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数学研发论坛»论坛 › 【数学研究】 › 难题征解 › 果树问题讨论:这两个问题等价么？
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[求助] 果树问题讨论:这两个问题等价么？

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数学星空
发表于 2020-1-7 18:48:36 | 显示全部楼层
17棵15行结果

初始数据：
[[A, 1, 0, 0], [B, 0, 1, 0], [C, 1, 0, 1], [D, 1, 1, 0], [E, 1, 1, 1], [F, 0, 1, 1], [G, 1, -1, 0], [H, 0, 0, 1], [I, 1/2, 1/2, 1], [J, 1/2, 3/2, 1], [K, 0, -1, 1], [L, 1, -1, 1], [M, 3/2, 1/2, 1], [N, -1, 0, 1], [O, -1, 1, 1], [P, 1/2, -1, 1], [Q, -1, 1/2, 1]]
[[A, B, D, G], [C, F, G, I], [D, E, H, I], [B, F, H, K], [B, C, E, L], [E, G, J, M], [C, D, K, M], [A, C, H, N], [D, F, J, N], [A, E, F, O], [G, H, L, O], [B, I, J, P], [A, K, L, P], [A, I, M, Q], [B, N, O, Q]]

变换基点
[1/2, 3/2, 1, J]-->[-1/4, sqrt(3)/6, 1],
[3/2, 1/2, 1, M]--> [-1/2, 0, 1],
[0, 1, 0, B]-->[1/2, 0, 1],
[1, 0, 0, A]-->[1/4, sqrt(3)/6, 1]


变换矩阵
[[-1/4, -1/4, 1/4], [-1/6*sqrt(3), 0, 1/4*sqrt(3)], [-1, -1/2, 9/4]]

变换后坐标
[1/4, sqrt(3)/6, A], [1/2, 0, B], [0, sqrt(3)/15, C], [1/3, sqrt(3)/9, D], [-1/3, sqrt(3)/9, E], [0, sqrt(3)/7, F], [0, sqrt(3)/3, G], [1/9, sqrt(3)/9, H], [0, sqrt(3)/9, I], [-1/4, sqrt(3)/6, J], [2/11, sqrt(3)/11, K], [1/7, sqrt(3)/21, L], [-1/2, 0, M], [2/13, (5*sqrt(3))/39, N], [1/11, (5*sqrt(3))/33, O], [1/6, (2*sqrt(3))/27, P], [1/8, (5*sqrt(3))/36, Q]

画图得到

1715-1-2.gif

初始数据：

[[A, 1, 0, 0], [B, 0, 1, 0], [C, 0, 0, 1], [D, 0, 1, 1], [E, 1, -1, 0], [F, 1, 1, 1], [G, 1, 1, 0], [H, 0, -1, 1], [I, 1, 0, 1], [J, 1, -1, 1], [K, -2, 0, 1], [L, -2, -1, 1], [M, -2, 1, 1], [N, -1/2, 1/2, 1], [O, -1/2, -1/2, 1], [P, 1/4, 3/4, 1], [Q, 1/4, -3/4, 1]]
[[A, B, E, G], [B, C, D, H], [B, F, I, J], [A, C, I, K], [A, H, J, L], [A, D, F, M], [B, K, L, M], [C, E, J, N], [D, G, L, N], [C, F, G, O], [E, H, M, O], [D, E, I, P], [F, K, N, P], [G, H, I, Q], [J, K, O, Q]]

变换基点
[1, 0, 0, A]-->[-1/4, sqrt(3)/6, 1],
[1, 1, 0, G]--> [-1/2, 0, 1],
[1, -1, 1, J]-->[1/2, 0, 1],
[1, 0, 1, I]-->[1/4, sqrt(3)/6, 1]

变换矩阵
[[-1/4, 0, 0], [1/6*sqrt(3), -1/6*sqrt(3), -1/3*sqrt(3)], [1, -1/2, -2]]

变换后坐标
[-1/4, sqrt(3)/6, A], [0, sqrt(3)/3, B], [0, sqrt(3)/6, C], [0, sqrt(3)/5, D], [-1/6, (2*sqrt(3))/9, E], [1/6, (2*sqrt(3))/9, F], [-1/2, 0, G], [0, sqrt(3)/9, H], [1/4, sqrt(3)/6, I], [1/2, 0, J], [-1/8, sqrt(3)/6, K], [-1/7, sqrt(3)/7, L], [-1/9, (5*sqrt(3))/27, M], [-1/22, (2*sqrt(3))/11, N], [-1/18, (4*sqrt(3))/27, O], [1/34, (10*sqrt(3))/51, P], [1/22, (4*sqrt(3))/33, Q]

画图得到


1715-2.gif


初始数据：
[[A, 1, 0, 0], [B, 0, 0, 1], [C, 0, 1, 0], [D, 1, -2, 0], [E, 0, 1, 1], [F, 1, 2, 0], [G, 1, 0, 1], [H, 1/2, -1, 1], [I, 0, -1, 1], [J, 1/2, 1, 1], [K, 1/4, 0, 1], [L, 1, -1, 1], [M, 1, 1, 1], [N, 1/4, 1/2, 1], [O, 1/4, -1/2, 1], [P, -1/2, 3, 1], [Q, -1/2, -3, 1]]
[[A, C, D, F], [B, C, E, I], [A, B, G, K], [A, H, I, L], [A, E, J, M], [C, G, L, M], [B, F, J, N], [D, E, L, N], [B, D, H, O], [F, I, M, O], [C, K, N, O], [D, G, J, P], [E, H, K, P], [F, G, H, Q], [I, J, K, Q]]

变换基点
[1, -2, 0, D]-->[-1/4, sqrt(3)/6, 1],
[-1/2, -3, 1, Q]-->[-1/2, 0, 1],
[-1/2, 3, 1, P]-->[1/2, 0, 1],
[0, -1, 1, I]-->[1/4, sqrt(3)/6, 1]

变换矩阵
[[1/2, 0, -1/4], [-1/3*sqrt(3), 0, -1/6*sqrt(3)], [-8/3, -1/3, -4/3]]

变换后坐标
[-3/16, sqrt(3)/8, A], [3/16, sqrt(3)/8, B], [0, 0, C], [-1/4, sqrt(3)/6, D], [3/20, sqrt(3)/10, E], [-3/20, sqrt(3)/10, F], [-1/16, sqrt(3)/8, G], [0, sqrt(3)/7, H], [1/4, sqrt(3)/6, I], [0, sqrt(3)/9, J], [1/16, sqrt(3)/8, K], [-3/44, (3*sqrt(3))/22, L], [-3/52, (3*sqrt(3))/26, M], [3/52, (3*sqrt(3))/26, N], [3/44, (3*sqrt(3))/22, O], [1/2, 0, P], [-1/2, 0, Q]

画图得到


1715-3.gif

初始数据
[[A, 0, 1, 0], [B, 0, 1, 1], [C, 1, 0, 1], [D, 0, 0, 1], [E, 1, -1, 0], [F, 1, 0, 0], [G, 1, 2, 1], [H, 1, 1, 0], [I, -1, 1, 1], [J, 0, 2, 1], [K, -1, 0, 1], [L, 1, 1, 1], [M, -1, -2, 1], [N, 1/3, 2/3, 1], [O, 3/2, 1/2, 1], [P, -2, 2, 1], [Q, -3, -1, 1]]
[[A, E, F, H], [A, B, D, J], [C, D, F, K], [B, G, H, K], [A, C, G, L], [B, F, I, L], [A, I, K, M], [B, C, E, N], [D, G, M, N], [E, J, L, O], [C, H, M, O], [D, E, I, P], [F, G, J, P], [H, I, J, Q], [K, L, N, Q]]

变换基点
[1, 0, 0, F]-->[-1/4, sqrt(3)/6, 1],
[1, 2, 1, G]-->[-1/2, 0, 1],
[3/2, 1/2, 1, O]-->[1/2, 0, 1],
[1, 1, 0, H]-->[1/4, sqrt(3)/6, 1]

