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Gothic Golden Cube: An Elegant Construction of Similarity Transformations and Co

Gothic Golden Cube: An Elegant Construction of Similarity Transformations and Co
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2020-11-13 05:28
PolygonSequenceSolid GeometryTransformationGeometric Construction
① This example demonstrates a complex three-dimensional geometric figure composed of a cube and multiple rotated, translated, and scaled polygons and line segments within it, presenting a gothic golden aesthetic with symmetry and layering. ② Core mathematical concepts include cubes, similarity transformations, dilation, combinatorial geometry of vertices and faces, isosceles triangles, and the interplay of algebra and geometry. ③ Construction highlights: from the summary, commands such as CurveCartesian, Sequence, Polygon, Rotate, Centroid, Line, Translate, Cube, Midpoint, Sphere, and Distance are used, with objects like B, a, n, d, b, l1, A, C, poly1, f, g, h, i, E, F, l1', D, j, l2, G, l1'_{1}, c, H, I, J, K, and named faces (faceAFKH, faceACIH, etc.) and edges (arêteAH, etc.), generated through rotations, translations, and dilations to create symmetric structures. ④ Suitable for high school to university-level teaching of geometry and algebra integration, useful for exploring transformations, symmetry, combinatorial counting in 3D space, and parametric construction with dynamic geometry software.

GeoGebra Commands & Parameters

  1. CurveCartesian(x(B) + sin(t / 2) - ((a * sin((n * t))) * cos(t)), y(B) + cos(t / 2) - ((a * sin((n * t))) * sin(t)), t, 0, d) → b
  2. Sequence(Sequence[Translate[b, Vector[(i * Vector[B, B + (2 + abs(a), 0)]) + (j * Vector[B, B + (0, 2 + abs(a))])]], i, 0, 2], j, 0, 2) → l1
  3. Polygon(A, C, 4) → poly1, f, g, h, i, E, F
  4. Rotate(l1, 90°, f) → l1'
  5. Centroid(poly1) → D
  6. Line(D, zAxis) → j
  7. Sequence(Rotate[l1', i, j], i, 0, (2 * π), (2 * π / 4)) → l2
  8. Translate(l1, Vector[D, G]) → l1'_{1}
  9. Cube(poly1, true) → c, H, arêteAH, arêteFK, arêteHK, arêteHI, arêteCI, arêteIJ, arêteEJ, arêteJK, I, J, K, faceAFKH, faceACIH, faceCEJI, faceEFKJ, faceHIJK
  10. Midpoint(D, G) → L
  11. Midpoint(F, A) → M
  12. Sphere(L, Distance[D, M]) → e
  13. Distance(D, M) → k

AI Drawing Prompt Reference

Create a complex three-dimensional geometric figure consisting of a cube and multiple rotated, translated, and scaled polygons and line segments inside, embodying a gothic golden aesthetic with symmetry and layering. Focus on cubes, similarity transformations, dilation, combinatorial geometry of vertices and faces, isosceles triangles, and the interplay of algebra and geometry. Start by constructing a cube, then generate multiple polygons and line segments through rotations, translations, and scalings, combining them to form symmetric structures.
The prompt above was compiled from the original drawing commands and is for reference only.