求最小面积

王守恩 · 发表于 2026-2-23 09:40:10

同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正六边形形中, 正六边形最小面积——

{0.866025, 1.61603, 2.47926, 2.59808, 4.57459, 5.81284, 7.18009, 8.97004, 10.8861, 12.0622, 15.0968, 17.3923, 19.8153, 22.6567, 25.6251, 27.8544, 31.9426, 35.2919, 38.7686,
42.6625, 46.6836, 49.9658, 55.1071, 59.5097, 64.0396, 68.9862, 74.0602, 78.3953, 84.5897, 90.0454, 95.6284, 101.628, 107.755, 113.143, 120.39, 126.899, 133.535, 140.587, 147.767,
154.208, 162.509, 170.07, 177.759, 185.865, 194.097, 201.591, 210.945, 219.559, 228.301, 237.46, 246.745, 255.292, 265.699, 275.366, 285.161, 295.372, 305.711, 315.311}
R[n_] := Csc[Pi/n]/2; p = {Pi/6, Pi/2, 5 Pi/6}; f[\[Alpha]_?NumericQ, n_] := Module[{w, u, v}, w = Table[\[Alpha] + 2 Pi k/n, {k, 0, n - 1}];
u = Mod[w, Pi]; v = Min[Table[Min[Table[With[{d = Abs[u[[i]] - p[[j]]]}, Min[d, Pi - d]], {j, 3}]], {i, n}]]; Return[v]];
Flatten@Table[{2 Sqrt[3] (R[n]*Cos[NMaximize[{f[\[Alpha], n], 0 <= \[Alpha] < Pi/3}, \[Alpha], Method -> "DifferentialEvolution"][[1]]])^2}, {n, 3, 60}]

王守恩 · 发表于 2026-2-23 09:47:44

同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正七边形形中, 正七边形最小面积——

{1.09871, 1.66438, 2.41969, 3.35220, 3.63391, 5.73660, 7.18651, 8.80769, 10.6000, 12.5632, 14.6974, 16.1773, 18.6532, 21.2999, 24.1175, 27.1059, 30.2651, 33.5952, 37.0960,
40.7676, 44.6100, 48.6232, 52.8072, 57.1619, 61.6875, 66.3838, 71.2510, 76.2889, 81.4976, 86.8770, 92.4273, 98.1483, 104.040, 110.103, 116.336, 122.740, 129.315, 136.061,
142.977, 150.065, 157.323, 164.752, 172.351, 180.122, 188.063, 196.175, 204.457, 212.911, 221.535, 230.330, 239.296, 248.433, 257.740, 267.218, 276.867, 286.687, 296.677, 306.839}

Table[N[(7 Cos[Which[n < 7, Pi/(7 n), n == 7, Pi/n, n < 14, Pi/(7 n), True, Pi/n]]^2)/( 4 Cot[Pi/7] Sin[Pi/n]^2), 6], {n, 3, 60}]

王守恩 · 发表于 2026-2-23 09:52:01

同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正十七边形形中, 正十七边形最小面积——

{1.05527, 1.58554, 2.29637, 3.17484, 4.21720, 5.42204, 6.78871, 8.31688, 10.0064, 11.8571, 13.8689, 16.0419, 18.3759, 20.8710, 22.7355, 26.3443, 29.3225, 32.4617, 35.7619,
39.2232, 42.8454, 46.6286, 50.5728, 54.6781, 58.9443, 63.3715, 67.9597, 72.7090, 77.6192, 82.6904, 87.9226, 92.5241, 98.8700, 104.585, 110.461, 116.499, 122.697, 129.056,
135.576, 142.257, 149.099, 156.102, 163.267, 170.592, 178.078, 185.725, 193.533, 201.502, 208.840, 217.923, 226.375, 234.988, 243.762, 252.697, 261.793, 271.050, 280.468, 290.047}

