The region examined here is the Euclidean Regular pentagon of side length 1 with Dirichlet Boundary conditions. Many eigenvalues of this region have multiplicity 2, due to the D_5 symmetry group.
Here are the counting graphs for the eigenvalue
spectrum, using the predicted values. The first approximately 135
values were used here. The equation for the predicted counting function is N'(t) = sqrt(5(5+2sqrt(5)))/16pi*t + 5/4pi*sqrt(t) + 2/9.
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 Pentagon of side length 1
under Dirichlet Boundary conditions.
| Euclidean Regular Pentagon with Dirichlet Boundary Conditions | ||||||||
|---|---|---|---|---|---|---|---|---|
| Refinement | ||||||||
| # | Initial | 1 | 2 | 3 | 4 | 5 | Predicition | |
| 1 | 11.147932 | 11.035256 | 11.006235 | 10.998889 | 10.997043 | 10.996581 | 10.996427 | |
| 2 | 28.800558 | 28.042734 | 27.850742 | 27.802376 | 27.790246 | 27.78721 | 27.786201 | |
| 3 | 28.817143 | 28.047633 | 27.852014 | 27.802698 | 27.790327 | 27.78723 | 27.786201 | |
| 4 | 52.411013 | 50.060611 | 49.470958 | 49.322998 | 49.285939 | 49.276667 | 49.273585 | |
| 5 | 52.571042 | 50.106111 | 49.482948 | 49.326056 | 49.286708 | 49.27686 | 49.273587 | |
| 6 | 61.587339 | 58.215666 | 57.375333 | 57.164765 | 57.112042 | 57.098853 | 57.09447 | |
| 7 | 84.938565 | 78.959641 | 77.472569 | 77.100698 | 77.007644 | 76.98437 | 76.976635 | |
| 8 | 85.066688 | 78.979804 | 77.477489 | 77.101934 | 77.007954 | 76.984447 | 76.976635 | |
| 9 | 99.438712 | 91.724 | 89.806892 | 89.327181 | 89.207103 | 89.177065 | 89.167082 | |
| 10 | 100.11915 | 91.892456 | 89.849077 | 89.337751 | 89.209748 | 89.177726 | 89.167084 | |
| 11 | 128.12691 | 115.30939 | 112.16324 | 111.38039 | 111.18478 | 111.13587 | 111.11961 | |
| 12 | 128.59471 | 115.40087 | 112.18495 | 111.38575 | 111.18611 | 111.1362 | 111.11961 | |
| 13 | 147.08556 | 131.58464 | 127.66452 | 126.68273 | 126.43695 | 126.37546 | 126.35502 | |
| 14 | 152.10324 | 135.56473 | 131.48367 | 130.46873 | 130.21516 | 130.15177 | 130.1307 | |
| 15 | 154.11254 | 136.10494 | 131.62215 | 130.50372 | 130.22395 | 130.15397 | 130.13071 | |
| 16 | 176.77087 | 154.46872 | 148.9629 | 147.59625 | 147.25503 | 147.16974 | 147.14139 | |
| 17 | 191.4466 | 165.1498 | 158.62787 | 157.00948 | 156.60535 | 156.50433 | 156.47076 | |
| 18 | 211.33903 | 182.43227 | 175.15507 | 173.34069 | 172.88707 | 172.77364 | 172.73594 | |
| 19 | 212.70812 | 182.75642 | 175.23342 | 173.36017 | 172.89194 | 172.77485 | 172.73594 | |
| 20 | 218.16895 | 187.31603 | 179.49452 | 177.54385 | 177.05598 | 176.93395 | 176.89339 | |
| 21 | 222.08195 | 187.94062 | 179.61993 | 177.57306 | 177.06313 | 176.93573 | 176.89339 | |
| 22 | 254.94657 | 214.61313 | 204.24677 | 201.67378 | 201.03156 | 200.87104 | 200.81769 | |
| 23 | 256.81087 | 214.90688 | 204.31886 | 201.69219 | 201.03621 | 200.87221 | 200.8177 | |
| 24 | 283.68156 | 237.35613 | 225.36422 | 222.38601 | 221.64258 | 221.45675 | 221.39499 | |
| 25 | 284.55206 | 237.63484 | 225.43183 | 222.40286 | 221.64679 | 221.45781 | 221.395 | |
| 26 | 303.44045 | 249.48699 | 235.61572 | 232.18298 | 231.32669 | 231.11269 | 231.04157 | |
