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Rotating Polygons by John Golden

Rotating Polygons by John Golden
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2016-03-23 01:38
PolygonTransformationGeometric Construction
This example demonstrates the process of constructing complex polygons by rotating a quadrilateral and using its intersections with itself. It involves core mathematical concepts such as rotation, symmetry, polygon construction, and intersection calculation. Key GeoGebra commands/objects used include Polygon, Rotate, Intersect, Sequence, PolyLine, and Rotgon, as outlined in the construction summary. It is suitable for high school and above students to understand transformation geometry and complex figure construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Rotate(A, (360 * ° / N), E) → A'
  3. Rotate(B, (360 * ° / N), E) → B'
  4. Rotate(C, (360 * ° / N), E) → C'
  5. Rotate(D, (360 * ° / N), E) → D'
  6. Polygon(A', B', C', D') → poly1', a', b', c', d'
  7. Intersect(poly1, poly1') → F, G, F_1, F_2, F_3, F_4, F_5, F_6, F_7, F_8
  8. Polygon({Intersect[poly1, poly1']}) → poly2
  9. Polygon(F_1, D, C) → poly3, c_1, f_1, d_1
  10. Polygon(F_2, A, G) → poly4, g, f_2, a_1
  11. Polygon(F, C, B) → poly5, b_1, f, c_2
  12. Sequence(Rotate[poly2, ((i * 360) * ° / N), E], i, 1, N) → list1
  13. Sequence(Polygon[Rotate[F_1, ((i * 360) * ° / N), E], Rotate[D, ((i * 360) * ° / N), E], Rotate[C, ((i * 360) * ° / N), E]], i, 1, N) → list2
  14. Sequence(Polygon[Rotate[F_2, ((i * 360) * ° / N), E], Rotate[A, ((i * 360) * ° / N), E], Rotate[G, ((i * 360) * ° / N), E]], i, 1, N) → list3
  15. Sequence(Polygon[Rotate[F, ((i * 360) * ° / N), E], Rotate[C, ((i * 360) * ° / N), E], Rotate[B, ((i * 360) * ° / N), E]], i, 1, N) → list4
  16. Sequence(Polygon[Rotate[A, ((i * 360) * ° / N), E], Rotate[B, ((i * 360) * ° / N), E], Rotate[C, ((i * 360) * ° / N), E], Rotate[D, ((i * 360) * ° / N), E]], i, 1, N) → list5
  17. Polygon(F_2, F_3, F_4) → poly6, f_4, f_3, f_5
  18. Sequence(Polygon[Rotate[F_2, ((i * 360) * ° / N), E], Rotate[F_3, ((i * 360) * ° / N), E], Rotate[F_4, ((i * 360) * ° / N), E]], i, 1, N) → list6
  19. PolyLine(A, B, C, D, A) → h
  20. Rotate(A', (360 * ° / N), E) → A''
  21. Rotate(B', (360 * ° / N), E) → B''
  22. Rotate(C', (360 * ° / N), E) → C''
  23. Rotate(D', (360 * ° / N), E) → D''
  24. Polygon(A'', B'', C'', D'') → poly1'', a'', b'', c'', d''
  25. Intersect(poly1'', poly1) → H, I, H_1, H_2, H_3, H_4, H_5, H_6, H_7, H_8
  26. Polygon(H, I, D) → poly7, i, j, k
  27. Polygon({Intersect[poly1, poly1'']}) → poly9
  28. Sequence(Rotate[poly9, ((i * 360) * ° / N), E], i, 1, N) → list8
  29. Polygon(F, H, B) → poly8, b_2, f_6, h_1
  30. Sequence(Polygon[Rotate[F, ((i * 360) * ° / N), E], Rotate[H, ((i * 360) * ° / N), E], Rotate[B, ((i * 360) * ° / N), E]], i, 1, N) → list9
  31. Polygon(G, H, D) → poly10, d_2, g_1, h_2
  32. Sequence(Polygon[Rotate[G, ((i * 360) * ° / N), E], Rotate[H, ((i * 360) * ° / N), E], Rotate[D, ((i * 360) * ° / N), E]], i, 1, N) → list10
  33. Sequence(Polygon[Rotate[H, ((i * 360) * ° / N), E], Rotate[I, ((i * 360) * ° / N), E], Rotate[B, ((i * 360) * ° / N), E]], i, 1, N) → list7
  34. Rotgon(E, N, F, F_2, H_1) → list11