a : (-0.360273816777749, -0.007819792170072892, -0.1641008310724844)
b : (-0.4656934010335664, -0.9616662754414407, 0.11708447631411056)
c : (-0.7593286535616818, -0.2253250041555212, 0.7266555050480045)
d : (-0.6522923166113972, 0.7285570872065646, 0.4462083681322201)
e : (-0.25161821380780597, 0.9460965192698826, -0.44381226631189075)
f : (0.042017634419342254, 0.20975541183613855, -1.0533838005406733)
g : (-0.06501952973300551, -0.7441256188083896, -0.7729359627002473)
h : (0.5275571894130376, -0.9667384498674942, 0.0012063767495918976)
i : (0.1266980422500269, -0.6672086793510945, 0.8669955198310774)
j : (-0.7174909301740264, -1.1661278201630592, 1.0630189596414192)
k : (0.018965053224728412, 0.3086426708066744, 1.0570155606764429)
l : (0.31209176721817644, 0.9849627132332283, 0.3812451605129288)
m : (0.7129513241066071, 0.6854337445827238, -0.48454407112071035)
n : (0.2893123622259781, 1.1568334554307436, -1.258050595215359)
o : (0.8206846206275237, -0.29041720203262117, -0.6745641087178186)
p : (0.4214388682138116, 0.00914723962373587, 0.19196170877338747)
			

This is the hexagonal antiprism with two tetrahdera attached.
			

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['e', 'f', 'n'], ['l', 'm', 'p'], ['m', 'n', 'f'], ['p', 'k', 'i'], ['d', 'e', 'l'], ['i', 'b', 'h'], ['a', 'g', 'f'], ['d', 'a', 'c'], ['b', 'j', 'c'], ['i', 'p', 'h'], ['d', 'k', 'c'], ['m', 'e', 'n'], ['l', 'k', 'p'], ['m', 'o', 'f'], ['i', 'k', 'c'], ['d', 'k', 'l'], ['i', 'j', 'c'], ['b', 'h', 'g'], ['b', 'j', 'i'], ['d', 'e', 'a'], ['o', 'g', 'f'], ['b', 'a', 'g'], ['m', 'o', 'p'], ['b', 'a', 'c'], ['p', 'o', 'h'], ['l', 'e', 'm'], ['e', 'a', 'f'], ['o', 'h', 'g']]
	Coordinate Data:
		i : [81382838207316047729 / 100000000000000000000, -1502694970137767623 / 12500000000000000000, 4399354913074816017 / 3125000000000000000]
		e : [43551212601532755433 / 100000000000000000000, 149308960100995579417 / 100000000000000000000, 9698578604097304333 / 100000000000000000000]
		l : [99922210704130995629 / 100000000000000000000, 19149447437166266991 / 12500000000000000000, 46102160643289628157 / 50000000000000000000]
		m : [140008166392974054307 / 100000000000000000000, 3851333832258740079 / 3125000000000000000, 225015924928613959 / 4000000000000000000]
		k : [17652384826196549843 / 25000000000000000000, 42781787627337376393 / 50000000000000000000, 19972670162866332951 / 12500000000000000000]
		g : [31105540504506400799 / 50000000000000000000, -19713253706831641539 / 100000000000000000000, -928551641389534003 / 4000000000000000000]
		f : [14582959484849515953 / 20000000000000000000, 18918712339405290903 / 25000000000000000000, -51258574818780940237 / 100000000000000000000]
		b : [17714955103165373 / 80000000000000000, -518341492126709407 / 1250000000000000000, 65788252866697442427 / 100000000000000000000]
		c : [-7219831373854826697 / 100000000000000000000, 32166807758455188071 / 100000000000000000000, 126745355740086832731 / 100000000000000000000]
		o : [6031259841802628891 / 4000000000000000000, 6414396992686298261 / 25000000000000000000, -6688302818247737 / 50000000000000000]
		j : [-1518029517544645331 / 50000000000000000000, -30956736921149298799 / 50000000000000000000, 160381701199428321199 / 100000000000000000000]
		d : [696760464234727197 / 20000000000000000000, 25511003378932755011 / 20000000000000000000, 98700642048508389883 / 100000000000000000000]
		a : [6537130460907691201 / 20000000000000000000, 26958664478500009881 / 50000000000000000000, 3766972212803793901 / 10000000000000000000]
		p : [55428460401847254133 / 50000000000000000000, 27807016068190448229 / 50000000000000000000, 73275976112625130653 / 100000000000000000000]
		n : [97644270204911165943 / 100000000000000000000, 17038265371708164677 / 10000000000000000000, -14345050857249900851 / 20000000000000000000]
		h : [12146875292361710981 / 10000000000000000000, -41974536812742119873 / 100000000000000000000, 5420044291024557677 / 10000000000000000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 16
	|E| = 42
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 7352926616066624116110835786727446902317622262333154954508190064995726619443761936526756127482678844400944154243809273 / 14705884977221341054077878511123575598888259344001964701954551551023193868403607500000000000000000000000000000000000000
	Collision distance in [70710601 / 100000000, 35355301 / 50000000] ~ [0.70711, 0.70711]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [3 / 500, 31 / 5000] ~ [0.006, 0.0062]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 9739391321439829145268626873822435488875056675381311827658841300662613 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000
	rho in [197376709 / 20000000000000, 197576709 / 20000000000000] ~ [1e-05, 1e-05]
	sigma_min ^ 2 / (16 * E ^ .5) in [5 / 14401646, 961 / 2592296276] ~ [0.0, 0.0]
	Failed: unable to verify rho < sigma_min ^ 2 / (16 * E ^ .5)