show/hide visualization coordinates
a : (0.06198891977777776, 0.927582519511111, -0.28673046722777784)
b : (-0.7989062485222223, 1.227434810911111, 0.12430251897222216)
c : (0.07006098717777776, 1.583018686911111, 0.46847696867222216)
d : (0.5481363577777778, 0.709426817711111, 0.5594779936722222)
e : (0.6139187472777777, 0.095806821811111, -0.2273785227277778)
f : (0.036413724077777765, 0.25661078461111103, -1.0277721953277776)
g : (-0.8074646998222221, 0.6995207118111111, -0.7249521164277778)
h : (-0.6115579098222221, 0.24534183391111103, 0.1441520754722222)
i : (-0.37434930042222225, 0.815605264211111, 0.9306199502722222)
j : (0.02457138587777774, -0.09647396318888896, 0.8358900946722222)
k : (1.0043931321777777, -0.17556368438888897, 0.6523304070722222)
l : (0.9856603792777778, -0.824822552588889, -0.10800636512777784)
m : (0.19989912887777778, -0.6768716464888891, -0.7085809141277778)
n : (-0.7246762772222222, -0.29638879068888896, -0.6887537869277779)
o : (-0.32007152902222225, -0.711066844188889, 0.12631498227222215)
p : (0.4386992428777778, -0.998087447888889, 0.7110252527722222)
q : (0.2436547385777778, -1.494682175988889, -0.13475917972777784)
r : (-0.5903707789222221, -1.286391145988889, -0.6456566962277778)
show/hide computer existence proof
(see shape-existence, preprint)
Attempting to prove existence
Starting realization:
Abstract data:
mode: maximal_simplices
data: [['e', 'k', 'l'], ['c', 'i', 'b'], ['e', 'a', 'f'], ['a', 'f', 'g'], ['e', 'm', 'f'], ['n', 'f', 'g'], ['q', 'p', 'o'], ['c', 'a', 'd'], ['h', 'j', 'i'], ['c', 'i', 'd'], ['q', 'r', 'o'], ['h', 'i', 'b'], ['h', 'n', 'o'], ['e', 'a', 'd'], ['h', 'n', 'g'], ['l', 'p', 'q'], ['h', 'j', 'o'], ['j', 'i', 'd'], ['h', 'g', 'b'], ['p', 'k', 'j'], ['l', 'q', 'm'], ['p', 'j', 'o'], ['q', 'r', 'm'], ['k', 'j', 'd'], ['l', 'p', 'k'], ['n', 'r', 'o'], ['e', 'k', 'd'], ['n', 'm', 'f'], ['a', 'g', 'b'], ['e', 'm', 'l'], ['c', 'a', 'b'], ['n', 'r', 'm']]
Coordinate Data:
k : [-4172462593 / 10000000000, 6263281691 / 10000000000, -2234942399 / 10000000000]
l : [-498141883 / 1250000000, 12755870373 / 10000000000, 5368425323 / 10000000000]
c : [5170858857 / 10000000000, -5661271011 / 5000000000, -79281603 / 2000000000]
i : [9614961733 / 10000000000, -729681559 / 2000000000, -5017837831 / 10000000000]
n : [13118231501 / 10000000000, 3735766377 / 5000000000, 11175899541 / 10000000000]
a : [5251579531 / 10000000000, -1192045087 / 2500000000, 894458293 / 1250000000]
f : [172104109 / 312500000, 1941537001 / 10000000000, 116528669 / 80000000]
r : [5887588259 / 5000000000, 17371556307 / 10000000000, 5372464317 / 5000000000]
b : [6930265607 / 5000000000, -3883351631 / 5000000000, 1522668241 / 5000000000]
e : [-33464843 / 1250000000, 3549576629 / 10000000000, 6562146899 / 10000000000]
d : [390105151 / 10000000000, -258662333 / 1000000000, -261283653 / 2000000000]
q : [3434921343 / 10000000000, 19454466607 / 10000000000, 5635953469 / 10000000000]
o : [9072184019 / 10000000000, 11618313289 / 10000000000, 3025211849 / 10000000000]
m : [3025373 / 7812500, 352386291 / 312500000, 11374170813 / 10000000000]
j : [562575487 / 1000000000, 5472384479 / 10000000000, -162821571 / 400000000]
g : [13946115727 / 10000000000, -2487562271 / 10000000000, 2884470709 / 2500000000]
p : [14844763 / 100000000, 7244259663 / 5000000000, -352736357 / 1250000000]
h : [11987047827 / 10000000000, 513556627 / 2500000000, 2846840917 / 10000000000]
Desired square lengths:
default : 1
Checking inequality 1:
d = 3
|V| = 18
|E| = 48
Success: d|V| >= |E|
Checking self-intersection:
Square collision distance = 491900838935048526179418469859075734064453881817026984228209 / 999753361438507743928478093603280865332500000000000000000000
Collision distance in [70144293 / 100000000, 35072147 / 50000000] ~ [0.70144, 0.70144]
Success: starting realization non-self-intersecting
Checking inequality 2:
sigma_min in [1889 / 5000, 189 / 500] ~ [0.3778, 0.378]
Success: sigma_min > 0
Checking inequality 3:
rho_squared = 56541151380133589740992971 / 10000000000000000000000000000000000000000
rho in [93992313 / 1250000000000000, 106492313 / 1250000000000000] ~ [0.0, 0.0]
sigma_min ^ 2 / (16 * E ^ .5) in [3568321 / 2771281296, 893025 / 692820323] ~ [0.00129, 0.00129]
Success: rho < sigma_min ^ 2 / (16 * E ^ .5)
Checking inequality 4:
LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-9449 / 50000000, 17 / 80000] ~ [-0.00019, 0.00021]
LHS DEN := 8 * |E| ^ .5 in [692820323 / 12500000, 173205081 / 3125000] ~ [55.42563, 55.42563]
LHS := (LHS NUM) / (LHS DEN) in [-9449 / 2771281292, 10625 / 2771281292] ~ [-0.0, 0.0]
CD / |V| ^ .5 in [70144293 / 424264069, 35072147 / 212132034] ~ [0.16533, 0.16533]
Success: LHS < CD / |V| ^ .5
Success: existence proven