サッカーボールと正20面体

正二十面体の頂点の周りを削るとサッカーボールの形になります。正二十面体のどの位置に点を取ればこのような形になるでしょうか。観察してみましょう。

サッカーボール

正二十面体

a = 1.618033988749895
b: a^(2)
A: (0, -1, (-a))
B: (0, 1, (-a))
C: (0, -1, a)
D: (0, 1, a)
E: ((-a), 0, -1)
F: ((-a), 0, 1)
G: (a, 0, -1)
H: (a, 0, 1)
I: (-1, (-a), 0)
J: (1, (-a), 0)
K: (-1, a, 0)
L: (1, a, 0)
poly2, j_1, c_1, h = Polygon(C, H, J)
poly4, d_1, c_3, f = Polygon(C, F, D)
poly5, i_1, c_4, f_1 = Polygon(C, F, I)
poly6, i_2, j_2, a_1 = Polygon(J, A, I)
poly7, f_2, i_3, e_{10} = Polygon(I, E, F)
poly8, a_2, i_4, e_1 = Polygon(I, E, A)
poly9, a_3, j_3, g = Polygon(J, G, A)
poly10, j_4, h_2, g_1 = Polygon(H, G, J)
poly11, b_1, e_2, a_4 = Polygon(E, A, B)
poly12, b_2, g_2, a_5 = Polygon(G, A, B)
poly13, g_3, h_3, l = Polygon(H, L, G)
poly14, g_4, l_1, b_3 = Polygon(L, B, G)
poly15, h_4, l_2, d_2 = Polygon(L, D, H)
poly16, f_3, k, d_3 = Polygon(K, D, F)
poly17, e_3, k_1, f_4 = Polygon(K, F, E)
poly20, d_4, k_4, l_4 = Polygon(K, L, D)
M: (((2 * x(C)) + x(H)) / 3, ((2 * y(C)) + y(H)) / 3, ((2 * z(C)) + z(H)) / 3)
N = ツール1(H, C)
O = ツール1(C, J)
P = ツール1(J, C)
Q = ツール1(J, H)
R = ツール1(H, J)
S = ツール1(J, G)
T = ツール1(G, J)
U = ツール1(G, H)
V = ツール1(H, G)
W = ツール1(J, A)
Z = ツール1(A, J)
A_1 = ツール1(A, G)
B_1 = ツール1(G, A)
C_1 = ツール1(A, I)
D_1 = ツール1(I, A)
E_1 = ツール1(I, J)
F_1 = ツール1(J, I)
G_1 = ツール1(I, C)
H_1 = ツール1(C, I)
poly18, l_3, k_2, b_4 = Polygon(K, B, L)
poly19, e_4, k_3, b_5 = Polygon(K, B, E)
I_1 = ツール1(K, D)
J_1 = ツール1(D, K)
K_1 = ツール1(D, F)
L_1 = ツール1(F, D)
M_1 = ツール1(F, K)
N_1 = ツール1(K, F)
O_1 = ツール1(K, E)
P_1 = ツール1(E, K)
Q_1 = ツール1(E, F)
R_1 = ツール1(F, E)
S_1 = ツール1(E, B)
T_1 = ツール1(B, E)
U_1 = ツール1(B, K)
V_1 = ツール1(K, B)
W_1 = ツール1(B, L)
Z_1 = ツール1(L, B)
A_2 = ツール1(L, K)
B_2 = ツール1(K, L)
C_2 = ツール1(L, D)
D_2 = ツール1(D, L)
E_2 = ツール1(D, H)
F_2 = ツール1(H, D)
G_2 = ツール1(H, L)
H_2 = ツール1(L, H)
I_2 = ツール1(L, G)
J_2 = ツール1(G, L)
K_2 = ツール1(G, B)
L_2 = ツール1(B, G)
M_2 = ツール1(B, A)
N_2 = ツール1(A, B)
O_2 = ツール1(A, E)
P_2 = ツール1(E, A)
Q_2 = ツール1(C, D)
R_2 = ツール1(D, C)
S_2 = ツール1(C, F)
T_2 = ツール1(F, C)
U_2 = ツール1(I, F)
V_2 = ツール1(F, I)
W_2 = ツール1(E, I)
Z_2 = ツール1(I, E)
poly21, l_5, m_1, r_1, v_2, t_2 = Polygon(L_1, M_1, R_1, V_2, T_2)
poly22, q_2, s_2, h_5, o, m = Polygon(Q_2, S_2, H_1, O, M)