| 27 | 305.76691 | 249.85391 | 235.69783 | 232.20331 | 231.33178 | 231.11396 | 231.04157 | |
| 28 | 308.4073 | 254.95136 | 241.1273 | 237.69733 | 236.84072 | 236.62654 | 236.55536 | |
| 29 | 345.25437 | 279.89671 | 262.73398 | 258.5031 | 257.44982 | 257.18675 | 257.09932 | |
| 30 | 349.13204 | 280.3067 | 262.82731 | 258.52789 | 257.45618 | 257.18836 | 257.09935 | |
| 31 | 371.08156 | 301.10514 | 282.39915 | 277.77795 | 276.62603 | 276.3382 | 276.24254 | |
| 32 | 379.04912 | 303.0304 | 282.85279 | 277.88935 | 276.65375 | 276.34512 | 276.24255 | |
| 33 | 395.4589 | 314.31783 | 292.91269 | 287.64218 | 286.32934 | 286.00136 | 285.89236 | |
| 34 | 404.72575 | 326.9108 | 303.88431 | 298.19101 | 296.77309 | 296.4189 | 296.30119 | |
| 35 | 409.84417 | 327.4443 | 306.02565 | 300.73538 | 299.41589 | 299.08612 | 298.97652 | |
| 36 | 415.6057 | 329.97661 | 306.62508 | 300.88192 | 299.45233 | 299.09523 | 298.97655 | |
| 37 | 451.20436 | 359.51539 | 332.29135 | 325.59605 | 323.93003 | 323.51394 | 323.37565 | |
| 38 | 460.36469 | 360.54283 | 332.52199 | 325.65228 | 323.94401 | 323.51743 | 323.37566 | |
| 39 | 471.85476 | 370.25031 | 341.73328 | 334.80614 | 333.08673 | 332.65757 | 332.51494 | |
| 40 | 476.63548 | 371.29521 | 342.05435 | 334.8887 | 333.10749 | 332.66276 | 332.51496 | |
| 41 | 506.49916 | 404.11585 | 371.23457 | 363.07208 | 361.03629 | 360.52759 | 360.35852 | |
| 42 | 523.07665 | 407.46177 | 371.89789 | 363.22647 | 361.07418 | 360.53701 | 360.35848 | |
| 43 | 527.82806 | 411.77154 | 377.91097 | 369.5196 | 367.42701 | 366.9041 | 366.73031 | |
| 44 | 532.06325 | 414.77956 | 380.57047 | 372.32176 | 370.27983 | 369.77047 | 369.60119 | |
| 45 | 544.1882 | 417.25726 | 381.26429 | 372.49802 | 370.32398 | 369.78151 | 369.60122 | |
| 46 | 558.31878 | 443.32182 | 403.49583 | 393.74379 | 391.319 | 390.71351 | 390.51228 | |
| 47 | 586.97482 | 451.72262 | 409.63318 | 399.39903 | 396.86198 | 396.22897 | 396.01859 | |
| 48 | 600.32183 | 454.44284 | 410.1802 | 399.5274 | 396.89355 | 396.23683 | 396.01857 | |
| 49 | 605.10313 | 466.05617 | 423.84035 | 413.47075 | 410.89245 | 410.24867 | 410.03471 | |
| 50 | 621.47907 | 491.51105 | 445.35852 | 433.90281 | 431.04831 | 430.33516 | 430.09815 | |
| 51 | 625.05329 | 493.45996 | 445.89083 | 434.03669 | 431.0818 | 430.34353 | 430.09817 | |
| 52 | 660.51532 | 506.48836 | 457.81085 | 445.78793 | 442.79073 | 442.04172 | 441.79279 | |
| 53 | 666.52046 | 510.35269 | 458.59347 | 445.97025 | 442.8355 | 442.05286 | 441.79275 | |
| 54 | 685.15589 | 514.91689 | 464.1731 | 451.93342 | 448.90319 | 448.14735 | 447.89615 | |
| 55 | 707.87242 | 524.0137 | 466.02845 | 452.36127 | 449.0077 | 448.17333 | 447.89603 | |
| 56 | 716.73804 | 549.41249 | 492.2403 | 477.89697 | 474.30157 | 473.40177 | 473.10272 | |
| 57 | 730.53327 | 553.60226 | 495.26575 | 481.18677 | 477.7088 | 476.84176 | 476.5536 | |
| 58 | 743.5592 | 553.92589 | 495.56558 | 481.29342 | 477.73798 | 476.84923 | 476.55386 | |
| 59 | 768.40717 | 565.23486 | 499.12703 | 483.46725 | 479.63811 | 478.68688 | 478.37074 | |
| 60 | 769.60511 | 598.98223 | 531.35264 | 514.78249 | 510.6733 | 509.64805 | 509.30731 | |
