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Geometric Visualization of Eratosthenes Sieve with Ellipse and Sequences

Geometric Visualization of Eratosthenes Sieve with Ellipse and Sequences
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2023-03-19 01:11
FunctionSequenceCoordinate GeometryProbability & StatisticsGeometric Construction
In collections geometriadinamica.es
This example demonstrates how the Eratosthenes sieve idea can be visualized through dynamic geometry and sequence-driven constructions. It links the prime-sieving concept with coordinate geometry: key mathematical notions include prime filtering, sequence generation (Sequence), constructing points and curves in the Cartesian plane, and using geometric primitives to encode sieve steps. Construction workflow: an Ellipse is created as the geometric baseline; Point and Element are used to define and manage key points and derived objects; Sequence generates the indexing/parameter values used to mark and progressively filter candidates; CurveCartesian draws the relationships in the Cartesian coordinate system; Segment is used to connect or highlight structural parts of the construction. Additional parameter/angle controls (e.g., a rotation angle variable) and parameter objects (such as p, h0, L, m2, m3, m5, m7, m11, etc.) guide how positions update across iterations. The resulting labeled indicators (e.g., primo and multiple picXX objects) reveal where primes are selected. Suitable for advanced secondary students and undergraduates for GeoGebra-based learning of number theory concepts, algorithm-to-visualization projects, and coordinate/geometry modeling.

GeoGebra Commands & Parameters

  1. Ellipse((-6.62, 5), (-5.38, 5), 0.65) → zElipse1
  2. Point(zElipse1) → XY
  3. CurveCartesian(((3 * cos(t)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(p / 30) * t) + 5, t, (-34 * p), 0) → h0
  4. Sequence(h0(((-i) * p / 3)), i, 0, 98) → L
  5. Sequence(Circle[h0(((-i) * p / 3)) + (0.19, 0.1), h0(((-i) * p / 3))], i, 0, 98) → L1
  6. Element(L1, 2) → p01
  7. Element(L1, 3) → p02
  8. Element(L1, 5) → p03
  9. Element(L1, 7) → p04
  10. Element(L1, 11) → p05
  11. Element(L1, 13) → p06
  12. Element(L1, 17) → p07
  13. Element(L1, 19) → p08
  14. Element(L1, 23) → p09
  15. Element(L1, 29) → p10
  16. Element(L1, 31) → p11
  17. Element(L1, 37) → p12
  18. Element(L1, 41) → p13
  19. Element(L1, 43) → p14
  20. Element(L1, 47) → p15
  21. Element(L1, 53) → p16
  22. Element(L1, 59) → p17
  23. Element(L1, 61) → p18
  24. Element(L1, 67) → p19
  25. Element(L1, 71) → p20
  26. Element(L1, 73) → p21
  27. Element(L1, 79) → p22
  28. Element(L1, 83) → p23
  29. Element(L1, 89) → p24
  30. Element(L1, 97) → p25
  31. Element(L1, 10) → m502
  32. Element(L1, 15) → m503
  33. Element(L1, 20) → m504
  34. Element(L1, 25) → m505
  35. Element(L1, 30) → m506
  36. Element(L1, 35) → m507
  37. Element(L1, 40) → m508
  38. Element(L1, 45) → m509
  39. Element(L1, 50) → m510
  40. Element(L1, 55) → m511
  41. Element(L1, 60) → m512
  42. Element(L1, 65) → m513
  43. Element(L1, 70) → m514
  44. Element(L1, 75) → m515
  45. Element(L1, 80) → m516
  46. Element(L1, 85) → m517
  47. Element(L1, 90) → m518
  48. Element(L1, 95) → m519
  49. Element(L1, 14) → m702
  50. Element(L1, 21) → m703
  51. Element(L1, 28) → m704
  52. Element(L1, 35) → m705
  53. Element(L1, 42) → m706
  54. Element(L1, 49) → m707
  55. Element(L1, 56) → m708
  56. Element(L1, 63) → m709
  57. Element(L1, 70) → m710
  58. Element(L1, 77) → m711
  59. Element(L1, 84) → m712
  60. Element(L1, 91) → m713
  61. Element(L1, 98) → m714
  62. Element(L1, 22) → m112
  63. Element(L1, 33) → m113
  64. Element(L1, 44) → m114
  65. Element(L1, 55) → m115
  66. Element(L1, 66) → m116
  67. Element(L1, 77) → m117
  68. Element(L1, 88) → m118
  69. Element(L1, 99) → m119
  70. Segment(h0(((-p)) / 3) + (0.2, -0.12), h0((-97 * p / 3)) + (0.2, -1)) → s2a
  71. Segment(h0((-p)) + (0.2, 0.1), h0((-93 * p / 3)) + (0.2, -1)) → s2b
  72. Segment(h0((-5 * p / 3)) + (0.15, 0.1), h0((-95 * p / 3)) + (0.15, -1)) → s2c
  73. Segment(h0((-2 * p / 3)) + (0.2, -0.12), h0((-98 * p / 3)) + (0.2, -1)) → s3a
  74. Segment(h0((-5 * p / 3)) + (0.2, 0.1), h0((-95 * p / 3)) + (0.2, -1)) → s3b
  75. CurveCartesian(0.2 + ((3 * cos(t + p)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(27.7°) * t) + 10.7, t, (-11 * p), (-11 * p / 3) - 0.1) → h5
  76. CurveCartesian(0.2 + ((3 * cos(t)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(36.4°) * t) + 13.7, t, (-17 * p / 2), (-4 * p) - 0.1) → h7
  77. CurveCartesian(0.2 + ((3 * cos(t + p)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(49°) * t) + 17.2, t, (-33 * p / 5), (-11 * p / 3) - 0.1) → h11
  78. Ellipse((-3, 5.5), (3.4, 5.5), 3.21) → base

AI Drawing Prompt Reference

Visualize the Eratosthenes sieve by drawing an ellipse in the Cartesian coordinate system. Generate a sequence of circles centered on the ellipse, with positions determined by a rotation angle and sequence parameters. Highlight specific circles from the sequence (e.g., at indices 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) using distinct colors or styles, and connect these highlighted circles with lines to illustrate the prime filtering process, emphasizing primes with a special marker.
The prompt above was compiled from the original drawing commands and is for reference only.