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Regular Tessellation {6,3}

Regular Tessellation {6,3}
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2022-12-02 06:28
PolygonVectorGeometric Construction
This example demonstrates the regular tessellation of hexagons using triplets. It involves core mathematical concepts like polygon construction, vector operations, and geometric transformations. The construction utilizes GeoGebra commands and objects such as Point, Polygon, Vector, and Translate. It is suitable for high school mathematics education, illustrating plane figure tiling properties and transformation rules.

GeoGebra Commands & Parameters

  1. Point(xAxis) → B
  2. Polygon(A, B, 6) → poly1, f, F, g, h, i, j, k, C, D, E
  3. Polygon(D, C, 6) → poly2, l, J, m, n, p, q, r, G, H, I
  4. Polygon(I, H, 6) → poly3, s, N, t, a, b, c, d, K, L, M
  5. Point(xAxis) → O
  6. Vector(B, O) → u
  7. Translate(poly1, u) → poly1'
  8. Point(yAxis) → P
  9. Vector(A, P) → v
  10. Translate(poly2, v) → poly2'
  11. Translate(poly2, u) → poly2'_{1}
  12. Translate(poly1', u) → poly1''
  13. Translate(poly2', u) → poly2''
  14. Translate(poly3, u) → poly3'
  15. Translate(poly1', v) → poly1''_{1}
  16. Translate(poly1'', v) → poly1'''
  17. Translate(poly1, v) → poly1'_{1}
  18. Translate(poly2', v) → poly2''_{1}
  19. Translate(poly2'_{1}, u) → poly4
  20. Translate(poly2'', u) → poly2'''
  21. Translate(poly2''_{1}, u) → poly5
  22. Translate(poly5, u) → poly5'
  23. Translate(poly1''', u) → poly6
  24. Translate(poly1'', u) → poly1'''_{1}
  25. Translate(poly4, u) → poly4'
  26. Translate(poly3', u) → poly3''
  27. Translate(poly2''', u) → poly7
  28. Translate(poly1'''_{1}, u) → poly8
  29. Translate(poly6, u) → poly6'
  30. Translate(poly1'_{1}, v) → poly9
  31. Translate(poly2''_{1}, v) → poly10
  32. Translate(poly1''_{1}, v) → poly11
  33. Translate(poly1''', v) → poly12
  34. Translate(poly5, v) → poly5'_{1}
  35. Translate(poly9, v) → poly9'
  36. Translate(poly10, v) → poly10'
  37. Translate(poly11, v) → poly11'
  38. Translate(poly5'_{1}, v) → poly13
  39. Translate(poly12, v) → poly12'
  40. Translate(poly5', v) → poly5''
  41. Translate(poly6, v) → poly6'_{1}
  42. Translate(poly7, v) → poly7'
  43. Translate(poly7', v) → poly7''
  44. Translate(poly6'_{1}, v) → poly14
  45. Translate(poly5'', v) → poly5'''
  46. Translate(poly6', v) → poly6''
  47. Translate(poly7'', v) → poly7'''
  48. Translate(poly6'', v) → poly6'''
  49. Translate(poly6''', v) → poly15
  50. Translate(poly9', v) → poly9''
  51. Translate(poly10', v) → poly10''
  52. Translate(poly11', v) → poly11''
  53. Translate(poly13, v) → poly13'
  54. Translate(poly12', v) → poly12''
  55. Translate(poly5''', v) → poly16
  56. Translate(poly14, v) → poly14'
  57. Translate(poly7''', v) → poly17
  58. Translate(poly4', u) → poly4''
  59. Translate(poly7, u) → poly7'_{1}
  60. Translate(poly6', u) → poly6''_{1}
  61. Translate(poly7', u) → poly7''_{1}
  62. Translate(poly6'', u) → poly6'''_{1}
  63. Translate(poly7''_{1}, v) → poly18
  64. Translate(poly18, v) → poly18'
  65. Translate(poly6'''_{1}, v) → poly19
  66. Translate(poly19, v) → poly19'
  67. Translate(poly18', v) → poly18''
  68. Translate(poly8, u) → poly8'
  69. Translate(poly9'', v) → poly9'''
  70. Translate(poly11'', v) → poly11'''
  71. Translate(poly12'', v) → poly12'''
  72. Translate(poly14', v) → poly14''
  73. Translate(poly15, v) → poly15'
  74. Translate(poly19', v) → poly19''
  75. Translate(poly7'_{1}, u) → poly20
  76. Translate(poly7''_{1}, u) → poly21
  77. Translate(poly18, u) → poly18'_{1}
  78. Translate(poly4'', u) → poly4'''
  79. Translate(poly18'_{1}, v) → poly22
  80. Translate(poly22, v) → poly22'
  81. Translate(poly8', u) → poly8''
  82. Translate(poly8'', v) → poly8'''
  83. Translate(poly8''', v) → poly23
  84. Translate(poly23, v) → poly23'
  85. Translate(poly23', v) → poly23''
  86. Translate(poly23'', v) → poly23'''
  87. Translate(poly3'', u) → poly3'''
  88. Translate(poly3''', u) → poly24

AI Drawing Prompt Reference

Create a regular tessellation {6,3} with hexagons. First, define a base setup on the plane using two reference points (A and B) determined by the axes, then construct a regular 6-sided polygon using A and B. Using the corresponding adjacent vertices, build a second and a third regular hexagon so they sit next to the first one in the same strip-like arrangement. Define two translation directions with vectors: one from B to a point O on the x-axis, and another from A to a point P on the y-axis. Repeatedly translate the constructed hexagons by these two vectors in a grid pattern to extend the tiling, ensuring the translated hexagons fit together edge-to-edge and form multiple rows and columns of the regular hexagon mosaic.
The prompt above was compiled from the original drawing commands and is for reference only.