Euclidean Regular Hexagon with Dirichlet Boundary Conditions

The region examined here is the Euclidean Regular Hexagon of side length 1 with Dirichlet Boundary conditions. All eigenvalues of the Euclidean Equilateral Triangle under Dirichlet Boundary conditions are also eigenvalue of this region, which is why we have divided the values in the table below by a factor of (4/3*pi)^2, so that we can observe the common values. This has a dual purpose of easily allowing us to check the accuracy of the Hexagon eigenvalues: since 43 = 6^2 + 1^2 + 6*1 is an eigenvalue of multiplicity two in the Equilateral Triangle, it should also be an eigenvalue of multiplicity two in the hexagon, and indeed we see that our predictions for eigenvalues 145 and 146 differ from 43 by less than .0002.

Here are the counting graphs for the eigenvalue spectrum, using the predicted values. The first approximately 150 values were used here. The equation for the predicted counting function is N'(t) = 3sqrt(3)/8pi*t - 3/2pi*sqrt(t) + 5/24.

Here are the difference graphs for the eigenvalue spectrum, using the predicted values. The first approximately 150 values were used here.

Below is the Eigenvalue spectrum for the Euclidean Regular Hexagon of side length 1 under Dirichlet Boundary conditions. We have removed a factor of (4/3*pi)^2 from the figures in order to easily compare the values to those of the Euclidean Equilateral Triangle. The calculated and predicted eigenvalue spectrum is presented here.

	Euclidean Regular Hexagon with Dirichlet Boundary Conditions
		Refinement
#	Initial Value	1	2	3	4	5		Predicted Value
1	0.413621295	0.40934487	0.40820039	0.407905331	0.407830539	0.407811725		0.407805456
2	1.06961183	1.042747246	1.035758516	1.033979817	1.033531679	1.033419232		1.03338173
3	1.070950087	1.043082879	1.035843265	1.034001189	1.033537037	1.0334206		1.033381787
4	1.96857763	1.88000552	1.857227118	1.851465907	1.850017995	1.849655177		1.849534237
5	1.969445294	1.880228933	1.857298473	1.851485285	1.850022954	1.849656431		1.849534237
6	2.28805852	2.175412895	2.146517531	2.139203883	2.137365398	2.136904665		2.136751069
7	2.976486835	2.782208759	2.731672691	2.71885356	2.715627405	2.714818444		2.714548809
8	3.328668637	3.081769911	3.02046601	3.00512174	3.00128097	3.000320294		3.000000049
9	3.808021113	3.52331808	3.450284643	3.431787397	3.427135615	3.425969706		3.425580956
10	3.816108614	3.524925515	3.450606997	3.431860463	3.427153226	3.425974037		3.42558107
11	4.573919326	4.14345251	4.034190855	4.006715589	3.999823235	3.998097197		3.997521851
12	4.592440908	4.148062915	4.035299999	4.006988871	3.999891227	3.998114181		3.997521851
13	5.809236449	5.197345967	5.041161895	5.001867843	4.992012813	4.989545578		4.988723167
14	5.834397792	5.203367978	5.042763517	5.002279333	4.992116597	4.989571624		4.988723281
15	5.988469127	5.352331358	5.188027983	5.146591387	5.136187569	5.133581272		5.132712525
16	6.410761699	5.639348314	5.441964282	5.392457339	5.380052777	5.376947903		5.37591285
17	6.486150549	5.65444757	5.445408094	5.393296507	5.380261144	5.376999938		5.375912964
18	7.654841578	6.638460225	6.377094126	6.311436859	6.294967544	6.290842378		6.289467133
19	8.541140019	7.321511348	6.997386085	6.915881298	6.895447539	6.890331832		6.888626597
20	8.645198141	7.419089347	7.104959546	7.0262782	7.006573953	7.001644044		7.000000361
21	8.768869892	7.482476007	7.119977815	7.02995483	7.007487554	7.001872017		7.000000931