| 61 | 778.18878 | 602.43882 | 532.02603 | 514.94933 | 510.71529 | 509.65858 | 509.30738 | |
| 62 | 810.82699 | 611.25966 | 545.5528 | 529.00038 | 524.71365 | 523.63074 | 523.27084 | |
| 63 | 820.5268 | 618.26216 | 546.75359 | 529.11611 | 524.73115 | 523.63478 | 523.2704 | |
| 64 | 823.73845 | 620.83898 | 547.74544 | 530.24242 | 526.0068 | 524.96294 | 524.61602 | |
| 65 | 844.71579 | 624.7731 | 548.42793 | 530.40954 | 526.0577 | 524.97625 | 524.61683 | |
| 66 | 861.84708 | 643.88498 | 566.521 | 547.8349 | 543.21723 | 542.06601 | 541.6834 | |
| 67 | 889.66461 | 666.67903 | 588.31543 | 569.0164 | 564.15755 | 562.93735 | 562.53182 | |
| 68 | 893.22765 | 671.30329 | 589.08502 | 569.15471 | 564.19311 | 562.94657 | 562.53229 | |
| 69 | 928.18531 | 678.37209 | 591.26498 | 570.21875 | 565.11864 | 563.86323 | 563.446 | |
| 70 | 944.59498 | 678.99618 | 591.54639 | 570.39753 | 565.17898 | 563.87942 | 563.44751 | |
| 71 | 973.89271 | 713.39483 | 624.05601 | 602.11726 | 596.66044 | 595.29751 | 594.84454 | |
| 72 | 984.22121 | 717.63701 | 625.02465 | 602.33612 | 596.71307 | 595.31053 | 594.8444 | |
| 73 | 1003.2488 | 720.8535 | 626.80434 | 604.21475 | 598.66635 | 597.28553 | 596.82662 | |
| 74 | 1022.2239 | 731.30889 | 636.98697 | 613.83748 | 608.09692 | 606.66461 | 606.18859 | |
| 75 | 1064.656 | 736.66857 | 638.19862 | 614.12104 | 608.16644 | 606.6819 | 606.18852 | |
| 76 | 0 | 758.55118 | 657.7531 | 633.21434 | 627.14005 | 625.62484 | 625.12126 | |
| 77 | 0 | 771.88794 | 669.01417 | 643.93149 | 637.72608 | 636.17848 | 635.66414 | |
| 78 | 0 | 781.52029 | 670.9575 | 644.36849 | 637.83183 | 636.20469 | 635.66391 | |
| 79 | 0 | 798.15073 | 688.65839 | 661.7951 | 655.13688 | 653.47569 | 652.9236 | |
| 80 | 0 | 807.9403 | 690.82713 | 662.29146 | 655.25745 | 653.50559 | 652.92336 | |
| 81 | 0 | 808.09953 | 701.63921 | 671.14335 | 664.49921 | 662.84227 | 662.29159 | |
| 82 | 0 | 822.00504 | 709.53687 | 672.0112 | 664.71279 | 662.89544 | 662.29145 | |
| 83 | 0 | 825.94639 | 710.7345 | 681.35208 | 674.40052 | 672.66853 | 672.09291 | |
| 84 | 0 | 830.51931 | 745.92704 | 681.6494 | 674.47447 | 672.68698 | 672.09291 | |
| 85 | 0 | 870.27049 | 747.71715 | 714.52227 | 706.60012 | 704.61583 | 703.95636 | |
| 86 | 0 | 877.46675 | 750.70408 | 715.97203 | 708.13077 | 706.17528 | 705.52538 | |
| 87 | 0 | 879.76135 | 751.8401 | 719.18717 | 711.48508 | 709.57101 | 708.93487 | |
| 88 | 0 | 885.38271 | 773.52586 | 719.5213 | 711.57193 | 709.59292 | 708.9352 | |
| 89 | 0 | 908.64629 | 776.22913 | 740.72548 | 732.60741 | 730.58125 | 729.90786 | |
| 90 | 0 | 917.23648 | 780.75222 | 741.39963 | 732.7745 | 730.62292 | 729.90785 | |
| 91 | 0 | 926.87141 | 802.14545 | 745.35762 | 736.71585 | 734.57073 | 733.8578 | |
| 92 | 0 | 945.03442 | 807.04188 | 766.29269 | 757.43314 | 755.22567 | 754.49202 | |
| 93 | 0 | 953.65073 | 809.64926 | 772.10121 | 763.43179 | 761.1821 | 760.43442 | |
| 94 | 0 | 961.85944 | 811.57947 | 772.72585 | 763.52197 | 761.18964 | 760.4145 | |
| 95 | 0 | 965.50857 | 812.71356 | 773.18339 | 763.80395 | 761.51062 | 760.74844 | |