变换矩阵
[[3/16, -7/16, 5/16], [-1/8*sqrt(3), -1/24*sqrt(3), 5/24*sqrt(3)], [-3/4, -1/4, 2]]

变换后坐标
[7/4, sqrt(3)/6, A], [-1/14, (2*sqrt(3))/21, B], [2/5, sqrt(3)/15, C], [5/32, (5*sqrt(3))/48, D], [-5/4, sqrt(3)/6, E], [-1/4, sqrt(3)/6, F], [-1/2, 0, G], [1/4, sqrt(3)/6, H], [-1/8, (7*sqrt(3))/60, I], [-3/8, sqrt(3)/12, J], [1/22, (4*sqrt(3))/33, K], [1/16, sqrt(3)/24, L], [4/13, (5*sqrt(3))/39, M], [1/19, (5*sqrt(3))/57, N], [1/2, 0, O], [-5/16, sqrt(3)/8, P], [1/24, (5*sqrt(3))/36, Q]

画图得到

1715-4.gif
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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数学星空
发表于 2020-1-7 18:57:23 | 显示全部楼层
17棵16行结果

初始数据：
[[A, 1, 0, 0], [B, 0, 1, 1], [C, 1 - t, t^2 - t + 1, 1], [D, -t^2 + t, 1 - t, 1], [E, 1, -t^2 + t - 2, 0], [F, 0, 0, 1], [G, t, 1, 1], [H, 1, 1 - t, 0], [I, 0, t^2 - t + 1, 1], [J, 0, 1, 0], [K, 1, 0, 1], [L, 1 - t, t^2 - 2*t + 2, 1], [M, 1 - t, 0, 1], [N, -t, t^2 - t + 1, 1], [O, 1, 1, 1], [P, -t^2 + t - 1, t^2 - 2*t + 2, 1], [Q, 1, t^2 - 2*t + 2, 1]]
[[A, E, H, J], [B, F, I, J], [C, E, G, K], [D, F, G, L], [B, D, E, M], [A, F, K, M], [C, J, L, M], [A, C, I, N], [B, H, L, N], [A, B, G, O], [C, D, H, O], [D, I, K, P], [E, F, N, P], [G, H, I, Q], [J, K, O, Q], [A, L, P, Q]]

变换基点
[1, 1 - t, 0, H]-->[-1/4, sqrt(3)/6, 1],
[1, -t^2 + t - 2, 0, E]-->[-1/2, 0, 1],
[1, t^2 - 2*t + 2, 1, Q]-->[1/2, 0, 1],
[-t^2 + t - 1, t^2 - 2*t + 2, 1, P]-->[1/4, sqrt(3)/6, 1]

变换矩阵
[[0, (-t^2 + 2*t - 3)/(4*t^2 - 4*t + 8), (t^2 - 2*t + 2)/(4*t^2 - 4*t + 8)],
[sqrt(3)/(6*t^2 - 6*t + 12), sqrt(3)/(12*t^2 - 18*t + 24), -1/6*sqrt(3)],
[1/(2*t^2 - 2*t + 4), (t^2 - 3*t + 4)/(2*t^2 - 2*t + 4), (-4*t^2 + 5*t - 8)/(2*t^2 - 2*t + 4)]]

变换后坐标
[0, sqrt(3)/3, A], [1/(6*t^2 - 4*t + 8), sqrt(3)*(2*t^2 - 4*t + 5)/(18*t^2 - 33*t + 45), B], [t/(6*t^2 - 8*t + 10), sqrt(3)*(2*t^2 - 3*t + 5)/(33*t^2 - 51*t + 78), C], [0, sqrt(3)*(3*t^2 - 5*t + 8)/(39*t^2 - 63*t + 96), D], [-1/2, 0, E], [(-t^2 + 2*t - 2)/(8*t^2 - 10*t + 16), sqrt(3)*(t^2 - t + 2)/(12*t^2 - 15*t + 24), F], [1/(6*t^2 - 6*t + 8), sqrt(3)*(2*t^2 - 3*t + 5)/(27*t^2 - 42*t + 66), G], [-1/4, sqrt(3)/6, H], [t/(6*t^2 - 10*t + 12), sqrt(3)*(2*t^2 - 3*t + 5)/(21*t^2 - 33*t + 51), I], [(-t^2 + 2*t - 3)/(2*t^2 - 6*t + 8), sqrt(3)*(t^2 - t + 2)/(15*t^2 - 21*t + 33), J], [(-t^2 + 2*t - 2)/(8*t^2 - 10*t + 14), sqrt(3)*(t^2 - t + 1)/(12*t^2 - 15*t + 21), K], [(t^2 - t + 2)/(2*t^2 + 4), sqrt(3)*(t^2 - 2*t + 3)/(18*t^2 - 30*t + 45), L], [(-t^2 + 2*t - 2)/(8*t^2 - 8*t + 14), sqrt(3)*(t^2 + 1)/(12*t^2 - 12*t + 21), M], [t/(6*t^2 - 8*t + 12), sqrt(3)*(t^2 - t + 2)/(9*t^2 - 12*t + 18), N], [1/(6*t^2 - 4*t + 6), sqrt(3)/(21*t^2 - 33*t + 54), O], [1/4, sqrt(3)/6, P], [1/2, 0, Q]

t^3 - 2*t^2 + 3*t - 1 = 0的唯一实根 t=0.4301597088 代入得

[[0., 0.577350269189626, A], [0.135325589164497, 0.185174569112185, B], [0.0560911077303695, 0.113660675372216, C], [0., 0.145732024860234, D], [-0.500000000000000, 0., E], [-0.100519608551767, 0.153760074359076, F], [0.153156572325283, 0.133500097553799, G], [-0.250000000000000, 0.288675134594814, H], [0.0488339103094002, 0.173653908875284, I], [-0.401566967695180, 0.113660675368244, J], [-0.118503737538779, 0.0779748538312558, K], [0.401566967753754, 0.113660675375214, L], [-0.110035352637172, 0.113660675354571, M], [0.0444887874018849, 0.209573831984808, N], [0.185543028443446, 0.0396436349614751, O], [0.250000000000000, 0.288675134594814, P], [0.500000000000000, 0., Q]]

画图得到


1716-1.gif
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2020-1-7 21:26:22 | 显示全部楼层
18棵18行实数解:
        Parameter [-3+5*t-4*t^2+1*t^3=0]{Real}
        A[0 ,+1 , 0]
        B[0 ,+1 , +1]
        C[+1 ,0 , +1]
        D[+3-3*t+1*t^2 ,-2+2*t-1*t^2 , +1]
        E[+1 ,0 , 0]
        F[+2-3*t+1*t^2 ,-2+3*t-1*t^2 , +1]
        G[0 ,0 , +1]
        H[+1 ,-1 , 0]
        I[+1 ,-2/3+1/3*t-1/3*t^2 , 0]
        J[+3-5*t+2*t^2 ,-2+3*t-1*t^2 , +1]
        K[0 ,+1-1*t+1*t^2 , +1]
        L[+3-3*t+1*t^2 ,0 , +1]
        M[+2-3*t+1*t^2 ,-1+1*t , +1]
        N[+1-1*t ,+1-1*t+1*t^2 , +1]
        O[+3-3*t+1*t^2 ,-2+3*t-1*t^2 , +1]
        P[+2-3*t+1*t^2 ,-2/3+4/3*t-1/3*t^2 , +1]
        Q[-1*t ,+1-1*t+1*t^2 , +1]
        R[+3-4*t+2*t^2 ,-3+4*t-2*t^2 , +1]
AEHIABGKCEGLCDKMBDFNBCHOEFJOADLOBJLPAFMPCINPDGIQHJMQEKNQFGHRIJKRLMNROPQR
（AL) (B) (CF) (DO) (EM) (GP) (HN) (IR) (JK) (Q)