Table[17 Cos[If[GCD[17, n] == 17, Pi/n, Pi/(17 n)]]^2/(4 Cot[Pi/17] Sin[Pi/n]^2), {n, 3, 60}]

王守恩 · 发表于 2026-2-24 09:16:08

拓展——面积为1的正三边形, 将其放入正n(n=3,4,5,6,7,...)边形, 正n边形最小面积S(n)。

S(3)=1.000。S(4)=2.155。S(5)=2.437。S(6)=2.000。S(7)=2.366。S(8)=2.408。

——可有通项公式？——谢谢各位！新年快乐！！！！！！

王守恩 · 发表于 2026-2-25 04:52:21

先整理整理——轻装上阵。

公式——边长为1的正A边形, 将其放入正W边形中,

正W边形最小面积S(W)——精确值。

\(S(W)=\frac{A*\cos^2(if(GCD(A,W)=A,\pi/W,\pi/(A*W)))}{4\cot(\pi/A)\sin^2(\pi/W)}\) ≥ 精确值。

其中W=A*n  (n=1,2,3,...)  时, \(S(W)=\frac{A*\cos^2(\pi/W)}{4\cot(\pi/A)\sin^2(\pi/W)}=\frac{A*\cot^2(\pi/W)}{4\cot(\pi/A)}\) = 精确值。

譬如——边长为1的正n(n=3,4,5,...)边形, 将其放入三角形中,
三角形最小面积——0.433013, 2.00000, 3.07768, 3.89711, 6.33114, 8.24264, 9.80596, 13.0373, 15.7567, 18.0933, 22.1151, 25.6343, 28.7523, 33.5635, 37.8786, 41.7815, 47.3817, 52.4910, 57.1804, 63.5696, 69.4717, 74.9486, 82.1269, 88.8212,
用公式得到的面积—0.433013, 2.42404, 3.59744, 3.89711, 6.74713, 8.71926, 9.80596, 13.4551, 16.2183, 18.0933, 22.5350, 26.0884, 28.7523, 33.9851, 38.3284, 41.7815, 47.8047, 52.9378, 57.1804, 63.9937, 69.9165, 74.9486, 82.5519, 89.2645,
                        {0.433013, 2.42404, 3.59744, 3.89711, 6.74713, 8.71926, 9.80596, 13.4551, 16.2183, 18.0933, 22.5350, 26.0884, 28.7523, 33.9851, 38.3284, 41.7815, 47.8047, 52.9378, 57.1804, 63.9937, 69.9165, 74.9486, 82.5519, 89.2645,

譬如——边长为1的正n(n=3,4,5,...)边形, 将其放入正方形中,
正方形最小面积——0.933013, 1.00000, 2.55397, 3.73205, 4.98562, 5.82843, 8.22788, 10.2159, 12.2807, 13.9282, 17.1441, 19.9425, 22.8181, 25.2741, 29.3027, 32.9115, 36.5978, 39.8635, 44.7035, 49.1229, 53.6197, 57.6955, 63.3466, 68.5765,
用公式得到的面积—1.244020, 1.00000, 2.82360, 3.93185, 5.24535, 5.82843, 8.48370, 10.4077, 12.5346, 13.9282, 17.3969, 20.1322, 23.0702, 25.2741, 29.5543, 33.1003, 36.8491, 39.8635, 44.9545, 49.3113, 53.8706, 57.6955, 63.5973, 68.7646,
                        {1.244020, 1.00000, 2.82360, 3.73205, 5.24535, 5.82843, 8.48370, 10.2159, 12.5346, 13.9282, 17.3969, 19.9425, 23.0702, 25.2741, 29.5543, 32.9115, 36.8491, 39.8635, 44.9545, 49.1229, 53.8706, 57.6955, 63.5973, 68.5765,

