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Math and Art Combined

Math and Art Combined
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2016-05-09 04:43
PolygonTransformationGeometric Construction
This example demonstrates the perfect combination of mathematics and art. It uses geometric shapes and transformations to illustrate concepts in geometry. Commands such as Line, Point, Polygon, Intersect, LineBisector, Mirror, Rotate, Circle, Segment, Dilate, and Midpoint are used in GeoGebra. Suitable for high school students and math enthusiasts to explore geometric transformations and construction.

GeoGebra Commands & Parameters

  1. Line(A, B) → l
  2. Point(l) → G
  3. Polygon(A, G, 8) → poly1, f, D, E, F, H, I, g, h, i, j, k, m, n, C
  4. Line(A, E) → p
  5. Line(D, I) → q
  6. Line(G, F) → r
  7. Intersect(p, q) → J
  8. Line(C, H) → s
  9. LineBisector(A, I) → t
  10. LineBisector(A, G) → a
  11. LineBisector(G, C) → b
  12. LineBisector(C, D) → c
  13. Polygon(I, G, 4) → poly2, d, e, f_1, g_1, K, L
  14. Polygon(A, C, 4) → poly3, h_1, i_1, j_1, k_1, M, N
  15. Intersect(t, d) → O
  16. Polygon(I, O, 4) → poly4, l_1, m_1, n_1, p_1, P, Q
  17. Intersect(a, d) → R
  18. Polygon(R, G, 4) → poly5, q_1, r_1, s_1, t_1, S, T
  19. Intersect(t, f_1) → U
  20. Polygon(D, U, 4) → poly6, a_1, b_1, c_1, d_1, V, W
  21. Intersect(a, f_1) → Z
  22. Polygon(Z, F, 4) → poly7, e_1, f_2, g_2, h_2, A_1, B_1
  23. Intersect(p, d) → C_1
  24. Mirror(C_1, A) → D_1
  25. Rotate(D_1, 60°, A) → E_1
  26. Mirror(E_1, p) → F_1
  27. Polygon(D_1, E_1, 6) → poly8, i_2, J_1, j_2, k_2, l_2, m_2, n_2, G_1, H_1, I_1
  28. LineBisector(T, R) → p_2
  29. Intersect(p, p_2) → K_1
  30. Intersect(a, p_2) → L_1
  31. Polygon(C_1, L_1, 4) → poly9, q_2, r_2, s_2, t_2, M_1, N_1
  32. Mirror(poly9, a) → poly9'
  33. Mirror(poly9', b) → poly9''
  34. Mirror(poly9'', c) → poly9'''
  35. Mirror(poly9''', t) → poly10
  36. Mirror(poly10, a) → poly10'
  37. Mirror(poly10', b) → poly10''
  38. Mirror(poly10'', c) → poly10'''
  39. Mirror(poly8, a) → poly8'
  40. Mirror(poly8', b) → poly8''
  41. Mirror(poly8'', c) → poly8'''
  42. Mirror(poly8''', t) → poly11
  43. Mirror(poly11, a) → poly11'
  44. Mirror(poly11', b) → poly11''
  45. Mirror(poly11'', c) → poly11'''
  46. Mirror(M_1, t) → O_1
  47. Mirror(O_1, c) → P_1
  48. Mirror(P_1, b) → Q_1
  49. Mirror(Q_1, a) → R_1
  50. Mirror(R_1, t) → S_1
  51. Mirror(S_1, c) → T_1
  52. Mirror(T_1, b) → U_1
  53. Mirror(U_1, a) → V_1
  54. Line(P_1, O_1) → a_2
  55. Intersect(t, a_2) → W_1
  56. Polygon(W_1, M_1, 4) → poly12, b_2, c_2, d_2, e_2, Z_1, A_2
  57. Mirror(poly12, a) → poly12'
  58. Mirror(poly12', r) → poly12''
  59. Mirror(poly12', s) → poly12''_1
  60. Mirror(poly12''_1, q) → poly13
  61. Mirror(poly13, p) → poly13'
