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Group 10: Rotation and Dilation Built Fractal-Like Pattern

Group 10: Rotation and Dilation Built Fractal-Like Pattern
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2023-11-09 06:54
PolygonTransformationGeometric Construction
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This example demonstrates how to generate a fractal-like, iteratively structured pattern by repeatedly applying geometric transformations—especially rotation combined with dilation/scaling effects—under parameter control. The key mathematical ideas include fractal geometry and self-similarity, together with an iterative/step-function perspective where angles and scaling ratios drive successive stages of the construction. In GeoGebra, the pattern is built by creating polygonal shapes using Polygon, then replicating and repositioning them via Rotate around selected points/anchors represented by constructed points (e.g., A, B, C). Parameterized point sets such as q1, q2, q3, q4, q5, q6 and their primed counterparts (q2', q4', q5', q6') encode “before/after” positions across transformation steps. Additional objects like l, w, h, and t (along with button1 and button2) indicate adjustable dimensions and iteration controls, enabling learners to observe how the overall figure evolves as parameters change. Suitable for advanced high school and pre-college/intro university students exploring transformation geometry, iterative construction, and fractal concepts.

GeoGebra Commands & Parameters

  1. Point(Ray[(1, 0), (3, 0)]) → A
  2. Point(Ray[A + (1, 0), A + (2, 0)]) → C
  3. Point(Ray[A + (0, 1), A + (0, 2)]) → B
  4. Polygon({(0, 0), A, B, (0, y(B))}) → q1
  5. Polygon({A, C, (x(C), y(B)), B}) → q2
  6. Polygon({((-w), w + h), ((-w), w), ((-w) - l, w), ((-w) - l, w + h)}) → q3
  7. Polygon({B, B + (0, h), B + ((-l), h), B + ((-l), 0)}) → q4
  8. Polygon({C, C + (l, 0), C + (l, w), C + (0, w)}) → q5
  9. Polygon({(0, w), (0, w + h), ((-w), w + h), ((-w), w)}) → q6
  10. Rotate(q4, (90° * t), Line[(0, w), B]) → q4'
  11. Rotate(q2, (90° * t), Line[B, A]) → q2'
  12. Rotate(Rotate[q5, (90° * t), Line[C + (0, w), C]], (90° * t), Line[B, A]) → q5'
  13. Rotate(Rotate[q6, (90° * t), Line[(0, w), (0, w + h)]], (90° * t), Line[(0, w), B]) → q6'
  14. Rotate(Rotate[Rotate[q3, (90° * t), Line[((-w), w), ((-w), w + h)]], (90° * t), Line[(0, w), (0, w + h)]], (90° * t), Line[(0, w), B]) → q3'

AI Drawing Prompt Reference

Draw in the plane a structured, fractal-like rotation pattern. First place a base point A from an intersection on a horizontal ray, then construct C to the right of A and B above A using vertical and horizontal rays. Using the height y(B), create polygon q1 from the points (0,0), A, B, and (0,y(B)). Create polygon q2 from A, C, (x(C),y(B)), and B. Next, using dimensions w, h, and l, draw four additional polygons q3, q4, q5, and q6 as rectilinear shapes aligned with the same reference location. Then rotate q4 by an angle 90°·t about the axis line through (0,w) and B to obtain q4′; rotate q2 by 90°·t about the axis line through B and A to get q2′. Apply one or two successive 90°·t rotations to q5, q6, and q3 around their corresponding axis lines so all pieces fit together around the chosen pivots, forming the iterative self-similar arrangement.
The prompt above was compiled from the original drawing commands and is for reference only.