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Non-Lamp Geometric Demonstration

Non-Lamp Geometric Demonstration
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2019-09-20 23:52
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the construction and properties of a non-traditional geometric shape, the dodecahedron. It involves core mathematical concepts including solid geometry, midpoints, line-plane relationships, normal distribution (for point generation), and coordinate geometry. GeoGebra commands used in the construction include Point, Dodecahedron, Midpoint, Line, Intersect, Translate, Plane, Ray, Rotate, OrthogonalLine, IntersectPath, Distance, PointIn, Circle, Mirror, Polygon, and Sequence. It is suitable for high school and above students for learning solid geometry and spatial visualization skills.

GeoGebra Commands & Parameters

  1. Point(Circle[((A * (1 - sqrt(5))) + (B * (3 + sqrt(5)))) / 4, (Distance[A, B] * sqrt(10 + (2 * sqrt(5))) / 4), Segment[A, B]]) → C
  2. Dodecahedron(A, B, C) → a, D, M, N, O, P, Q, R, S, T, 면ABCDE, 면AEIOJ, E, 면ABFKJ, 면BCGLF, 면CDHMG, 면DEINH, 면INSTO, 면JKPTO, 면FKPQL, 면GLQRM, 면HMRSN, 면PQRST, F, 모서리AE, 모서리AB, 모서리BC, 모서리CD, 모서리DE, 모서리AJ, 모서리EI, 모서리IO, 모서리JO, 모서리JK, G, 모서리FK, 모서리BF, 모서리FL, 모서리GL, 모서리CG, 모서리GM, 모서리HM, 모서리DH, 모서리HN, 모서리IN, H, 모서리NS, 모서리ST, 모서리OT, 모서리PT, 모서리KP, 모서리PQ, 모서리LQ, 모서리QR, 모서리MR, 모서리RS, I, J, K, L
  3. Midpoint(A, R) → U
  4. Line(P, L) → h
  5. Line(Q, G) → g
  6. Line(Q, K) → f
  7. Intersect(h, f) → V
  8. Translate(V, Vector[(0.9 * u)]) → W
  9. Translate(W, Vector[(-5.2 * u) - (3.3 * v)]) → W'_1
  10. Translate(W, Vector[(2 * v)]) → W'
  11. Translate(W', Vector[(0.6 * u)]) → W''
  12. Plane(g, f) → p
  13. Ray(U, l3(A12)) → B12
  14. Ray(U, Centroid[l2(A6)]) → C6
  15. Rotate(Rotate[W'', (-(72°)), C6], 120°, B12) → P_1
  16. Rotate(Rotate[W', (-(72°)), C6], 120°, B12) → O_1
  17. Rotate(Rotate[p, (-(72°)), C6], 120°, B12) → q
  18. OrthogonalLine(O_1, q) → n
  19. Plane(P_1, n) → r
  20. IntersectPath(p, r) → s
  21. Point(s) → U_1
  22. OrthogonalLine(U_1, s, p) → c
  23. Distance(U, J) → b
  24. Translate(W, Vector[(0.2 * u)]) → Z
  25. Translate(W, Vector[(0.6 * u)]) → A_1
  26. Translate(W, Vector[(1.7 * u)]) → C_1
  27. Translate(V, Vector[(3.4 * v)]) → B_1
  28. Translate(C_1, Vector[(3.4 * v)]) → D_1
  29. Translate(C_1, Vector[(-1.1 * v)]) → E_1
  30. Translate(V, Vector[(-1.1 * v)]) → F_1
  31. Translate(V, Vector[(-0.5 * v)]) → V'
