面积为1的正多边形, 将其放入三角形中, 三角形最小面积

王守恩

面积为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——{1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716}

问题1——来个反例。

问题2——提供资料。

问题3——通项公式。

谢谢各位！新年快乐！！！

王守恩

题目——面积为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——

1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716, 1.66455, 1.66275, 1.64163, 1.66180, 1.66064, 1.64453, 1.66001, 1.65921, 1.64652, 1.65876,
1.65819, 1.64794, 1.65787, 1.65745, 1.64899, 1.65721, 1.65689, 1.64979, 1.65670, 1.65645, 1.65041, 1.65630, 1.65611, 1.65090, 1.65599, 1.65583, 1.65130, 1.65573, 1.65560, 1.65162, 1.65552, 1.65542, 1.65189, 1.65535, 1.65526, 1.65212,

这个没找到通项公式——但若把n=3,6,9,...拉出来,

1.00000, 1.50000, 1.58626, 1.61603, 1.62973, 1.63716, 1.64163, 1.64453, 1.64652, 1.64794, 1.64899, 1.64979, 1.65041, 1.65090, 1.65130, 1.65162, 1.65189, 1.65212, 1.65231, 1.65247, 1.65262, 1.65274, 1.65284, 1.65294, 1.65302, 1.65309,
1.65316, 1.65322, 1.65327, 1.65331, 1.65336, 1.65340, 1.65343, 1.65346, 1.65349, 1.65352, 1.65355, 1.65357, 1.65359, 1.65361, 1.65363, 1.65364, 1.65366, 1.65367, 1.65369, 1.65370, 1.65371, 1.65372, 1.65373, 1.65374, 1.65375, 1.65376,

这串数在慢慢长大,  极限 = \(\frac{3\sqrt{3}}{\pi}\) = 1.6539866862653761485。

这串数可以有一个漂亮的公式—— \(\cot(\frac{\pi}{3n})\frac{\sqrt{3}}{n}\)

王守恩

题目——面积为1的正n(n=3,4,5,...)边形, 将其放入三角形中, 三角形最小面积是这样一串数。
1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716, 1.66455, 1.66275, 1.64163, 1.66180, 1.66064, 1.64453, 1.66001, 1.65921, 1.64652, 1.65876,
1.65819, 1.64794, 1.65787, 1.65745, 1.64899, 1.65721, 1.65689, 1.64979, 1.65670, 1.65645, 1.65041, 1.65630, 1.65611, 1.65090, 1.65599, 1.65583, 1.65130, 1.65573, 1.65560, 1.65162, 1.65552, 1.65542, 1.65189, 1.65535, 1.65526, 1.65212,
1.65520, 1.65513, 1.65231, 1.65508, 1.65502, 1.65247, 1.65498, 1.65492, 1.65262, 1.65488, 1.65484, 1.65274, 1.65481, 1.65476, 1.65284, 1.65474, 1.65470, 1.65294, 1.65468, 1.65464, 1.65302, 1.65462, 1.65459, 1.65309, 1.65457, 1.65455,
1.65316, 1.65453, 1.65451, 1.65322, 1.65449, 1.65447, 1.65327, 1.65446, 1.65444, 1.65331, 1.65443, 1.65441, 1.65336, 1.65440, 1.65439, 1.65340, 1.65438, 1.65436, 1.65343, 1.65435, 1.65434, 1.65346, 1.65433, 1.65432, 1.65349, 1.65431}

通项公式是这样——Table[Piecewise[{{3 Sqrt[3] Cot[Pi/n]/n, Mod[n, 3] == 0}, {8 Cos[(n + 6 - 4 Mod[n, 3]) Pi/(6 n)] (Cos[Pi/n] + Sin[(n + 6 - 4 Mod[n, 3]) Pi/(6 n)])/(n Sin[2 Pi/n]), Mod[n, 3] ≠ 0}}], {n, 3, 60}]

