14 vertices

bcdefg aghijc abjed ace adcjkf aeklmg afmhb bgmi bhmlnj binkec ejnlf fknim flihg ilkj

show/hide visualization coordinates

a : (0.9092978986214285, 0.2715986672214288, -0.14286040983571424)
b : (0.01736684502142849, 0.41109654322142875, -0.5729847016357142)
c : (0.35772102772142855, 1.0703685767214286, 0.09747230606428575)
d : (1.2497432299214286, 0.9309902814214288, 0.5274251459642857)
e : (0.48364321562142853, 0.3316540983214288, 0.7596470646642858)
f : (0.16791815372142854, -0.35354192067857126, 0.10272681526428573)
g : (0.5691799676214285, -0.38759313987857125, -0.8129955387357143)
h : (-0.3227168172785715, -0.2484677777785712, -1.2432923631357142)
i : (-0.7486470966785715, -0.18823335567857125, -0.3405440621357143)
j : (-0.4084257055785715, 0.47118790962142876, 0.32984052276428577)
k : (-0.28207506067857147, -0.2674489157785712, 0.9920029780642857)
l : (-0.6221684030785716, -0.9266307000785713, 0.32184812976428573)
m : (-0.19654498007857146, -0.9866859525785713, -0.5806441573357143)
n : (-1.1742922748785714, -0.12829431407857128, 0.5623582702642858)
			
show/hide manual existence proof

This is part of the standard sphere packing arrangement (link).
			
show/hide computer existence proof (failed) (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['c', 'b', 'j'], ['h', 'g', 'b'], ['i', 'b', 'j'], ['h', 'm', 'g'], ['k', 'n', 'l'], ['k', 'e', 'j'], ['c', 'a', 'b'], ['b', 'a', 'g'], ['f', 'a', 'g'], ['k', 'f', 'l'], ['f', 'l', 'm'], ['i', 'l', 'n'], ['d', 'e', 'c'], ['c', 'a', 'd'], ['d', 'e', 'a'], ['k', 'n', 'j'], ['e', 'c', 'j'], ['k', 'e', 'f'], ['i', 'm', 'h'], ['f', 'g', 'm'], ['i', 'm', 'l'], ['i', 'j', 'n'], ['h', 'i', 'b'], ['e', 'a', 'f']]
	Coordinate Data:
		e : [505972193 / 5000000000, 335376819 / 2000000000, -962026093 / 5000000000]
		g : [78288433 / 5000000000, 8869356477 / 10000000000, 1725296731 / 1250000000]
		i : [13334847509 / 10000000000, 1375151727 / 2000000000, 4538929541 / 5000000000]
		c : [454233253 / 2000000000, -5710260689 / 10000000000, 23488477 / 50000000]
		k : [8669127149 / 10000000000, 1916978559 / 2500000000, -106190283 / 250000000]
		m : [7813826343 / 10000000000, 3715071151 / 2500000000, 5739430017 / 5000000000]
		a : [-811150611 / 2500000000, 1138719203 / 5000000000, 7101022559 / 10000000000]
		h : [1815108943 / 2000000000, 934762857 / 1250000000, 4526335523 / 2500000000]
		n : [17591299291 / 10000000000, 6276368219 / 10000000000, 24417879 / 5000000000]
		j : [4966316799 / 5000000000, 140772991 / 5000000000, 2374013233 / 10000000000]
		l : [12070060573 / 10000000000, 14259732079 / 10000000000, 2453937163 / 10000000000]
		f : [833839001 / 2000000000, 1705768857 / 2000000000, 1161287577 / 2500000000]
		b : [1418677023 / 2500000000, 441229823 / 5000000000, 11402265477 / 10000000000]
		d : [-6649055757 / 10000000000, -539559717 / 1250000000, 398167001 / 10000000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 14
	|E| = 36
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 124961337914075464511578539045965933769899746691331911000689 / 250017552981430223899136339308511088266200000000000000000000
	Collision distance in [3534863 / 5000000, 70697261 / 100000000] ~ [0.70697, 0.70697]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [1361 / 10000, 1363 / 10000] ~ [0.1361, 0.1363]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 15712700049062176435222544995914213 / 2500000000000000000000000000000000000000
	rho in [2507 / 1000000, 250701 / 100000000] ~ [0.00251, 0.00251]
	sigma_min ^ 2 / (16 * E ^ .5) in [1852321 / 9600000016, 1857769 / 9600000000] ~ [0.00019, 0.00019]
	Failed: unable to verify rho < sigma_min ^ 2 / (16 * E ^ .5)