  32. Translate(V, Vector[(-0.9 * v)]) → V'_1
  33. Translate(C_1, Vector[(-0.9 * v)]) → G_1
  34. Translate(C_1, Vector[(-0.5 * v)]) → H_1
  35. Translate(Q, Vector[(-((Distance[Q, W] * v)))]) → I_1
  36. Translate(I_1, Vector[(0.3 * u)]) → J_1
  37. Translate(I_1, Vector[(0.3 * u) + (0.2 * v)]) → K_1
  38. Translate(I_1, Vector[(0.2 * v)]) → L_1
  39. Translate(J_1, Vector[(-((0.3 * v)))]) → M_1
  40. Translate(J_1, Vector[(1.1 * v)]) → N_1
  41. Ray(U, l3(A1)) → B1
  42. Ray(U, l3(A2)) → B2
  43. Ray(U, l3(A3)) → B3
  44. Ray(U, l3(A4)) → B4
  45. Ray(U, l3(A5)) → B5
  46. Ray(U, l3(A6)) → B6
  47. Ray(U, l3(A7)) → B7
  48. Ray(U, l3(A8)) → B8
  49. Ray(U, l3(A9)) → B9
  50. Ray(U, l3(A10)) → B10
  51. Ray(U, l3(A11)) → B11
  52. Ray(U, l3(A13)) → B13
  53. Ray(U, l3(A14)) → B14
  54. Ray(U, l3(A15)) → B15
  55. Ray(U, l3(A16)) → B16
  56. Ray(U, l3(A17)) → B17
  57. Ray(U, l3(A18)) → B18
  58. Ray(U, l3(A19)) → B19
  59. Ray(U, l3(A20)) → B20
  60. Ray(U, Centroid[l2(A1)]) → C1
  61. Ray(U, Centroid[l2(A2)]) → C2
  62. Ray(U, Centroid[l2(A3)]) → C3
  63. Ray(U, Centroid[l2(A4)]) → C4
  64. Ray(U, Centroid[l2(A5)]) → C5
  65. Ray(U, Centroid[l2(A7)]) → C7
  66. Ray(U, Centroid[l2(A8)]) → C8
  67. Ray(U, Centroid[l2(A9)]) → C9
  68. Ray(U, Centroid[l2(A10)]) → C10
  69. Ray(U, Centroid[l2(A11)]) → C11
  70. Ray(U, Centroid[l2(A12)]) → C12
  71. Translate(W, Vector[(-3.5 * u) - (1.4 * v)]) → W'_2
  72. Translate(W'_1, Vector[(0.4 * UnitVector[c])]) → R_1
  73. Line(R_1, W'_1) → t
  74. Translate(W'_1, Vector[(0.3 * UnitVector[t])]) → S_1
  75. Translate(R_1, Vector[(-0.3 * UnitVector[t])]) → T_1
  76. Translate(U_1, Vector[(0.3 / tan(36°) * UnitVector[c])]) → V_1
  77. Point(s) → W_1
  78. Translate(V_1, Vector[U_1, W_1]) → Z_1
  79. Point(Circle[W'_2, 0.4, p]) → Q_1
  80. OrthogonalLine(S_1, t, p) → d
  81. Translate(S_1, Vector[(1.3 * UnitVector[d])]) → A_2
  82. Translate(T_1, Vector[(1.3 * UnitVector[d])]) → B_2
  83. PointIn(p) → C_2
  84. PointIn(p) → D_2
  85. PointIn(p) → E_2
  86. Point(d) → F_2
  87. PointIn(p) → G_2
  88. PointIn(p) → H_2
  89. PointIn(p) → I_2
  90. PointIn(p) → J_2
  91. PointIn(p) → K_2
  92. PointIn(p) → L_2
  93. Translate(N_1, Vector[(1.25 * u) - (0.5 * v)]) → M_2
  94. Translate(N_1, Vector[(1.25 * u) - (0.9 * v)]) → N_2
  95. PointIn(p) → O_2
  96. PointIn(p) → P_2
  97. PointIn(p) → Q_2
  98. PointIn(p) → R_2
  99. PointIn(p) → S_2
  100. PointIn(p) → V_2