22	8.792514646	7.550939047	7.232969615	7.153350163	7.133397995	7.128403114		7.126737774
23	8.953552826	7.556389873	7.23409922	7.153607202	7.133460118	7.128418502		7.126738344
24	10.50304882	8.873230647	8.437618773	8.328100135	8.300632849	8.293754344		8.291460939
25	10.58593968	8.896012526	8.443269645	8.329510146	8.300985067	8.293842113		8.291461509
26	11.36475243	9.409693069	8.877553579	8.744964678	8.711847659	8.703567692		8.700807513
27	11.54251868	9.436927823	8.883508795	8.746431112	8.712213555	8.703658881		8.700807513
28	12.4633128	10.21556617	9.628546825	9.482048742	9.445414105	9.436251884		9.43319762
29	12.61928372	10.25078624	9.636106969	9.483863975	9.445863782	9.436364161		9.43319762
30	12.73302498	10.41786171	9.819804491	9.671167455	9.634018169	9.624725434		9.621627855
31	13.85598754	11.40198265	10.69275798	10.51429585	10.46955508	10.45835478		10.45462002
32	14.13311168	11.46252934	10.70593822	10.51745897	10.47033702	10.45854913		10.45462116
33	14.36462704	11.72321551	10.93782404	10.74225997	10.69347724	10.68128526		10.67722051
34	14.61114528	11.79774091	10.95995961	10.74763043	10.69480803	10.68161753		10.67722108
35	15.25658376	11.86587396	11.05889747	10.86549116	10.81713531	10.8050311		10.80099655
36	16.9258452	13.37981249	12.34003601	12.08468041	12.02115241	12.00528722		11.99999883
37	17.36999522	13.73933108	12.72460481	12.47453818	12.41211413	12.39650484		12.39130194
38	17.70223004	13.76419036	12.73106441	12.47617046	12.41252391	12.39660743		12.39130194
39	17.82783215	13.98927062	12.89977102	12.62749643	12.55959535	12.54262335		12.53696621
40	18.53077902	14.42995603	13.2039635	12.90423245	12.82973041	12.81112613		12.80492471
41	18.73458829	14.55529198	13.3822877	13.09513847	13.02375966	13.00593846		12.99999693
42	19.03053841	14.66567007	13.40536139	13.10064173	13.02512179	13.00627814		12.99999807
43	19.40472533	15.07339348	13.83838463	13.53206289	13.45564589	13.43653722		13.43016766
44	19.67803207	15.10591264	13.84655745	13.5341648	13.45617535	13.43667001		13.43016823
45	21.73533945	17.1128261	15.54692405	15.15308531	15.05476868	15.03019893		15.02200901
46	21.99401832	17.18028435	15.5585176	15.15609398	15.05553353	15.030391		15.02201015
47	22.30365022	17.33618915	15.71074577	15.30857122	15.2092042	15.18442301		15.17616242
48	22.87879447	17.41692168	15.72212332	15.31238064	15.2102164	15.18467947		15.17616698
49	23.43513554	17.71758628	16.11804743	15.72912949	15.63245768	15.60830854		15.60025882
50	23.65215241	17.82311254	16.19606709	15.79758398	15.69837541	15.67359022		15.66532222
51	24.06308225	17.98166012	16.22191349	15.80301657	15.69966573	15.67390824		15.66532849
52	24.39719215	18.59554719	16.73367678	16.27561644	16.16176119	16.13333072		16.1238539
53	24.97470162	18.66185874	16.74584083	16.2784718	16.1624662	16.13350683		16.1238539
54	25.37276696	19.42393658	17.53558658	17.06992164	16.95384195	16.92481761		16.91514302
55	26.34463153	20.72776867	18.54815509	18.00396497	17.86760939	17.83347789		17.82209522
56	26.59373896	20.87378459	18.57893653	18.0107728	17.86923597	17.83388026		17.82210091
57	27.81036524	20.99465456	18.66929578	18.1094559	17.97135633	17.9369353		17.92545631
58	28.21575933	21.13276381	18.70084206	18.11672994	17.97311457	17.93737073		17.92546144
59	28.97271407	21.57838198	19.19199549	18.60117811	18.45397787	18.41718252		18.40491759
