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正方体的Snub操作：从正方体到八面体、菱形二十面体、Snub立方体

正方体的Snub操作：从正方体到八面体、菱形二十面体、Snub立方体
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2020-03-18 01:00
立体几何变换几何作图
本范例演示了正方体经过Snub操作后变成八面体、菱形二十面体和Snub立方体的过程。涉及的核心数学概念包括立体几何中的多面体变换。通过使用GeoGebra命令如Zip、Length、Sequence、Join等，可以直观地展示多面体的生成过程。适合高中或大学阶段的学生和教师使用。

GeoGebra 命令及参数

  1. Zip(Vector[(zoom * (Element[a, 1], Element[a, 2], Element[a, 3]))], a, m_octahedron) → V6zoom
  2. Length(V6zoom(1)) → r
  3. Zip(Vector[(zoom * (Element[a, 1], Element[a, 2], Element[a, 3]))], a, m_cube) → V8zoom
  4. Sequence(Sequence[V8zoom(faces(i, k)), k, 1, Length[faces(1)]], i, 1, Length[faces]) → V8cube
  5. Zip(Polygon[a], a, V8cube) → poly_cube0
  6. Zip(Sum[a] / 4, a, V8cube) → Cmi
  7. Zip(Rotate[a, β, Line[O, b]], a, V8cube, b, Cmi) → V8'
  8. Zip(Dilate[a, q, b] + ((c - b) * t), a, V8', b, Cmi, Vector[c], V6zoom) → V4
  9. Join(V4) → lP0
  10. Textfield(t) → InputBox1
  11. Zip(Polygon[a], a, V4) → poly4
  12. Length(lP0) → läA0
  13. Length(lP0(1)) → R_A0
  14. Sphere((0, 0, 0), R_A0) → a
  15. Length(V6zoom(1)) → rA6
  16. Zip(Polygon[a], a, V4) → poly4_cube'
  17. Sequence(Polygon[Sequence[lP0(m_3(f, k)), k, 1, Length[m_3(1)]]], f, 1, Length[m_3]) → poly3
  18. Sequence(Polygon[Sequence[lP0(m3_4(f, k)), k, 1, Length[m3_4(1)]]], f, 1, Length[m3_4]) → poly_{3}'
  19. Sequence(Polygon[Sequence[lP0(m_4(f, k)), k, 1, Length[m_4(1)]]], f, 1, Length[m_4]) → poly_{4}
  20. Length(Unique[lP0]) → lä
  21. Length(lP0) → n
  22. If(j, "\fgcolor{orange}{\; ∙---}", "") → txt1
  23. If(j, "\fgcolor{#318CE7}{\; ∙---}", "") → txt2
  24. If(j, "\fgcolor{magenta}{\; ∙---}", "") → txt3
  25. If(j, "\fgcolor{green}{\; ∙---}", "") → txt2a
  26. Sequence(Segment[lP0(m_3(2, i)), lP0(m_3(2, If[i ≟ 3, 0, i] + 1))], i, 1, 3) → seg3
  27. Sequence(Segment[lP0(m_4(2, i)), lP0(m_4(2, If[i ≟ 4, 0, i] + 1))], i, 1, 4) → seg4'
  28. Sequence(Segment[V4(1, i), V4(1, If[i ≟ 4, 1, i + 1])], i, 1, 4) → seg4
  29. Sequence(Segment[lP0(m3_4(4, i)), lP0(m3_4(4, If[i ≟ 3, 0, i] + 1))], i, 1, 3) → seg3'
  30. TableText(Append["segments", 1…4], Append["\fgcolor{#FB607F}{3-gon}" + txt1, seg3], Append["\fgcolor{#B2BEB5}{3-gon'}'" + txt2, seg3'], Append["\fgcolor{Turquoise}{4-gon}" + txt3, seg4], "v") → text2b
  31. Textfield(dα) → InputBox2
  32. Textfield(dt) → InputBox3
  33. TableText(Append["segments", 1…4], Append["\fgcolor{#FB607F}{3-gon}" + txt1, seg3], Append["\fgcolor{#C19A6B}{4'-gon}" + txt2a, seg4], Append["\fgcolor{Turquoise}{4-gon}" + txt3, seg4'], "v") → text2a