譬如——边长为1的正n(n=3,4,5,...)边形, 将其放入正五边形中,
正五边形最小面积—1.05047, 1.51007, 1.72048, 3.36936, 4.70078, 6.11731, 7.50981, 8.60239, 11.1712, 13.4755, 15.7387, 18.0917, 20.1012, 23.5935, 26.7802, 30.0007, 33.2554, 37.1113, 40.6168, 44.7232, 48.8640, 53.0391, 56.9066, 62.2601,
用公式得到的面积—1.15856, 1.77191, 1.72048, 3.59302, 4.78543, 6.16325, 7.72590, 8.60239, 11.4046, 13.5203, 15.8203, 18.3043, 20.1012, 23.8248, 26.8612, 30.0816, 33.4861, 36.2031, 40.8473, 44.8040, 48.9447, 53.2694, 56.9066, 62.4711,
                        {1.15856, 1.77191, 1.72048, 3.59302, 4.78543, 6.16325, 7.72590, 8.60239, 11.4046, 13.5203, 15.8203, 18.3043, 20.1012, 23.8248, 26.8612, 30.0816, 33.4861, 36.2031, 40.8473, 44.8040, 48.9447, 53.2694, 56.9066, 62.4711,

譬如——边长为1的正n(n=3,4,5,...)边形, 将其放入正六边形中,
正六边形最小面积—0.866025, 1.61603, 2.47926, 2.59808, 4.57459, 5.81284, 7.18009, 8.97004, 10.8861, 12.0622, 15.0968, 17.3923, 19.8153, 22.6567, 25.6251, 27.8544, 31.9426, 35.2919, 38.7686, 42.6625, 46.6836, 49.9658, 55.1071, 59.5097,
用公式得到的面积—1.119880, 1.70254, 2.47926, 2.59808, 4.57459, 5.88830, 7.37830, 9.04429, 10.8861, 12.0622, 15.0968, 17.4655, 20.0098, 22.7297, 25.6251, 27.8544, 31.9426, 35.3645, 38.9620, 42.7351, 46.6836, 49.9658, 55.1071, 59.5821,
                        {0.866025, 1.61603, 2.47926, 2.59808, 4.57459, 5.81284, 7.18009, 8.97004, 10.8861, 12.0622, 15.0968, 17.3923, 19.8153, 22.6567, 25.6251, 27.8544, 31.9426, 35.2919, 38.7686, 42.6625, 46.6836, 49.9658, 55.1071, 59.5097,

譬如——边长为1的正n(n=3,4,5,...)边形, 将其放入正八边形中,
正八边形最小面积—1.04284, 1.41421, 2.34889, 3.25726, 4.17841, 4.82843, 6.86159, 8.62209, 10.3927, 12.1562, 14.4212, 16.6780, 18.9548, 20.9378, 24.3144, 27.4212, 30.5354, 33.6440, 37.2624, 40.8519, 44.4589, 47.7965, 52.5205, 56.9672,
用公式得到的面积—1.08575, 1.64094, 2.38306, 3.29953, 4.38672, 4.82843, 7.06844, 8.66203, 10.4238, 12.3537, 14.4516, 16.7175, 19.1513, 20.9378, 24.5228, 27.4604, 30.5659, 33.8393, 37.2807, 40.8899, 44.6669, 47.7965, 52.7248, 57.0055,
                        {1.08575, 1.41421, 2.38306, 3.25725, 4.38672, 4.82843, 7.06844, 8.62200, 10.4238, 12.1562, 14.4516, 16.6780, 19.1513, 20.9378, 24.5228, 27.4212, 30.5659, 33.6440, 37.2807, 40.8508, 44.6669, 47.7965, 52.7248, 56.9665,

下面的公式——\(S(W)=\frac{A*\cos^2(\pi/LCM(A,W))}{4\cot(\pi/A)\sin^2(\pi/W)}\) ≥ 精确值。——精确值多一些。

王守恩 · 发表于 2026-2-25 11:33:51

下面的公式——\(S(W)=\frac{A*\cos^2(\pi/LCM(A,W))}{4\cot(\pi/A)\sin^2(\pi/W)}\) ≥ 精确值。——精确值多一些。

