21 vertices

bcdefg aghijc abjkld aclmne adnopf aepqg afqrhb bgri bhrsj bistkc cjtoml ckmd dlkon dmoe enmktp eotuqf fpusrg gqsih irqutj jsupok ptsq

show/hide visualization coordinates

a : (-0.5581586196027034, -0.010055095368362799, 0.18617584397679943)
b : (-0.45098483568159237, -1.0018177421838916, 0.11602862434916617)
c : (-0.10448083552206638, -0.5170206389775044, 0.9190894854276667)
d : (-0.26459610463575733, 0.4653566140076914, 1.0155157962246215)
e : (-0.5502205752094943, 0.9895470278967187, 0.21324210738999272)
f : (-0.8762181683938067, 0.5140612456164875, -0.603849588864101)
g : (-0.8206870769619736, -0.4830926408383094, -0.6548439474980089)
h : (-0.7554446110433768, -1.4795433559306914, -0.7080359866837889)
i : (0.22160315248280643, -1.361646997151336, -0.530615254030998)
j : (0.4863548991971321, -0.653490462001504, 0.12392338832428595)
k : (0.686240489693382, 0.07812150090666797, 0.7756823057482188)
l : (0.29277794723966105, -0.018961001476536732, 1.6898826204587691)
m : (0.576541842181703, 0.8816691534244825, 1.3607270705886694)
n : (-0.15735349242221036, 1.4595189663593415, 1.0036735290325627)
o : (0.43636101809251077, 0.9589786547285415, 0.3736239465100386)
p : (0.07499926025829201, 0.8174860982461891, -0.54800370956103)
q : (-0.17881701124138516, 0.08066206086705568, -1.1746356894841068)
r : (-0.20227572799500385, -0.909132938360852, -1.3151901206191652)
s : (0.580614895043803, -0.4874996451769606, -0.8576886032043967)
t : (0.8105821093438657, 0.23372965679420127, -0.20427956261400237)
u : (0.7531614451762143, 0.4431295386185716, -1.1804222554711918)
			
show/hide computer existence proof (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['n', 'o', 'm'], ['t', 'k', 'o'], ['s', 'r', 'i'], ['g', 'b', 'h'], ['q', 'g', 'f'], ['t', 'p', 'o'], ['j', 's', 'i'], ['a', 'b', 'c'], ['b', 'j', 'i'], ['b', 'j', 'c'], ['a', 'd', 'c'], ['t', 'p', 'u'], ['h', 'r', 'i'], ['o', 'p', 'e'], ['k', 'o', 'm'], ['t', 'j', 's'], ['k', 'l', 'm'], ['g', 'a', 'b'], ['t', 'k', 'j'], ['d', 'n', 'm'], ['l', 'd', 'c'], ['q', 'p', 'u'], ['a', 'e', 'd'], ['a', 'f', 'e'], ['b', 'h', 'i'], ['f', 'p', 'e'], ['g', 'h', 'r'], ['d', 'n', 'e'], ['q', 'p', 'f'], ['k', 'j', 'c'], ['g', 'a', 'f'], ['q', 's', 'u'], ['g', 'r', 'q'], ['d', 'l', 'm'], ['q', 'r', 's'], ['o', 'n', 'e'], ['l', 'k', 'c'], ['t', 's', 'u']]
	Coordinate Data:
		s : [20356816717677989627 / 20000000000000000000, -990264731706716301 / 100000000000000000000, -6477300057215441453 / 20000000000000000000]
		g : [-38346113612187714799 / 100000000000000000000, -27478214892079741 / 5000000000000000000, -12102034715438422161 / 100000000000000000000]
		l : [14600077761595152251 / 20000000000000000000, 45863599638335672971 / 100000000000000000000, 222370622080239373251 / 100000000000000000000]
		o : [87358695893260726411 / 100000000000000000000, 143657565258843492747 / 100000000000000000000, 45372377342683160809 / 50000000000000000000]
		e : [-11299463436939772781 / 100000000000000000000, 4584825080489412813 / 3125000000000000000, 74706570773361731831 / 100000000000000000000]
		j : [92358084003722861719 / 100000000000000000000, -8794673207080526767 / 50000000000000000000, 32887349433395527383 / 50000000000000000000]
		c : [33274510531803015219 / 100000000000000000000, -1971182055880548353 / 50000000000000000000, 145291308577129131711 / 100000000000000000000]
		u : [119038738601631085077 / 100000000000000000000, 92072653647846508423 / 100000000000000000000, -12931973102551346297 / 20000000000000000000]
		d : [3452596724086783967 / 20000000000000000000, 9429536118675848393 / 10000000000000000000, 77466969828412305579 / 50000000000000000000]
		k : [112346643053347846323 / 100000000000000000000, 55571849876656142081 / 100000000000000000000, 16368823826148044537 / 12500000000000000000]
		r : [11747510642254633877 / 50000000000000000000, -5394199256261980401 / 12500000000000000000, -9767081503444256729 / 12500000000000000000]
		f : [-43899222755371020479 / 100000000000000000000, 99165824347638084739 / 100000000000000000000, -7002598852047632403 / 100000000000000000000]
		q : [25840892959871134107 / 100000000000000000000, 27912952936347458767 / 50000000000000000000, -200253777856400673 / 312500000000000000]
		i : [65882909332290292391 / 100000000000000000000, -88404999929144263703 / 100000000000000000000, 320834631262672517 / 100000000000000000000]
		p : [25611260054919425687 / 50000000000000000000, 64754154805304136969 / 50000000000000000000, -4431284130439167 / 312500000000000000]
		a : [-755829242266293071 / 6250000000000000000, 46754190249153065639 / 100000000000000000000, 35999972216021203371 / 50000000000000000000]
		n : [3498405605223577271 / 12500000000000000000, 48427899105480875767 / 25000000000000000000, 153749712937618712757 / 100000000000000000000]
		t : [124780805018396221191 / 100000000000000000000, 71132665465409473917 / 100000000000000000000, 16477201886481112743 / 50000000000000000000]
		m : [50688389151089977837 / 50000000000000000000, 2123853361381837557 / 1562500000000000000, 18945506709322939401 / 10000000000000000000]
		h : [-7955466755082007529 / 25000000000000000000, -100194635807079781421 / 100000000000000000000, -3484247726803283671 / 20000000000000000000]
		b : [-171986185518698283 / 12500000000000000000, -3355012763673589 / 6400000000000000, 32492611234639542003 / 50000000000000000000]

