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Hyperbolic Cube

Hyperbolic Cube
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2023-03-15 19:39
Coordinate GeometrySolid GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the geometric construction of a hyperbolic cube. It involves key concepts from hyperbolic and solid geometry. Using GeoGebra commands such as Segment, Point, and OrthogonalLine, the hyperbolic cube's faces and edges can be constructed. Suitable for high school or university solid geometry courses.

GeoGebra Commands & Parameters

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q5, Q6) → p5
  6. Point(p5) → P5
  7. Segment(Q1, Q2) → p1
  8. Point(p1) → P1
  9. Segment(P1, Q2) → p3
  10. Point(p3) → P2
  11. OrthogonalLine(C, xAxis) → zz2
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz2, 1) → ZZ4
  14. Intersect(zz0, zz2, 2) → ZZ3
  15. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  16. Point(zElipse2) → Z
  17. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  18. Angle(ZZ8, C, Z) → zAngInc
  19. OrthogonalLine(C, yAxis) → zz1
  20. Intersect(zz0, zz1, 2) → ZZ2
  21. Intersect(zz0, zz1, 1) → ZZ1
  22. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  23. Point(zElipse1) → XY
  24. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  25. Angle(ZZ6, C, XY) → zAngRot
  26. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r / 1.2 * (((((cos(v))^(3) + (sin(v))^(3)) * ((cos(α))^(3) + (sin(α))^(3))) * sin(zAngRot)) + ((((-(cos(v))^(3)) + (sin(v))^(3)) * ((cos(α))^(3) + (sin(α))^(3))) * cos(zAngRot)))), (r / 1.2 * (((((-((cos(v))^(3) + (sin(v))^(3))) * ((cos(α))^(3) + (sin(α))^(3))) * cos(zAngRot)) * sin(zAngInc)) + (((((-(cos(v))^(3)) + (sin(v))^(3)) * ((cos(α))^(3) + (sin(α))^(3))) * sin(zAngRot)) * sin(zAngInc)) + (((-(cos(α))^(3)) + (sin(α))^(3)) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian((r / 1.2 * (((-((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) + (((cos(u))^(3) + (sin(u))^(3)) * cos(zAngRot)))), (r / 1.2 * (((((cos(u))^(3) + (sin(u))^(3)) * cos(zAngRot)) * sin(zAngInc)) + ((((cos(u))^(3) + (sin(u))^(3)) * sin(zAngRot)) * sin(zAngInc)) + (((-(cos(u))^(3)) + (sin(u))^(3)) * cos(zAngInc)))), u, u1, u2) → curva1
  33. CurveCartesian((r / 1.2 * (((((cos(β))^(3) + (sin(β))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) + ((((-(cos(β))^(3)) + (sin(β))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * cos(zAngRot)))), (r / 1.2 * (((((-((cos(β))^(3) + (sin(β))^(3))) * ((cos(u))^(3) + (sin(u))^(3))) * cos(zAngRot)) * sin(zAngInc)) + (((((-(cos(β))^(3)) + (sin(β))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) * sin(zAngInc)) + (((-(cos(u))^(3)) + (sin(u))^(3)) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 1.2 * (((((cos(v))^(3) + (sin(v))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) + ((((-(cos(v))^(3)) + (sin(v))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * cos(zAngRot)))), (r / 1.2 * (((((-((cos(v))^(3) + (sin(v))^(3))) * ((cos(u))^(3) + (sin(u))^(3))) * cos(zAngRot)) * sin(zAngInc)) + (((((-(cos(v))^(3)) + (sin(v))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) * sin(zAngInc)) + (((-(cos(u))^(3)) + (sin(u))^(3)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list3
  35. Sequence(CurveCartesian[(r / 1.2 * (((((cos(v))^(3) + (sin(v))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) + ((((-(cos(v))^(3)) + (sin(v))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * cos(zAngRot)))), (r / 1.2 * (((((-((cos(v))^(3) + (sin(v))^(3))) * ((cos(u))^(3) + (sin(u))^(3))) * cos(zAngRot)) * sin(zAngInc)) + (((((-(cos(v))^(3)) + (sin(v))^(3)) * ((cos(u))^(3) + (sin(u))^(3))) * sin(zAngRot)) * sin(zAngInc)) + (((-(cos(u))^(3)) + (sin(u))^(3)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Draw a three-dimensional geometric construction of a hyperbolic cube. Start with coordinate axes and a central point. Create two perpendicular ellipses centered at this point, representing two rotation planes. Place points on each ellipse and connect them to form the edges and faces of the cube. Then generate a series of curved grid lines that cover the surface, illustrating the hyperbolic curvature. The final figure includes the central point, two ellipses, several edges, and dense curved meshes, showing a symmetric yet distorted solid shape.
The prompt above was compiled from the original drawing commands and is for reference only.