{0.43301, 2.42404, 3.59744, 3.89711, 6.74713, 8.71926, 9.80596, 13.4551, 16.2183, 18.0933, 22.5350, 26.0884, 28.7523, 33.9851, 38.3284, 41.7815, 47.8047, 52.9378, 57.1804, 63.9937, 69.9165, 74.9486, 82.5519, 89.2645, 95.0862, 103.479},
{1.24402, 1.00000, 2.82360, 3.73205, 5.24535, 5.82843, 8.48370, 10.2159, 12.5346, 13.9282, 17.3969, 19.9425, 23.0702, 25.2741, 29.5543, 32.9115, 36.8491, 39.8635, 44.9545, 49.1229, 53.8706, 57.6955, 63.5973, 68.5765, 74.1346, 78.7700},
{1.15856, 1.77191, 1.72048, 3.59302, 4.78543, 6.16325, 7.72590, 8.60239, 11.4046, 13.5203, 15.8203, 18.3043, 20.1012, 23.8248, 26.8612, 30.0816, 33.4861, 36.2031, 40.8473, 44.8040, 48.9447, 53.2694, 56.9066, 62.4711, 67.3480, 72.4089},
{0.86602, 1.61603, 2.47926, 2.59808, 4.57459, 5.81284, 7.18009, 8.97004, 10.8861, 12.0622, 15.0968, 17.3923, 19.8153, 22.6567, 25.6251, 27.8544, 31.9426, 35.2919, 38.7686, 42.6625, 46.6836, 49.9658, 55.1071, 59.5097, 64.0396, 68.9862},
{1.09871, 1.66438, 2.41969, 3.35220, 3.63391, 5.73660, 7.18651, 8.80769, 10.6000, 12.5632, 14.6974, 16.1773, 19.4785, 22.1253, 24.9429, 27.9313, 31.0905, 34.4206, 37.0960, 41.5930, 45.4354, 49.4486, 53.6326, 57.9874, 62.5130, 66.3838},
{1.08575, 1.41421, 2.38306, 3.25725, 4.38672, 4.82843, 7.06844, 8.62200, 10.4238, 12.1562, 14.4516, 16.6780, 19.1513, 20.9378, 24.5228, 27.4212, 30.5659, 33.6440, 37.2807, 40.8508, 44.6669, 47.7965, 52.7248, 56.9665, 61.4541, 65.8759},
{0.96418, 1.62542, 2.35881, 3.17696, 4.33932, 5.58138, 6.18182, 8.56553, 10.3071, 12.1323, 14.2887, 16.5286, 18.8526, 21.5065, 24.2445, 26.3397, 30.2183, 33.4542, 36.7747, 40.4239, 44.1577, 47.9762, 52.1232, 56.3549, 59.9438, 65.3161},
{1.07123, 1.58484, 2.12663, 3.21370, 4.30620, 5.51258, 6.93559, 7.69421, 10.2256, 12.0930, 14.1749, 16.3719, 18.5860, 21.3096, 24.0500, 26.9058, 29.9755, 32.3811, 36.5595, 40.0739, 43.8020, 47.6456, 51.5072, 55.8758, 60.2623, 64.7645},
{1.06690, 1.60673, 2.32955, 3.22258, 4.28211, 5.50675, 6.89584, 8.44907, 9.36564, 12.0473, 14.0921, 16.3007, 18.6729, 21.2089, 23.9085, 26.7718, 29.7988, 32.9894, 36.3437, 39.0608, 43.5431, 47.3883, 51.3971, 55.5696, 59.9057, 64.4054},
{1.00000, 1.50000, 2.32031, 3.00000, 4.26402, 5.39550, 6.81960, 8.39494, 10.1217, 11.1962, 14.0299, 16.2115, 18.5449, 21.0301, 23.8023, 26.4558, 29.6661, 32.7580, 36.1366, 39.6668, 43.3487, 46.3784, 51.1674, 55.3043, 59.5929, 64.0333},
{1.06116, 1.59627, 2.31318, 3.19902, 4.25008, 5.46494, 6.84297, 8.38384, 10.0874, 11.9534, 13.1858, 16.1730, 18.5264, 21.0422, 23.7204, 26.5609, 29.5638, 32.7290, 36.0566, 39.5465, 43.1988, 47.0134, 50.9904, 54.3334, 59.4313, 63.8952},
{1.05919, 1.57768, 2.30757, 3.17756, 4.03334, 5.43775, 6.82485, 8.34885, 10.0603, 11.9088, 13.9442, 15.3345, 18.4761, 20.9726, 23.6558, 26.4762, 29.4832, 32.6274, 35.7615, 39.4262, 43.0808, 46.8726, 50.8509, 54.9666, 59.2686, 62.9256},
{1.01684, 1.58981, 2.20738, 3.15351, 4.23029, 5.43912, 6.78085, 8.25600, 10.0386, 11.8665, 13.9140, 16.0941, 17.6424, 20.9392, 23.6041, 26.4020, 29.4186, 32.4825, 35.8507, 39.3519, 42.9861, 46.7534, 50.6537, 54.8579, 59.1098, 63.5801},
