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Properties of N-gonal Numbers (Peacock Theorem)

Properties of N-gonal Numbers (Peacock Theorem)
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2020-04-01 14:30
FunctionSequenceGeometric Construction
This example demonstrates the properties of N-gonal numbers, involving concepts such as sequences, rotation, and sum. Using GeoGebra commands like Sequence, Sum, and Rotate, the construction of N-gonal numbers is illustrated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Sequence((1; (i * α)), i, 0, n - 2) → vrepart
  2. Sequence(Sum[vrepart, i], i, 1, n - 1) → srepart
  3. Sequence(((n - 2) * i) - n + 3, i, 1, k) → suite
  4. Sum(suite) → a
  5. Sequence(A + (i; 0), i, 0, k - 1) → lhoriz
  6. Sequence(Polygon[A + (0, 0), A + (i; 0), n], i, 1, k - 1) → l1
  7. Sequence(Sequence[Sequence[A + (i * Element[srepart, j - 1]) + (t * Element[vrepart, j]), t, 1, i], j, 2, n - 1], i, 0, k - 1) → l2
  8. Rotate(l1, π / n, A) → l1'
  9. Sequence(Rotate[Element[l2, i], π / n, A], i, 1, Length[l2]) → l2'
  10. Rotate(lhoriz, π / n, A) → lhoriz'
  11. Sequence(Sequence[Polygon[Element[Element[Element[l2, 2], i], 1], Element[Element[Element[l2, m], i], 1], Element[Element[Element[l2, m], i], m - 1]], m, 2, k], i, 1, n - 2) → l4
  12. Sequence(Rotate[Element[l4, i], π / n, A], i, 1, Length[l4]) → l4'
  13. Element(l4', A1) → B1
  14. Element(l4', A2) → B2
  15. Element(l4', A3) → B3
  16. Element(l4', A4) → B4
  17. Element(l4', A5) → B5
  18. Element(l4', A6) → B6
  19. Element(l4', A7) → B7
  20. Element(l4', A8) → B8
  21. Element(l4', A9) → B9
  22. Element(l4', A10) → B10
  23. Element(l4', A11) → B11
  24. Element(l4', A12) → B12
  25. Element(l4', A13) → B13
  26. Element(l4', A14) → B14
  27. Element(l4', A15) → B15
  28. Sequence(F - (0, (1.2 * i)), i, 0, k) → l3
  29. Sequence(Text[" +" + (Element[suite, i]), Element[l3, i + 1] + (0.2, -0.1)], i, 1, n - 1) → l5

AI Drawing Prompt Reference

Draw a diagram illustrating the properties of N-gonal numbers, involving sequences and rotation, with sums and rotations constructing the N-gonal numbers, and label the key steps for high school mathematics education.
The prompt above was compiled from the original drawing commands and is for reference only.