彭罗斯三角形

彭罗斯三角形
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2018-01-16 16:26
①本范例演示了彭罗斯三角形的构造与性质,这是一个著名的视觉悖论,由三个互相连接但无法拼接成三维空间的三角形组成;②涉及的核心数学概念包括多边形构造、立体几何、向量运算和空间几何关系;③作图要点提炼自构造摘要,使用了Polygon(多边形)、Prism(棱柱)、Vector(向量)、Plane(平面)、IntersectPath(路径交点)、Segment(线段)等GeoGebra命令与对象;④适合高中及以上学生学习立体几何、空间想象能力和几何作图技巧,也可用于数学竞赛或兴趣拓展。

GeoGebra 命令及参数

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Prism(poly1, l) → e, E, edgeAE, edgeBF, edgeCG, edgeDH, edgeEF, edgeFG, edgeGH, edgeEH, F, G, H, faceABFE, faceBCGF, faceCDHG, faceADHE, faceEFGH
  3. Polygon(I, H, G, J) → poly2, i, h, g, j
  4. Prism(poly2, l - 1) → f, K, edge1, edge2, edge3, edge4, edge5, edge6, edge7, edge8, L, M, N, face1, face2, face3, face4, face5
  5. Polygon(C, O, P, B) → poly3, c_1, o, p, b_1
  6. Prism(poly3, l) → k, Q, edgeCQ, edgeOR, edgePS, edgeBT, edgeQR, edgeRS, edgeST, edgeQT, R, S, T, faceCORQ, faceOPSR, faceBPST, faceBCQT, faceQRST
  7. Vector(N, Q) → u
  8. Plane(N, edgeCQ) → m
  9. Plane(S, m) → n
  10. IntersectPath(n, f) → poly4, U, V, W, Z, q, r, s, t
  11. Polygon(A, E, V, U, I, D) → poly5, a_1, e_1, v, u_1, i_1, d_1
  12. Polygon(E, F, W, V) → poly6, e_2, f_1, w, v_1
  13. Polygon(Q, D, I, J, O, R) → poly7, q_1, d_2, i_2, j_1, o_1, r_1
  14. Polygon(O, P, S, R) → poly8, o_2, p_1, s_1, r_2
  15. Polygon(W, Z, J, O, P, F) → poly9, w_1, z_1, j_2, o_3, p_2, f_2
  16. Polygon(R, S, T, Q) → poly10, r_3, s_2, t_1, q_2
  17. Polygon(T, A, D, Q) → poly11, t_2, a_2, d_3, q_3
  18. Polygon(E, A, T, S, P, F) → poly12, e_3, a_3, t_3, s_3, p_3, f_3
  19. Polygon(J, I, U, Z) → poly13, j_3, i_3, u_2, z_2
  20. Polygon(Q, R, O, D) → poly14, q_4, r_4, o_4, d_4
  21. Polygon(D, I, J, O) → poly15, d_5, i_4, j_4, o_5
  22. Polygon(A, E, I, D) → poly16, a_4, e_4, i_5, d_6
  23. Polygon(I, E, V, U) → poly17, i_6, e_5, v_2, u_3
  24. Polygon(W, Z, J, F) → poly18, w_2, z_3, j_5, f_4
  25. Polygon(J, O, P, F) → poly19, j_6, o_6, p_4, f_5
  26. Polygon(F, E, A, P) → poly20, f_6, e_6, a_5, p_5
  27. Polygon(P, A, T, S) → poly21, p_6, a_6, t_4, s_4
  28. Segment(P, A_1) → g_1