Roundest Polyhedra, Monte Carlo Search

This table is the result of runs made in 2015 and 2016 using a program based on the outline given in A defect-correction algorithm for minimizing the volume of a simple polyhedron which circumscribes a sphere. It is noted in the table where better polyhedra have since been found.

Table 1-1: Summary of Monte Carlo runs for *n=4* to *200*
n	area	IQ	upper bound	runs	occurrs	types	hepts	symm
4	41.569219381653	0.302299894039	0.302299894039	1000000	1000000	1	0	T_d
5	31.176914536240	0.403066525385	0.434798216690	1000000	1000000	1	0	D_3h
6	24.000000000000	0.523598775598	0.523598775598	1000000	1000000	1	0	O_h
7	21.796275840161	0.576537510651	0.588174165689	1000000	1000000	1	0	D_5h
8	20.102751310900	0.625107002520	0.637349714015	1000000	969804	2	0	D_2d
9	18.789570476614	0.668794990817	0.676058307811	1000000	1000000	1	0	D_3h
10	17.923571509859	0.701108627120	0.707318712042	1000000	996844	2	0	D_4d
11	17.337252743755	0.724819024103	0.733089302208	1000000	1000000	1	0	C_2v
12	16.650873085547	0.754697399337	0.754697399337	1000000	943455	2	0	I_h
13	16.351703362777	0.768505294865	0.773074693125	1000000	1000000	1	0	C_2v
14	15.993891308361	0.785698137625	0.788894402368	1000000	999367	2	0	D_6d
15	15.721247710048	0.799324000622	0.802655044081	1000000	1000000	1	0	D_3h
16	15.472222719977	0.812189097959	0.814733609959	1000000	650601	4	0	T_d
17	15.283302027801	0.822228769117	0.825420441696	1000000	919318	2	0	C_2v
18	15.103395340104	0.832022888323	0.834942754338	1000000	537674	7	0	D₃
19	14.953719941256	0.840350806604	0.843480895546	1000000	903271	5	0	C₂
20	14.806461242007	0.848708574518	0.851179828648	1000000	310138	6	0	D_2d
21	14.682796384536	0.855856764968	0.858157399216	1000000	754662	4	0	C₂
22	14.570889153520	0.862429909524	0.864510388893	1000000	551351	8	0	D₂
23	14.474303627945	0.868184814784	0.870319017645	1000000	592574	6	0	D₃
24	14.380527444730	0.873846294071	0.875650339164	1000000	665625	10	0	D₂
25	14.302385479643	0.878620607188	0.880560834232	1000000	542917	6	0	C₁
26	14.225104931341	0.883393878289	0.885098414627	1000000	949968	11	0	C₂
27	14.151081376354	0.888014864741	0.889303988096	1000000	428654	7	0	D_5h
28	14.089487889163	0.891896903082	0.893212692575	1000000	880678	11	0	T
29	14.035465557164	0.895329803146	0.896854878407	1000000	254623	13	0	D₃
30	13.980298018310	0.898862856707	0.900256896589	1000000	161571	31	0	C_2v
31	13.927129345375	0.902294385492	0.903441736299	1000000	682509	21	0	C_3v
32	13.873255410368	0.905798260224	0.906429544276	1000000	622863	24	0	I_h
33	13.838989137211	0.908041077984	0.909238050826	1000000	311434	26	1	C_s
34	13.797546530468	0.910768489645	0.911882921464	1000000	281205	56	0	D₂
35	13.761042003617	0.913184525638	0.914378048900	1000000	55161	65	0	C_2v
36	13.724499252860	0.915615964039	0.916735796857	1000000	207497	71	0	C_s
37	13.688116101506	0.918049680553	0.918967204731	1000000	104484	75	0	D_5h
38	13.656468619987	0.920177167615	0.921082160244	1000000	388024	95	0	C_2v
39	13.624913616191	0.922308277935	0.923089545780	1000000	417406	95	0	D_3h
