12 vertices

bcdefg aghc abhd ache adhijf aejklg aflihb bgiedc ehglkj eikf fjil fkig

show/hide visualization coordinates

a : (0.04934001977499991, -0.07905574141666677, 0.4980998772666667)
b : (0.939662406275, 0.08380838488333325, 0.9243971358666667)
c : (0.12365149777499995, -0.00486137891666677, 1.4932472617666668)
d : (-0.7751579069250001, -0.08687825651666681, 1.0658512861666667)
e : (-0.8540746208250001, -0.09304594271666677, 0.0704753169666667)
f : (-0.03245771542500009, -0.07717526231666677, -0.49981786523333327)
g : (0.8539570771749999, 0.09428380578333323, -0.07047430403333327)
h : (0.033118853474999876, 0.07726241018333319, 0.4998621382666667)
i : (-0.05004614082500014, 0.07923071918333319, -0.4981041572333333)
j : (-0.9381739561250001, -0.09716167271666681, -0.9246157433333333)
k : (-0.12327286972500007, 0.0010612548833331958, -1.4932301602333333)
l : (0.7734533553749999, 0.10253167968333315, -1.0656907862333331)
			
show/hide manual existence proof

This is a flat doubled portion of the plane hexagonal lattice.
			
show/hide computer existence proof (failed) (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['c', 'h', 'd'], ['g', 'i', 'l'], ['a', 'c', 'd'], ['j', 'i', 'e'], ['k', 'l', 'i'], ['a', 'b', 'c'], ['i', 'h', 'e'], ['g', 'b', 'h'], ['c', 'b', 'h'], ['a', 'g', 'f'], ['g', 'l', 'f'], ['a', 'g', 'b'], ['k', 'l', 'f'], ['h', 'e', 'd'], ['j', 'f', 'e'], ['j', 'k', 'f'], ['g', 'i', 'h'], ['a', 'e', 'd'], ['a', 'f', 'e'], ['j', 'k', 'i']]
	Coordinate Data:
		d : [-121602221 / 625000000, 16049353 / 40000000, 14414204541 / 10000000000]
		b : [3800641899 / 2500000000, 714900583 / 1250000000, 6499831519 / 5000000000]
		c : [7042458511 / 10000000000, 2416253513 / 5000000000, 18688164297 / 10000000000]
		g : [2869102861 / 2000000000, 5823958873 / 10000000000, 3050948639 / 10000000000]
		h : [1534283017 / 2500000000, 5653744917 / 10000000000, 4377156531 / 5000000000]
		i : [42443857 / 80000000, 5673428007 / 10000000000, -1225349893 / 10000000000]
		f : [5481366379 / 10000000000, 32104439 / 78125000, -1242486973 / 10000000000]
		e : [-109392107 / 400000000, 987665347 / 2500000000, 4460444849 / 10000000000]
		k : [1143303709 / 2500000000, 1222933341 / 2500000000, -11176609923 / 10000000000]
		l : [13540477087 / 10000000000, 1476609403 / 2500000000, -6901216183 / 10000000000]
		j : [-893949007 / 2500000000, 488688011 / 1250000000, -2745232877 / 5000000000]
		a : [6299343731 / 10000000000, 4090563401 / 10000000000, 2184172613 / 2500000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 12
	|E| = 30
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 11469585629453954679874392060940266752011239831611226961 / 477208972015872950180612462390441290377500000000000000000
	Collision distance in [484473 / 3125000, 15503137 / 100000000] ~ [0.15503, 0.15503]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [651 / 5000, 163 / 1250] ~ [0.1302, 0.1304]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 276795497071987800539613563910373087 / 2500000000000000000000000000000000000000
	rho in [1052227 / 100000000, 263057 / 25000000] ~ [0.01052, 0.01052]
	sigma_min ^ 2 / (16 * E ^ .5) in [47089 / 243432248, 106276 / 547722557] ~ [0.00019, 0.00019]
	Failed: unable to verify rho < sigma_min ^ 2 / (16 * E ^ .5)