a : (0.34444991157916427, -0.2608876234497072, -0.03087154087446542)
b : (0.27293662534822055, -0.41382502296534296, 0.954774384255142)
c : (-0.1267260740464891, -1.0716737524921807, 0.3164207658681511)
d : (-0.0567836548606267, -0.9178207643122477, -0.6691930770862594)
e : (0.4128245082831914, -0.1061039983146802, -1.016450366770317)
f : (0.8124886249351488, 0.5517495347946124, -0.37809884357506773)
g : (0.7425465398813016, 0.39789092006005367, 0.60751372544944)
h : (-0.2043733991638843, 0.4643042048010939, 0.922051997451655)
i : (-0.3778783621872024, -0.171371612784468, 1.6742577433102128)
j : (-0.7062495356559606, -0.384154861245658, 0.7539884515797575)
k : (-0.8966122393519225, -0.6751322441201716, -0.1836093727499286)
l : (-0.5851016670274898, -0.11766004738776503, -0.9531426032695197)
m : (-0.08322837057083032, 0.730792058104436, -0.7850820774763531)
n : (0.7408761384975019, 0.7778767807607725, -1.3495594372916768)
o : (0.10713722178258286, 1.0217767444558425, 0.15251530171223188)
p : (-0.3963062674427056, 0.17423968409541035, -0.015515050533002839)
			

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'], ['d', 'e', 'l'], ['i', 'b', 'h'], ['a', 'g', 'f'], ['d', 'a', 'c'], ['b', 'j', 'c'], ['k', 'j', 'p'], ['d', 'k', 'c'], ['m', 'e', 'n'], ['l', 'k', 'p'], ['m', 'o', 'f'], ['d', 'k', 'l'], ['b', 'h', 'g'], ['b', 'j', 'i'], ['d', 'e', 'a'], ['p', 'j', 'h'], ['o', 'g', 'f'], ['b', 'a', 'g'], ['i', 'j', 'h'], ['k', 'j', 'c'], ['m', 'o', 'p'], ['b', 'a', 'c'], ['p', 'o', 'h'], ['l', 'e', 'm'], ['e', 'a', 'f'], ['o', 'h', 'g']]
	Coordinate Data:
		i : [2896772573838220783 / 4000000000000000000, 1466571230996738111 / 2000000000000000000, -27321105371962725899 / 25000000000000000000]
		e : [-1662743175270963707 / 25000000000000000000, 66801800102858124671 / 100000000000000000000, 499332467250631459 / 312500000000000000]
		l : [9314164482998426497 / 10000000000000000000, 67957405010166601459 / 100000000000000000000, 76727806585061163469 / 50000000000000000000]
		m : [42954315184318315763 / 100000000000000000000, -16887805539053502411 / 100000000000000000000, 4270298768462676981 / 3125000000000000000]
		g : [-9905793965223718589 / 25000000000000000000, 8201154132692366141 / 50000000000000000000, -522003940354728299 / 20000000000000000000]
		k : [62146351031213762743 / 50000000000000000000, 30926156170851813531 / 25000000000000000000, 38251145059081607861 / 50000000000000000000]
		f : [-46617384366279600539 / 100000000000000000000, 1016446791928855093 / 100000000000000000000, 9595123720067712579 / 10000000000000000000]
		b : [917226949051653499 / 12500000000000000000, 3902956102716975947 / 4000000000000000000, -3733608558234384971 / 10000000000000000000]
		c : [23652042765942097249 / 50000000000000000000, 32671755104121633161 / 20000000000000000000, 26499276256355247533 / 100000000000000000000]
		o : [747429873405531087 / 3125000000000000000, -9197254834838831733 / 20000000000000000000, 42889822671947169903 / 100000000000000000000]
		j : [105256431692831353263 / 100000000000000000000, 23651721598988974707 / 25000000000000000000, -690299692592215697 / 4000000000000000000]
		d : [20154921806648976241 / 50000000000000000000, 147973476702614878379 / 100000000000000000000, 31265165137949075193 / 25000000000000000000]
		a : [46621742329714121 / 25000000000000000000, 41140081308180411347 / 50000000000000000000, 15307126732654225681 / 25000000000000000000]
		p : [74262104871505846049 / 100000000000000000000, 38767431861849068779 / 100000000000000000000, 11938571579294129289 / 20000000000000000000]
		n : [-4932016965314363503 / 12500000000000000000, -10798138902343570233 / 50000000000000000000, 38619459314467611593 / 20000000000000000000]
		h : [55068818043623714199 / 100000000000000000000, 4880489895640355963 / 50000000000000000000, -34063846901995150929 / 100000000000000000000]

Desired square lengths:
	default : 1

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

Checking self-intersection:
	Square collision distance = 41666450447500845812219221468292809916449627138263112192561546540782055743630188226383830535629996458871150146644926227 / 83333504715959869508902019000052353391642239850634224394391807746458893789920970000000000000000000000000000000000000000
	Collision distance in [70710421 / 100000000, 35355211 / 50000000] ~ [0.7071, 0.7071]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [31 / 5000, 4 / 625] ~ [0.0062, 0.0064]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 85590473705654311702502827995406071563754937852022146015279663637341953 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000
	rho in [9142453 / 312500000000, 4572789 / 156250000000] ~ [3e-05, 3e-05]
	sigma_min ^ 2 / (16 * E ^ .5) in [961 / 2592296280, 256 / 648074069] ~ [0.0, 0.0]
	Failed: unable to verify rho < sigma_min ^ 2 / (16 * E ^ .5)