40	13.596091510221	0.924263462401	0.924997362965	1000000	283440	141	0	T_d
41	13.569460065276	0.926077423413	0.926812839171	1000000	556466	166	0	D_3h
42	13.543958060472	0.927821140486	0.928542518938	1000000	593461	220	0	D_5h
43	13.519967642979	0.929467506595	0.930192342738	1000000	24842	307	0	C_2v
44	13.495892960540	0.931125539533	0.931767715087	1000000	368366	391	0	D_2d
45	13.474895887385	0.932576453234	0.933273563645	1000000	312541	540	0	C₂
46	13.454777570032	0.933970892417	0.934714390669	1000000	14113	661	0	T
47	13.434848798549	0.935356311246	0.936094317938	1000000	40227	825	0	C₂
48	13.413935826134	0.936814576813	0.937417126110	1000000	30090	1267	0	D₂
49	13.396992508493	0.937999376083	0.938686289293	1000000	16427	1728	0	C₃
50	13.378236095681	0.939314460029	0.939905005491	1000000	179823	2077	0	D_6d
51	13.361998862967	0.940455896100	0.941076223495	1000000	3572	2205	0	D₃
52	13.346164030611	0.941571719450	0.942202666696	1000000	53960	2640	0	C₃
53	13.330401722055	0.942685065040	0.943286854212	1000000	71036	3093	0	C₂
54	13.315450573632	0.943743551513	0.944331119684	1000000	57210	3819	0	D₃
55	13.301142258210	0.944758756084	0.945337628041	1000000	62360	4720	0	C₂
56	13.287447483322	0.945732476469	0.946308390471	1000000	31799	5876	0	D₂
57	13.274182399219	0.946677560729	0.947245277828	1000000	66203	7029	0	C₂
58	13.261331710856	0.947594923975	0.948150032663	1000000	34484	8499	0	C₂
59	13.248808147065	0.948490647224	0.949024280037	1000000	51058	10189	0	C₂
60	13.236278919084	0.949388471728	0.949869537255	1000000	31177	11607	0	D₃
61	13.225276047939	0.950178322842	0.950687222653	1000000	26653	13227	0	C₂
62	13.214033353030	0.950986748605	0.951478663539	1000000	7986	15052	0	D₅
63	13.203386268922	0.951753615202	0.952245103375	1000000	10553	18643	0	C₂
64	13.192741313317	0.952521565906	0.952987708298	1000000	10377	22789	0	D₂
65	13.182847199482	0.953236461305	0.953707573039	1000000	4689	26680	0	C₂
66	13.173098142638	0.953941926060	0.954405726300	1000000	3520	30994	0	C₂
67	13.162878524114	0.954682563646	0.955083135666	1000000	34397	34328	0	D₅
68	13.154129603909	0.955317530901	0.955740712069	1000000	4325	38049	0	D₃
69	13.145256431011	0.955962379305	0.956379313881	1000000	8403	43258	0	D₃
70	13.136535817314	0.956596989428	0.956999750645	1000000	19159	49818	0	C₃
71	13.128072172808	0.957213705786	0.957602786503	1000000	31597	59329	0	C₂
72	13.118923767044	0.957881213238	0.958189143332	1000000	16585	68489	0	I
73	13.112086175107	0.958380721919	0.958759503626	1000000	18545	77852	0	C₂
74	13.104717674348	0.958919598776	0.959314513146	1000000	4300	87154	0	C₂
75	13.096972746903	0.959486658268	0.959854783359	1000000	1393	95910	0	D₃
76	13.089676302286	0.960021495120	0.960380893684	1000000	21125	106406	0	C₂
77	13.082084359143	0.960578625651	0.960893393563	1000000	29222	114860	0	D₅
78	13.075098033184	0.961091884930	0.961392804377	1000000	4028	124472	0	T_h
79	13.068752038542	0.961558577078	0.961879621214	1000000	16438	137676	0	C₂
80	13.062472687315	0.962020814525	0.962354314504	1000000	1644	151000	0	D₂