  62. Mirror(poly13', r) → poly13''
  63. Mirror(poly13'', s) → poly13'''
  64. Circle(J, D_1) → c_3
  65. Line(J, L_1) → f_3
  66. Mirror(B, f_3) → B'
  67. LineBisector(G_1, E_1) → g_3
  68. Intersect(f_3, g_3) → B_2
  69. Mirror(G_1, f_3) → C_2
  70. Mirror(E_1, f_3) → D_2
  71. Line(N_1, O) → h_3
  72. Intersect(c_3, h_3, 2) → E_2
  73. Segment(F_1, E_2) → i_3
  74. Mirror(F_1, h_3) → F_2
  75. Mirror(E_2, h_3) → G_2
  76. Segment(F_2, G_2) → j_3
  77. Mirror(I_1, h_3) → H_2
  78. Point(g_3) → I_2
  79. Line(I_2, j_3) → k_3
  80. Intersect(h_3, k_3) → J_2
  81. Mirror(I_2, h_3) → K_2
  82. Line(K_2, m_2) → l_3
  83. Line(I_1, H_2) → m_3
  84. Intersect(l_3, m_3) → L_2
  85. Line(L_2, l_2) → n_3
  86. Point(n_3) → M_2
  87. Line(M_2, t_2) → p_3
  88. Intersect(p_3, h_3) → N_2
  89. Mirror(C_1, h_3) → O_2
  90. Line(N_1, O_2) → q_3
  91. Mirror(p_3, h_3) → r_3
  92. Mirror(n_3, h_3) → s_3
  93. Intersect(m_3, s_3) → P_2
  94. Line(N_1, J_2) → t_3
  95. Mirror(M_2, t_3) → Q_2
  96. Polygon(M_2, N_2, Q_2, P_2, I_2, J_2, K_2, L_2) → poly14, m_4, n_4, q_4, p_4, i_4, j_4, k_4, l_4
  97. Mirror(M_2, p) → R_2
  98. Mirror(N_2, p) → S_2
  99. Mirror(Q_2, p) → T_2
  100. Mirror(P_2, p) → U_2
  101. Mirror(I_2, p) → V_2
  102. Mirror(J_2, p) → W_2
  103. Mirror(K_2, p) → Z_2
  104. Mirror(L_2, p) → A_3
  105. Polygon(R_2, S_2, T_2, U_2, V_2, W_2, Z_2, A_3) → poly14', r_4, s_4, t_4, u_2, v_2, w_2, z_2, a_3
  106. Mirror(R_2, r) → B_3
  107. Mirror(S_2, r) → C_3
  108. Mirror(T_2, r) → D_3
  109. Mirror(U_2, r) → E_3
  110. Mirror(V_2, r) → F_3
  111. Mirror(W_2, r) → G_3
  112. Mirror(Z_2, r) → H_3
  113. Mirror(A_3, r) → I_3
  114. Polygon(B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3) → poly14'', b_3, c_4, d_3, e_3, f_4, g_4, h_4, i_5
  115. Mirror(B_3, s) → J_3
  116. Mirror(C_3, s) → K_3
  117. Mirror(D_3, s) → L_3
  118. Mirror(E_3, s) → M_3
  119. Mirror(F_3, s) → N_3
  120. Mirror(G_3, s) → O_3
  121. Mirror(H_3, s) → P_3
  122. Mirror(I_3, s) → Q_3
  123. Polygon(J_3, K_3, L_3, M_3, N_3, O_3, P_3, Q_3) → poly14''', j_5, k_5, l_5, m_5, n_5, o_3, p_5, q_5
  124. Mirror(J_3, q) → R_3
  125. Mirror(K_3, q) → S_3
  126. Mirror(L_3, q) → T_3
  127. Mirror(M_3, q) → U_3
  128. Mirror(N_3, q) → V_3
  129. Mirror(O_3, q) → W_3
  130. Mirror(P_3, q) → Z_3
  131. Mirror(Q_3, q) → A_4
  132. Polygon(R_3, S_3, T_3, U_3, V_3, W_3, Z_3, A_4) → poly15, r_5, s_5, t_5, u_3, v_3, w_3, z_3, a_4
  133. Mirror(R_3, p) → B_4
  134. Mirror(S_3, p) → C_4
  135. Mirror(T_3, p) → D_4
  136. Mirror(U_3, p) → E_4
  137. Mirror(V_3, p) → F_4
  138. Mirror(W_3, p) → G_4
  139. Mirror(Z_3, p) → H_4
  140. Mirror(A_4, p) → I_4
  141. Polygon(B_4, C_4, D_4, E_4, F_4, G_4, H_4, I_4) → poly15', b_4, c_5, d_4, e_4, f_5, g_5, h_5, i_6
