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Icosahedron with Pyramids: Geometric Construction

Icosahedron with Pyramids: Geometric Construction
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2016-01-22 00:10
PolygonAngleCoordinate GeometrySolid GeometryGeometric Construction
This GeoGebra example demonstrates the geometric construction of a regular icosahedron adorned with pyramids on its faces, showcasing a well-known spatial construction method. The core mathematical concepts involved include solid geometry, coordinate geometry, spatial vectors, and the calculation of angles and segments in three-dimensional space. To build this model, the user employs key GeoGebra commands such as Angle, Pyramid, Segment, and Intersect, working with variables and object names like tau, vertices I_1 to I_{12}, faces Fläche01 to Fläche06, and edges Kante01 to Kante09. It is highly suitable for high school or university students and educators learning advanced spatial geometry, 3D modeling, and interactive geometry software applications.

GeoGebra Commands & Parameters

  1. Angle(Segment[(0, 0, 0), ((-b), 0, a)], zAxis) → w_{2max}
  2. Angle(Segment[(0, 0, 0), ((-b), 0, a)], zAxis) → w_{1max}
  3. Pyramid(I_1, I_5, I_{10}, I_{11}, I_6, I_4) → c, Fläche01, Kante04, Kante05, Kante06, Kante07, Kante08, Kante09, Kante10, Fläche02, Fläche03, Fläche04, Fläche05, Fläche06, Kante01, Kante02, Kante03
  4. Pyramid(I_3, I_7, I_{12}, I_9, I_8, I_2) → d, Fläche07, Kante20, Kante16, Kante22, Kante23, Kante24, Kante25, Kante21, Fläche09, Fläche10, Fläche11, Fläche12, Fläche08, Kante17, Kante18, Kante19
  5. Segment(I_7, I_4) → d_{13}
  6. Segment(I_4, I_3) → d_{12}
  7. Segment(I_4, I_8) → d_{11}
  8. Segment(I_4, I_9) → d_{15}
  9. Segment(I_4, I_{12}) → d_{14}
  10. Segment(I_2, I_5) → d_{17}
  11. Segment(I_2, I_{10}) → d_{18}
  12. Segment(I_2, I_{11}) → d_{19}
  13. Segment(I_2, I_6) → d_{20}
  14. Segment(I_2, I_1) → d_{16}
  15. Segment(I_5, I_8) → d_3
  16. Segment(I_8, I_{10}) → d_4
  17. Segment(I_{10}, I_3) → d_5
  18. Intersect(Kante03, Kante09) → A
  19. Segment(I_3, I_{11}) → d_6
  20. Segment(I_{11}, I_7) → d_7
  21. Segment(I_7, I_6) → d_8
  22. Segment(I_6, I_{12}) → d_9
  23. Segment(I_{12}, I_1) → d_{10}
  24. Segment(I_1, I_9) → d_1
  25. Segment(I_9, I_5) → d_2