 / 2000000000]
m : [66459701 / 5000000000, 2727185149 / 10000000000, 2043240029 / 2000000000]
a : [2522611119 / 2000000000, 6799464517 / 5000000000, 1054488227 / 2000000000]
b : [12663811651 / 10000000000, 949863863 / 1000000000, -1924071197 / 5000000000]
e : [9756844131 / 10000000000, 4445977989 / 10000000000, 2028108279 / 2500000000]
r : [-2328195063 / 5000000000, 978025189 / 5000000000, 147160947 / 1000000000]
j : [-2171699969 / 5000000000, 561531361 / 500000000, -2254549523 / 10000000000]
k : [-2896031911 / 10000000000, 582525419 / 312500000, 430245187 / 1000000000]
l : [-1591402889 / 2500000000, 4900097223 / 5000000000, 148676419 / 200000000]
h : [371333537 / 312500000, -151741941 / 10000000000, -33653697 / 250000000]
f : [18466107929 / 10000000000, 1429539097 / 2500000000, 3365824021 / 10000000000]
g : [2062058527 / 1000000000, 14629139123 / 10000000000, -314124151 / 5000000000]
o : [16110081721 / 10000000000, -322011737 / 1000000000, 7181107891 / 10000000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 19
|E| = 51
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 488281221757118167923742615658062671744250410502434099121 / 976562510952402457323136212115728181506445312500000000000
Collision distance in [2828427 / 4000000, 17677669 / 25000000] ~ [0.70711, 0.70711]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [3183 / 10000, 637 / 2000] ~ [0.3183, 0.3185]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 55447637910888137706994211 / 312500000000000000000000000000000000000
rho in [421227303 / 1000000000000000, 431227303 / 1000000000000000] ~ [0.0, 0.0]
sigma_min ^ 2 / (16 * E ^ .5) in [10131489 / 11426285488, 1449175 / 1632326496] ~ [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 [-3111 / 25000000, 13871 / 50000000] ~ [-0.00012, 0.00028]
LHS DEN := 8 * |E| ^ .5 in [357071421 / 6250000, 714142843 / 12500000] ~ [57.13143, 57.13143]
LHS := (LHS NUM) / (LHS DEN) in [-1037 / 476095228, 13871 / 2856571368] ~ [-0.0, 0.0]
CD / |V| ^ .5 in [2020305 / 12453997, 35355338 / 217944947] ~ [0.16222, 0.16222]
Success: LHS < CD / |V| ^ .5
Success: existence proven