        Parameter [+1/4+3/2*t+1*t^2=0]{Real}
        A[0 ,0 , +1]
        B[+1 ,+2+4*t , 0]
        C[0 ,+1 , +1]
        D[+1 ,0 , 0]
        E[-1-2*t ,+2+4*t , +1]
        F[+1 ,-2 , 0]
        G[+1 ,0 , +1]
        H[0 ,+2 , +1]
        I[+1/2 ,+1+2*t , +1]
        J[0 ,+1 , 0]
        K[+1/2 ,0 , +1]
        L[-1/2-1*t ,+1+2*t , +1]
        M[+1 ,+2+4*t , +1]
        N[+1/5-2/5*t ,+6/5+8/5*t , +1]
        O[-1/2-1*t ,+2+2*t , +1]
        P[+1 ,+1+2*t , +1]
        Q[-1*t ,+2+4*t , +1]
        R[+1/2 ,+1 , +1]
BDFJACHJADGKAEFLABIMBEHOCFKOGINODILPGJMPCENPBCLQDEMQHKNQFGHRIJKRLMNROPQR
P<6>
        AEGHMN
        EANMHG
        HANEGM
        GANHME
        MEGANH
        NEGMHA
        AHMGEN
        EMHNAG
        HEGNAM
        GHMNAE
        MANGEH
        NMHGEA
        AGEMHN
        ENAHMG
        HNAGEM
        GNAMHE
        MGENAH
        NGEHMA
        AMHEGN
        EHMANG
        HGEANM
        GMHANE
        MNAEGH
        NHMEGA
P<12>
        BCDFIJKLOPQR
        BDCLOQPFIKJR
        FBKCROQJLIDP
        JFIKPRODCLBQ
        LBPDRIJQFOCK
        QLOPKRICDFBJ
        FKBJLDICRQOP
        LPBQFCODRJIK
        CQFOJKRBPDLI
        KOJRDIPFQBCL
        DJLIQPRBKCFO
        PIQRCOKLJBDF
        JIFDCBLKPORQ
        QOLCDBFPKIRJ
        ORCKBFJQILPD
        RPKIFJDOLCQB
        IRDPBLQJOFKC
        RKPOLQCIFDJB
        DLJBKFCIQRPO
        CFQBPLDOJRKI
        KJOFQCBRDPIL
        IDRJOKFPBQLC
        PQILJDBRCKOF
        OCRQIPLKBJFD

毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2020-1-7 21:56:19 | 显示全部楼层
第一个18棵18行对称图比较容易给出
s18.png
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2020-1-7 22:27:42 | 显示全部楼层
第二个注意到第六行变换是二阶变换，可以得出对称图: s18.2.png
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2020-1-8 08:11:31 | 显示全部楼层
如果注意到第6行和第13行都是二阶变换，而它们分别以AN为对称轴交换EG以及以EG为对称轴交换AN
可以猜测可以同时以AN和EG为两个正交方向的对称轴，将AENG映射为正方形，得到
s18.3.png

点评

mathe
这些复数解肯定无法映射为实数解，不然我们就可以计算出对应的实数解了。不过复数解我也不知道是否有比较好的图示方案  发表于 2020-1-8 10:38
数学星空
下一步可以研究一下如何将复数解映射为实数解~  发表于 2020-1-8 10:16
数学星空
唯一不足就是H,M为无穷远点  发表于 2020-1-8 09:37
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2020-1-8 10:34:52 | 显示全部楼层
注意第三个变换是3阶的，A->H->E->A, G->N->M->G
而且观察到AM, HG, EN三线共点X,  在原坐标中计算出X(0.61803398874989484820458683436563811772, 0.76393202250021030359082633126872376456)
我们可以尝试将A,H,E映射到正三角形顶点而X映射到正三角形中心:
变换阵
[-0.69098300562505257589770658281718094115 -0.75000000000000000000000000000000000000 1.0000000000000000000000000000000000000]
[-0.53523313465963489791295476456913181097 0.43301270189221932338186158537646809173 0]
[3.2360679774997896964091736687312762354 -1.0285575569695015695 E-38 1.0000000000000000000000000000000000000]
得到包含3个无穷远点的对称图:
s18.4.png

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数学星空
发表于 2020-1-8 18:48:45 | 显示全部楼层
18棵18行结果：

初始数据：

[[A, 0, 1, 0], [B, 0, 1, 1], [C, 1, 0, 1], [D, t^2 - 3*t + 3, -t^2 + 2*t - 2, 1], [E, 1, 0, 0], [F, t^2 - 3*t + 2, -t^2 + 3*t - 2, 1], [G, 0, 0, 1], [H, 1, -1, 0], [I, 1, -2/3 + 1/3*t - 1/3*t^2, 0], [J, 2*t^2 - 5*t + 3, -t^2 + 3*t - 2, 1], [K, 0, t^2 - t + 1, 1], [L, t^2 - 3*t + 3, 0, 1], [M, t^2 - 3*t + 2, t - 1, 1], [N, 1 - t, t^2 - t + 1, 1], [O, t^2 - 3*t + 3, -t^2 + 3*t - 2, 1], [P, t^2 - 3*t + 2, -2/3 + 4/3*t - 1/3*t^2, 1], [Q, -t, t^2 - t + 1, 1], [R, 2*t^2 - 4*t + 3, -2*t^2 + 4*t - 3, 1]]
[[A, E, H, I], [A, B, G, K], [C, E, G, L], [C, D, K, M], [B, D, F, N], [B, C, H, O], [E, F, J, O], [A, D, L, O], [B, J, L, P], [A, F, M, P], [C, I, N, P], [D, G, I, Q], [H, J, M, Q], [E, K, N, Q], [F, G, H, R], [I, J, K, R], [L, M, N, R], [O, P, Q, R]]

第一类基点变换
C[1, 0, 1]-->[-1, 1, 1],
A[0, 1, 0]-->[-1, -1, 1],
L[t^2 - 3*t + 3, 0, 1]-->[1, -1, 1],
F[t^2 - 3*t + 2, -t^2 + 3*t - 2, 1]-->[1, 1, 1]

变换矩阵
[[(-t^2 + 3*t - 2)/(t^2 - 3*t + 1), 1/(t^2 - 3*t + 2), (t - 2)/(t^2 - 3*t + 1)], [(t^2 - 3*t + 2)/(t^2 - 3*t + 1), 1/(t^2 - 3*t + 2), (-t + 2)/(t^2 - 3*t + 1)], [t*(t - 3)/(t^2 - 3*t + 1), -1/(t^2 - 3*t + 2), (-t + 4)/(t^2 - 3*t + 1)]]

变换后坐标
[-1, -1, A], [0, (t^2 - 3*t + 1)/(t^2 - 3*t + 2), B], [-1, 1, C], [(t - 3)/(t - 1), -1, D], [(-t^2 + 3*t - 2)/(t*(t - 3)), (t^2 - 3*t + 2)/(t*(t - 3)), E], [1, 1, F], [(-t + 2)/(t - 4), (t - 2)/(t - 4), G], [(t^2 - 2*t - 1)/(t^2 - 4*t + 5), (t^2 - 4*t + 3)/(t^2 - 4*t + 5), H], [(t^2 - t - 4)/(5*t^2 - 17*t + 16), (t^2 - 7*t + 8)/(5*t^2 - 17*t + 16), I], [t/(3*t - 2), (t^2 - 4*t + 2)/(3*t^2 - 8*t + 4), J], [-t/(3*t - 2), (t^2 - 4*t + 2)/(3*t^2 - 8*t + 4), K], [1, -1, L], [(t^2 - 3*t + 1)/(t^2 - t - 3), (t^2 - 3*t + 1)/(t^2 - t - 3), M], [(-t + 2)/(2*t^2 - 5*t + 4), -t/(2*t^2 - 5*t + 4), N], [2*t^2 - 6*t + 3, -1, O], [(2*t^2 - 8*t + 7)/(4*t^2 - 16*t + 17), (2*t^2 - 8*t + 7)/(4*t^2 - 16*t + 17), P], [0, (1 - t)/(2*t^2 - 6*t + 5), Q], [(t - 2)/(3*t - 2), (-2*t^2 + 5*t - 4)/(3*t - 2), R]