  101. PointIn(p) → W_2
  102. PointIn(p) → Z_2
  103. PointIn(p) → A_3
  104. PointIn(p) → B_3
  105. Rotate(Rotate[Rotate[W', 72°, C6], (-(120°)), B16], (-(144°)), C2) → C_3
  106. Rotate(Rotate[Rotate[W'', 72°, C6], (-(120°)), B16], (-(144°)), C2) → D_3
  107. Line(C_3, D_3) → e
  108. Line(O_1, P_1) → i_1
  109. Translate(P_1, Vector[(0.2 * UnitVector[i_1])]) → H_3
  110. Translate(O_1, Vector[(0.2 * UnitVector[i_1])]) → I_3
  111. OrthogonalLine(P_1, i_1, r) → j_1
  112. Translate(P_1, Vector[(-0.3 * UnitVector[j_1])]) → J_3
  113. Translate(H_3, Vector[(-0.3 * UnitVector[j_1])]) → K_3
  114. Translate(I_3, Vector[(-0.3 * UnitVector[j_1])]) → L_3
  115. Translate(O_1, Vector[(-0.3 * UnitVector[j_1])]) → M_3
  116. Circle(V, 4.2, r) → c_1
  117. Translate(O_1, Vector[(-0.3 * UnitVector[i_1])]) → N_3
  118. Translate(H_3, Vector[(0.3 * UnitVector[i_1])]) → O_3
  119. Translate(I_3, Vector[(1.25 * UnitVector[j_1])]) → P_3
  120. Translate(P_1, Vector[(1.25 * UnitVector[j_1])]) → Q_3
  121. Translate(Q_3, Vector[(-0.4 * UnitVector[j_1]) + (0.4 * UnitVector[i_1])]) → H_4
  122. Translate(P_3, Vector[(-0.4 * UnitVector[j_1]) - (0.4 * UnitVector[i_1])]) → I_4
  123. Midpoint(D_3, P_1) → E_3
  124. Translate(V_1, Vector[(-0.3 * UnitOrthogonalVector[p])]) → F_3
  125. Line(Q_3, j_1) → k_1
  126. Mirror(W_1, E_3) → G_3
  127. Line(G_3, F_3) → l_1
  128. Line(W_1, l_1) → a_1
  129. Line(P_3, j_1) → f_3
  130. Line(P_3, Q_3) → g_3
  131. Intersect(s, g_3) → R_3
  132. Intersect(k_1, l_1) → S_3
  133. Intersect(a_1, f_3) → T_3
  134. Mirror(G_3, E_3) → U_3
  135. Mirror(F_3, E_3) → V_3
  136. Mirror(S_3, E_3) → W_3
  137. Mirror(Q_3, E_3) → Z_3
  138. Mirror(H_4, E_3) → A_4
  139. Mirror(O_3, E_3) → B_4
  140. Mirror(P_1, E_3) → C_4
  141. Mirror(J_3, E_3) → D_4
  142. Mirror(K_3, E_3) → E_4
  143. Mirror(H_3, E_3) → F_4
  144. Mirror(O_1, E_3) → G_4
  145. Mirror(M_3, E_3) → J_4
  146. Mirror(L_3, E_3) → K_4
  147. Mirror(I_3, E_3) → L_4
  148. Mirror(N_3, E_3) → M_4
  149. Mirror(I_4, E_3) → N_4
  150. Mirror(P_3, E_3) → O_4
  151. Mirror(T_3, E_3) → P_4
  152. Mirror(W_1, E_3) → Q_4
  153. Translate(M_2, Vector[(-0.4 * u) + (0.4 * v)]) → T_2
  154. Translate(N_2, Vector[(-0.4 * u) - (0.4 * v)]) → U_2
  155. Translate(J_1, Vector[(0.6 * v)]) → R_4
  156. Translate(I_1, Vector[(0.6 * v)]) → S_4
  157. Translate(L_1, Vector[(0.6 * v)]) → T_4
  158. Translate(K_1, Vector[(0.6 * v)]) → U_4