Desired square lengths:
	default : 1

Checking inequality 1:
	 d  = 3
	|V| = 21
	|E| = 57
	Success: d|V| >= |E|

Checking self-intersection:
	Square collision distance = 524778856726904755326216751251836752043920849081006892589317609224119726849208710052894963801638287994760499598960594481 / 996888515454931449829734125302085465705392257016559905988109136229533946974480190000000000000000000000000000000000000000
	Collision distance in [362772929822186225381019670673 / 500000000000000000000000000000, 725545859644372450762039341347 / 1000000000000000000000000000000] ~ [0.72555, 0.72555]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [213 / 1250, 853 / 5000] ~ [0.1704, 0.1706]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 2125770495528830889167505743584356317068077 / 50000000000000000000000000000000000000000000000000000000000000000000000000000000
	rho in [515481631041401766403800440841 / 2500000000000000000000000000000000000000000000000, 515481631043901766403800440841 / 2500000000000000000000000000000000000000000000000] ~ [0.0, 0.0]
	sigma_min ^ 2 / (16 * E ^ .5) in [1814760000000000000000000000 / 7549834435270749697236684806947, 909511250000000000000000000 / 3774917217635374848618342403473] ~ [0.00024, 0.00024]
	Success: rho < sigma_min ^ 2 / (16 * E ^ .5)

Checking inequality 4:
	LHS NUM := sigma_min - [sigma_min ^ 2 - 16 * rho * |E| ^ .5 ] ^ .5 in [-39999999999985400043258951 / 200000000000000000000000000000, 200000000000073085464202907 / 1000000000000000000000000000000] ~ [-0.0002, 0.0002]
	LHS DEN := 8 * |E| ^ .5 in [3774917217635374848618342403473 / 62500000000000000000000000000, 7549834435270749697236684806947 / 125000000000000000000000000000] ~ [60.39868, 60.39868]
	LHS     := (LHS NUM) / (LHS DEN) in [-66666666666642333405431585 / 20132891827388665859297826151856, 200000000000073085464202907 / 60398675482165997577893478455568] ~ [-0.0, 0.0]
	CD / |V| ^ .5 in [725545859644372450762039341346 / 4582575694955840006588047193729, 725545859644372450762039341347 / 4582575694955840006588047193728] ~ [0.15833, 0.15833]
	Success: LHS < CD / |V| ^ .5

Success: existence proven