取t^3 - 4*t^2 + 5*t - 3 = 0 的实根t=2.465571232

代入得到
[[-1., -1., A], [0., -0.465571231365610, B], [-1., 1., C], [-0.364655607541292, -1., D], [0.517828186836976, -0.517828186836976, E], [1., 1., F], [0.303416647099776, -0.303416647099776, G], [0.121551869504082, -0.643714154429498, H], [-0.0862693960534162, -0.709733192282313, I], [0.456865301901685, -0.709733192092709, J], [-0.456865301901685, -0.709733192092709, K], [1., -1., L], [-0.517828185766002, -0.517828185766002, M], [-0.121551869231838, -0.643714154515984, N], [0.36465560813200, -1., O], [-0.303416646797583, -0.303416646797583, P], [0., -0.619782105673203, Q], [0.0862693961966294, -0.709733192437267, R]]

画图得到：

1818-1-1.gif

第二类基点变换

M[t^2 - 3*t + 2, t - 1, 1]-->[-cos(1/10*Pi), sin(1/10*Pi), 1]
B[0, 1, 1]-->[0, 1, 1],
E[1, 0, 0]--> [cos(1/10*Pi), sin(1/10*Pi), 1]
P[t^2 - 3*t + 2, -2/3 + 4/3*t - 1/3*t^2, 1]--> [cos(3/10*Pi), -sin(3/10*Pi), 1]

变换矩阵
[[(150*s^3 + 351*s^2 + 93*s - 24)/((33*s^3 + 73*s^2 + 19*s - 5)*(2*t^2 - 5*t + 4)), 1/4*((225*t^2 - 2025*t + 3375)*s^3 + (561*t^2 - 4773*t + 7863)*s^2 + (153*t^2 - 1269*t + 2079)*s - 39*t^2 + 327*t - 537)/((t^2 - 7*t + 8)*(33*s^3 + 73*s^2 + 19*s - 5)), 1/4*((-225*t^2 + 2025*t - 3375)*s^3 + (-561*t^2 + 4773*t - 7863)*s^2 + (-153*t^2 + 1269*t - 2079)*s + 39*t^2 - 327*t + 537)/((t^2 - 7*t + 8)*(33*s^3 + 73*s^2 + 19*s - 5))],
[-3/(4*t^2 - 10*t + 8), 1/8*((6147*t^2 - 31617*t + 44793)*s^3 + (14241*t^2 - 73371*t + 104019)*s^2 + (3747*t^2 - 19317*t + 27393)*s - 975*t^2 + 5025*t - 7125)/((t^2 - 7*t + 8)*(317*s^3 + 731*s^2 + 192*s - 50)), 1/8*((18441*t^2 - 66735*t + 41265)*s^3 + (42723*t^2 - 154485*t + 95355)*s^2 + (11241*t^2 - 40635*t + 25065)*s - 2925*t^2 + 10575*t - 6525)/((t^2 - 7*t + 8)*(317*s^3 + 731*s^2 + 192*s - 50))],
[-(6147*s^3 + 14241*s^2 + 3747*s - 975)/((1268*s^3 + 2924*s^2 + 768*s - 200)*(2*t^2 - 5*t + 4)), (-3*t^2 + 3*t + 3)/(t^2 - 7*t + 8), 1/4*((16098*t^2 - 52980*t + 39225)*s^3 + (37254*t^2 - 122700*t + 90915)*s^2 + (9798*t^2 - 32280*t + 23925)*s - 2550*t^2 + 8400*t - 6225)/((t^2 - 7*t + 8)*(317*s^3 + 731*s^2 + 192*s - 50))]]

