10 vertices

bcde aefgc abghd achije adjifb beihg bfhc cgfid dhfej die

show/hide visualization coordinates

a : (-0.9212535743000001, 0.29610193868, -0.04238642442000001)
b : (-0.5251502686, -0.57365835482, -0.33669782942000004)
c : (-0.4831908452, -0.30534305532, 0.6257190795800001)
d : (-0.04649911029999998, 0.55611596598, 0.36651128068)
e : (-0.08845853370000001, 0.28780066668, -0.59590562842)
f : (0.4714338667, -0.51632065012, -0.39613223722)
g : (0.044417971899999964, -1.0908021784200002, 0.30217170748)
h : (0.5133932901, -0.24800535072000002, 0.56628467178)
i : (0.7975998018999999, 0.39499485067999995, -0.14489561441999999)
j : (0.23770740149999997, 1.19911616738, -0.34466900562)
			
show/hide computer existence proof (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['a', 'e', 'b'], ['g', 'h', 'c'], ['j', 'e', 'd'], ['a', 'e', 'd'], ['f', 'e', 'b'], ['g', 'c', 'b'], ['h', 'c', 'd'], ['i', 'j', 'e'], ['f', 'i', 'h'], ['i', 'j', 'd'], ['f', 'g', 'h'], ['i', 'h', 'd'], ['a', 'c', 'd'], ['f', 'i', 'e'], ['a', 'c', 'b'], ['f', 'g', 'b']]
	Coordinate Data:
		j : [300753549 / 1000000000, -1435142581 / 2000000000, 7418998781 / 10000000000]
		b : [10636112191 / 10000000000, 10552032317 / 10000000000, 7339287019 / 10000000000]
		e : [3134597421 / 5000000000, 968721051 / 5000000000, 9931365009 / 10000000000]
		i : [-1295694257 / 5000000000, 432750131 / 5000000000, 5421264869 / 10000000000]
		g : [2470214893 / 5000000000, 15723470553 / 10000000000, 19011833 / 200000000]
		h : [62669151 / 2500000000, 1823875569 / 2500000000, -1690537993 / 10000000000]
		c : [10216517957 / 10000000000, 3934439661 / 5000000000, -2284882071 / 10000000000]
		f : [335135419 / 5000000000, 997865527 / 1000000000, 7933631097 / 10000000000]
		a : [456160789 / 312500000, 927214691 / 5000000000, 4396172969 / 10000000000]
		d : [182800019 / 312500000, -745710891 / 10000000000, 153597959 / 5000000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 10
	|E| = 24
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 31249999986497285967623791292564302147654082675565401387689 / 62499999991853896764050043266699985702500000000000000000000
	Collision distance in [35355339 / 50000000, 70710679 / 100000000] ~ [0.70711, 0.70711]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [573 / 500, 5731 / 5000] ~ [1.146, 1.1462]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 151528904863599532101 / 312500000000000000000000000000000000000
	rho in [27853689 / 40000000000000000, 427853689 / 40000000000000000] ~ [0.0, 0.0]
	sigma_min ^ 2 / (16 * E ^ .5) in [8208225 / 489897949, 32844361 / 1959591792] ~ [0.01675, 0.01676]
	Success: rho < sigma_min ^ 2 / (16 * E ^ .5)

Checking inequality 4:
	LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-9999 / 50000000, 20037 / 100000000] ~ [-0.0002, 0.0002]
	LHS DEN := 8 * |E| ^ .5 in [122474487 / 3125000, 489897949 / 12500000] ~ [39.19184, 39.19184]
	LHS     := (LHS NUM) / (LHS DEN) in [-3333 / 653197264, 6679 / 1306394528] ~ [-1e-05, 1e-05]
	CD / |V| ^ .5 in [70710678 / 316227767, 70710679 / 316227766] ~ [0.22361, 0.22361]
	Success: LHS < CD / |V| ^ .5

Success: existence proven