a : (-0.4027583402166667, 0.28216096726666673, 0.18127952451666662)
b : (0.5840403123833334, 0.43837320236666677, 0.13854596651666662)
c : (0.07177337488333335, 0.2447187614666667, -0.6981621918833334)
d : (-0.6611082885166667, -0.40534011643333323, -0.49739386098333344)
e : (0.2153830009833334, -0.4773300997333333, -0.021389174383333376)
f : (0.1926699404833334, -0.08258271493333325, 0.8971197362166666)
			

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['e', 'c', 'd'], ['f', 'a', 'b'], ['f', 'a', 'e'], ['a', 'e', 'd'], ['f', 'e', 'b'], ['a', 'c', 'd'], ['a', 'c', 'b'], ['c', 'e', 'b']]
	Coordinate Data:
		b : [346419427 / 5000000000, 405018531 / 5000000000, 735966473 / 2500000000]
		e : [8553539 / 19531250, 9967070083 / 10000000000, 4543217301 / 10000000000]
		d : [13144324863 / 10000000000, 36988681 / 40000000, 9303264167 / 10000000000]
		c : [5815508229 / 10000000000, 2746581471 / 10000000000, 2827736869 / 2500000000]
		a : [528041269 / 500000000, 2372159413 / 10000000000, 314566289 / 1250000000]
		f : [4606542573 / 10000000000, 1203919247 / 2000000000, -928374361 / 2000000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 6
	|E| = 12
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 166666666656318432619984285778277378193477782215698379236923 / 333333333348027356809177505006945752534300000000000000000000
	Collision distance in [35355339 / 50000000, 70710679 / 100000000] ~ [0.70711, 0.70711]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [5611 / 5000, 1403 / 1250] ~ [1.1222, 1.1224]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 761533046357790961663 / 5000000000000000000000000000000000000000
	rho in [390264793 / 1000000000000000000, 10390264793 / 1000000000000000000] ~ [0.0, 0.0]
	sigma_min ^ 2 / (16 * E ^ .5) in [31483321 / 1385640648, 7873636 / 346410161] ~ [0.02272, 0.02273]
	Success: rho < sigma_min ^ 2 / (16 * E ^ .5)

Checking inequality 4:
	LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-1 / 5000, 10013 / 50000000] ~ [-0.0002, 0.0002]
	LHS DEN := 8 * |E| ^ .5 in [346410161 / 12500000, 173205081 / 6250000] ~ [27.71281, 27.71281]
	LHS     := (LHS NUM) / (LHS DEN) in [-2500 / 346410161, 10013 / 1385640644] ~ [-1e-05, 1e-05]
	CD / |V| ^ .5 in [70710678 / 244948975, 70710679 / 244948974] ~ [0.28868, 0.28868]
	Success: LHS < CD / |V| ^ .5

Success: existence proven