变换后坐标
[((-75*t^2 + 675*t - 1125)*s^3 + (-187*t^2 + 1591*t - 2621)*s^2 + (-51*t^2 + 423*t - 693)*s + 13*t^2 - 109*t + 179)/(4*(t^2 - t - 1)*(33*s^3 + 73*s^2 + 19*s - 5)), ((-2049*t^2 + 10539*t - 14931)*s^3 + (-4747*t^2 + 24457*t - 34673)*s^2 + (-1249*t^2 + 6439*t - 9131)*s + 325*t^2 - 1675*t + 2375)/(8*(t^2 - t - 1)*(317*s^3 + 731*s^2 + 192*s - 50)), A], [0, 1, B], [((-26306795*t^2 + 46783760*t + 26306795)*s^3 + (-60867300*t^2 + 108246000*t + 60867300)*s^2 + (-16006395*t^2 + 28465680*t + 16006395)*s + 4166580*t^2 - 7409820*t - 4166580)/((18789424*t^2 + 37578848*t - 123301996)*s^3 + (43474078*t^2 + 86948156*t - 285289912)*s^2 + (11432472*t^2 + 22864944*t - 75023268)*s - 2975954*t^2 - 5951908*t + 19529096), ((-3611*t^2 + 18248*t - 33271)*s^3 + (-8393*t^2 + 42344*t - 77053)*s^2 + (-2211*t^2 + 11148*t - 20271)*s + 575*t^2 - 2900*t + 5275)/((6634*t^2 + 13268*t - 43516)*s^3 + (15342*t^2 + 30684*t - 100788)*s^2 + (4034*t^2 + 8068*t - 26516)*s - 1050*t^2 - 2100*t + 6900), C], [((-4834005*t^2 + 4834005*t + 28298445)*s^3 + (-11184300*t^2 + 11184300*t + 65475900)*s^2 + (-2941125*t^2 + 2941125*t + 17218365)*s + 765600*t^2 - 765600*t - 4482060)/((52979736*t^2 + 16695312*t - 28307820)*s^3 + (122582442*t^2 + 38628114*t - 65496540)*s^2 + (32235808*t^2 + 10158056*t - 17223700)*s - 8391206*t^2 - 2644222*t + 4483460), ((-14049*t^2 + 38637*t - 34245)*s^3 + (-32507*t^2 + 89471*t - 79255)*s^2 + (-8549*t^2 + 23537*t - 20845)*s + 2225*t^2 - 6125*t + 5425)/((18726*t^2 + 5862*t - 9960)*s^3 + (43138*t^2 + 13826*t - 23320)*s^2 + (11326*t^2 + 3662*t - 6160)*s - 2950*t^2 - 950*t + 1600), D], [(-5522075*s^3 - 12776700*s^2 - 3359915*s + 874610)/(5806254*s^3 + 13434213*s^2 + 3532822*s - 919619), (634*s^3 + 1462*s^2 + 384*s - 100)/(2049*s^3 + 4747*s^2 + 1249*s - 325), E], [((175*t^2 - 775*t + 950)*s^3 + (375*t^2 - 1779*t + 2154)*s^2 + (95*t^2 - 467*t + 562)*s - 25*t^2 + 121*t - 146)/(12*(t^2 - t + 2)*(33*s^3 + 73*s^2 + 19*s - 5)), ((-487*t^2 - 8003*t + 7516)*s^3 + (-1101*t^2 - 18609*t + 17508)*s^2 + (-287*t^2 - 4903*t + 4616)*s + 75*t^2 + 1275*t - 1200)/(24*(t^2 - t + 2)*(317*s^3 + 731*s^2 + 192*s - 50)), F], [((-4218495*t^2 + 37350945*t - 62046405)*s^3 + (-9760500*t^2 + 86420700*t - 143559900)*s^2 + (-2566735*t^2 + 22726225*t - 37752245)*s + 668140*t^2 - 5915800*t + 9827180)/((30401932*t^2 - 100076980*t + 74110640)*s^3 + (70342504*t^2 - 231553060*t + 171473330)*s^2 + (18498116*t^2 - 60891980*t + 45092680)*s - 4815192*t^2 + 15850620*t - 11737950), ((6147*t^2 - 22245*t + 13755)*s^3 + (14241*t^2 - 51495*t + 31785)*s^2 + (3747*t^2 - 13545*t + 8355)*s - 975*t^2 + 3525*t - 2175)/((10732*t^2 - 35320*t + 26150)*s^3 + (24836*t^2 - 81800*t + 60610)*s^2 + (6532*t^2 - 21520*t + 15950)*s - 1700*t^2 + 5600*t - 4150), G], [((26306795*t^2 - 46783760*t - 26306795)*s^3 + (60867300*t^2 - 108246000*t - 60867300)*s^2 + (16006395*t^2 - 28465680*t - 16006395)*s - 4166580*t^2 + 7409820*t + 4166580)/((4435592*t^2 - 95641388*t + 100076980)*s^3 + (10262774*t^2 - 221290286*t + 231553060)*s^2 + (2698816*t^2 - 58193164*t + 60891980)*s - 702522*t^2 + 15148098*t - 15850620), ((11807*t^2 - 38738*t + 25075)*s^3 + (27381*t^2 - 89814*t + 58065)*s^2 + (7207*t^2 - 23638*t + 15275)*s - 1875*t^2 + 6150*t - 3975)/((1562*t^2 - 33758*t + 35320)*s^3 + (3646*t^2 - 78154*t + 81800)*s^2 + (962*t^2 - 20558*t + 21520)*s - 250*t^2 + 5350*t - 5600), H], [((7205970*t^2 - 93947835*t + 177086715)*s^3 + (16673400*t^2 - 217372500*t + 409734900)*s^2 + (4384690*t^2 - 57162955*t + 107748875)*s - 1141360*t^2 + 14879920*t - 28047800)/((65639300*t^2 + 121855096*t - 185400284)*s^3 + (151873850*t^2 + 281940862*t - 428968748)*s^2 + (39938660*t^2 + 74142448*t - 112806692)*s - 10396310*t^2 - 19299826*t + 29364404), ((-24000*t^2 + 78441*t - 65559)*s^3 + (-55520*t^2 + 181523*t - 151597)*s^2 + (-14600*t^2 + 47741*t - 39859)*s + 3800*t^2 - 12425*t + 10375)/((23210*t^2 + 42946*t - 65384)*s^3 + (53390*t^2 + 99958*t - 151832)*s^2 + (14010*t^2 + 26346*t - 39984)*s - 3650*t^2 - 6850*t + 10400), I], [-15*(t^2 - t + 2)*(843699*s^3 + 1952100*s^2 + 513347*s - 133628)/((44755764*t^2 - 79593288*t + 124349052)*s^3 + (103553808*t^2 - 184159086*t + 287712894)*s^2 + (27231772*t^2 - 48428704*t + 75660476)*s - 7088624*t^2 + 12606338*t - 19694962), (t^2 - 7*t + 8)*(2049*s^3 + 4747*s^2 + 1249*s - 325)/((15804*t^2 - 28098*t + 43902)*s^3 + (36532*t^2 - 65014*t + 101546)*s^2 + (9604*t^2 - 17098*t + 26702)*s - 2500*t^2 + 4450*t - 6950), J], [((-20476965*t^2 + 74086380*t - 74086380)*s^3 + (-47378700*t^2 + 171417600*t - 171417600)*s^2 + (-12459285*t^2 + 45078060*t - 45078060)*s + 3243240*t^2 - 11734140*t + 11734140)/((1694268*t^2 - 49838568*t + 38226060)*s^3 + (3919896*t^2 - 115313496*t + 88445070)*s^2 + (1030804*t^2 - 30324184*t + 23258540)*s - 268328*t^2 + 7893608*t - 6054370), ((5853*t^2 - 19608*t + 15510)*s^3 + (13519*t^2 - 45304*t + 35810)*s^2 + (3553*t^2 - 11908*t + 9410)*s - 925*t^2 + 3100*t - 2450)/((588*t^2 - 17568*t + 13470)*s^3 + (1444*t^2 - 40864*t + 31370)*s^2 + (388*t^2 - 10768*t + 8270)*s - 100*t^2 + 2800*t - 2150), K], [5*(t^2 + 2*t - 1)*(843699*s^3 + 1952100*s^2 + 513347*s - 133628)/((4435592*t^2 - 95641388*t + 100076980)*s^3 + (10262774*t^2 - 221290286*t + 231553060)*s^2 + (2698816*t^2 - 58193164*t + 60891980)*s - 702522*t^2 + 15148098*t - 15850620), 3*(t^2 - 4*t + 5)*(2049*s^3 + 4747*s^2 + 1249*s - 325)/((1562*t^2 - 33758*t + 35320)*s^3 + (3646*t^2 - 78154*t + 81800)*s^2 + (962*t^2 - 20558*t + 21520)*s - 250*t^2 + 5350*t - 5600), L], [(125*s^3 + 281*s^2 + 73*s - 19)/(132*s^3 + 292*s^2 + 76*s - 20), (781*s^3 + 1823*s^2 + 481*s - 125)/(2536*s^3 + 5848*s^2 + 1536*s - 400), M], [((4598810*t^2 + 235195*t - 19100825)*s^3 + (10641000*t^2 + 543300*t - 44193900)*s^2 + (2798330*t^2 + 142795*t - 11621705)*s - 728420*t^2 - 37180*t + 3025220)/((109348008*t^2 - 208777776*t + 188941296)*s^3 + (253004676*t^2 - 483060822*t + 437163762)*s^2 + (66533224*t^2 - 127031608*t + 114961928)*s - 17319068*t^2 + 33067226*t - 29925406), ((-2242*t^2 - 101*t + 9271)*s^3 + (-5126*t^2 - 343*t + 21533)*s^2 + (-1342*t^2 - 101*t + 5671)*s + 350*t^2 + 25*t - 1475)/((38628*t^2 - 73746*t + 66726)*s^3 + (89164*t^2 - 170278*t + 154178)*s^2 + (23428*t^2 - 44746*t + 40526)*s - 6100*t^2 + 11650*t - 10550), N], [((20476965*t^2 - 119874315*t + 140351280)*s^3 + (47378700*t^2 - 277359300*t + 324738000)*s^2 + (12459285*t^2 - 72937755*t + 85397040)*s - 3243240*t^2 + 18986220*t - 22229460)/((9918240*t^2 + 59756808*t - 49838568)*s^3 + (22948530*t^2 + 138262026*t - 115313496)*s^2 + (6034840*t^2 + 36359024*t - 30324184)*s - 1570910*t^2 - 9464518*t + 7893608), ((-1755*t^2 + 873*t - 2628)*s^3 + (-4025*t^2 + 1859*t - 5884)*s^2 + (-1055*t^2 + 473*t - 1528)*s + 275*t^2 - 125*t + 400)/((3510*t^2 + 21078*t - 17568)*s^3 + (8050*t^2 + 48914*t - 40864)*s^2 + (2110*t^2 + 12878*t - 10768)*s - 550*t^2 - 3350*t + 2800), O], [(-75*s^3 - 187*s^2 - 51*s + 13)/(132*s^3 + 292*s^2 + 76*s - 20), (-2049*s^3 - 4747*s^2 - 1249*s + 325)/(2536*s^3 + 5848*s^2 + 1536*s - 400), P], [((15642960*t^2 - 77073855*t + 69252375)*s^3 + (36194400*t^2 - 178330500*t + 160233300)*s^2 + (9518160*t^2 - 46896015*t + 42136935)*s - 2477640*t^2 + 12207360*t - 10968540)/((97735500*t^2 - 127490220*t + 96041232)*s^3 + (226136250*t^2 - 294981840*t + 222216354)*s^2 + (59467580*t^2 - 77572100*t + 58436776)*s - 15479830*t^2 + 20192560*t - 15211502), ((-3510*t^2 + 8775*t - 873)*s^3 + (-8050*t^2 + 20125*t - 1859)*s^2 + (-2110*t^2 + 5275*t - 473)*s + 550*t^2 - 1375*t + 125)/((34530*t^2 - 45060*t + 33942)*s^3 + (79670*t^2 - 103820*t + 78226)*s^2 + (20930*t^2 - 27260*t + 20542)*s - 5450*t^2 + 7100*t - 5350), Q], [((-17181735*t^2 + 29837220*t - 9052500)*s^3 + (-39753900*t^2 + 69035400*t - 20944800)*s^2 + (-10454135*t^2 + 18154340*t - 5507860)*s + 2721290*t^2 - 4725710*t + 1433740)/((86446598*t^2 - 125396108*t + 94994176)*s^3 + (200016281*t^2 - 290135876*t + 219793372)*s^2 + (52598734*t^2 - 76297684*t + 57799568)*s - 13691823*t^2 + 19860828*t - 15045636), -6*(t^2 - t + 2)*(317*s^3 + 731*s^2 + 192*s - 50)/((30533*t^2 - 44288*t + 33556)*s^3 + (70519*t^2 - 102304*t + 77468)*s^2 + (18533*t^2 - 26888*t + 20356)*s - 4825*t^2 + 7000*t - 5300), R]

