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Twelve Rhombuses in a Cube

Twelve Rhombuses in a Cube
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2019-07-21 22:52
PolygonSolid GeometryGeometric Construction
This example demonstrates the construction of twelve interrelated rhombuses within a cube. Key mathematical concepts include solid geometry, properties of polygons, angle calculations, and geometric transformations. The construction utilizes GeoGebra commands such as Intersect, Point, Circle, Rotate, Angle, Cube, Midpoint, and Polygon, along with objects like points, edges, faces, and angles. It is suitable for high school students learning spatial geometry and polygon construction, as well as for mathematical demonstrations and educational purposes.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Point(c, 1 - u) → C
  5. Rotate(C, 90°, A) → C'
  6. Angle(C, A, C') → α
  7. Rotate(C', 90°, A) → C''
  8. Angle(C', A, C'') → β
  9. Cube(C, C', C'') → a, D, faceC''DHG, faceEFGH, edgeCD, edgeCC', edgeC'C'', edgeC''D, edgeCE, edgeDH, edgeEH, edgeEF, E, edgeC'F, edgeFG, edgeC''G, edgeGH, F, G, H, faceCC'C''D, faceCDHE, faceCC'FE, faceC'C''GF
  10. Midpoint(edgeCD) → I
  11. Midpoint(edgeC''D) → J
  12. Midpoint(edgeGH) → K
  13. Midpoint(edgeFG) → L
  14. Midpoint(edgeEH) → M
  15. Midpoint(edgeC''G) → N
  16. Midpoint(edgeC'C'') → O
  17. Midpoint(edgeCC') → P
  18. Midpoint(edgeDH) → Q
  19. Midpoint(edgeCE) → R
  20. Midpoint(edgeEF) → S
  21. Polygon(C, J, G, S) → q1, c_1, j, g, s
  22. Polygon(C', I, H, L) → q2, c', i, h, l
  23. Polygon(E, K, C'', P) → q3, e, k, c'', p
  24. Polygon(F, R, D, N) → q4, f, r, d, n
  25. Midpoint(edgeC'F) → T
  26. Polygon(E, Q, C'', T) → q5, e_1, q, m, t
  27. Polygon(H, J, C', S) → q6, h_1, j_1, b, s_1
  28. Polygon(E, L, C'', I) → q7, e_2, l_1, f_1, i_1
  29. Polygon(F, K, D, P) → q8, f_2, k_1, d_1, p_1
  30. Polygon(G, M, C, O) → q9, g_1, m_1, c_2, o
  31. Polygon(F, M, D, O) → q10, f_3, m_2, d_2, o_1
  32. Polygon(C', N, H, R) → q11, n_1, n_2, h_2, r_1
  33. Polygon(G, T, C, Q) → q12, g_2, t_1, c_3, q_1