MathPlotAI Logo MathPlotAI

Pascal's Triangle and Metatron's Cube

Pascal's Triangle and Metatron's Cube
Loading the math board and drawing, please wait…
2019-10-06 03:34
TriangleCirclePolygonVectorSequence
This example demonstrates the geometric connection and visual correspondence between Pascal's Triangle and Metatron's Cube. It involves core mathematical concepts such as binomial distribution, integers, polygon construction, vector operations, and geometric transformations. The construction utilizes GeoGebra commands and objects like Intersect, Point, Polygon, Vector, Sequence, Element, Translate, Circle, Mirror, and Midpoint. It is suitable for high school students and above, mathematics enthusiasts, geometers, or as an educational case study combining probability statistics with geometry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(yAxis) → B
  3. Polygon(B, A, 6) → poly1, f, F, g, h, i, j, k, C, D, E
  4. Vector(F, A) → u
  5. Vector(A, D) → v
  6. Sequence(Sequence[nCr[P, Q], Q, 0, P], P, 0, rows) → pasclist
  7. Element(Element[pasclist, 3], 2) → a
  8. Sequence(Sequence[Polygon[A + (P * u) + (Q * v) + (0, 1), A + (P * u) + (Q * v), 6], Q, 0, P], P, 0, rows) → sierp
  9. Sequence(Sequence[Text[Element[Element[pasclist, P + 1], Q + 1], A + (P * u) + (Q * v) + (1; 30°) + (0, adjusty) + ((adjustx * (1, 0)) * ceil(log(10, 0.01 + Element[Element[pasclist, P + 1], Q + 1])))], Q, 0, P], P, 0, rows) → L_1
  10. Sequence(Sequence[If[Mod[Element[Element[pasclist, P + 1], Q + 1], n] ≟ 0, Polygon[A + (P * u) + (Q * v) + B, A + (P * u) + (Q * v), 6]], Q, 0, P], P, 0, rows) → sierp2
  11. Translate(sierp, (0, -1)) → sierp'
  12. Point(u) → G
  13. Point(v) → H
  14. Point(v) → I
  15. Circle(G, H) → p
  16. Mirror(G, H) → G'
  17. Mirror(G, I) → G'_1
  18. Circle(H, G') → q
  19. Circle(I, G'_1) → r
  20. Circle(G', H) → s
  21. Circle(G'_1, I) → t
  22. Mirror(H, G') → H'
  23. Mirror(I, G'_1) → I'
  24. Midpoint(G', G'_1) → J
  25. Circle(J, H) → c_1
  26. Circle(H', G') → d_1
  27. Circle(I', G'_1) → e_1
  28. Intersect(s, c_1, 2) → K
  29. Intersect(t, c_1, 1) → L
  30. Circle(K, J) → f_1
  31. Circle(L, J) → g_1
  32. Mirror(G', H') → G''
  33. Mirror(G'_1, I') → M
  34. Mirror(H', G'') → H''
  35. Mirror(I', M) → I''
  36. Circle(G'', H') → h_1
  37. Circle(M, I') → k_1
  38. Intersect(d_1, f_1, 2) → N
  39. Intersect(e_1, g_1, 1) → O
  40. Circle(N, K) → p_1
  41. Circle(O, L) → q_1
  42. Intersect(f_1, g_1, 2) → P
  43. Circle(P, L) → r_1
  44. Intersect(h_1, p_1, 2) → Q
  45. Intersect(p_1, r_1, 2) → R
  46. Intersect(q_1, r_1, 2) → S