取{t^3 - 4*t^2 + 5*t - 3 = 0, s^4 - 4*s^3 - 14*s^2 - 4*s + 1 = 0}的根{s = 6.313751515, t = 2.465571232}得

[[0.234616058495904, -0.211507467000040, A], [0., 1., B], [-0.218545932874459, -0.346768763285241, C], [0.0323392394302098, -0.206099664300224, D], [-0.951056516295156, 0.309016994374962, E], [0.0363466927495114, -0.355558593215524, F], [0.361347766757931, -0.865923076272166, G], [-0.167919712838658, -0.0347894422763860, H], [-0.0208974647710039, -0.0993339700298028, I], [-0.354888959699818, -0.0922359087933712, J], [0.213859124085480, -0.104323069066427, K], [-0.388232192815577, -0.194855828429939, L], [0.951056516294995, 0.309016994374906, M], [0.0278405294947495, -0.0383192014529248, N], [-0.194733833784535, -0.200028942010060, O], [-0.587785252291691, -0.809016994374908, P], [-0.0683378112793189, -0.00419281487409369, Q], [-0.128270766443560, -0.0970520488229928, R]]

画图得到

1818-1-2.gif

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数学星空
发表于 2020-1-8 19:09:20 | 显示全部楼层
18棵18行第二种结果：

初始数据：

[[A, 0, 0, 1], [B, 1, 2 + 4*t, 0], [C, 0, 1, 1], [D, 1, 0, 0], [E, -1 - 2*t, 2 + 4*t, 1], [F, 1, -2, 0], [G, 1, 0, 1], [H, 0, 2, 1], [I, 1/2, 1 + 2*t, 1], [J, 0, 1, 0], [K, 1/2, 0, 1], [L, -1/2 - t, 1 + 2*t, 1], [M, 1, 2 + 4*t, 1], [N, 1/5 - (2*t)/5, 6/5 + (8*t)/5, 1], [O, -1/2 - t, 2 + 2*t, 1], [P, 1, 1 + 2*t, 1], [Q, -t, 2 + 4*t, 1], [R, 1/2, 1, 1]]
[[B, D, F, J], [A, C, H, J], [A, D, G, K], [A, E, F, L], [A, B, I, M], [B, E, H, O], [C, F, K, O], [G, I, N, O], [D, I, L, P], [G, J, M, P], [C, E, N, P], [B, C, L, Q], [D, E, M, Q], [H, K, N, Q], [F, G, H, R], [I, J, K, R], [L, M, N, R], [O, P, Q, R]]

第一类基点变换

F[1, -2, 0]-->[-1, 1, 1],
M[1, 2 + 4*t, 1]-->[-1, -1, 1],
H[0, 2, 1]-->[1, -1, 1],
D[1, 0, 0]-->[1, 1, 1]

变换矩阵
[[-1, 1/4*(-1 - 4*t)/t, 0], [-1, 1/4*1/t, 2], [-1, 1/4*1/t, (-1 - 2*t)/t]]

变换后坐标
[0, -2*t/(1 + 2*t), A], [1 + 4*t, 1, B], [(1 + 4*t)/(3 + 8*t), (-8*t - 1)/(3 + 8*t), C], [1, 1, D], [-t/(3*t + 1), -t/(3*t + 1), E], [-1, 1, F], [t/(3*t + 1), -t/(3*t + 1), G], [1, -1, H], [(-1 - 4*t)/(3 + 8*t), (-8*t - 1)/(3 + 8*t), I], [-1 - 4*t, 1, J], [t/(5*t + 2), -3*t/(5*t + 2), K], [-t/(5*t + 2), -3*t/(5*t + 2), L], [-1, -1, M], [0, (-8*t - 1)/(4 + 12*t), N], [(1 + 2*t)/(3 + 10*t), (-1 - 6*t)/(3 + 10*t), O], [(-1 - 2*t)/(3 + 10*t), (-1 - 6*t)/(3 + 10*t), P], [(-1 - 6*t)/(3 + 10*t), (-1 - 6*t)/(3 + 10*t), Q], [(1 + 6*t)/(3 + 10*t), (-1 - 6*t)/(3 + 10*t), R]

取1/4 + 3/2*t + t^2 = 0的实根t = -0.1909830058得到

[[0., 0.618033989665930, A], [0.2360679768, 1., B], [0.160357456268025, 0.358570174927898, C], [1., 1., D], [0.447213596459244, 0.447213596459244, E], [-1., 1., F], [-0.447213596459244, 0.447213596459244, G], [1., -1., H], [-0.160357456268025, 0.358570174927898, I], [-0.2360679768, 1., J], [-0.182743997951914, 0.548231993855742, K], [0.182743997951914, 0.548231993855742, L], [-1., -1., M], [0., 0.309016995574055, N], [0.566915271270614, 0.133830542541229, O], [-0.566915271270614, 0.133830542541229, P], [0.133830542541229, 0.133830542541229, Q], [-0.133830542541229, 0.133830542541229, R]]

画图得到：

1818-2-1.gif


第二类基点变换
A[0, 0, 1]-->[0, sqrt(3)/3, 1],
K[1/2, 0, 1]-->[-1/2, -sqrt(3)/6, 1],
L[-1/2 - t, 1 + 2*t, 1]-->[1/2, -sqrt(3)/6, 1],
N[1/5 - (2*t)/5, 6/5 + (8*t)/5, 1]-->[0, 0, 1]

变换矩阵

[[5/(24 + 12*t), (2*t - 1)/(24 + 24*t), 0], [(-5 - 10*t)*sqrt(3)/(90 + 72*t), -5*sqrt(3)*(1 + t)/(90 + 72*t), (5 + 10*t)*sqrt(3)/(63 + 54*t)], [(-13 - 14*t)/(12*t + 15), (-16 - 13*t)/(12*t + 15), (5 + 10*t)/(21 + 18*t)]]

变换后坐标

[0, sqrt(3)/3, A], [(-4*t - 5)/(74 + 56*t), 5*sqrt(3)*(1 + t)/(96 + 78*t), B], [(4 + 3*t)/(50 + 38*t), -sqrt(3)/(186 + 144*t), C], [(-4*t - 5)/(18 + 16*t), (5 + 10*t)*sqrt(3)/(78 + 84*t), D], [(47 + 36*t)/(130 + 100*t), -sqrt(3)*(1 + t)/(33 + 24*t), E], [(13 + 10*t)/(46 + 36*t), 5*sqrt(3)/(114 + 72*t), F], [(-29 - 22*t)/(80 + 60*t), -sqrt(3)*(1 + t)/(33 + 24*t), G], [(4 + 3*t)/(55 + 42*t), sqrt(3)*(4*t + 5)/(204 + 156*t), H], [(-29 - 22*t)/(362 + 276*t), -sqrt(3)/(186 + 144*t), I], [(7 + 6*t)/(102 + 76*t), 5*sqrt(3)*(1 + t)/(96 + 78*t), J], [-1/2, -sqrt(3)/6, K], [1/2, -sqrt(3)/6, L], [(-29 - 22*t)/(398 + 304*t), sqrt(3)*(4*t + 5)/(204 + 156*t), M], [0, 0, N], [(47 + 36*t)/(326 + 248*t), sqrt(3)*(3 + 2*t)/(270 + 204*t), O], [(-38 - 29*t)/(264 + 202*t), sqrt(3)*(3 + 2*t)/(270 + 204*t), P], [(9 + 7*t)/(264 + 202*t), sqrt(3)*(3 + 2*t)/(270 + 204*t), Q], [(-9 - 7*t)/(264 + 202*t), sqrt(3)*(3 + 2*t)/(270 + 204*t), R]

