21 vertices

bcdefg aghic abijkd acklme admnf aenopg afpqhb bgqri bhrsjc cistok cjonld dknm dlne emlkof fnkjtp fotqg gpturh hqusi irutj jsuqpo qtsr

show/hide visualization coordinates

a : (0.40658035177114743, 0.4874317995340909, 0.7934728615140851)
b : (0.606556990661502, -0.4823188543112116, 0.9334479066640158)
c : (0.7199815842653641, -0.07407690885452023, 0.02764805264794523)
d : (0.8056808102189863, 0.9213626176810277, -0.01425411533456189)
e : (0.20522944935148463, 1.4455141848418518, 0.5896691396202522)
f : (-0.45572756412879317, 0.7339982383193322, 0.3511720030013927)
g : (-0.32289625545252243, -0.1910807663804308, 0.7069594093257947)
h : (-0.13712181092520753, -1.1506835327643827, 0.9182657899886473)
i : (0.4445700095934159, -1.0315506231039187, 0.11362799832238374)
j : (0.05542256691135722, -0.4624534799033421, -0.6107287573270382)
k : (0.49449473336277644, 0.41208733256015595, -0.816626790302931)
l : (0.8838583027700502, 1.3267688123852996, -0.9250417398408626)
m : (0.7203379693249068, 1.91096025469357, -0.13006912441076351)
n : (0.06377763263572961, 1.1952352040782084, -0.36811570852453046)
o : (-0.4778713310366712, 0.38308439930169935, -0.5849738890330896)
p : (-1.0963882056734882, 0.006766028730562423, 0.10482416819829216)
q : (-0.7050002936779465, -0.9131102581774668, 0.1301839617319001)
r : (-0.21344305448987538, -1.7837740868060479, 0.1479597351221676)
s : (-0.5365560962142047, -0.8385253051843237, 0.10210449430612117)
t : (-0.9348225579657399, -0.4915261929950621, -0.7469982330383252)
u : (-0.5266632313022732, -1.4041088636450916, -0.7225271626308953)
			
show/hide computer existence proof (see shape-existence, preprint)

Attempting to prove existence

Starting realization:
	Abstract data:	
		mode: maximal_simplices
		data: [['g', 'p', 'q'], ['d', 'l', 'k'], ['o', 't', 'j'], ['r', 'i', 's'], ['a', 'b', 'c'], ['r', 'q', 'h'], ['t', 'j', 's'], ['r', 'u', 's'], ['o', 'p', 't'], ['d', 'a', 'e'], ['o', 'n', 'k'], ['d', 'l', 'm'], ['f', 'a', 'e'], ['p', 't', 'q'], ['p', 'g', 'f'], ['d', 'm', 'e'], ['r', 'u', 'q'], ['j', 'k', 'c'], ['j', 'i', 's'], ['r', 'i', 'h'], ['i', 'b', 'c'], ['g', 'b', 'h'], ['j', 'i', 'c'], ['i', 'b', 'h'], ['o', 'p', 'f'], ['f', 'n', 'e'], ['l', 'n', 'k'], ['d', 'a', 'c'], ['d', 'k', 'c'], ['m', 'l', 'n'], ['o', 'n', 'f'], ['u', 't', 's'], ['u', 't', 'q'], ['m', 'n', 'e'], ['o', 'k', 'j'], ['g', 'q', 'h'], ['g', 'a', 'f'], ['g', 'a', 'b']]
	Coordinate Data:
		n : [51887528621734866429 / 100000000000000000000, 178931293820133873981 / 100000000000000000000, 9869020741093910543 / 100000000000000000000]
		s : [-254557633226830307 / 3125000000000000000, -4888951421223868537 / 20000000000000000000, 56891041024159076079 / 100000000000000000000]
		c : [7344245236543644789 / 6250000000000000000, 26000041263430501327 / 50000000000000000000, 24722698429170740047 / 50000000000000000000]
		r : [2416545990917436877 / 10000000000000000000, -59484817634145884723 / 50000000000000000000, 30738282552881855839 / 50000000000000000000]
		g : [826258738306854121 / 6250000000000000000, 40299696774269943267 / 100000000000000000000, 23475306505225287979 / 20000000000000000000]
		a : [43083900267638325903 / 50000000000000000000, 108150953365722109269 / 100000000000000000000, 63013938872477734833 / 50000000000000000000]
		h : [31797584265641155977 / 100000000000000000000, -55660579864125244541 / 100000000000000000000, 138507170592411677901 / 100000000000000000000]
		k : [11869904836804943859 / 12500000000000000000, 100616506668328630031 / 100000000000000000000, -34982087436746133911 / 100000000000000000000]
		t : [-23986245219206036691 / 50000000000000000000, 10255154112806814401 / 100000000000000000000, -14009615855142783671 / 50000000000000000000]
		j : [25526011024648815177 / 50000000000000000000, 6581212710989406977 / 50000000000000000000, -14392284139156871261 / 100000000000000000000]
		o : [-1138683872752603813 / 50000000000000000000, 97716213342482957823 / 100000000000000000000, -11816797309762000499 / 100000000000000000000]
		d : [15759730797507568637 / 12500000000000000000, 37886008795103947741 / 25000000000000000000, 22627590030045382347 / 50000000000000000000]
		u : [-7156557772065410283 / 100000000000000000000, -5062694559512258253 / 6250000000000000000, -25572124669542582773 / 100000000000000000000]
		q : [-2499026400963274499 / 10000000000000000000, -6380650481086731381 / 20000000000000000000, 3731186735421060723 / 6250000000000000000]
		i : [89966766317503498681 / 100000000000000000000, -8749457779615766441 / 20000000000000000000, 7255423928223165831 / 12500000000000000000]
		l : [133895595635166936219 / 100000000000000000000, 192084654650842999877 / 100000000000000000000, -45823582390539304351 / 100000000000000000000]
		b : [106165464424312122559 / 100000000000000000000, 2235177596238373357 / 20000000000000000000, 140025382259948528413 / 100000000000000000000]
		p : [-64129055209186902209 / 100000000000000000000, 12016875257073853993 / 20000000000000000000, 14290752103344042889 / 25000000000000000000]
		e : [33016355146655187151 / 50000000000000000000, 203959191896498189941 / 100000000000000000000, 26411876388893043997 / 25000000000000000000]
		m : [117543562290652587113 / 100000000000000000000, 125251899440835003559 / 50000000000000000000, 6734735830494121243 / 20000000000000000000]
		f : [-62991054717409963 / 100000000000000000000, 16600949655530779673 / 12500000000000000000, 81797791893686222677 / 100000000000000000000]

