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Euler Relation and Polyhedra: Exploring V - E + F = 2

Euler Relation and Polyhedra: Exploring V - E + F = 2
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2020-11-12 05:11
PolygonVectorSequenceSolid GeometryGeometric Construction
① This interactive 3D GeoGebra example demonstrates Euler's formula V - E + F = 2 for polyhedra, visually showing the invariant relationship among vertices, edges, and faces. ② Core mathematical concepts include Euler's formula, classification of convex polyhedra, and spatial relationships of points, lines, and planes in 3D. ③ Construction highlights: using GeoGebra commands such as Vector, Rotate, Random, Sequence, Join, Midpoint, CurveCartesian, Polygon, Textfield, Point, and Segment, within a 3D view (View3D), the model generates polyhedra with adjustable parameters, manages vertices/edges/faces via lists (liste1 to liste12), and dynamically updates counts. ④ Suitable for upper secondary to undergraduate geometry courses, ideal for verifying Euler's formula, understanding polyhedral structure, and supporting inquiry-based math labs.

GeoGebra Commands & Parameters

  1. Vector(C, Z) → w
  2. Vector(C, Y) → v
  3. Vector(C, X) → u
  4. Rotate(w, β) → w'
  5. Rotate(v, β) → v'
  6. Rotate(u, β) → u'
  7. Random(3, 10) → Sides
  8. Sequence(A + ((R * cos(((i * 360) * ° / Sides))) * u') + ((R * sin(((i * 360) * ° / Sides))) * v') + (h / 2 * w'), i, 1, Sides) → liste1
  9. Sequence(A + ((R * cos(((i * 360) * ° / Sides))) * u') + ((R * sin(((i * 360) * ° / Sides))) * v') - (h / 2 * w'), i, 1, Sides) → liste2
  10. Sequence(A + ((R * cos(((i * 360) * ° / Sides))) * u') + ((R * sin(((i * 360) * ° / Sides))) * v') + (h / 2 * w'), i, 1, Sides) → liste3
  11. Sequence(A + ((R * cos(((i * 360) * ° / Sides))) * u') + ((R * sin(((i * 360) * ° / Sides))) * v') - (h / 2 * w'), i, 1, Sides) → liste4
  12. Join({liste1, First[liste1]}) → liste5
  13. Join({liste2, First[liste2]}) → liste6
  14. Join({liste3, First[liste3]}) → liste7
  15. Join({liste4, First[liste4]}) → liste8
  16. Sequence(If[Angle[Element[liste5, i], Element[liste6, i], Element[liste6, i + 1]] < 180°, Polygon[Element[liste5, i], Element[liste5, i + 1], Element[liste6, i + 1], Element[liste6, i]]], i, 1, Sides) → liste9
  17. Sequence(If[Angle[Element[liste5, i], Element[liste5, i + 1], Element[liste7, i + 1]] < 180°, Polygon[Element[liste5, i], Element[liste5, i + 1], Element[liste7, i + 1], Element[liste7, i]]], i, 1, Sides) → liste10
  18. Sequence(If[Angle[Element[liste8, i], Element[liste8, i + 1], Element[liste6, i + 1]] < 180°, Polygon[Element[liste6, i], Element[liste6, i + 1], Element[liste8, i + 1], Element[liste8, i]]], i, 1, Sides) → liste11
  19. Sequence(If[Angle[Element[liste7, i], Element[liste7, i + 1], Element[liste8, i + 1]] < 180°, Polygon[Element[liste8, i], Element[liste8, i + 1], Element[liste7, i + 1], Element[liste7, i]]], i, 1, Sides) → liste12
  20. Midpoint(A, B) → I
  21. CurveCartesian(x(I + ((2 * cos(i)) * u) + ((2 * sin(i)) * v)), y(I + ((2 * cos(i)) * u) + ((2 * sin(i)) * v)), i, -0.5235987755982988, 0.5235987755982988) → c
  22. Polygon(liste1) → poly1
  23. Polygon(liste2) → poly2
  24. Polygon(liste2) → poly3
  25. Polygon(liste1) → poly4
  26. Sequence(If[Angle[Element[liste5, i], Element[liste6, i], Element[liste6, i + 1]] < 180°, Polygon[Element[liste5, i], Element[liste5, i + 1], Element[liste6, i + 1], Element[liste6, i]]], i, 1, Sides) → liste13
  27. Midpoint(L, D) → N
  28. Midpoint(D, M) → O
  29. Vector(D, O) → z
  30. Vector(D, Q + (0.25 * u)) → b
  31. Midpoint(D, Q) → S
  32. Polygon(View3D, AA, BB, CC) → poly5, view3d, aa, bb, cc
  33. Textfield(Faces) → ChampTexte1
  34. Textfield(Sommets) → ChampTexte2
  35. Textfield(Aretes) → ChampTexte3
  36. Polygon(T, U, V, W) → poly6, t, u_1, v_1, w_1
  37. Point(t) → A_1
  38. Point(v_1) → B_1
  39. Segment(A_1, B_1) → f
  40. Point(t) → C_1
  41. Segment(C_1, D_1) → g
  42. Point(w_1) → E_1
  43. Point(u_1) → F_1
  44. Segment(E_1, F_1) → i
  45. Polygon(View3D, AA, BB, CC) → q1, j, k, l, m

AI Drawing Prompt Reference

Create a convex polyhedron by adjusting parameters to generate polyhedra with different numbers of sides, showing the relationship between vertices, edges, and faces, ensuring each face is convex. Use vector rotation and random point generation to build the polyhedron, and dynamically update the counts of vertices, edges, and faces.
The prompt above was compiled from the original drawing commands and is for reference only.