取1/4 + 3/2*t + t^2 = 0的实根t = -0.1909830058得到

[[0., 0.577350269189626, A], [-0.0669152706811006, 0.0863872429817105, B], [0.0801787282956556, -0.0109278724085414, C], [-0.283457635347166, 0.0863872429574391, D], [0.361803398875274, -0.0493116000095872, E], [0.283457635341814, 0.0863872429967109, F], [-0.361803398874245, -0.0493116000095872, G], [0.0729490168753955, 0.0421171345296315, H], [-0.0801787282955373, -0.0109278724085414, I], [0.0669152706799704, 0.0863872429817105, J], [-0.500000000000000, -0.288675134594814, K], [0.500000000000000, -0.288675134594814, L], [-0.0729490168752486, 0.0421171345296315, M], [0., 0., N], [0.144003577762967, 0.0196268107190822, O], [-0.144003577763216, 0.0196268107190822, P], [0.0339946333551765, 0.0196268107190822, Q], [-0.0339946333551765, 0.0196268107190822, R]]

画图得到

1818-2-2.gif

第三类（mathe 已得到的超级嵌套三角形，唯一不足就是要去掉三个J,O,L无穷远点)

设AM,HG,EN相交于S,可以求得[1/(2*(t + 1)), (1 + 2*t)/(t + 1), 1]

基点变换

H[0, 2, 1]-->[0, sqrt(3)/3, 1],
E[-1 - 2*t, 2 + 4*t, 1]-->[-1/2, -sqrt(3)/6, 1],
A[0, 0, 1]-->[1/2, -sqrt(3)/6, 1],
S[1/(2*(t + 1)), (1 + 2*t)/(t + 1), 1]-->[0, 0, 1]

变换矩阵
[[1/6, 1/12, -1/6], [1/18*sqrt(3)*(4*t + 1), -1/12*sqrt(3), 1/18*sqrt(3)], [-4/3 - 4/3*t, 0, -1/3]]

变换后坐标
[1/2, -sqrt(3)/6, A], [-1/4, sqrt(3)/12, B], [1/4, sqrt(3)/12, C], [-1/(8*t + 8), (-4*t - 1)*sqrt(3)/(24*t + 24), D], [-1/2, -sqrt(3)/6, E], [0, -sqrt(3)/6, F], [0, (-1 - 2*t)*sqrt(3)/(15 + 12*t), G], [0, sqrt(3)/3, H], [-t/(6 + 4*t), sqrt(3)*t/(18 + 12*t), I], [1/(12 + 8*t), (-3 - 4*t)*sqrt(3)/(36 + 24*t), K], [(-1 - 2*t)/(10 + 8*t), sqrt(3)*(1 + 2*t)/(30 + 24*t), M], [(-2*t + 1)/(22 + 16*t), (-2*t + 1)*sqrt(3)/(66 + 48*t), N], [(-1 - 2*t)/(20 + 16*t), (-1 - 2*t)*sqrt(3)/(60 + 48*t), P], [-t/(4*t + 4), sqrt(3)/12, Q], [0, -sqrt(3)*t/(9 + 6*t), R], [0, 0, S]

取1/4 + 3/2*t + t^2 = 0的实根t = -0.1909830058得到

[[0.500000000000000, -0.288675134594814], [-0.250000000000000, 0.144337567297407], [0.250000000000000, 0.144337567297407], [-0.154508497220886, -0.0210585672071243], [-0.500000000000000, -0.288675134594814], [0., -0.288675134594814], [0., -0.0842342690262086], [0., 0.577350269189626], [0.0364745084758656, -0.0210585672871003], [0.0954915028252885, -0.123279000010306], [-0.0729490168459093, 0.0421171345131043], [0.0729490169044062, 0.0421171345468775], [-0.0364745084229546, -0.0210585672565521], [0.0590169944417714, 0.144337567297407], [0., 0.0421171345742006], [0., 0.]]

画图得到

1818-2-3.gif
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数学星空
发表于 2020-1-9 18:42:06 | 显示全部楼层
现给出669# 有关14棵10行实数解的结果

第一个初始数据

[[A, 1, t^2 + 3*t + 2, 0], [B, 1/2 - t - 1/2*t^2, 1/2 + t + 1/2*t^2, 1], [C, 0, 1, 0], [D, 1, 0, 0], [E, 1, t^2 + 2*t, 0], [F, 0, 0, 1], [G, 1, t^2 + 3*t + 2, 1], [H, t^2, 1 + t, 1], [I, 0, 1, 1], [J, 1, 0, 1], [K, -t, 1 + t, 1], [L, 0, 1 + t, 1], [M, 1, t^2 + 2*t + 1, 1], [N, -t^2 - 2*t, 0, 1]]
[[A, C, D, E], [A, F, G, H], [B, I, J, K], [B, L, M, N], [C, F, I, L], [C, G, J, M], [D, F, J, N], [D, H, K, L], [E, G, K, N], [E, H, I, M]]

变换基点
[1, t^2 + 2*t, 0, E]--> [-1/4, sqrt(3)/6, 1],
[-t^2 - 2*t, 0, 1, N]--> [-1/2, 0, 1],
[0, 1, 1, I]--> [1/2, 0, 1],
[0, 0, 1, F]-->[1/4, sqrt(3)/6, 1]

变换矩阵
[[-1/4*t^2 - 1/2*t, 1/(8 + 4*t), -1/4], [-1/6*sqrt(3)/(t*(2 + t)), 1/6*sqrt(3), -1/6*sqrt(3)], [1/2*(t - 1)/(t*(2 + t)), (t + 3)/(4 + 2*t), -1]]

变换后坐标
[(t^2 + t - 1)/(6*t^2 + 16*t + 6), sqrt(3)*(2*t^2 + 5*t + 2)/(9*t^2 + 24*t + 9), A], [0, -sqrt(3)/(6*t - 3), B], [1/(2*t + 6), sqrt(3)*(2 + t)/(3*t + 9), C], [(-2 - t)/(2*t - 2), -sqrt(3)/(3*t - 3), D], [-1/4, sqrt(3)/6, E], [1/4, sqrt(3)/6, F], [-1/(2*t + 6), sqrt(3)*(2 + t)/(3*t + 9), G], [(-t - 1)/(4*t - 2), sqrt(3)*t/(6*t - 3), H], [1/2, 0, I], [(1 + t)^2/(2 + 4*t), sqrt(3)*(1 + t)/(3 + 6*t), J], [1/(2*t), sqrt(3)*(1 + t)/(3*t), K], [-1/(2*t^2 + 4*t - 2), sqrt(3)*t*(2 + t)/(3*t^2 + 6*t - 3), L], [t*(2 + t)/(2*t - 4), -sqrt(3)/(3*t - 6), M], [-1/2, 0, N]

取t^3 + 2*t^2 - 1 = 0 实根t=-1.618033988得

[[4.01119536712019*10^(-10), 0.235921486642479, A], [0., 0.136293910404810, B], [0.361803398678664, 0.159575689947929, C], [0.0729490170392700, 0.220528179479702, D], [-0.250000000000000, 0.288675134594814, E], [0.250000000000000, 0.288675134594814, F], [-0.361803398678664, 0.159575689947929, G], [-0.0729490168124724, 0.220528179392409, H], [0.500000000000000, 0., I], [-0.0854101964749895, 0.159575689634642, J], [-0.309016994518164, 0.220528179251163, K], [0.309016994197921, 0.220528179620948, L], [0.0854101967707684, 0.159575689754307, M], [-0.500000000000000, 0., N]]

