show/hide visualization coordinates
a : (0.2613710385699999, -0.050348907639999885, 0.61430196366)
b : (-0.05232211913000012, -0.52119061814, -0.21026136794000005)
c : (0.9169884723699999, -0.30282447363999987, -0.09733127514000006)
d : (1.20210424407, -0.28311755573999997, 0.8609591999599999)
e : (1.0083737409699998, 0.5715374261600001, 0.37926079586)
f : (0.2557179256699999, 0.4286935889600001, -0.26347154364)
g : (-0.6071439079300002, 0.20759384986000007, 0.19104398025999997)
h : (-1.0490529045299999, -0.5530380196399999, -0.28451421554)
i : (-0.5305364409300001, 0.12708012276000008, -0.8027612499400001)
j : (-1.40550004913, 0.37561458706000006, -0.3872262875400001)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['f', 'i', 'b'], ['a', 'g', 'b'], ['f', 'c', 'b'], ['f', 'e', 'c'], ['e', 'c', 'd'], ['g', 'h', 'b'], ['i', 'h', 'j'], ['i', 'h', 'b'], ['f', 'g', 'i'], ['g', 'j', 'h'], ['f', 'a', 'e'], ['a', 'e', 'd'], ['i', 'g', 'j'], ['a', 'c', 'd'], ['a', 'c', 'b'], ['f', 'a', 'g']]
Coordinate Data:
j : [4600655751 / 2500000000, 347619831 / 2000000000, 2591526699 / 2500000000]
b : [608855463 / 1250000000, 10706151207 / 10000000000, 2686393 / 3125000]
e : [-5736114897 / 10000000000, -55282309 / 2500000000, 1350617981 / 5000000000]
i : [4826493461 / 5000000000, 2111721899 / 5000000000, 726072821 / 500000000]
g : [1302382699 / 1250000000, 3418306527 / 10000000000, 2291702059 / 5000000000]
d : [-959177491 / 1250000000, 8325420583 / 10000000000, -2115748079 / 10000000000]
h : [7419075779 / 5000000000, 5512312611 / 5000000000, 2334746519 / 2500000000]
c : [-4822262211 / 10000000000, 4261244881 / 5000000000, 116674323 / 156250000]
a : [1733912127 / 10000000000, 2998867051 / 5000000000, 87706071 / 2500000000]
f : [223805407 / 1250000000, 75456821 / 625000000, 9128559357 / 10000000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 10
|E| = 24
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 31249999988698149880564001260912796989830549925612701555201 / 62500000164437045871629441062784907341650000000000000000000
Collision distance in [35355339 / 50000000, 70710679 / 100000000] ~ [0.70711, 0.70711]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [3699 / 5000, 37 / 50] ~ [0.7398, 0.74]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 28435691115256110419477 / 1000000000000000000000000000000000000000
rho in [8332051 / 1562500000000000, 23957051 / 1562500000000000] ~ [0.0, 0.0]
sigma_min ^ 2 / (16 * E ^ .5) in [13682601 / 1959591796, 855625 / 122474487] ~ [0.00698, 0.00699]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-4993 / 25000000, 10041 / 50000000] ~ [-0.0002, 0.0002]
LHS DEN := 8 * |E| ^ .5 in [122474487 / 3125000, 489897949 / 12500000] ~ [39.19184, 39.19184]
LHS := (LHS NUM) / (LHS DEN) in [-4993 / 979795896, 3347 / 653197264] ~ [-1e-05, 1e-05]
CD / |V| ^ .5 in [70710678 / 316227767, 70710679 / 316227766] ~ [0.22361, 0.22361]
Success: LHS < CD / |V| ^ .5
Success: existence proven