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Ideal Hyperbolic Tetrahedra

Ideal Hyperbolic Tetrahedra
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2016-10-07 21:05
Solid GeometryGeometric Construction
This example demonstrates the geometric construction and properties of ideal tetrahedra in hyperbolic geometry. Key math concepts include hyperbolic geometry, ideal tetrahedra, and geometric construction. Construction highlights involve using GeoGebra commands like Plane, PointIn, LineBisector, Intersect, Surface, and Face. Suitable for high school students and math enthusiasts studying hyperbolic and solid geometry constructions.

GeoGebra Commands & Parameters

  1. Plane(xAxis, yAxis) → P
  2. PointIn(P) → A
  3. PointIn(P) → B
  4. PointIn(P) → C
  5. LineBisector(B, A, xOyPlane) → f
  6. LineBisector(B, C, xOyPlane) → g
  7. Intersect(f, g) → O
  8. Plane(A, C, A + (0, 0, 1)) → b
  9. Plane(A, B, A + (0, 0, 1)) → c
  10. Plane(C, B, C + (0, 0, 1)) → a
  11. Surface((((u * x(A)) + ((1 - u) * x(B))) * v) + ((1 - v) * x(C)), (((u * y(A)) + ((1 - u) * y(B))) * v) + ((1 - v) * y(C)), sqrt(Distance[O, A]^(2) - ((((u * x(A)) + ((1 - u) * x(B))) * v) + ((1 - v) * x(C)) - x(O))^(2) - ((((u * y(A)) + ((1 - u) * y(B))) * v) + ((1 - v) * y(C)) - y(O))^(2)), u, 0, 1, v, 0, 1) → i
  12. PointIn(P) → D
  13. Face(A, B, D) → d
  14. Face(A, C, D) → e
  15. Face(B, C, D) → h
  16. PointIn(P) → E
  17. Face(B, C, E) → k
  18. Face(A, B, E) → l
  19. Face(A, C, E) → m
  20. Face(A, D, E) → j
  21. Face(B, D, E) → n
  22. Face(C, D, E) → o

AI Drawing Prompt Reference

In 3D space, start with a horizontal base plane and pick three random points on it to form a triangle. Construct the perpendicular bisectors of two sides; their intersection is the circumcenter. From each of the three points, create three slanted planes extending upward, forming the side faces of an ideal tetrahedron. Then, add a curved surface over the triangle region, with height determined by the distance to the circumcenter. Finally, choose two additional points on the base plane and connect them with the original three points to form multiple triangular faces, completing a polyhedral structure that illustrates the properties of an ideal hyperbolic tetrahedron.
The prompt above was compiled from the original drawing commands and is for reference only.