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Folding and Unfolding a Net

Folding and Unfolding a Net
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2019-05-12 12:12
PolygonVectorSolid Geometry
This example demonstrates the process of constructing 3D shapes by folding and unfolding 2D nets. It involves core mathematical concepts such as solid geometry, polygons, transformations, and vectors. The construction utilizes GeoGebra commands like Sequence, Insert, Div, Segment, and objects like t, N, l, a, z, R_z, LiRa, LiR, V_0, RV_0, RV_x, RV_y, SumRV, s, R_x, α, f, Cav, altezza, S_R, VQ, VQ_x, VQ_y, SumVQ, R, V_{0y}, V_{0x}, SumV_0, P, Q, M, R1, b, Rb, RP, p, PQ, PP, lista1, S. It is suitable for high school students learning about the construction of 3D shapes and for mathematical teaching demonstrations.

GeoGebra Commands & Parameters

  1. Sequence({{cos(((k * a) * t)), (-sin(((k * a) * t))), 0}, {sin(((k * a) * t)), cos(((k * a) * t)), 0}, {0, 0, 1}}, k, 0, N) → LiRa
  2. Sequence({{cos((k * a)), sin((k * a)), 0}, {(-sin((k * a))), cos((k * a)), 0}, {0, 0, 1}}, k, 0, N) → LiR
  3. Sequence(Element[({{l, 0, 0}} * Element[LiR, k]), 1], k, 1, N + 1) → V_0
  4. Sequence(Element[Element[RV_0, k], 1], k, 1, N + 1) → RV_x
  5. Sequence(Element[Element[RV_0, k], 2], k, 1, N + 1) → RV_y
  6. Sequence({Sum[RV_x, k], Sum[RV_y, k], 0}, k, 1, N + 1) → SumRV
  7. Sequence(Element[({Element[V_0, k]} * Element[LiRa, k]), 1], k, 1, N) → VQ
  8. Sequence(Element[Element[VQ, k], 1], k, 1, N) → VQ_x
  9. Sequence(Element[Element[VQ, k], 2], k, 1, N) → VQ_y
  10. Sequence({Sum[VQ_x, k], Sum[VQ_y, k], 0}, k, 1, N) → SumVQ
  11. Sequence(Element[SumVQ, k] + {0, 0, (-1 * altezza)}, k, 1, N) → R
  12. Sequence(Element[Element[V_0, k], 2], k, 1, N + 1) → V_{0y}
  13. Sequence(Element[Element[V_0, k], 1], k, 1, N + 1) → V_{0x}
  14. Sequence({Sum[V_{0x}, k], Sum[V_{0y}, k], 0}, k, 1, N + 1) → SumV_0
  15. Sequence(Element[SumV_0, k], k, 1, N + 1) → P
  16. Insert({Element[P, N]}, SumVQ, 1) → Q
  17. Div(N, 2) → M
  18. Insert({Element[P, N] + {0, 0, (-1 * altezza)}}, R, 1) → R1
  19. Sequence((Element[Element[(Q * Cav), k], 1], Element[Element[(Q * Cav), k], 2]), k, 1, N + 1) → PQ
  20. Sequence((Element[Element[(RP * Cav), k], 1], Element[Element[(RP * Cav), k], 2]), k, 1, N + 1) → PP
  21. Sequence(Element[R, N] + Element[S_R, k], k, 1, N + 1) → S
  22. Sequence((Element[Element[(R1 * Cav), k], 1], Element[Element[(R1 * Cav), k], 2]), k, 1, N + 1) → PR
  23. Sequence((Element[Element[(S * Cav), k], 1], Element[Element[(S * Cav), k], 2]), k, 1, N + 1) → PS
  24. Sequence(Segment[Element[PS, k], Element[PS, k + 1]], k, 1, N) → SpigoliSotto
  25. Sequence(Segment[Element[PP, k], Element[PP, k + 1]], k, 1, N) → SpigoliSopra
  26. Sequence(Polygon[Element[PQ, k], Element[PQ, k + 1], Element[PR, k + 1], Element[PR, k]], k, 1, N) → FacceLat
  27. Sequence(Segment[Element[PQ, k], Element[PR, k]], k, 1, N + 1) → SpigoliLat
  28. Sequence(Segment[Element[PS, k], Element[PS, k + 1]], k, 1, N) → p_{base}
  29. Sequence(Segment[Element[PR, k], Element[PR, k + 1]], k, 1, N + 1) → SpigoliLatSotto
  30. Segment(Element[PQ, 1], Element[PR, 1]) → h

AI Drawing Prompt Reference

Draw a 3D shape formed by folding a 2D net. First, create an N-vertex square net, then rotate this net around its center to generate a series of vertices and edges. Next, connect these vertices and edges in a specific order to form the side and bottom faces of the 3D shape. Finally, combine the side and bottom faces to create the complete 3D shape.
The prompt above was compiled from the original drawing commands and is for reference only.