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Series of Cubes

Series of Cubes
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2020-01-16 20:16
SequenceSolid GeometryTransformation
This example demonstrates the construction of a series of cubes. It involves key concepts in sequences and solid geometry. Using GeoGebra commands such as Sequence, Join, Rotate, Translate, etc., it dynamically shows the arrangement and rotation of cubes. Suitable for high school mathematics education, particularly in solid geometry and sequences.

GeoGebra Commands & Parameters

  1. Sequence(1 / sqrt(i), i, 1, m_1) → l1
  2. Join({{0}, Sequence[Sum[Sequence[Element[l1, j], j, 1, i]], i, 1, m_1]}) → l4
  3. Sequence(Translate[(Element[l1, i] / 2, Element[l1, i] / 2, 0), Vector[(0, 0, 0), (0, 0, Element[l4, i])]], i, 1, m_1 - 1) → l51
  4. Rotate(l51, π / 2, zAxis) → l52
  5. Rotate(l51, (2 * π / 2), zAxis) → l53
  6. Rotate(l51, (3 * π / 2), zAxis) → l54
  7. Sequence(Translate[Element[l51, i], Vector[(0, 0, 0), (0, 0, Element[l1, i])]], i, 1, Length[l51]) → l61
  8. Sequence(Translate[Element[l52, i], Vector[(0, 0, 0), (0, 0, Element[l1, i])]], i, 1, Length[l52]) → l62
  9. Sequence(Translate[Element[l53, i], Vector[(0, 0, 0), (0, 0, Element[l1, i])]], i, 1, Length[l53]) → l63
  10. Sequence(Translate[Element[l54, i], Vector[(0, 0, 0), (0, 0, Element[l1, i])]], i, 1, Length[l54]) → l64
  11. Sequence(Polygon[Element[l51, i], Element[l52, i], Element[l62, i], Element[l61, i]], i, 1, Length[l51]) → l72
  12. Sequence(Polygon[Element[l51, i], Element[l52, i], Element[l53, i], Element[l54, i]], i, 1, Length[l51]) → l71
  13. Sequence(Polygon[Element[l61, i], Element[l62, i], Element[l63, i], Element[l64, i]], i, 1, Length[l51]) → l73
  14. Rotate(l72, π / 2, zAxis) → l74
  15. Rotate(l72, (2 * π / 2), zAxis) → l75
  16. Rotate(l72, (3 * π / 2), zAxis) → l76
  17. Translate(l71, Vector[(0, 0, 0), (0, 0, 0)]) → l81
  18. Translate(l72, Vector[(0, 0, 0), (0, 0, 0)]) → l82
  19. Translate(l73, Vector[(0, 0, 0), (0, 0, 0)]) → l83
  20. Translate(l74, Vector[(0, 0, 0), (0, 0, 0)]) → l84
  21. Translate(l75, Vector[(0, 0, 0), (0, 0, 0)]) → l85
  22. Translate(l76, Vector[(0, 0, 0), (0, 0, 0)]) → l86
  23. Sequence(Element[l1, i]^(3), i, 1, Length[l1]) → l9
  24. Sum(l9) → la
  25. Sum(l71) → lb
  26. Sequence(Centroid[Element[l76, i]], i, 1, Length[l76]) → l2
  27. Sequence(Text["\sqrt" + (k), Element[l2, k] + (0.1, 0, 0.2), true, true], k, 1, n) → l3
  28. Sequence(CurveCartesian[x(O) + ((R * a / i) * cos(j)), y(O) + ((R * a / i) * sin(j)), z(O) + (R * i) - 0.7, i, π / m, n], j, 0, (2 * π), π / m) → l5
  29. Surface(x(O) + ((R * a / i) * cos(j)), y(O) + ((R * a / i) * sin(j)), z(O) + (R * i) - 0.5, i, π / m, n, j, 0, 6.28319) → b

AI Drawing Prompt Reference

Draw a series of cubes that gradually decrease in size, demonstrating their construction and rotation. Start by generating a sequence to represent the heights of the cubes, then connect these height points to form the bases of the cubes. Next, translate and rotate these bases along the height to create multiple cubes. Finally, connect the faces of these cubes using polygons and display their dynamic rotation.
The prompt above was compiled from the original drawing commands and is for reference only.