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Golden and Root Rectangles Demo

Golden and Root Rectangles Demo
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2019-04-22 05:10
Coordinate GeometryCalculusTransformationParametric EquationGeometric Construction
This example demonstrates the geometric construction and properties of golden and root rectangles. It involves core math concepts like geometric transformations, upper and lower sums or Riemann sums, parametric curves, triangles, and random variables. The construction uses GeoGebra commands and objects such as points, segments, intersections, polygons, circles, and lines. It is suitable for high school and above students to visualize geometric transformations and the connection to integral concepts.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(xAxis) → D
  4. Point(yAxis) → E
  5. Segment(O, P) → n
  6. Intersect(xAxis, yAxis) → G
  7. Polygon(G, C, 4) → poly1, f, g, h, i, H, I
  8. Segment(G, H) → j
  9. Circle(G, H) → c
  10. Intersect(c, xAxis, 2) → J
  11. Line(G, C) → k
  12. Line(I, H) → l
  13. Line(J, g) → m
  14. Intersect(l, m) → A_1
  15. Circle(G, A_1) → d
  16. Intersect(d, k, 2) → B_1
  17. Line(B_1, m) → p
  18. Intersect(l, p) → C_1
  19. Circle(G, C_1) → e
  20. Intersect(e, k, 2) → D_1
  21. Line(D_1, p) → q
  22. Intersect(l, q) → E_1
  23. Circle(G, E_1) → r
  24. Intersect(r, k, 2) → F_1
  25. Line(F_1, q) → s
  26. Intersect(l, s) → G_1
  27. Circle(G, G_1) → t
  28. Intersect(t, k, 2) → H_1
  29. Line(H_1, s) → a
  30. Intersect(l, a) → I_1
  31. Circle(G, I_1) → c_1
  32. Intersect(c_1, k, 2) → J_1
  33. Line(J_1, a) → b
  34. Intersect(l, b) → K_1
  35. Circle(G, K_1) → d_1
  36. Intersect(d_1, k, 2) → L_1
  37. Line(L_1, b) → f_1
  38. Intersect(l, f_1) → M_1
  39. Circle(G, M_1) → e_1
  40. Intersect(e_1, k, 2) → N_1
  41. Line(N_1, f_1) → g_1
  42. Intersect(l, g_1) → O_1

AI Drawing Prompt Reference

On a coordinate plane, place three points on the horizontal axis and one on the vertical axis. Draw a segment from the origin to a point. Construct a square with the origin as one corner, then draw a circle centered at the origin with radius equal to the square's side, intersecting the horizontal axis. From this intersection, draw a line through the origin)Skip, then create parallel lines to form successive intersections. Repeatedly draw circles centered at the origin through these intersection points, each time intersecting a diagonal line and drawing a new parallel line, generating a nested sequence of rectangles and arcs that illustrate the geometric progression of golden and root rectangles.
The prompt above was compiled from the original drawing commands and is for reference only.