  47. Intersect(k_1, q_1, 1) → T
  48. Circle(H'', Q) → s_1
  49. Circle(Q, R) → t_1
  50. Circle(R, S) → c_2
  51. Circle(S, T) → d_2
  52. Circle(T, I'') → e_2
  53. Circle(I'', T) → f_2
  54. Intersect(s_1, t_1) → U, V, ,
  55. Intersect(t_1, c_2, 2) → W
  56. Intersect(c_2, d_2, 2) → Z
  57. Intersect(d_2, e_2, 2) → A_1
  58. Intersect(e_2, f_2, 2) → B_1
  59. Circle(V, Q) → g_2
  60. Circle(W, R) → h_2
  61. Circle(Z, S) → k_2
  62. Circle(A_1, T) → p_2
  63. Circle(B_1, I'') → q_2
  64. Intersect(s_1, g_2, 2) → C_1
  65. Intersect(f_2, q_2, 1) → D_1
  66. Intersect(p_2, q_2, 2) → E_1
  67. Intersect(k_2, p_2, 2) → F_1
  68. Intersect(h_2, k_2, 2) → G_1
  69. Intersect(g_2, h_2, 2) → H_1
  70. Circle(C_1, V) → r_2
  71. Circle(D_1, B_1) → s_2
  72. Circle(H_1, V) → t_2
  73. Circle(G_1, Z) → c_3
  74. Circle(F_1, A_1) → d_3
  75. Circle(E_1, B_1) → e_3
  76. Intersect(g_2, r_2, 1) → I_1
  77. Intersect(q_2, s_2, 2) → J_1
  78. Circle(I_1, V) → f_3
  79. Circle(J_1, B_1) → g_3
  80. Intersect(r_2, f_3, 2) → K_1
  81. Intersect(s_2, g_3, 2) → M_1
  82. Circle(K_1, I_1) → h_3
  83. Circle(M_1, J_1) → k_3
  84. Intersect(f_3, h_3, 1) → N_1
  85. Intersect(t_2, f_3, 2) → O_1
  86. Intersect(t_2, c_3, 2) → P_1
  87. Intersect(c_3, d_3, 2) → Q_1
  88. Intersect(d_3, e_3, 2) → R_1
  89. Intersect(e_3, g_3, 2) → S_1
  90. Intersect(g_3, k_3, 2) → T_1
  91. Circle(N_1, I_1) → p_3
  92. Circle(O_1, H_1) → q_3
  93. Circle(P_1, G_1) → r_3
  94. Circle(Q_1, F_1) → s_3
  95. Circle(R_1, E_1) → t_3
  96. Circle(S_1, J_1) → c_4
  97. Circle(T_1, J_1) → d_4
  98. Intersect(h_3, p_3, 2) → U_1
  99. Intersect(k_3, d_4, 2) → V_1
  100. Circle(U_1, K_1) → e_4
  101. Circle(V_1, M_1) → f_4
  102. Intersect(p_3, e_4, 1) → W_1
  103. Intersect(p_3, q_3, 2) → Z_1
  104. Intersect(q_3, r_3, 2) → A_2
  105. Intersect(r_3, s_3, 2) → B_2
  106. Intersect(s_3, t_3, 2) → C_2
  107. Intersect(t_3, c_4, 2) → D_2
  108. Intersect(c_4, d_4, 2) → E_2
  109. Intersect(d_4, f_4, 2) → F_2
  110. Circle(W_1, N_1) → g_4
  111. Circle(Z_1, O_1) → h_4
  112. Circle(A_2, P_1) → k_4
  113. Circle(B_2, Q_1) → p_4
  114. Circle(C_2, R_1) → q_4
  115. Circle(D_2, S_1) → r_4
  116. Circle(E_2, T_1) → s_4
  117. Circle(F_2, T_1) → t_4
  118. Intersect(e_4, f_4) → G_2, , ,
  119. Mirror(Z_1, W_1) → H_2
  120. Mirror(E_2, F_2) → I_2
  121. Circle(H_2, W_1) → c_5
  122. Circle(I_2, F_2) → d_5
  123. Mirror(U_1, H_2) → J_2
  124. Mirror(N_1, W_1) → L_2
  125. Mirror(O_1, Z_1) → M_2
  126. Mirror(P_1, A_2) → Q_2
  127. Mirror(Q_1, B_2) → S_2
  128. Mirror(R_1, C_2) → U_2
  129. Mirror(S_1, D_2) → W_2
