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Truncated Tetrahedron: 3D Construction and Geometric Analysis

Truncated Tetrahedron: 3D Construction and Geometric Analysis
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2017-11-20 14:56
PolygonAngleSolid GeometryTransformationGeometric Construction
① This GeoGebra example demonstrates the construction of a truncated tetrahedron, a semi-regular polyhedron obtained by cutting off the vertices of a regular tetrahedron, and visualizes its faces, edges, and vertices in 3D. ② Core mathematical concepts include solid geometry, polyhedra, regular tetrahedron, truncation, symmetry, and the interplay of planes and solids; it involves basic geometric elements such as polygons, circles, lines, perpendiculars, angles, and midpoints. ③ Construction highlights: from the summary, the commands used are Circle, Intersect, Rotate, Polygon, Line, OrthogonalLine, Angle, and Midpoint; objects include multiple points (A, B, C, D, E, F, etc.), multiple polygons (veelhoek1 to veelhoek4), and auxiliary lines (a, b, c_1, d, e, f, etc.). The construction uses rotation and intersection to generate new vertices after truncation, then connects them to form faces. ④ Suitable for high school or early college solid geometry lessons, as well as mathematical modeling or computer-aided geometric design, helping students intuitively understand polyhedron construction and spatial transformations.

GeoGebra Commands & Parameters

  1. Circle(OO, 1, xOyPlane) → c
  2. Intersect(yAxis, c, 2) → A
  3. Rotate(A, 60°) → B
  4. Rotate(A, 120°) → C
  5. Rotate(A, 180°) → D
  6. Rotate(A, 240°) → E
  7. Rotate(A, 300°) → F
  8. Polygon(A, B, C, D, E, F) → veelhoek1, a, b, c_1, d, e, f
  9. Polygon(B, A, 6, xOyPlane) → veelhoek2, g, J_1, h, i, j, k, l, G_1, H_1, I_1
  10. Polygon(D, C, 6, xOyPlane) → veelhoek3, m, N, n, p, q, r, s, K_1, L_1, M
  11. Polygon(F, E, 6, xOyPlane) → veelhoek4, t, R, f_1, g_1, h_1, i_1, j_1, O, P, Q
  12. Circle(c_1, K_1) → d_1
  13. Circle(a, J_1) → e_1
  14. Intersect(e_1, d_1) → J, T, ,
  15. Line(D, C) → k_1
  16. OrthogonalLine(K_1, k_1, xOyPlane) → l_1
  17. Intersect(k_1, l_1) → KK
  18. Angle(K_1, KK, J) → α
  19. Rotate(veelhoek3, (-α), c_1) → veelhoek3'
  20. Line(OO, OOO) → as
  21. Rotate(veelhoek3', 120°, as) → veelhoek3''
  22. Rotate(veelhoek3', (-(120°)), as) → veelhoek3''_1
  23. Rotate(J, (-(120°)), as) → L
  24. Rotate(J, 120°, as) → G
  25. Midpoint(L, J) → U
  26. OrthogonalLine(U, veelhoek3''_1) → mm
  27. Rotate(J, (-(60°)), mm) → I
  28. Rotate(I, 120°, as) → H
  29. Rotate(I, (-(120°)), as) → K
  30. Polygon(D, E, G) → driehoek1, s'_1, d_2, e_2
  31. Rotate(driehoek1, 120°, as) → driehoek1'
  32. Rotate(driehoek1, (-(120°)), as) → driehoek1'_1