81	13.056326127842	0.962473707482	0.962817331539	1000000	6996	165822	0	C₂
82	13.050123468771	0.962931166470	0.963269097873	1000000	5973	183826	0	C₂
83	13.044049522776	0.963379554211	0.963710018632	1000000	11117	198284	0	C₂
84	13.038141706543	0.963816078794	0.964140479721	1000000	4179	212243	0	D₂
85	13.032327559921	0.964246068600	0.964560848953	1000000	7121	225369	0	C₂
86	13.026707550117	0.964662065684	0.964971477096	1000000	1039	237933	0	D₃
87	13.021226627808	0.965068113285	0.965372698849	1000000	13463	253141	0	C₁
88	13.015695942555	0.965478194160	0.965764833752	1000000	9972	269577	0	D₂
89	13.010504554829	0.965863434535	0.966148187033	1000000	15298	288554	0	C₂
90	13.005372323902	0.966244587344	0.966523050404	1000000	5619	310157	0	D₃
91	13.000386732712	0.966615137897	0.966889702796	1000000	9379	334939	0	C₂
92	12.995420823318	0.966984508252	0.967248411057	1000000	7766	359837	0	D₂
93	12.990814914820	0.967327353731	0.967599430594	1000000	8613	382972	0	C₁
94	12.986055657280	0.967681869384	0.967943005983	1000000	1373	404663	0	D₂
95	12.981527490717	0.968019412457	0.968279371536	1000000	4844	422011	0	C₂
96	12.977081239834	0.968351078499	0.968608751832	1000000	2598	437325	0	C₁
97	12.972732846147	0.968675664827	0.968931362223	1000000	1879	453176	0	C₂
98	12.968346610201	0.969003296417	0.969247409294	1000000	757	471072	0	D₃
99	12.964278584455	0.969307357328	0.969557091314	1000000	3367	491306	0	C₁
100	12.960166193037	0.969614928326	0.969860598643	1000000	548	515957	0	D₂
101	12.956177946293	0.969913400885	0.970158114127	1000000	186	543312	0	D₃
102	12.952296267374	0.970204074625	0.970449813463	1000000	69	569493	0	D₃
103	12.948411399969	0.970495161622	0.970735865545	1000000	458	592553	0	C₂
104	12.944600665876	0.970780863676	0.971016432792	1000000	746	612131	0	D₂
105	12.940968610994	0.971053326231	0.971291671451	1000000	97	630355	0	D₃
106	12.937364247501	0.971323862725	0.971561731894	1000000	144	644332	0	D₂
107	12.933748231817	0.971595425327	0.971826758886	1000000	1204	659490	0	C₂
108	12.930169686052	0.971864323476	0.972086891845	1000000	1302	674169	0	C₂
109	12.926824115714	0.972115850101	0.972342265088	1000000	898	690960	0	C₂
110	12.923399539216	0.972373451446	0.972593008064	1000000	1277	707142	0	C₂
111	12.919958611778	0.972632420270	0.972839245566	1000000	957	725788	0	D₃
112	12.916737176216	0.972874994894	0.973081097945	1000000	219	745915	0	T
113	12.913637070797	0.973108547613	0.973318681304	1000000	189	764809	0	D₃
114	12.910550387987	0.973341200546	0.973552107682	1000000	421	782706	0	C₂
115	12.907494579868	0.973571635967	0.973781485231	1000000	388	800158	0	C₃
116	12.904489889897	0.973798323032	0.974006918386	1000000	152	815009	0	T_d
117	12.901549954212	0.974020226946	0.974228508018	1000000	196	825247	0	C₂
118	12.898658080610	0.974238601863	0.974446351590	1000000	150	833097	0	C₂
119	12.895689854229	0.974462844284	0.974660543297	1000000	197	838326	0	D_3h
120	12.892889838754	0.974674473413	0.974871174201	1000000	754	844976	0	C₁
121	12.889986064346	0.974894041904	0.975078332363	1000000	1291	854893	0	C₃
