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Dome of Speyer Cathedral

Dome of Speyer Cathedral
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2014-11-13 22:09
PolygonSolid GeometryParametric EquationGeometric Construction
In collections Selected Examples
This example demonstrates the geometric construction and 3D form of the dome of Speyer Cathedral. Core math concepts include polygons, surfaces, transformations, parametric equations, and solid geometry. Construction highlights involve GeoGebra commands such as Rotate, Polygon, Sequence, CurveCartesian, Invert, and Surface, with objects like H, r, A, A', p, Vieleck1, a, b, c, d, e, f, g, h, B, C, D, E, F, G, Liste1, s_2, s_1, s_3, s_4, s_5, s_6, s_7, s_8, n, Δz, q, LPunkte, LAchtecke, LPrismen, i, j, fl_1, fl_2, fl_3. Suitable for high school and above learning solid geometry, geometric construction, and 3D modeling, as well as mathematical modeling and visualization education.

GeoGebra Commands & Parameters

  1. Rotate(A, 45*°) → A'
  2. Polygon(A, A', 8) → Vieleck1, a, C, D, E, F, G, b, c, d, e, f, g, h, B
  3. Sequence(CurveCartesian[(t * cos((k * π / 4))), (t * sin((k * π / 4))), p(t), t, 0, r], k, 0, 7) → Liste1
  4. CurveCartesian((t * cos(45°)), (t * sin(45°)), p(t), t, 0, r) → s_2
  5. CurveCartesian(t, (t * 0), p(t), t, 0, r) → s_1
  6. CurveCartesian((t * 0), t, p(t), t, 0, r) → s_3
  7. CurveCartesian((t * cos(((3 * 45))°)), (t * sin(((3 * 45))°)), p(t), t, 0, r) → s_4
  8. CurveCartesian((t * (-1)), (t * 0), p(t), t, 0, r) → s_5
  9. CurveCartesian((t * cos(((5 * 45))°)), (t * sin(((5 * 45))°)), p(t), t, 0, r) → s_6
  10. CurveCartesian((t * 0), (t * (-1)), p(t), t, 0, r) → s_7
  11. CurveCartesian((t * cos(((7 * 45))°)), (t * sin(((7 * 45))°)), p(t), t, 0, r) → s_8
  12. Invert(p) → q
  13. Sequence((q((k * Δz)), 0, (k * Δz)), k, 0, n - 1) → LPunkte
  14. Sequence(Polygon[Element[LPunkte, j], Rotate[Element[LPunkte, j], 45*°], 8, xOyPlane], j, 1, n) → LAchtecke
  15. Sequence(Prism[Element[LAchtecke, m], Δz], m, 1, n) → LPrismen
  16. Surface(t + ((u * t) * (cos(π / 4) - cos(0))), (t * 0) + ((u * t) * (sin(π / 4) - sin(0))), p(t), t, 0, r, u, 0, 1) → fl_1
  17. Surface((t * cos(π / 4)) + ((u * t) * (cos((2 * π / 4)) - cos(π / 4))), (t * sin(π / 4)) + ((u * t) * (sin((2 * π / 4)) - sin(π / 4))), p(t), t, 0, r, u, 0, 1) → fl_2
  18. Surface((t * 0) + ((u * t) * (cos((3 * π / 4)) - cos((2 * π / 4)))), t + ((u * t) * (sin((3 * π / 4)) - sin((2 * π / 4)))), p(t), t, 0, r, u, 0, 1) → fl_3
  19. Surface((t * cos((3 * π / 4))) + ((u * t) * (cos((4 * π / 4)) - cos((3 * π / 4)))), (t * sin((3 * π / 4))) + ((u * t) * (sin((4 * π / 4)) - sin((3 * π / 4)))), p(t), t, 0, r, u, 0, 1) → fl_4
  20. Surface((t * (-1)) + ((u * t) * (cos((5 * π / 4)) - cos((4 * π / 4)))), (t * 0) + ((u * t) * (sin((5 * π / 4)) - sin((4 * π / 4)))), p(t), t, 0, r, u, 0, 1) → fl_5
  21. Surface((t * cos((5 * π / 4))) + ((u * t) * (cos((6 * π / 4)) - cos((5 * π / 4)))), (t * sin((5 * π / 4))) + ((u * t) * (sin((6 * π / 4)) - sin((5 * π / 4)))), p(t), t, 0, r, u, 0, 1) → fl_6
  22. Surface((t * 0) + ((u * t) * (cos((7 * π / 4)) - cos((6 * π / 4)))), (t * (-1)) + ((u * t) * (sin((7 * π / 4)) - sin((6 * π / 4)))), p(t), t, 0, r, u, 0, 1) → fl_7
  23. Surface((t * cos((7 * π / 4))) + ((u * t) * (cos((8 * π / 4)) - cos((7 * π / 4)))), (t * sin((7 * π / 4))) + ((u * t) * (sin((8 * π / 4)) - sin((7 * π / 4)))), p(t), t, 0, r, u, 0, 1) → fl_8

AI Drawing Prompt Reference

Draw the geometric construction and 3D form of the dome of Speyer Cathedral, showcasing core concepts of polygons, surfaces, transformations, parametric equations, and solid geometry. Begin by rotating a point, then construct an octagon using the original and rotated points. Generate multiple curves based on parametric equations rotating around the center at different angles. Extend these curves vertically to form multiple octagonal prisms. Finally, combine these prisms into a smooth surface to simulate the dome's shape.
The prompt above was compiled from the original drawing commands and is for reference only.