{1.05633, 1.53073, 2.29940, 3.16898, 4.22312, 5.22625, 6.79848, 8.31931, 10.0210, 11.8268, 13.8893, 16.0560, 18.4030, 20.1094, 23.5620, 26.3739, 29.3660, 32.4629, 35.8150, 39.2720, 42.9090, 46.5013, 50.6480, 54.7500, 59.0320, 63.4190},
{1.05527, 1.58554, 2.29637, 3.17484, 4.21720, 5.42204, 6.78871, 8.31688, 10.0064, 11.8571, 13.8689, 16.0419, 18.3759, 20.8710, 22.7355, 26.3443, 29.3225, 32.4617, 35.7619, 39.2232, 42.8454, 46.6286, 50.5728, 54.6781, 58.9443, 63.3715},
{1.02606, 1.57489, 2.29385, 3.07818, 4.21225, 5.40785, 6.57856, 8.29922, 9.99420, 11.7551, 13.8519, 16.0147, 18.3334, 20.8379, 23.4981, 25.5208, 29.2862, 32.4141, 35.6979, 39.1671, 42.7922, 46.4846, 50.5100, 54.6028, 58.6745, 63.2854},
{1.05364, 1.58256, 2.29172, 3.16813, 4.20808, 5.41014, 6.77366, 8.29831, 9.98392, 11.8304, 13.8376, 16.0056, 18.3342, 20.8236, 23.4736, 26.2843, 28.4652, 32.3876, 35.6802, 39.1335, 42.7473, 46.5219, 50.4570, 54.5528, 58.8092, 63.2263},
{1.05300, 1.54508, 2.23607, 3.15901, 4.20453, 5.37429, 6.76779, 8.09017, 9.97516, 11.7896, 13.8254, 15.9853, 18.2698, 20.7750, 23.4527, 26.2549, 29.2295, 31.5688, 35.6483, 39.0925, 42.7091, 46.4504, 50.3641, 54.4980, 58.7566, 63.1398},
{1.03164, 1.58041, 2.28834, 3.14756, 4.11002, 5.40151, 6.74781, 8.28485, 9.96764, 11.7963, 13.8149, 15.8918, 18.2894, 20.7892, 23.4348, 26.2263, 29.2071, 32.3339, 34.8315, 39.0685, 42.6763, 46.4301, 50.3731, 54.4620, 58.6969, 63.0345},
{1.05199, 1.57351, 2.28699, 3.15596, 4.19883, 5.39291, 6.75839, 8.27441, 9.76103, 11.7982, 13.8058, 15.9637, 18.2919, 20.7705, 23.4192, 26.2184, 29.1877, 32.3074, 35.5973, 38.2533, 42.6479, 46.4087, 50.3396, 54.4208, 58.6722, 63.0740},
{1.05158, 1.57880, 2.28581, 3.15964, 4.19653, 5.39506, 6.75459, 8.27479, 9.95547, 11.7965, 13.7979, 15.9595, 18.2814, 20.7634, 23.4057, 26.2082, 29.1708, 32.2937, 35.5767, 39.0198, 41.8344, 46.3867, 50.3103, 54.3941, 58.6381, 63.0423},
{1.03528, 1.55291, 2.28478, 3.10583, 4.19451, 5.30198, 6.73984, 8.26643, 9.95050, 11.5911, 13.7909, 15.9473, 18.2610, 20.6656, 23.3939, 26.1465, 29.1560, 32.2565, 35.5475, 38.9959, 42.6015, 45.5745, 50.2847, 54.3623, 58.5972, 62.9895},
{1.05090, 1.57756, 2.24942, 3.15685, 4.19274, 5.39011, 6.74833, 8.23577, 9.94612, 11.7854, 13.7848, 15.9444, 18.2333, 20.7437, 23.3834, 26.1832, 29.1430, 32.2323, 35.5427, 38.9825, 42.5824, 46.3423, 49.4738, 54.3420, 58.5819, 62.9818},
{1.05062, 1.57273, 2.28307, 3.15185, 4.19116, 5.38437, 6.74573, 8.26023, 9.94224, 11.7772, 13.5804, 15.9345, 18.2568, 20.7319, 23.3742, 26.1693, 29.1314, 32.2465, 35.5285, 38.9635, 42.5654, 46.3203, 50.2421, 53.5323, 58.5586, 62.9532},
{1.03777, 1.57659, 2.28236, 3.14518, 4.18976, 5.38623, 6.65364, 8.26100, 9.93879, 11.7678, 13.7746, 15.9325, 18.2416, 20.7282, 23.3659, 26.0762, 29.1212, 32.2386, 35.5073, 38.9532, 42.5504, 46.2987, 50.2243, 54.3011, 57.7499, 62.9344},
{1.05014, 1.55765, 2.28172, 3.15043, 4.13706, 5.36872, 6.74135, 8.25533, 9.93570, 11.7576, 13.7703, 15.7289, 18.2447, 20.7064, 23.3586, 26.1523, 29.1120, 32.2132, 35.4561, 38.9379, 42.5369, 46.2776, 50.2084, 54.2809, 58.5192, 62.1267}}