122	12.887030589938	0.975117621291	0.975282102963	1000000	279	867553	0	I_h
123	12.884681996485	0.975295363734	0.975482568418	1000000	633	880778	0	C₂
124	12.882116009967	0.975489632653	0.975679808494	1000000	92	894587	0	D₂
125	12.879520800058	0.975686192789	0.975873900410	1000000	148	906035	0	C_2v
126	12.877021971493	0.975875527912	0.976064918939	1000000	48	914824	0	D₃
127	12.874394584793	0.976074683092	0.976252936507	1000000	15	921916	0	D₅
128	12.871999265376	0.976256318485	0.976438023278	1000000	122	928681	0	C₂
129	12.869548512675	0.976442227323	0.976620247250	1000000	161	933021	0	C₂
130	12.867170786657	0.976622664198	0.976799674332	1000000	156	937223	0	C₂
131	12.864814011164	0.976801577034	0.976976368424	1000000	102	939960	0	C₂
132	12.862290487260	0.976993221138	0.977150391497	1000000	2	943721	0	I
133	12.860177727221	0.977153728425	0.977321803659	1000000	63	948364	0	C₃
134	12.857912989159	0.977325840123	0.977490663230	1000000	333	952552	0	C₂
135	12.855618866175	0.977500246793	0.977657026806	1000000	218	957276	0	D₃
136	12.853418621072	0.977667575049	0.977820949322	1000000	152	961308	0	T
137	12.851277291548	0.977830477802	0.977982484113	1000000	169	965961	0	D₃
138	12.849249036411	0.977984828432	0.978141682969	1000000	162	970277	0	C₂
139	12.847238053880	0.978137912729	0.978298596197	1000000	122	974489	0	C₂
140	12.845259735377	0.978288557276	0.978453272668	1000000	32	978735	1	C₁
141	12.843181526647	0.978446858225	0.978605759867	1000000	18	980765	0	C_2v
142	12.841175593252	0.978599702426	0.978756103949	1000000	57	981925	0	C₂
143	12.839173280051	0.978752318413	0.978904349777	1000000	53	981490	0	C₂
144	12.837206187408	0.978902296256	0.979050540972	1000000	107	981384	0	C₂
145	12.835288823192	0.979048526875	0.979194719955	1000000	168	981672	0	C_s
146	12.833365538559	0.979195252921	0.979336927986	1000000	153	983661	0	D₂
147	12.831608010577	0.979329371970	0.979477205204	1000000	50	986028	0	C₂
148	12.829785318449	0.979468502586	0.979615590668	1000000	105	988863	0	C₂
149	12.828069080407	0.979599543438	0.979752122387	1000000	19	991242	0	C₁
150	12.826229061590	0.979740074344	0.979886837362	1000000	6	993071	0	T
151	12.824495574808	0.979872505788	0.980019771614	1000000	26	994057	0	C₂
152	12.822767627788	0.980004549652	0.980150960218	1000000	8	994916	0	D₂
153	12.821005873424	0.980139213602	0.980280437334	1000000	27	995379	0	D₃
154	12.819367051381	0.980264514152	0.980408236238	1000000	9	995805	0	C₂
155	12.817696442463	0.980392278033	0.980534389345	1000000	28	996003	0	C₂
156	12.816041446592	0.980518880711	0.980658928246	2000000	6	1987989	0	D₃
157	12.814419458137	0.980642990142	0.980781883724	2000000	31	1988322	0	C₂
158	12.812817546248	0.980765594219	0.980903285786	2000000	29	1988659	0	C₂
159	12.811194664930	0.980889834479	0.981023163686	2000000	76	1989831	0	C₂
160	12.809625758574	0.981009972594	0.981141545948	2000000	52	1990955	0	D₂
161	12.808099302953	0.981126888317	0.981258460387	4000000	100	3975187	0	C₂
162	12.806576034791	0.981243587687	0.981373934136	4000000	20	3978463	0	D₃