  142. Mirror(B_4, r) → J_4
  143. Mirror(C_4, r) → K_4
  144. Mirror(D_4, r) → L_4
  145. Mirror(E_4, r) → M_4
  146. Mirror(F_4, r) → N_4
  147. Mirror(G_4, r) → O_4
  148. Mirror(H_4, r) → P_4
  149. Mirror(I_4, r) → Q_4
  150. Polygon(J_4, K_4, L_4, M_4, N_4, O_4, P_4, Q_4) → poly15'', j_6, k_6, l_6, m_6, n_6, o_4, p_6, q_6
  151. Mirror(J_4, s) → R_4
  152. Mirror(K_4, s) → S_4
  153. Mirror(L_4, s) → T_4
  154. Mirror(M_4, s) → U_4
  155. Mirror(N_4, s) → V_4
  156. Mirror(O_4, s) → W_4
  157. Mirror(P_4, s) → Z_4
  158. Mirror(Q_4, s) → A_5
  159. Polygon(R_4, S_4, T_4, U_4, V_4, W_4, Z_4, A_5) → poly15''', r_6, s_6, t_6, u_4, v_4, w_4, z_4, a_5
  160. Mirror(F_2, q) → B_5
  161. Mirror(G_2, q) → C_5
  162. Segment(B_5, C_5) → b_5
  163. Mirror(B_5, c) → D_5
  164. Mirror(C_5, c) → E_5
  165. Segment(D_5, E_5) → d_5
  166. Mirror(Z_4, s) → F_5
  167. Mirror(D_5, s) → G_5
  168. Mirror(E_5, s) → H_5
  169. Segment(G_5, H_5) → e_5
  170. Mirror(G_5, b) → I_5
  171. Mirror(H_5, b) → J_5
  172. Segment(I_5, J_5) → f_6
  173. Mirror(I_5, r) → K_5
  174. Mirror(J_5, r) → L_5
  175. Segment(K_5, L_5) → g_6
  176. Mirror(K_5, f_3) → M_5
  177. Mirror(L_5, f_3) → N_5
  178. Segment(M_5, N_5) → h_6
  179. Mirror(M_5, p) → O_5
  180. Mirror(N_5, p) → P_5
  181. Segment(O_5, P_5) → a_6
  182. Mirror(O_5, t_3) → Q_5
  183. Mirror(P_5, t_3) → R_5
  184. Segment(Q_5, R_5) → b_6
  185. Mirror(Q_5, q) → S_5
  186. Mirror(R_5, q) → T_5
  187. Segment(S_5, T_5) → c_6
  188. Mirror(S_5, c) → U_5
  189. Mirror(T_5, c) → V_5
  190. Segment(U_5, V_5) → d_6
  191. Mirror(U_5, s) → W_5
  192. Mirror(V_5, s) → Z_5
  193. Segment(W_5, Z_5) → e_6
  194. Mirror(W_5, b) → A_6
  195. Mirror(Z_5, b) → B_6
  196. Segment(A_6, B_6) → f_7
  197. Mirror(A_6, r) → C_6
  198. Mirror(B_6, r) → D_6
  199. Segment(C_6, D_6) → g_7
  200. Mirror(C_6, f_3) → E_6
  201. Mirror(D_6, f_3) → F_6
  202. Segment(E_6, F_6) → h_7
  203. Dilate(poly11''', 0.85, I) → poly16
  204. Dilate(poly11'', 0.85, H) → poly11'''_1
  205. Dilate(poly11', 0.85, F) → poly11''_1
  206. Dilate(poly11, 0.85, E) → poly11'_1
  207. Dilate(poly8''', 0.85, D) → poly17
  208. Dilate(poly8'', 0.85, C) → poly8'''_1
  209. Midpoint(U_1, Q_1) → G_6
  210. Mirror(poly11''_1, G_6) → poly18
  211. Mirror(poly11'_1, G_6) → poly19

AI Drawing Prompt Reference

Start with a line segment and a point on it, construct an octagon, then draw diagonals to create auxiliary lines. Use their intersections to build several squares and a hexagon, guided by angle bisectors and perpendicular bisectors. Mirror and rotate these polygons across the symmetry axes to produce a symmetric geometric artwork that blends mathematics with visual art.
The prompt above was compiled from the original drawing commands and is for reference only.