公式——Table[N[A*Cos[Pi/LCM[A, w]]^2/(4 Cot[Pi/A] Sin[Pi/w]^2), 6], {A, 3, 28}, {w, 3, 28}]

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穷举——Table[Module[{R, v, m, a, \[CurlyPhi], w, s}, R = Csc[Pi/n]/2; v = Csc[Pi/A]/2*Table[{Sin[2 Pi k/A], Cos[2 Pi k/A]}, {k, 0, A - 1}];
m = Table[d = v[[Mod[i, A] + 1]] - v[[i]]; {-d[[2]], d[[1]]}, {i, A}];
Min[Table[w = Flatten@Table[(R*Cos[\[CurlyPhi] + 2 Pi k/n] - a*v[[i, 1]]) m[[i, 1]] + (R Sin[\[CurlyPhi] + 2 Pi k/n] - a*v[[i, 2]]) m[[i, 2]] <= 0, {k, 0, n - 1}, {i, A}];
w = Join[w, {a > 0}]; s = FindMinimum[{a, w}, {{a, R}, {\[CurlyPhi], \[CurlyPhi]0}}];
(A/4) Cot[Pi/A]*(a /. s[[2]])^2, {\[CurlyPhi]0, {0, Pi/6, Pi/10}}]]], {A, 3, 28}, {n, 3, 28}]

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公式,穷举——答案都是一样的。只是让穷举=公式,有几处要微调。——怎么多数据摆在一起, 为的是规律好找一些。

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[讨论] 求最小面积