163	12.805088236194	0.981357596494	0.981487993659	4000000	27	3981238	0	C₂
164	12.803603063688	0.981471430491	0.981600664779	4000000	43	3983588	0	C₂
165	12.802134202656	0.981584040242	0.981711972693	4000000	20	3986137	0	C₂
166	12.800696713327	0.981694269912	0.981821941992	4000000	25	3987974	0	C₂
167	12.799278055778	0.981803079798	0.981930596679	4000000	16	3989880	0	C₂
168	12.797836704634	0.981913654970	0.982037960187	4000000	3	3990865	0	D₃
169	12.796501698498	0.982016093964	0.982144055394	4000000	20	3991685	0	C₁
170	12.795096325156	0.982123955539	0.982248904644	4000000	10	3992094	0	C₂
171	12.793734345983	0.982228509247	0.982352529755	4000000	15	3992318	0	C₃
172	12.792379224010	0.982332558651	0.982454952041	4000000	27	3992525	0	C₂
173	12.791039826186	0.982435422383	0.982556192324	4000000	39	3993489	0	C_s
174	12.789675345331	0.982540234608	0.982656270947	4000000	13	3994617	0	T_h
175	12.788440168222	0.982635133688	0.982755207790	4000000	45	3995825	0	C₁
176	12.787169017264	0.982732815793	0.982853022282	4000000	9	3996754	0	C₂
177	12.785892601684	0.982830922004	0.982949733412	4000000	5	3997444	0	D₅
178	12.784673844660	0.982924614820	0.983045359746	4000000	3	3997998	0	C₂
179	12.783447179350	0.983018933630	0.983139919432	4000000	3	3998363	0	C₂
180	12.782186696434	0.983115871548	0.983233430219	4000000	3	3998610	0	C₂
181	12.780953902103	0.983210698561	0.983325909462	4000000	3	3998732	0	C₂
182	12.779698559862	0.983307278767	0.983417374137	4000000	1	3998875	0	D₅
183	12.778578565377	0.983393461962	0.983507840848	4000000	9	3999047	0	C₁
184	12.777371960354	0.983486326715	0.983597325839	4000000	1	3999113	0	T
185	12.776271812445	0.983571013425 ^a	0.983685845004	4000000	2	3999178	0	C₂
186	12.775112225012	0.983660291434 ^a	0.983773413896	4000000	1	3999302	1	C₁
187	12.773891164101	0.983754319880	0.983860047735	4000000	6	3999258	0	D₅
188	12.772815252573	0.983837185919	0.983945761417	4000000	5	3999409	0	D₂
189	12.771711810774	0.983922186825	0.984030569525	4000000	10	3999533	0	C₁
190	12.770586592545	0.984008880351	0.984114486334	10000000	18	9997646	0	C₃
191	12.769495624856	0.984092949599	0.984197525823	10000000	7	9998053	1	C₁
192	12.768324092586	0.984183243097	0.984279701676	10000000	3	9998445	0	I
193	12.767387855101	0.984255413635	0.984361027299	10000000	11	9998614	0	C₂
194	12.766359437844	0.984334702116	0.984441515817	10000000	2	9998952	0	C₂
195	12.765319648475	0.984414880348 ^a	0.984521180089	10000000	7	9999118	0	C₁
196	12.764275073201	0.984495440775	0.984600032712	10000000	3	9999270	1	C₁
197	12.763290043066	0.984571421002 ^a	0.984678086027	10000000	3	9999392	0	C₁
198	12.762275197137	0.984649713335	0.984755352127	10000000	1	9999477	0	D₃
199	12.761290461195	0.984725694676 ^a	0.984831842859	10000000	1	9999508	0	C₁
200	12.760269576679	0.984804477589	0.984907569839	10000000	3	9999495	0	C₁

^a A better value has since been found.

Note for 1/18/2021: Added notes to flag values known not to be best.

Note for 1/22/2021: Corrected symmetry for n=165, was C₁, should be C₂.

Wayne Deeter - wrd@deetour.net

Last modified: January 31, 2021