60	29.41972001	22.15401067	19.77966627	19.19397201	19.04843768	19.01210625		18.99999406
61	30.71709135	22.30072875	19.88796696	19.22250735	19.05555955	19.01388557		18.99999577
62	31.2866195	22.43568002	19.95661295	19.36318985	19.21775013	19.1814717		19.16937889
63	31.89798234	22.71764174	19.9714762	19.36859337	19.21913563	19.1818205		19.16938231
64	33.01597168	23.93193585	21.2298397	20.55767712	20.39000722	20.34808818		20.33411516
65	33.79430056	24.03002109	21.24754976	20.56181768	20.39103367	20.34834465		20.33411516
66	34.06367989	24.61327946	21.57113302	20.81923871	20.63297251	20.58651282		20.57102606
67	34.98896565	24.68769372	21.69927132	20.96157059	20.77770038	20.7317542		20.716439
68	36.35625962	25.18599865	22.01099544	21.2495753	21.06220571	21.01554027		20.99998513
69	37.2944948	25.3121781	22.04698891	21.25893471	21.06455496	21.01612787		20.9999857
70	38.36976434	25.96488834	22.58000837	21.74499352	21.53794419	21.48627532		21.46904914
71	39.65872691	26.03497626	22.59073278	21.74717693	21.5384577	21.48640128		21.46905256
72	40.33380811	27.42789322	24.04895593	23.20350677	22.99313701	22.9406047		22.92309412
73	41.13230117	27.5218248	24.06651438	23.20896728	22.99460459	22.94097858		22.92310324
74	41.92219567	27.65307265	24.24714054	23.40781473	23.19840417	23.14602346		23.12856303
75	42.69030738	28.44026852	24.66258025	23.72470528	23.49246155	23.43454851		23.4152398
76	44.31152244	28.57529787	24.69666616	23.73273504	23.49442097	23.43503523		23.41524436
77	44.73636773	29.10804548	25.2810245	24.29525759	24.04876101	23.98717077		23.96663899
78	0	29.3043174	25.31019189	24.30135586	24.05025024	23.9875418		23.9666407
79	0	29.52942502	25.59720263	24.61320537	24.37113886	24.31086118		24.29076881
80	0	29.74105888	25.60544441	24.62050562	24.37320372	24.31139122		24.29078705
81	0	29.80370007	25.64380993	24.66932027	24.43200927	24.3728498		24.35313016
82	0	30.44433517	26.00314044	24.90295976	24.63135655	24.56372333		24.54117911
83	0	30.59256242	26.02271703	24.90712083	24.63239781	24.5639855		24.54118139
84	0	32.64530807	28.00010061	26.83141015	26.54057061	26.46789634		26.4436714
85	0	33.20453927	28.6072015	27.40972494	27.10356335	27.02597399		27.00011107
86	0	33.32435543	28.64456223	27.47227038	27.17942182	27.10622607		27.08170305
87	0	33.55137232	28.6997755	27.4847781	27.18248179	27.10698522		27.08177144
88	0	33.70897438	28.79412427	27.50797318	27.18745558	27.10814104		27.08181989
89	0	33.96199041	28.80367576	27.50810825	27.18817369	27.10837186		27.0818273
90	0	34.35302565	29.2609154	27.96865463	27.63000523	27.54552597		27.51736622
91	0	34.68785195	29.32758714	28.00830079	27.70075541	27.62419022		27.598655
92	0	34.902242	29.35515759	28.02669135	27.70518036	27.6252862		27.59866868
93	0	35.09133678	29.77338028	28.44047085	28.10991676	28.02746702		27.99994217
94	0	35.65639609	29.88763675	28.46184272	28.11476915	28.02864906		27.99998378
95	0	35.82236418	30.3481991	29.02942511	28.70282008	28.62130161		28.59412898
96	0	35.94862171	30.40916127	29.04807955	28.70765823	28.62252184		28.59414323
97	0	38.48524389	32.52704099	30.98327451	30.59673432	30.50015029		30.46795542
98	0	38.66758783	32.54497503	30.9889744	30.59876613	30.50069571		30.46800557
99	0	39.06861397	32.6416816	31.02980943	30.63624598	30.53868963		30.50617104
100	0	39.30053398	32.82412869	31.20760702	30.80527574	30.70472441		30.67120729
101	0	39.72733813	33.0911525	31.40749516	30.98551548	30.8800331		30.84483412
