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Step 7: GeoGebra Polygonal Pyramid with Vector Transformations and Cosine

Step 7: GeoGebra Polygonal Pyramid with Vector Transformations and Cosine
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2015-12-09 07:52
PolygonVectorSolid GeometryTransformationGeometric Construction
In collections Hands in the Air: Square Translation Tessellation
Step 7 demonstrates constructing a polygonal pyramid in GeoGebra using polygons, circles, vectors, and translations to illustrate cosine-related geometric relationships. The core concepts include polygonal geometry, vector operations, spatial transformations, the geometric interpretation of cosine, and the analysis of solid structures. The construction uses GeoGebra commands such as Polygon, Segment, Vector, Translate, and Circle, with objects like poly1, A, B, C, etc., to build the layers of the pyramid. This example is suitable for high school or early university mathematics, computer-aided geometry design, and STEM education.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Segment(B, E) → e
  3. Segment(E, F) → f
  4. Segment(F, G) → g
  5. Segment(G, H) → h
  6. Segment(H, I) → i
  7. Segment(I, J) → j
  8. Segment(J, K) → k
  9. Segment(K, L) → l
  10. Segment(L, M) → m
  11. Segment(M, N) → n
  12. Segment(N, O) → p
  13. Segment(O, P) → q
  14. Segment(P, Q) → r
  15. Segment(Q, R) → s
  16. Segment(R, S) → t
  17. Segment(S, T) → a_1
  18. Segment(T, U) → b_1
  19. Segment(U, V) → c_1
  20. Segment(V, W) → d_1
  21. Segment(W, Z) → e_1
  22. Segment(Z, A_1) → f_1
  23. Segment(A_1, B_1) → g_1
  24. Segment(B_1, C_1) → h_1
  25. Segment(C_1, C) → i_1
  26. Vector(B, A) → u
  27. Translate(B, u) → B'
  28. Translate(C, u) → C'
  29. Segment(B', C') → b'
  30. Translate(B, u) → B'_1
  31. Translate(E, u) → E'
  32. Segment(B'_1, E') → e'
  33. Translate(E, u) → E'_1
  34. Translate(F, u) → F'
  35. Segment(E'_1, F') → f'
  36. Translate(F, u) → F'_1
  37. Translate(G, u) → G'
  38. Segment(F'_1, G') → g'
  39. Translate(G, u) → G'_1
  40. Translate(H, u) → H'
  41. Segment(G'_1, H') → h'
  42. Translate(H, u) → H'_1
  43. Translate(I, u) → I'
  44. Segment(H'_1, I') → i'
  45. Translate(I, u) → I'_1
  46. Translate(J, u) → J'
  47. Segment(I'_1, J') → j'
  48. Translate(J, u) → J'_1
  49. Translate(K, u) → K'
  50. Segment(J'_1, K') → k'
  51. Translate(K, u) → K'_1
  52. Translate(L, u) → L'
  53. Segment(K'_1, L') → l'
  54. Translate(L, u) → L'_1
  55. Translate(M, u) → M'
  56. Segment(L'_1, M') → m'
  57. Translate(M, u) → M'_1
  58. Translate(N, u) → N'
  59. Segment(M'_1, N') → n'
  60. Translate(N, u) → N'_1
  61. Translate(O, u) → O'
  62. Segment(N'_1, O') → p'
  63. Translate(O, u) → O'_1
  64. Translate(P, u) → P'
  65. Segment(O'_1, P') → q'
  66. Translate(P, u) → P'_1
  67. Translate(Q, u) → Q'
  68. Segment(P'_1, Q') → r'
  69. Translate(Q, u) → Q'_1
  70. Translate(R, u) → R'
  71. Segment(Q'_1, R') → s'
  72. Translate(R, u) → R'_1
  73. Translate(S, u) → S'
  74. Segment(R'_1, S') → t'
  75. Translate(S, u) → S'_1
  76. Translate(T, u) → T'
  77. Segment(S'_1, T') → j_1
  78. Translate(T, u) → T'_1
  79. Translate(U, u) → U'
  80. Segment(T'_1, U') → k_1
  81. Translate(U, u) → U'_1
  82. Translate(V, u) → V'
  83. Segment(U'_1, V') → l_1
  84. Translate(V, u) → V'_1
  85. Translate(W, u) → W'
  86. Segment(V'_1, W') → m_1