  159. Polygon(V', B_1, U_2, M_1, J_1, I_1, L_1, K_1, R_4, S_4, T_4, U_4, D_1, N_1, T_2, M_2, S_2, R_2, Q_2, P_2, O_2, G_1, E_1, H_1, F_1, V'_1, L_2, Q_1, R_1, T_1, B_2, K_2, J_2, I_2, B_3, H_2, G_2, F_2, E_2, W_1, Z_1, V_1, U_1, D_2, A_2, A_3, S_1, W'_1, W'_2, C_2, Z_2, W_2, V_2, N_2) → 다각형1, i, n_1, p_1, q_1, r_1, s_1, t_1, b_1, d_1, e_1, f_2, j, g_2, h_2, i_2, j_2, k_2, l_2, m_2, n_2, p_2, q_2, k, r_2, s_2, t_2, a_2, b_2, c_2, d_2, e_2, h_3, a_3, l, b_3, r_4, s_4, t_4, l_5, m_5, n_5, p_5, q_5, r_5, m, s_5, t_5, a_5, b_5, c_5, f_1, g_1, h_1, m_1
  160. Polygon(O_3, H_4, L_4, K_4, J_4, C_3, F_4, E_4, D_4, C_4, B_4, A_4, Q_3, Z_3, W_3, V_3, W_1, T_3, P_3, I_4, N_3, I_3, L_3, S_3, M_3, O_1, H_3, K_3, J_3, P_1, F_3, G_3, P_4, O_4, N_4, M_4) → 다각형2, i_3, s_3, t_3, c_3, d_3, e_3, f_4, g_4, h_4, i_4, j_4, j_3, k_4, l_4, m_4, n_4, p_4, q_4, a_4, b_4, c_4, d_4, k_3, e_4, f_5, g_5, h_5, i_5, j_5, k_5, l_3, m_3, n_3, p_3, q_3, r_3
  161. Sequence(Rotate[Prism[다각형1, -0.3], ((ω * 2) * π / 5), C6], ω, 0, 4) → l4
  162. Rotate(Prism[다각형2, 0.3], 120°, B11) → o
  163. Rotate(l4, 120°, B17) → l4'
  164. Rotate(l4', 120°, B17) → l4''
  165. Rotate(l4', 120°, B13) → l4''_1
  166. Rotate(l4''_1, 120°, B13) → l5
  167. Rotate(l5, 120°, B14) → l5'
  168. Rotate(l5', 120°, B14) → l5''
  169. Rotate(l5'', 120°, B15) → l5'''
  170. Rotate(l5''', 120°, B15) → l6
  171. Rotate(l4, 120°, B6) → l4'_1
  172. Rotate(l4'_1, 120°, B6) → l7
  173. Rotate(l7, 120°, B3) → l7'
  174. Rotate(o, 120°, B16) → o'
  175. Rotate(o', 120°, B16) → o''
  176. Rotate(o', 120°, B17) → o''_1
  177. Rotate(o''_1, 120°, B17) → w
  178. Rotate(w, 120°, B18) → w'
  179. Rotate(w', 120°, B18) → w''
  180. Rotate(w'', 120°, B19) → w'''
  181. Rotate(w''', 120°, B19) → o_1
  182. Rotate(w''', 120°, B14) → u_1
  183. Rotate(u_1, 120°, B14) → v_1
  184. Rotate(v_1, 120°, B9) → w_1
  185. Rotate(w_1, 120°, B9) → o_2
  186. Rotate(o_2, 120°, B15) → u_2
  187. Rotate(u_2, 120°, B15) → v_2
  188. Rotate(u_2, 120°, B10) → w_2
  189. Rotate(w_2, 120°, B10) → o_3
  190. Rotate(o_3, 120°, B11) → u_3
  191. Rotate(u_3, 120°, B6) → v_3
  192. Rotate(v_3, 120°, B6) → w_3
  193. Rotate(w_3, 120°, B12) → o_4
  194. Rotate(o_4, 120°, B7) → u_4
  195. Rotate(u_4, 120°, B7) → v_4
  196. Rotate(v_4, 120°, B13) → w_4
  197. Rotate(w_4, 120°, B8) → d_5
  198. Rotate(w_1, 120°, B5) → e_5
  199. Rotate(e_5, 120°, B5) → o_5
  200. Rotate(o_5, 120°, B1) → u_5
  201. Rotate(u_5, 120°, B2) → v_5
  202. Rotate(v_5, 120°, B3) → w_5