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Ford Circles: Visualizing Fractions via Tangent Circles

Ford Circles: Visualizing Fractions via Tangent Circles
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2020-06-10 01:10
CircleFunctionSequenceCoordinate GeometryGeometric Construction
This GeoGebra example demonstrates the construction of Ford circles, where for each reduced fraction p/q, a circle of radius 1/(2q²) is drawn tangent to the number line at (p/q, 0). All Ford circles are either mutually tangent or disjoint, creating a beautiful fractal-like arrangement. Core mathematical concepts include rational numbers, fractions, sequences, tangent properties of circles, and connections to continued fractions in number theory. Construction highlights: the example uses the Circle command to generate multiple circles and the Segment command to draw auxiliary segments; objects such as c, d, e, f, g, h, k and their indexed variants (c_1, d_1, etc.) are created systematically to build families of circles with different denominators. This is suitable for high school to undergraduate mathematics education, offering an intuitive visualization of rational number distribution and circle tangency, ideal for interdisciplinary lessons in number theory, geometry, and sequences.

GeoGebra Commands & Parameters

  1. Circle((0, 0.5), 0.5) → c
  2. Circle((0.5, 0.125), 0.125) → d
  3. Circle((0.25, 0.03125), 0.03125) → e
  4. Circle((0.75, 0.03125), 0.03125) → f
  5. Circle((1, 0.5), 0.5) → g
  6. Circle((0.3333333333333333, 0.05555555555555555), 0.05555555555555555) → h
  7. Circle((0.6666666666666666, 0.05555555555555555), 0.05555555555555555) → k
  8. Circle((0.2, 0.02), 0.02) → p
  9. Circle((0.4, 0.02), 0.02) → q
  10. Circle((0.6, 0.02), 0.02) → r
  11. Circle((0.8, 0.02), 0.02) → s
  12. Circle((0.16666666666666666, 0.013888888888888888), 0.013888888888888888) → t
  13. Circle((0.8333333333333334, 0.013888888888888888), 0.013888888888888888) → c_1
  14. Segment((0, 0), (1, 0)) → a
  15. Circle((0.14285714285714285, 0.01020408163265306), 0.01020408163265306) → d_1
  16. Circle((0.2857142857142857, 0.01020408163265306), 0.01020408163265306) → e_1
  17. Circle((0.42857142857142855, 0.01020408163265306), 0.01020408163265306) → f_1
  18. Circle((0.5714285714285714, 0.01020408163265306), 0.01020408163265306) → g_1
  19. Circle((0.7142857142857143, 0.01020408163265306), 0.01020408163265306) → h_1
  20. Circle((0.8571428571428571, 0.01020408163265306), 0.01020408163265306) → k_1
  21. Circle((0.125, 0.0078125), 0.0078125) → p_1
  22. Circle((0.875, 0.0078125), 0.0078125) → q_1
  23. Circle((0.1111111111111111, 0.006172839506172839), 0.006172839506172839) → r_1
  24. Circle((0.2222222222222222, 0.006172839506172839), 0.006172839506172839) → s_1
  25. Circle((0.4444444444444444, 0.006172839506172839), 0.006172839506172839) → t_1
  26. Circle((0.5555555555555556, 0.006172839506172839), 0.006172839506172839) → c_2
  27. Circle((0.7777777777777778, 0.006172839506172839), 0.006172839506172839) → d_2
  28. Circle((0.8888888888888888, 0.006172839506172839), 0.006172839506172839) → e_2
  29. Circle((0.1, 0.0050), 0.0050) → f_2
  30. Circle((0.3, 0.0050), 0.0050) → g_2
  31. Circle((0.7, 0.0050), 0.0050) → h_2
  32. Circle((0.9, 0.0050), 0.0050) → k_2
  33. Circle((0.09090909090909091, 0.004132231404958678), 0.004132231404958678) → p_2
  34. Circle((0.18181818181818182, 0.004132231404958678), 0.004132231404958678) → q_2
  35. Circle((0.2727272727272727, 0.004132231404958678), 0.004132231404958678) → r_2
  36. Circle((0.36363636363636365, 0.004132231404958678), 0.004132231404958678) → s_2
  37. Circle((0.45454545454545453, 0.004132231404958678), 0.004132231404958678) → t_2
  38. Circle((0.5454545454545454, 0.004132231404958678), 0.004132231404958678) → c_3
  39. Circle((0.6363636363636364, 0.004132231404958678), 0.004132231404958678) → d_3
  40. Circle((0.7272727272727273, 0.004132231404958678), 0.004132231404958678) → e_3
  41. Circle((0.8181818181818182, 0.004132231404958678), 0.004132231404958678) → f_3
  42. Circle((0.08333333333333333, 0.003472222222222222), 0.003472222222222222) → h_3
  43. Circle((0.4166666666666667, 0.003472222222222222), 0.003472222222222222) → k_3
  44. Circle((0.5833333333333334, 0.003472222222222222), 0.003472222222222222) → p_3
  45. Circle((0.9166666666666666, 0.003472222222222222), 0.003472222222222222) → q_3
  46. Circle((0.9090909090909091, 0.004132231404958678), 0.004132231404958678) → r_3
  47. Circle((0.07692307692307693, 0.0029585798816568047), 0.0029585798816568047) → s_3
  48. Circle((0.15384615384615385, 0.0029585798816568047), 0.0029585798816568047) → t_3
  49. Circle((0.23076923076923078, 0.0029585798816568047), 0.0029585798816568047) → c_4
  50. Circle((0.3076923076923077, 0.0029585798816568047), 0.0029585798816568047) → d_4
  51. Circle((0.38461538461538464, 0.0029585798816568047), 0.0029585798816568047) → e_4
  52. Circle((0.46153846153846156, 0.0029585798816568047), 0.0029585798816568047) → f_4
  53. Circle((0.5384615384615384, 0.0029585798816568047), 0.0029585798816568047) → g_4
  54. Circle((0.6153846153846154, 0.0029585798816568047), 0.0029585798816568047) → h_4
  55. Circle((0.6923076923076923, 0.0029585798816568047), 0.0029585798816568047) → k_4
  56. Circle((0.7692307692307693, 0.0029585798816568047), 0.0029585798816568047) → p_4
  57. Circle((0.8461538461538461, 0.0029585798816568047), 0.0029585798816568047) → q_4
  58. Circle((0.9230769230769231, 0.0029585798816568047), 0.0029585798816568047) → r_4

AI Drawing Prompt Reference

Draw a horizontal number line from 0 to 1. Above this line, construct a family of circles each tangent to the line at a rational point (p/q, 0), with radius equal to 1/(2q²). Start with the two largest circles of radius 0.5 tangent at 0 and 1, then add a circle of radius 0.125 at 1/2, circles of radius 0.03125 at 1/4 and 3/4, and circles of radius 0.0556 at 1/3 and 2/3. Continue adding circles for fractions with denominators 5, 6, 7, 8, 9, 10, 11, 12, and 13, each radius shrinking as the denominator squared increases. Ensure all circles are either mutually tangent or disjoint, creating a fractal-like arrangement. Finally, draw a segment connecting the endpoints 0 and 1 as the baseline.
The prompt above was compiled from the original drawing commands and is for reference only.