10 vertices

bcde aefghc abhd ache adhifb beijg bfjih bgiedc ehgjf fig

show/hide visualization coordinates

a : (-0.8510867180999999, 0.31313735985999996, -0.68624319679)
b : (-0.29568938879999995, 0.50755437526, 0.12229612220999997)
c : (-0.00021112589999994213, 0.83064805336, -0.77676064529)
d : (-0.05787962829999993, -0.10268007844000004, -1.13112337229)
e : (-0.35335789119999994, -0.42577375644000004, -0.23206660479)
f : (-0.34598812249999994, -0.26525449284, 0.7549385890100001)
g : (0.5048874699000001, 0.25225620065999993, 0.6644211405099999)
h : (0.49751770120000005, 0.09173693695999996, -0.32258405329)
i : (0.4472189675000001, -0.68107193104, 0.31005841350999996)
j : (0.45458873620000007, -0.52055266734, 1.2970636072100001)
			
show/hide computer existence proof (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['f', 'g', 'j'], ['a', 'e', 'b'], ['h', 'c', 'b'], ['i', 'g', 'h'], ['f', 'i', 'j'], ['g', 'h', 'b'], ['h', 'e', 'd'], ['i', 'h', 'e'], ['a', 'e', 'd'], ['f', 'e', 'b'], ['h', 'c', 'd'], ['i', 'g', 'j'], ['a', 'c', 'd'], ['f', 'i', 'e'], ['a', 'c', 'b'], ['f', 'g', 'b']]
	Coordinate Data:
		j : [1708086569 / 2000000000, -1644424091 / 10000000000, 17206849517 / 10000000000]
		b : [207530319 / 2000000000, 1727329267 / 2000000000, 5459174667 / 10000000000]
		e : [460966571 / 10000000000, -348317491 / 5000000000, 1915547397 / 10000000000]
		i : [4233367579 / 5000000000, -406202091 / 1250000000, 366839879 / 500000000]
		g : [4521710091 / 5000000000, 6083664589 / 10000000000, 217608497 / 200000000]
		d : [8539373 / 25000000, 1267150899 / 5000000000, -3537510139 / 5000000000]
		h : [1793944499 / 2000000000, 279904497 / 625000000, 63148307 / 625000000]
		c : [249527139 / 625000000, 2966895779 / 2500000000, -220712063 / 625000000]
		a : [-2258160849 / 5000000000, 6692476181 / 10000000000, -2626218523 / 10000000000]
		f : [267332129 / 5000000000, 454278827 / 5000000000, 2357119867 / 2000000000]

Desired square lengths:
	default : 1

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

Checking self-intersection:
	Square collision distance = 166666666639326157011738768576562738459324388584864813885227 / 333333333337369396439609116314541821964600000000000000000000
	Collision distance in [35355339 / 50000000, 70710679 / 100000000] ~ [0.70711, 0.70711]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [9751 / 10000, 9753 / 10000] ~ [0.9751, 0.9753]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 6291485902739662453971 / 10000000000000000000000000000000000000000
	rho in [396594437 / 500000000000000000, 5396594437 / 500000000000000000] ~ [0.0, 0.0]
	sigma_min ^ 2 / (16 * E ^ .5) in [95082001 / 7838367184, 31707003 / 2612789056] ~ [0.01213, 0.01214]
	Success: rho < sigma_min ^ 2 / (16 * E ^ .5)

Checking inequality 4:
	LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-19997 / 100000000, 5011 / 25000000] ~ [-0.0002, 0.0002]
	LHS DEN := 8 * |E| ^ .5 in [122474487 / 3125000, 489897949 / 12500000] ~ [39.19184, 39.19184]
	LHS     := (LHS NUM) / (LHS DEN) in [-19997 / 3919183584, 5011 / 979795896] ~ [-1e-05, 1e-05]
	CD / |V| ^ .5 in [70710678 / 316227767, 70710679 / 316227766] ~ [0.22361, 0.22361]
	Success: LHS < CD / |V| ^ .5

Success: existence proven