show/hide visualization coordinates
a : (-0.4212716513210527, -0.6000071500105263, -1.0342487929578947)
b : (0.5272918061789473, -0.6990471986105261, -0.7335522085578948)
c : (-0.21779537302105267, -0.7748457543105262, -0.07090624395789469)
d : (-1.1589853610210528, -0.5985374897105261, -0.3591367114578947)
e : (-0.7887943471210528, 0.2667519507894738, -0.6971201390578948)
f : (0.1703486977789473, 0.2038770940894738, -0.9729667843578947)
g : (1.0222540634789472, 0.12413637518947379, -0.4553775017578947)
h : (0.7001975225789473, -0.5230954958105262, 0.23554252954210528)
i : (-0.05243631442105268, -0.5646566596105261, 0.8926688314421052)
j : (-0.9644021074210527, -0.8426025488105263, 0.5908999216421053)
k : (-0.8618702834210528, 0.10454456598947381, 0.28692694924210527)
l : (-0.2135993826210527, 0.7815882249894738, -0.06143596935789475)
m : (-0.3040945036210527, 1.0833622237894738, -1.0105108276578947)
n : (0.6009549083789473, 1.0210415667894739, -0.5897951311578947)
o : (0.6955308275789474, 0.6462107589894738, 0.33246148224210526)
p : (1.4957067245789473, 0.053255232989473766, 0.4225851633421053)
q : (0.7526150775789473, -0.004966474310526214, 1.0892372740421052)
r : (-0.1195725927210527, 0.4328530595894738, 0.8710567904421053)
s : (-0.8620777114210527, -0.10986228201052617, 1.2636713683421053)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['q', 'p', 'o'], ['d', 'a', 'c'], ['b', 'g', 'f'], ['h', 'i', 'c'], ['d', 'a', 'e'], ['k', 's', 'r'], ['r', 'q', 'o'], ['b', 'c', 'h'], ['l', 'k', 'e'], ['l', 'k', 'r'], ['d', 'k', 'e'], ['r', 's', 'i'], ['m', 'n', 'f'], ['r', 'i', 'q'], ['d', 'k', 'j'], ['j', 's', 'i'], ['h', 'p', 'q'], ['i', 'h', 'q'], ['d', 'j', 'c'], ['j', 'i', 'c'], ['b', 'a', 'c'], ['g', 'n', 'o'], ['g', 'p', 'o'], ['g', 'n', 'f'], ['k', 's', 'j'], ['b', 'g', 'h'], ['m', 'e', 'f'], ['l', 'r', 'o'], ['a', 'e', 'f'], ['b', 'a', 'f'], ['g', 'h', 'p'], ['l', 'm', 'e'], ['l', 'n', 'o'], ['l', 'm', 'n']]
Coordinate Data:
d : [8448692179 / 5000000000, 5730510319 / 5000000000, 8201658461 / 10000000000]
c : [3742742239 / 5000000000, 3306025821 / 2500000000, 2659676893 / 5000000000]
n : [-21938073 / 312500000, -4734769927 / 10000000000, 5254121329 / 5000000000]
s : [6964153931 / 5000000000, 6574268561 / 10000000000, -8026422337 / 10000000000]
q : [-554655007 / 2500000000, 1381327621 / 2500000000, -3141040697 / 5000000000]
p : [-4824768249 / 5000000000, 4943093411 / 10000000000, 384439713 / 10000000000]
i : [1457973473 / 2500000000, 11122212337 / 10000000000, -539549621 / 1250000000]
m : [1043559473 / 1250000000, -5357976497 / 10000000000, 14715399623 / 10000000000]
a : [9520247261 / 10000000000, 11475717241 / 10000000000, 3738194819 / 2500000000]
b : [17306343 / 5000000000, 12466117727 / 10000000000, 1493226679 / 1250000000]
e : [13195474219 / 10000000000, 2808126233 / 10000000000, 11581492737 / 10000000000]
r : [260130267 / 400000000, 229423029 / 2000000000, -2050138279 / 5000000000]
j : [7475775911 / 5000000000, 13901671229 / 10000000000, -129870787 / 1000000000]
k : [6963116791 / 5000000000, 4430200081 / 10000000000, 870510927 / 5000000000]
l : [3721762287 / 5000000000, -2340236509 / 10000000000, 32654069 / 62500000]
h : [-847222239 / 5000000000, 10706600699 / 10000000000, 2254866051 / 10000000000]
f : [360404377 / 1000000000, 8592187 / 25000000, 1433995919 / 1000000000]
g : [-4915009887 / 10000000000, 4234281989 / 10000000000, 2291016591 / 2500000000]
o : [-205972191 / 1250000000, -986461849 / 10000000000, 321419131 / 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 = 7594912775933577441409706076518249210343559806394754401 / 13383476885021862999291657701160926353000000000000000000
Collision distance in [2354111 / 3125000, 75331553 / 100000000] ~ [0.75332, 0.75332]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [4521 / 10000, 4523 / 10000] ~ [0.4521, 0.4523]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 2887153110161638057363 / 2500000000000000000000000000000000000000
rho in [10746447 / 10000000000000000, 110746447 / 10000000000000000] ~ [0.0, 0.0]
sigma_min ^ 2 / (16 * E ^ .5) in [20439441 / 11426285488, 20457529 / 11426285472] ~ [0.00179, 0.00179]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-19987 / 100000000, 1007 / 5000000] ~ [-0.0002, 0.0002]
LHS DEN := 8 * |E| ^ .5 in [357071421 / 6250000, 714142843 / 12500000] ~ [57.13143, 57.13143]
LHS := (LHS NUM) / (LHS DEN) in [-19987 / 5713142736, 5035 / 1428285684] ~ [-0.0, 0.0]
CD / |V| ^ .5 in [75331552 / 435889895, 6848323 / 39626354] ~ [0.17282, 0.17282]
Success: LHS < CD / |V| ^ .5
Success: existence proven