17 vertices

bcdefg aghijc abjkd acklme admlnf aenopg afphb bgpqji bhj bihqkc cjqld dkqnem dle elqpof fnp fonqhg hpnlkj

show/hide visualization coordinates

a : (0.16168489727647062, 0.03991843566470593, -0.42024183564705875)
b : (0.31948136937647065, -0.9408769333352941, -0.5349309616470588)
c : (-0.5339723241235294, -0.530795258135294, -0.8565433176470587)
d : (-0.6940325092235293, 0.4490477004647059, -0.7370626900470587)
e : (-0.0006385190235293492, 1.018807014064706, -0.2959697457470587)
f : (0.8528160581764707, 0.608724604464706, 0.025643081252941236)
g : (1.0128751625764707, -0.37111721523529406, -0.09383823004705871)
h : (0.3848245762764706, -0.9151116064352941, 0.46259900655294117)
i : (0.1413594601764706, -1.767224863135294, -0.000681165647058779)
j : (-0.5082483949235294, -1.007301878735294, 0.022251059452941258)
k : (-1.0934008545235294, -0.20482837553529407, -0.09445018114705872)
l : (-0.7854801686235293, 0.689834139664706, 0.22919734555294125)
m : (-0.9123026301235293, 1.388945375464706, -0.47447838954705873)
n : (0.1075930275764706, 0.7820246370647058, 0.6695457000529413)
o : (1.0572870919764705, 0.8939968039647058, 0.9620251442529412)
p : (0.6927458802764707, -0.020449188035294052, 0.7862462135529412)
q : (-0.20259212312352937, -0.11359339223529408, 0.35068896645294123)
			
show/hide manual existence proof

This is the hexagonal antiprism with three tetrahdera attached.
			
show/hide computer existence proof (failed) (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['q', 'l', 'k'], ['l', 'm', 'd'], ['a', 'e', 'd'], ['l', 'e', 'm'], ['p', 'h', 'g'], ['p', 'f', 'g'], ['q', 'p', 'h'], ['q', 'l', 'n'], ['a', 'f', 'g'], ['o', 'f', 'n'], ['i', 'h', 'j'], ['a', 'c', 'd'], ['l', 'e', 'n'], ['b', 'a', 'g'], ['c', 'd', 'k'], ['a', 'e', 'f'], ['q', 'h', 'j'], ['b', 'h', 'g'], ['e', 'm', 'd'], ['b', 'i', 'h'], ['q', 'j', 'k'], ['b', 'a', 'c'], ['c', 'j', 'k'], ['b', 'i', 'j'], ['q', 'p', 'n'], ['b', 'c', 'j'], ['l', 'd', 'k'], ['p', 'o', 'n'], ['p', 'o', 'f'], ['e', 'f', 'n']]
	Coordinate Data:
		q : [2652245003 / 10000000000, 3136453937 / 10000000000, 9121104157 / 10000000000]
		p : [11605625037 / 10000000000, 4067895979 / 10000000000, 3369169157 / 2500000000]
		h : [8526411997 / 10000000000, -975745641 / 2000000000, 5120102279 / 5000000000]
		i : [1522940209 / 2500000000, -3349965193 / 2500000000, 1401850709 / 2500000000]
		f : [412697713 / 312500000, 647477119 / 625000000, 1174129061 / 2000000000]
		k : [-6255842311 / 10000000000, 278013013 / 1250000000, 4669712681 / 10000000000]
		e : [1167945261 / 2500000000, 7230229 / 5000000, 530903407 / 2000000000]
		d : [-1131079429 / 5000000000, 273839527 / 312500000, -219551551 / 1250000000]
		j : [-80863543 / 2000000000, -362539433 / 625000000, 5836725087 / 10000000000]
		l : [-794158863 / 2500000000, 1396341157 / 1250000000, 1976546987 / 2500000000]
		a : [6295015207 / 10000000000, 583946527 / 1250000000, 176474517 / 1250000000]
		g : [740345893 / 500000000, 561215707 / 10000000000, 36529939 / 78125000]
		n : [575409651 / 1000000000, 1209263423 / 1000000000, 12309671493 / 10000000000]
		m : [-4444860067 / 10000000000, 9080920807 / 5000000000, 869430597 / 10000000000]
		o : [7625518577 / 5000000000, 13212355899 / 10000000000, 3046893187 / 2000000000]
		b : [984122491 / 1250000000, -2568190737 / 5000000000, 66226219 / 2500000000]
		c : [-661557007 / 10000000000, -517782361 / 5000000000, -737804671 / 2500000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 17
	|E| = 45
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 24691140224618500097765344351023066860736776342738841 / 49382718063535843394369598772138558274000000000000000
	Collision distance in [17677591 / 25000000, 14142073 / 20000000] ~ [0.7071, 0.7071]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [37 / 2000, 187 / 10000] ~ [0.0185, 0.0187]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 25483187779761483709127544358557 / 5000000000000000000000000000000000000000
	rho in [142781477 / 2000000000000, 142801477 / 2000000000000] ~ [7e-05, 7e-05]
	sigma_min ^ 2 / (16 * E ^ .5) in [34225 / 10733126304, 34969 / 10733126288] ~ [0.0, 0.0]
	Failed: unable to verify rho < sigma_min ^ 2 / (16 * E ^ .5)