画图得到

1410-1.gif


第二个初始数据

[[A, 1, -1, 0], [B, 0, 0, 1], [C, 1 + t, 1, 1], [D, -t, 1 - t, 1], [E, 0, 1, 1], [F, 1, 0, 1], [G, 1 + t, -t, 1], [H, 1, -2 + t, 0], [I, 0, 1 - t, 1], [J, -t, 0, 1], [K, -t, 1, 1], [L, 1 + t, 1 - t, 1], [M, 0, 1, 0], [N, 1, 0, 0]]
[[A, H, M, N], [A, E, F, G], [B, E, I, M], [B, F, J, N], [C, E, K, N], [C, G, L, M], [D, I, L, N], [D, J, K, M], [F, H, K, L], [G, H, I, J]]

变换基点
[0, 1, 0, M]-->[0, sqrt(3)/3, 1],
[1, -1, 0, A]-->[-1/2, -sqrt(3)/6, 1],
[1 + t, 1, 1, C]--> [1/2, -sqrt(3)/6, 1],
[0, 1, 1, E]-->[0, 0, 1]

变换矩阵
[[-1/(6 + 6*t), 0, 0], [sqrt(3)/(18 + 18*t), sqrt(3)/(9 + 9*t), -sqrt(3)/(9 + 9*t)], [2/(3 + 3*t), 1/(3 + 3*t), (-4 - 3*t)/(3 + 3*t)]]

变换后坐标
[-1/2, -sqrt(3)/6, A], [0, sqrt(3)/(12 + 9*t), B], [1/2, -sqrt(3)/6, C], [-t/(6 + 12*t), t*sqrt(3)/(6 + 12*t), D], [0, 0, E], [1/(4 + 6*t), sqrt(3)/(12 + 18*t), F], [1/4, sqrt(3)/12, G], [-1/(2*t), sqrt(3)*(-3 + 2*t)/(6*t), H], [0, t*sqrt(3)/(9 + 12*t), I], [-t/(8 + 10*t), (2 + t)*sqrt(3)/(24 + 30*t), J], [-t/(6 + 10*t), t*sqrt(3)/(18 + 30*t), K], [(1 + t)/(2 + 4*t), (t - 1)*sqrt(3)/(6 + 12*t), L], [0, sqrt(3)/3, M], [-1/4, sqrt(3)/12, N]

取t^2 - t - 1 = 0= 0 实根t = 1.618033988得

[[-0.500000000000000, -0.288675134594814, A], [0., 0.0652070951449489, B], [0.500000000000000, -0.288675134594814, C], [-0.0636610018680525, 0.110264089696204, D], [0., 0., E], [0.0729490168991014, 0.0421171345438148, F], [0.250000000000000, 0.144337567297407, G], [-0.309016994518164, 0.0421171342819314, H], [0., 0.0986232000114525, I], [-0.0669152706715386, 0.0863872429947423, J], [-0.0729490168660121, 0.0421171345247108, K], [0.309016994395843, 0.0421171344937995, L], [0., 0.577350269189626, M], [-0.250000000000000, 0.144337567297407, N]]

画图得到

1410-2.gif

第三个初始数据

[[A, 1, -t - 1, 0], [B, 1, -t, 0], [C, 0, 0, 1], [D, 1, 1 - t, 1], [E, 1 - t, 1, 1], [F, 0, 1, 1], [G, 1, -t, 1], [H, 1, 0, 1], [I, -t, 1, 1], [J, 0, 1 - t, 1], [K, 1 - t, 0, 1], [L, 0, 1, 0], [M, 1, 0, 0], [N, 1 - t, 1 - t, 1]]
[[A, B, L, M], [A, F, G, N], [B, H, I, N], [C, F, J, L], [C, H, K, M], [D, G, H, L], [D, J, M, N], [E, F, I, M], [E, K, L, N], [G, I, J, K]]

变换基点
[0, 1, 0, L]-->[0, sqrt(3)/3, 1],
[0, 1, 1, F]-->[-1/2, -sqrt(3)/6, 1],
[1 - t, 0, 1, K]-->[1/2, -sqrt(3)/6, 1],
[1, -t, 0, B]-->[0, 0, 1]

变换矩阵
[[0, 0, 1/6*t - 1/6], [sqrt(3)*(-1 + 2*t)/(-18 + 18*t), -1/18*sqrt(3), (1 - 2*t)*sqrt(3)/(-18 + 18*t)], [(5 - 4*t)/(6*t - 6), -1/6, (-5 + 7*t)/(6*t - 6)]]

变换后坐标
[0, -sqrt(3)/15, A], [0, 0, B], [(-3*t + 2)/(-5 + 7*t), (1 - 2*t)*sqrt(3)/(-15 + 21*t), C], [1 - (3*t)/2, -sqrt(3)*(3*t - 2)/6, D], [(3*t - 2)/(-5 + 7*t), (1 - 2*t)*sqrt(3)/(-15 + 21*t), E], [-1/2, -sqrt(3)/6, F], [(-3*t + 2)/(1 + t), (1 - 2*t)*sqrt(3)/(3 + 3*t), G], [(-3*t + 2)/(3*t), 0, H], [(3*t - 2)/(3*t), 0, I], [(-3*t + 2)/(4*t - 3), (-5*t + 3)*sqrt(3)/(12*t - 9), J], [1/2, -sqrt(3)/6, K], [0, sqrt(3)/3, L], [0, (1 - 2*t)*sqrt(3)/(-15 + 12*t), M], [1/3, 0, N]

取t^2 + t - 1 = 0的实根t=0.6180339880得到
[[0., -0.115470053837925, A], [0., 0., B], [-0.216542366311014, 0.202287888747760, C], [0.072949018000000, 0.0421171351794189, D], [0.216542366311013, 0.202287888747760, E], [-0.500000000000000, -0.288675134594814, F], [0.0901699451816460, -0.0842342685638631, G], [0.0786893271420967, 0., H], [-0.0786893271420963, 0., I], [-0.276393204941284, 0.0986231953645240, J], [0.500000000000000, -0.288675134594814, K], [0., 0.577350269189626, L], [0., 0.0539166295744756, M], [0.333333333333333, 0., N]]

画图得到

1410-3.gif

第四个初始数据

[[A, 1/2, 7/6, 1], [B, 0, 0, 1], [C, 1, 0, 1], [D, 1, 1, 1], [E, 5/8, 9/8, 1], [F, 1/4, 5/4, 1], [G, 3/8, 9/8, 1], [H, 0, 1, 1], [I, 3/4, 5/4, 1], [J, 3/4, 3/4, 1], [K, 1, 1, 0], [L, 1/4, 3/4, 1], [M, 1, -1, 0], [N, 1/2, 3/2, 1]]
[[A, D, E, F], [A, G, H, I], [B, D, J, K], [B, G, L, N], [C, E, J, N], [C, H, L, M], [D, I, M, N], [E, I, K, L], [F, G, J, M], [F, H, K, N]]

变换基点
[1, 0, 1, C]--> [-1/4, sqrt(3)/6, 1],
[1, 1, 0, K]-->[-1/2, 0, 1],
[1, -1, 0, M]-->[1/2, 0, 1],
[0, 0, 1, B]-->[1/4, sqrt(3)/6, 1]

变换矩阵
[[1/2, 0, -1/4], [0, 0, -1/6*sqrt(3)], [0, -1, -1]]

变换后坐标
[0, sqrt(3)/13, A], [1/4, sqrt(3)/6, B], [-1/4, sqrt(3)/6, C], [-1/8, sqrt(3)/12, D], [-1/34, (4*sqrt(3))/51, E], [1/18, (2*sqrt(3))/27, F], [1/34, (4*sqrt(3))/51, G], [1/8, sqrt(3)/12, H], [-1/18, (2*sqrt(3))/27, I], [-1/14, (2*sqrt(3))/21, J], [-1/2, 0, K], [1/14, (2*sqrt(3))/21, L], [1/2, 0, M], [0, sqrt(3)/15, N]

画图得到


1410-4.gif

毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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