102	0	39.83987854	33.1747649	31.42155994	30.98853954	30.88076034		30.84487231
103	0	40.18202619	33.24997935	31.54978373	31.13661349	31.03410273		30.99993248
104	0	40.4892564	33.32427507	31.57017133	31.14182209	31.03541186		30.9999416
105	0	41.07020255	34.32261637	32.67247843	32.26061446	32.15774123		32.12345015
106	0	41.09184685	34.34215648	32.67921844	32.26265767	32.15826728		32.12347067
107	0	41.33206392	34.41876783	32.71366739	32.30129961	32.19909034		32.1650204
108	0	41.99401048	35.07378174	33.27726597	32.83167744	32.72054703		32.68350375
109	0	42.27623038	35.12677285	33.29440609	32.83620782	32.72169488		32.6835237
110	0	42.77689311	35.80612568	34.01824893	33.57057046	33.45854584		33.42120449
111	0	44.31346078	36.71035873	34.77719215	34.29944865	34.18033294		34.14061326
112	0	44.89402907	36.85708422	34.81244414	34.30811845	34.18248956		34.14062751
113	0	45.4548213	37.18895314	35.07973126	34.55716719	34.42686542		34.38341554
114	0	45.56437869	37.24093889	35.09087	34.55978659	34.42750831		34.3834315
115	0	0	37.57795943	35.51710194	35.00585501	34.87761696		34.83481396
116	0	0	37.67203862	35.53997501	35.0115452	34.87899677		34.83487095
117	0	0	37.77013297	35.57930086	35.04935846	34.91898089		34.87552189
118	0	0	37.8104089	35.59755463	35.05476198	34.92035671		34.87555495
119	0	0	38.13523746	36.02430008	35.50455148	35.37502254		35.33184622
120	0	0	39.93617876	37.73519559	37.18382718	37.04596014		37.00000427
121	0	0	40.05826154	37.80881993	37.20329376	37.050853		37.00003961
122	0	0	40.10157007	37.90276804	37.36051568	37.22532846		37.18022692
123	0	0	40.32499639	37.91804335	37.36337845	37.22601466		37.18026625
124	0	0	41.08619997	38.74494465	38.16051621	38.01436009		37.96564119
125	0	0	41.12058622	38.7572603	38.1643074	38.01589719		37.9658475
126	0	0	41.30218013	38.79524112	38.17268881	38.01755797		37.96642713
127	0	0	41.63920237	38.96276056	38.30422733	38.14060052		38.08605806
128	0	0	41.82598551	39.33336774	38.71305754	38.55778821		38.50595977
129	0	0	41.94706692	39.35430874	38.71735198	38.55880782		38.50603158
130	0	0	42.38443931	39.78523977	39.15009938	38.99208583		38.93941445
131	0	0	42.41441088	39.80070087	39.15439267	38.99317895		38.93944124
132	0	0	42.57092095	39.86808047	39.21512003	39.05366067		38.99984089
133	0	0	42.65897026	39.89793292	39.22302669	39.05565429		38.99986368
134	0	0	43.65459755	40.90646917	40.22736938	40.05816693		40.00176592
135	0	0	43.75132749	40.93050603	40.23337019	40.05966927		40.00176877
136	0	0	0	42.95453568	42.17888206	41.98531503		41.9207925
137	0	0	0	42.95994889	42.18030575	41.98567979		41.92080447
138	0	0	0	43.19175833	42.45999458	42.27715309		42.21618238
139	0	0	0	43.22069604	42.46691127	42.2788646		42.21620574
140	0	0	0	43.44524	42.68063906	42.48926456		42.42547325
141	0	0	0	43.46299565	42.73281801	42.55046268		42.48967776
142	0	0	0	43.73699371	43.00324546	42.82026434		42.75927082
143	0	0	0	43.76400163	43.0113704	42.82315047		42.76040727
144	0	0	0	43.82540037	43.0246008	42.82645551		42.76041069
145	0	0	0	44.059204	43.26311321	43.06566494		42.99981162
146	0	0	0	44.0743659	43.26640741	43.06646057		42.99984866
147	0	0	0	44.83127902	44.03047717	43.83141714		43.76506399
148	0	0	0	44.84211741	44.03442053	43.83248348		43.76517113
149	0	0	0	45.12969524	44.34298495	44.14683613		44.081453
150	0	0	0	45.44473351	44.58100778	44.36636981		44.29474593
151	0	0	0	45.46831329	44.58601577	44.36756325		44.29482401