a : (-0.14555181808571427, 0.17701483214285713, 0.37547308761428577)
b : (-0.5379687122857144, 0.3398126685428571, -0.5297925339857144)
c : (-0.16717179498571433, -0.5600298431571429, -0.30002504198571417)
d : (-0.23836845228571435, -0.7727049056571429, 0.6745006650142857)
e : (0.6289385985142857, -0.4504469385571429, 0.29512185421428583)
f : (0.4491784426142857, 0.21706360594285712, -0.4274540651857142)
g : (0.010943736514285707, 1.049290580742857, -0.08782396568571421)
			

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['a', 'g', 'b'], ['f', 'c', 'b'], ['f', 'e', 'c'], ['e', 'c', 'd'], ['f', 'a', 'e'], ['a', 'e', 'd'], ['a', 'c', 'd'], ['a', 'c', 'b'], ['f', 'g', 'b'], ['f', 'a', 'g']]
	Coordinate Data:
		b : [11240333519 / 10000000000, 43082371 / 250000000, 5133675129 / 5000000000]
		e : [-428739589 / 10000000000, 9625890911 / 10000000000, 252275797 / 1250000000]
		g : [5751209031 / 10000000000, -2685742141 / 5000000000, 233906583 / 400000000]
		d : [8244330919 / 10000000000, 6424235291 / 5000000000, -443895433 / 2500000000]
		c : [3766182173 / 5000000000, 10721719957 / 10000000000, 3984837669 / 5000000000]
		a : [7316164577 / 10000000000, 837818301 / 2500000000, 607347021 / 5000000000]
		f : [136886197 / 1000000000, 1475392733 / 5000000000, 924396557 / 1000000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 7
	|E| = 15
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 124999999955186858145140571774271636582456928182781314655569 / 249999999961742860320031298267730591347800000000000000000000
	Collision distance in [35355339 / 50000000, 70710679 / 100000000] ~ [0.70711, 0.70711]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [5077 / 5000, 2539 / 2500] ~ [1.0154, 1.0156]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 226937550111071628653 / 500000000000000000000000000000000000000
	rho in [67370253 / 100000000000000000, 1067370253 / 100000000000000000] ~ [0.0, 0.0]
	sigma_min ^ 2 / (16 * E ^ .5) in [25775929 / 1549193340, 6446521 / 387298334] ~ [0.01664, 0.01664]
	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, 20033 / 100000000] ~ [-0.0002, 0.0002]
	LHS DEN := 8 * |E| ^ .5 in [193649167 / 6250000, 77459667 / 2500000] ~ [30.98387, 30.98387]
	LHS     := (LHS NUM) / (LHS DEN) in [-9999 / 1549193336, 871 / 134712464] ~ [-1e-05, 1e-05]
	CD / |V| ^ .5 in [35355339 / 132287566, 70710679 / 264575131] ~ [0.26726, 0.26726]
	Success: LHS < CD / |V| ^ .5

Success: existence proven