  130. Mirror(R_1, D_2) → A_3
  131. Mirror(S_1, E_2) → C_3
  132. Mirror(T_1, F_2) → E_3
  133. Mirror(V_1, I_2) → G_3
  134. Circle(J_2, H_2) → e_5
  135. Circle(L_2, W_1) → f_5
  136. Circle(M_2, Z_1) → g_5
  137. Circle(Q_2, A_2) → h_5
  138. Circle(S_2, B_2) → k_5
  139. Circle(U_2, C_2) → p_5
  140. Circle(W_2, D_2) → q_5
  141. Circle(A_3, E_2) → r_5
  142. Circle(C_3, F_2) → s_5
  143. Circle(E_3, I_2) → t_5
  144. Circle(G_3, I_2) → c_6
  145. Intersect(e_5, f_5, 2) → K_2
  146. Circle(K_2, J_2) → d_6
  147. Intersect(e_5, d_6, 2) → N_2
  148. Circle(N_2, J_2) → e_6
  149. Intersect(f_5, g_5, 2) → O_2
  150. Circle(O_2, M_2) → f_6
  151. Intersect(g_5, h_5, 2) → P_2
  152. Circle(P_2, Q_2) → g_6
  153. Intersect(h_5, k_5, 2) → R_2
  154. Circle(R_2, S_2) → h_6
  155. Intersect(k_5, p_5, 2) → T_2
  156. Circle(T_2, U_2) → k_6
  157. Intersect(p_5, q_5, 2) → V_2
  158. Circle(V_2, W_2) → p_6
  159. Intersect(q_5, r_5, 2) → Z_2
  160. Circle(Z_2, A_3) → q_6
  161. Intersect(r_5, s_5, 2) → B_3
  162. Circle(B_3, C_3) → r_6
  163. Intersect(s_5, t_5, 2) → D_3
  164. Circle(D_3, E_3) → s_6
  165. Intersect(t_5, c_6, 2) → F_3
  166. Circle(F_3, G_3) → t_6
  167. Intersect(c_6, t_6, 1) → H_3
  168. Circle(H_3, G_3) → c_7
  169. Intersect(d_6, e_6, 2) → I_3
  170. Circle(I_3, N_2) → d_7
  171. Intersect(e_6, d_7, 2) → J_3
  172. Circle(J_3, N_2) → e_7
  173. Intersect(d_6, f_6, 2) → K_3
  174. Circle(K_3, O_2) → f_7
  175. Intersect(f_6, g_6, 2) → L_3
  176. Circle(L_3, P_2) → g_7
  177. Intersect(g_6, h_6, 2) → M_3
  178. Circle(M_3, R_2) → h_7
  179. Intersect(h_6, k_6, 2) → N_3
  180. Circle(N_3, T_2) → k_7
  181. Intersect(k_6, p_6, 2) → O_3
  182. Circle(O_3, T_2) → p_7
  183. Intersect(p_6, q_6, 2) → P_3
  184. Circle(P_3, Z_2) → q_7
  185. Intersect(q_6, r_6, 2) → Q_3
  186. Circle(Q_3, B_3) → r_7
  187. Intersect(r_6, s_6, 2) → R_3
  188. Circle(R_3, B_3) → s_7
  189. Intersect(s_6, t_6, 2) → S_3
  190. Circle(S_3, D_3) → t_7
  191. Intersect(t_6, c_7, 2) → T_3
  192. Circle(T_3, H_3) → c_8
  193. Intersect(c_7, c_8, 1) → U_3
  194. Circle(U_3, H_3) → d_8

AI Drawing Prompt Reference

Draw the geometric connection between Pascal's Triangle and Metatron's Cube. Start by creating a hexagon, then use vectors and sequences to generate a Sierpinski triangle. Place text based on the values of Pascal's Triangle on the Sierpinski triangle. Finally, construct the geometric shapes of Metatron's Cube, including multiple circles and their intersections, forming a complex geometric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.