| 96 | 0 | 971.09648 | 836.09531 | 773.45271 | 763.83752 | 761.52822 | 760.76073 | |
| 97 | 0 | 1001.3103 | 852.72118 | 796.19243 | 786.4015 | 783.96579 | 783.15629 | |
| 98 | 0 | 1010.6476 | 854.80185 | 813.87912 | 804.25538 | 801.85387 | 801.05573 | |
| 99 | 0 | 1016.323 | 865.52242 | 814.41392 | 804.38918 | 801.88731 | 801.05582 | |
| 100 | 0 | 1029.3135 | 868.10342 | 824.25445 | 814.05261 | 811.51059 | 810.66576 | |
| 101 | 0 | 1039.4047 | 889.94976 | 824.92115 | 814.2204 | 811.55261 | 810.66598 | |
| 102 | 0 | 1067.3435 | 891.22619 | 845.97179 | 835.16217 | 832.47234 | 831.57838 | |
| 103 | 0 | 1075.2421 | 897.82084 | 846.26961 | 835.23671 | 832.49101 | 831.57848 | |
| 104 | 0 | 1076.6712 | 900.51668 | 854.38783 | 843.70641 | 841.04735 | 840.16362 | |
| 105 | 0 | 1086.4376 | 916.95161 | 855.04837 | 843.87072 | 841.08838 | 840.16368 | |
| 106 | 0 | 1097.5024 | 930.48898 | 871.46325 | 860.15673 | 857.33374 | 856.39553 | |
| 107 | 0 | 0 | 932.71696 | 882.46711 | 870.52451 | 867.54488 | 866.55461 | |
| 108 | 0 | 0 | 941.25934 | 882.90874 | 870.62781 | 867.57027 | 866.5541 | |
| 109 | 0 | 0 | 943.71288 | 893.07723 | 881.14388 | 878.10232 | 877.09146 | |
| 110 | 0 | 0 | 946.72952 | 893.41574 | 881.15763 | 878.11137 | 877.09895 | |
| 111 | 0 | 0 | 979.38826 | 894.57002 | 882.0115 | 878.96988 | 877.959 | |
| 112 | 0 | 0 | 982.05872 | 928.68062 | 916.11159 | 912.97702 | 911.93525 | |
| 113 | 0 | 0 | 996.34157 | 929.41533 | 916.30184 | 913.02504 | 911.936 | |
| 114 | 0 | 0 | 998.12696 | 941.67749 | 928.30946 | 924.98802 | 923.88415 | |
| 115 | 0 | 0 | 1006.5521 | 942.38134 | 928.50356 | 925.03767 | 923.88579 | |
| 116 | 0 | 0 | 1009.7396 | 952.84507 | 939.61655 | 936.31543 | 935.21831 | |
| 117 | 0 | 0 | 1012.7349 | 953.68224 | 939.82838 | 936.36843 | 935.21852 | |
| 118 | 0 | 0 | 1015.8933 | 955.99942 | 942.19645 | 938.7714 | 937.63309 | |
| 119 | 0 | 0 | 1045.3799 | 956.56931 | 942.31192 | 938.79834 | 937.63061 | |
| 120 | 0 | 0 | 1060.8607 | 984.79387 | 969.86934 | 966.15466 | 964.9201 | |
| 121 | 0 | 0 | 1066.7616 | 1001.9749 | 986.87039 | 983.06591 | 981.8015 | |
| 122 | 0 | 0 | 1067.7955 | 1005.6615 | 990.85781 | 987.20332 | 985.98876 | |
| 123 | 0 | 0 | 1074.1477 | 1006.1546 | 991.04118 | 987.25283 | 985.99378 | |
| 124 | 0 | 0 | 1078.9298 | 1010.8641 | 994.7053 | 990.69369 | 989.36044 | |
| 125 | 0 | 0 | 1079.7583 | 1013.4197 | 998.11513 | 994.3316 | 993.07415 | |
| 126 | 0 | 0 | 0 | 1014.2269 | 998.33225 | 994.38624 | 993.07479 | |
| 127 | 0 | 0 | 0 | 1049.5474 | 1033.2772 | 1029.2087 | 1027.8565 | |
| 128 | 0 | 0 | 0 | 1053.4326 | 1036.794 | 1032.6432 | 1031.2637 | |
| 129 | 0 | 0 | 0 | 1054.2913 | 1036.9941 | 1032.6924 | 1031.2627 | |
| 130 | 0 | 0 | 0 | 1059.073 | 1042.6888 | 1038.6055 | 1037.2484 | |
| 131 | 0 | 0 | 0 | 1059.4054 | 1042.7675 | 1038.6249 | 1037.2481 | |
| 132 | 0 | 0 | 0 | 1067.5981 | 1050.413 | 1046.1397 | 1044.7195 | |
| 133 | 0 | 0 | 0 | 1068.6298 | 1050.6307 | 1046.1914 | 1044.716 | |
| 134 | 0 | 0 | 0 | 1097.4033 | 1079.5061 | 1075.0575 | 1073.579 | |
| 135 | 0 | 0 | 0 | 0 | 1098.6675 | 1094.1013 | 1092.5837 | |
| 136 | 0 | 0 | 0 | 0 | 1098.9231 | 1094.1606 | 1092.5778 |