Desired square lengths:
	default : 1

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

Checking self-intersection:
	Square collision distance = 17787606307893113286591821772231375108831667455386889418566311072715790782787888181487438627947107945395825404397241 / 732421874999999999921214580911911209915096948033361187408724384878131471128626025390625000000000000000000000000000000
	Collision distance in [77919849544863616023349519329 / 500000000000000000000000000000, 155839699089727232046699038659 / 1000000000000000000000000000000] ~ [0.15584, 0.15584]
	Success: starting realization non-self-intersecting

Checking inequality 2:
	sigma_min in [427 / 2500, 171 / 1000] ~ [0.1708, 0.171]
	Success: sigma_min > 0

Checking inequality 3:
	rho_squared = 7420663724504602330502081782628139527361417 / 100000000000000000000000000000000000000000000000000000000000000000000000000000000
	rho in [2724089522116445117050107415181 / 10000000000000000000000000000000000000000000000000, 2724089522126445117050107415181 / 10000000000000000000000000000000000000000000000000] ~ [0.0, 0.0]
	sigma_min ^ 2 / (16 * E ^ .5) in [1823290000000000000000000000 / 7549834435270749697236684806947, 304593750000000000000000000 / 1258305739211791616206114134491] ~ [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 [-99999999999951891403792199 / 500000000000000000000000000000, 100000000000048164929458807 / 500000000000000000000000000000] ~ [-0.0002, 0.0002]
	LHS DEN := 8 * |E| ^ .5 in [3774917217635374848618342403473 / 62500000000000000000000000000, 7549834435270749697236684806947 / 125000000000000000000000000000] ~ [60.39868, 60.39868]
	LHS     := (LHS NUM) / (LHS DEN) in [-99999999999951891403792199 / 30199337741082998788946739227784, 100000000000048164929458807 / 30199337741082998788946739227784] ~ [-0.0, 0.0]
	CD / |V| ^ .5 in [155839699089727232046699038658 / 4582575694955840006588047193729, 155839699089727232046699038659 / 4582575694955840006588047193728] ~ [0.03401, 0.03401]
	Success: LHS < CD / |V| ^ .5

Success: existence proven