poly23, g_5, u_2, z_2, d_5, e_5 = Polygon(G_1, U_2, Z_2, D_1, E_1)
poly24, p, f_5, w, s, q = Polygon(P, F_1, W, S, Q)
poly25, c_5, o_2, n_2, a_6, z = Polygon(C_1, O_2, N_2, A_1, Z)
poly26, u, t, b_6, k_5, j_5 = Polygon(U, T, B_1, K_2, J_2)
poly27, n, r, v, g_6, f_6 = Polygon(N, R, V, G_2, F_2)
poly28, c_6, h_6, i_5, z_1, a_7 = Polygon(C_2, H_2, I_2, Z_1, A_2)
poly29, i_6, b_7, v_1, o_1, n_1 = Polygon(I_1, B_2, V_1, O_1, N_1)
poly30, w_1, l_6, m_2, t_1, u_1 = Polygon(W_1, L_2, M_2, T_1, U_1)
poly31, r_2, e_6, d_6, j_6, k_6 = Polygon(R_2, E_2, D_2, J_1, K_1)
poly32, q_1, p_1, s_1, p_2, w_2 = Polygon(Q_1, P_1, S_1, P_2, W_2)
poly33, j_7, i_7, n_3, m_3, l_7, k_7 = Polygon(J_1, I_1, N_1, M_1, L_1, K_1)
poly34, r_3, k_8, l_8, t_3, s_3, q_3 = Polygon(R_2, K_1, L_1, T_2, S_2, Q_2)
poly35, s_4, t_4, v_3, u_3, g_7, h_7 = Polygon(S_2, T_2, V_2, U_2, G_1, H_1)
poly36, v_4, u_4, z_3, w_3, q_4, r_4 = Polygon(V_2, U_2, Z_2, W_2, Q_1, R_1)
poly37, m_4, r_5, q_5, p_3, o_3, n_4 = Polygon(M_1, R_1, Q_1, P_1, O_1, N_1)
poly38, c_7, d_7, e_7, f_7, w_4, z_4 = Polygon(C_1, D_1, E_1, F_1, W, Z)
poly39, c_8, d_8, z_5, w_5, p_4, o_4 = Polygon(C_1, D_1, Z_2, W_2, P_2, O_2)
poly40, z_6, w_6, s_5, t_5, b_8, a_8 = Polygon(Z, W, S, T, B_1, A_1)
poly41, n_5, a_9, b_9, k_9, l_9, m_5 = Polygon(N_2, A_1, B_1, K_2, L_2, M_2)
poly42, s_6, p_5, o_5, n_6, m_6, t_6 = Polygon(S_1, P_2, O_2, N_2, M_2, T_1)
poly44, q_6, p_7, o_7, m_7, n_7, r_6 = Polygon(Q, P, O, M, N, R)
poly45, q_7, r_7, v_5, u_5, t_7, s_7 = Polygon(Q, R, V, U, T, S)
poly46, f_9, e_8, r_8, q_8, m_8, n_8 = Polygon(F_2, E_2, R_2, Q_2, M, N)
poly47, v_6, u_6, t_8, s_8, p_8, o_8 = Polygon(V_1, U_1, T_1, S_1, P_1, O_1)
poly48, c_9, d_9, e_9, f_{10}, g_9, h_9 = Polygon(C_2, D_2, E_2, F_2, G_2, H_2)
poly49, h_{10}, g_{10}, v_7, u_7, j_8, i_8 = Polygon(H_2, G_2, V, U, J_2, I_2)
poly50, z_7, i_9, j_9, k_{10}, l_{10}, w_7 = Polygon(Z_1, I_2, J_2, K_2, L_2, W_1)
poly51, a_{10}, z_8, w_8, u_8, v_8, b_{11} = Polygon(A_2, Z_1, W_1, U_1, V_1, B_2)
poly52, c_{10}, a_{11}, b_{12}, i_{10}, j_{10}, d_{10} = Polygon(C_2, A_2, B_2, I_1, J_1, D_2)
d = true
c = false

Tool: 3等分点 (command ツール1)
Input: C, H
Output: M

M: (((2 * x(C)) + x(H)) / 3, ((2 * y(C)) + y(H)) / 3, ((2 * z(C)) + z(H)) / 3)