复杂的数据是这样:
   {1,
2,
4/Sqrt[5],
   3/2,
8/7 (Cos[\[Pi]/7] + Sin[(3 \[Pi])/14]),
1 + 1/Sqrt[2],
   Cot[\[Pi]/9]/Sqrt[3],
2/5 (2 + Sqrt[5]),
4/11 (Cos[\[Pi]/22] + 2 Cos[(5 \[Pi])/22]) Csc[(2 \[Pi])/11],
   1/4 (3 + 2 Sqrt[3]),
4/13 Csc[(2 \[Pi])/13] (2 Cos[(3 \[Pi])/26] + Sin[(3 \[Pi])/13]),
4/7 Cot[\[Pi]/7] (Cos[\[Pi]/14] + Sin[\[Pi]/7]),
   1/5 Sqrt[3] Cot[\[Pi]/15],
1/8 (Sqrt[2] + 4 Cos[\[Pi]/8]) Csc[\[Pi]/8],
4/17 (Cos[(3 \[Pi])/34] + 2 Cos[(7 \[Pi])/34]) Csc[(2 \[Pi])/17],
   Cot[\[Pi]/18]/(2 Sqrt[3]),
4/19 (2 Cos[(5 \[Pi])/38] + Cos[(9 \[Pi])/38]) Csc[(2 \[Pi])/19],
1/5 (3 + Sqrt[5] + Sqrt[5 + 2 Sqrt[5]]),
   1/7 Sqrt[3] Cot[\[Pi]/21],
2/11 (2 Cos[(3 \[Pi])/22] + Cos[(5 \[Pi])/22]) Csc[\[Pi]/11],
4/23 (Cos[(5 \[Pi])/46] + 2 Cos[(9 \[Pi])/46]) Csc[(2 \[Pi])/23],
   1/8 Sqrt[3] Cot[\[Pi]/24],
4/25 (2 Cos[(7 \[Pi])/50] + Cos[(11 \[Pi])/50]) Csc[(2 \[Pi])/25],
2/13 (Cos[(3 \[Pi])/26] + 2 Cos[(5 \[Pi])/26]) Csc[\[Pi]/13],
   Cot[\[Pi]/27]/(3 Sqrt[3]),
1/7 (2 Cos[\[Pi]/7] + Cos[(3 \[Pi])/14]) Csc[\[Pi]/14],
4/29 (Cos[(7 \[Pi])/58] + 2 Cos[(11 \[Pi])/58]) Csc[(2 \[Pi])/29],
   1/10 Sqrt[3] Cot[\[Pi]/30],
4/31 (2 Cos[(9 \[Pi])/62] + Cos[(13 \[Pi])/62]) Csc[(2 \[Pi])/31],
1/8 (Cos[\[Pi]/8] + 2 Cos[(3 \[Pi])/16]) Csc[\[Pi]/16],
   1/11 Sqrt[3] Cot[\[Pi]/33],
2/17 (2 Cos[(5 \[Pi])/34] + Cos[(7 \[Pi])/34]) Csc[\[Pi]/17],
4/35 (Cos[(9 \[Pi])/70] + 2 Cos[(13 \[Pi])/70]) Csc[(2 \[Pi])/35],
   Cot[\[Pi]/36]/(4 Sqrt[3]),
4/37 (2 Cos[(11 \[Pi])/74] + Cos[(15 \[Pi])/74]) Csc[(2 \[Pi])/37],
2/19 (Cos[(5 \[Pi])/38] + 2 Cos[(7 \[Pi])/38]) Csc[\[Pi]/19],
   1/13 Sqrt[3] Cot[\[Pi]/39],
1/40 (1 + Sqrt[5] + 8 Cos[(3 \[Pi])/20]) Csc[\[Pi]/20],
4/41 (Cos[(11 \[Pi])/82] + 2 Cos[(15 \[Pi])/82]) Csc[(2 \[Pi])/41],
   1/14 Sqrt[3] Cot[\[Pi]/42],
4/43 (2 Cos[(13 \[Pi])/86] + Cos[(17 \[Pi])/86]) Csc[(2 \[Pi])/43],
1/11 (Cos[(3 \[Pi])/22] + 2 Cos[(2 \[Pi])/11]) Csc[\[Pi]/22],
   Cot[\[Pi]/45]/(5 Sqrt[3]),
2/23 (2 Cos[(7 \[Pi])/46] + Cos[(9 \[Pi])/46]) Csc[\[Pi]/23],
4/47 (Cos[(13 \[Pi])/94] + 2 Cos[(17 \[Pi])/94]) Csc[(2 \[Pi])/47],
   1/16 Sqrt[3] Cot[\[Pi]/48],
4/49 (2 Cos[(15 \[Pi])/98] + Cos[(19 \[Pi])/98]) Csc[(2 \[Pi])/49],
2/25 (Cos[(7 \[Pi])/50] + 2 Cos[(9 \[Pi])/50]) Csc[\[Pi]/25],
   1/17 Sqrt[3] Cot[\[Pi]/51],
1/13 (2 Cos[(2 \[Pi])/13] + Cos[(5 \[Pi])/26]) Csc[\[Pi]/26],
4/53 (Cos[(15 \[Pi])/106] + 2 Cos[(19 \[Pi])/106]) Csc[(2 \[Pi])/53],
   Cot[\[Pi]/54]/(6 Sqrt[3]),
4/55 (2 Cos[(17 \[Pi])/110] + Cos[(21 \[Pi])/110]) Csc[(2 \[Pi])/55],
1/14 (Cos[\[Pi]/7] + 2 Cos[(5 \[Pi])/28]) Csc[\[Pi]/28],
   1/19 Sqrt[3] Cot[\[Pi]/57],
2/29 (2 Cos[(9 \[Pi])/58] + Cos[(11 \[Pi])/58]) Csc[\[Pi]/29],
4/59 (Cos[(17 \[Pi])/118] + 2 Cos[(21 \[Pi])/118]) Csc[(2 \[Pi])/59],
   1/20 Sqrt[3] Cot[\[Pi]/60]}

问题1——来个反例。

问题2——提供资料。

问题3——通项公式。

谢谢各位！新年快乐！！！

王守恩

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。