  87. Translate(W, u) → W'_1
  88. Translate(Z, u) → Z'
  89. Segment(W'_1, Z') → n_1
  90. Translate(Z, u) → Z'_1
  91. Translate(A_1, u) → D_1
  92. Segment(Z'_1, D_1) → p_1
  93. Translate(A_1, u) → E_1
  94. Translate(B_1, u) → F_1
  95. Segment(E_1, F_1) → q_1
  96. Translate(B_1, u) → G_1
  97. Translate(C_1, u) → H_1
  98. Segment(G_1, H_1) → r_1
  99. Translate(C_1, u) → I_1
  100. Translate(C, u) → C'_1
  101. Segment(I_1, C'_1) → s_1
  102. Segment(B, J_1) → t_1
  103. Segment(J_1, K_1) → a_2
  104. Segment(K_1, L_1) → b_2
  105. Segment(L_1, M_1) → c_2
  106. Segment(M_1, N_1) → d_2
  107. Segment(N_1, O_1) → e_2
  108. Segment(O_1, P_1) → f_2
  109. Segment(P_1, Q_1) → g_2
  110. Segment(Q_1, R_1) → h_2
  111. Segment(R_1, A) → i_2
  112. Vector(A, D) → v
  113. Translate(A, v) → A'
  114. Translate(B, v) → B'_2
  115. Segment(A', B'_2) → a'
  116. Translate(B'_1, v) → S_1
  117. Translate(B, v) → B'_3
  118. Translate(J_1, v) → U_1
  119. Segment(B'_3, U_1) → j_2
  120. Translate(J_1, v) → V_1
  121. Translate(K_1, v) → W_1
  122. Segment(V_1, W_1) → k_2
  123. Translate(K_1, v) → Z_1
  124. Translate(L_1, v) → A_2
  125. Segment(Z_1, A_2) → l_2
  126. Translate(L_1, v) → B_2
  127. Translate(M_1, v) → C_2
  128. Segment(B_2, C_2) → m_2
  129. Translate(M_1, v) → D_2
  130. Translate(N_1, v) → E_2
  131. Segment(D_2, E_2) → n_2
  132. Translate(N_1, v) → F_2
  133. Translate(O_1, v) → G_2
  134. Segment(F_2, G_2) → p_2
  135. Translate(O_1, v) → H_2
  136. Translate(P_1, v) → I_2
  137. Segment(H_2, I_2) → q_2
  138. Translate(P_1, v) → J_2
  139. Translate(Q_1, v) → K_2
  140. Segment(J_2, K_2) → r_2
  141. Translate(Q_1, v) → L_2
  142. Translate(R_1, v) → M_2
  143. Segment(L_2, M_2) → s_2
  144. Translate(R_1, v) → N_2
  145. Translate(A, v) → A'_1
  146. Segment(N_2, A'_1) → t_2
  147. Translate(u, v) → u'
  148. Translate(B', v) → B''
  149. Circle(T_1, O_2) → c_3
  150. Circle(P_2, Q_2) → d_3
  151. Segment(Q_1, J') → a_3
  152. Polygon(A, R_1, Q_1, J', I', H', G', F', E') → poly2, a_4, r_3, q_3, j'_1, i'_1, h'_1, g'_1, f'_1, e'_1
  153. Polygon(R_2, S_2, T_2, U_2, V_2, W_2) → poly3, r_4, s_3, t_3, u_2, v_2, w_2
  154. Polygon(Q_1, P_1, F, G, H, I, J, K, L, M, N, O, O_1, P, Q, R, S, T, U, V, W, Z, A_1, N_1, B_1, C_1, C, U_1, W_1, A_2, C_2, E_2, G_2, I_2, M_1, K_2, M_2, A', H_1, F_1, D_1, Z', W', V', T', L_1, S', R', Q', P', O', N', M', L', K', J', K_1, J_1, B, E) → poly4, q_4, e_3, f_3, g_3, h_3, i_3, j_4, k_4, l_4, m_4, n_4, p_3, o, p_4, q_5, r_5, s_4, t_4, u_1, v_1, w, z_1, o_1, a_5, b_4, c_4, c_5, u_3, w_1, a_6, c_6, e_4, g_4, n_3, i_4, k_5, m_5, a'_1, h_4, f_4, d_4, z', w', v', m_3, t'_1, s'_1, r'_1, q'_1, p'_1, o', n'_1, m'_1, l'_1, k'_1, l_3, j'_2, k_3, j_3, b_3
  155. Polygon(H_1, M_2, A') → poly5, a'_2, h_5, m_6
  156. Polygon(Q_1, R_1, A, E', F', G', H', I', J') → poly6, q_6, r_6, a_7, e'_2, f'_2, g'_2, h'_2, i'_2, j'_3

AI Drawing Prompt Reference

Draw a polyhedron with a quadrilateral base. The top is formed by translating vertices along a vector and connecting them. Key objects include the base polygon, translated points, and connected edges. The construction order is: draw the base polygon, determine a vector, translate the base vertices along it, then connect the translated points to form the polyhedron's sides.
The prompt above was compiled from the original drawing commands and is for reference only.