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Escherized Hexagon 2

Escherized Hexagon 2
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2016-06-27 10:10
PolygonVectorTransformationGeometric Construction
This example demonstrates the construction and properties of Escherized Hexagon 2, a type of isohedral convex hexagon tiling. Key math concepts include reflection, rotation, vectors, and geometric transformations. The construction utilizes commands/objects like Angle, Point, Segment, Intersect, PointIn, Midpoint, Rotate, ptinsq, Vector, Translate, AngularBisector, Mirror, Polygon, glrefl from GeoGebra. Suitable for high school and above learners to understand complex geometric constructions and transformations.

GeoGebra Commands & Parameters

  1. Angle(D, C, B) → γ
  2. Angle(C, B, A) → β
  3. Point(Circle[D, Distance[A, B]]) → E
  4. Segment(D, E) → f
  5. Intersect(Circle[E, Distance[C, D]], Ray[Rotate[E, (-(360° - γ - β)), E], Rotate[D, (-(360° - γ - β)), E]]) → F,
  6. PointIn(Polygon[Midpoint[C, B], B, 4]) → G
  7. PointIn(Polygon[Midpoint[C, B], B, 4]) → H
  8. Midpoint(C, B) → I
  9. Rotate(H, 180°, I) → H'
  10. Rotate(G, 180°, I) → G'
  11. ptinsq(D, E) → J
  12. ptinsq(D, E) → K
  13. Midpoint(F, A) → L
  14. ptinsq(F, L) → M
  15. ptinsq(F, L) → N
  16. Rotate(N, 180°, L) → N'
  17. Rotate(M, 180°, L) → M'
  18. Vector(D, A) → u
  19. Translate(E, u) → E'
  20. AngularBisector(E', A, B) → g
  21. Mirror(Translate[J, u], g) → Q
  22. Mirror(Translate[K, u], g) → O
  23. Polygon(B, G, F, M, N, N', M', A, O, Q, H, H', G', C, D, K, J, E) → poly1, b, e, f_1, m, n, n', m', a, o, q, g_1, h, h', g', c, d, k, j
  24. Rotate(B, 180°, I) → B'
  25. Rotate(G, 180°, I) → G'_1
  26. Rotate(H, 180°, I) → H'_1
  27. Rotate(H', 180°, I) → H''
  28. Rotate(G', 180°, I) → G''
  29. Rotate(C, 180°, I) → C'
  30. Rotate(D, 180°, I) → D'
  31. Rotate(K, 180°, I) → K'
  32. Rotate(J, 180°, I) → J'
  33. Rotate(E, 180°, I) → E'_1
  34. Rotate(F, 180°, I) → F'
  35. Rotate(M, 180°, I) → M'_1
  36. Rotate(N, 180°, I) → N'_1
  37. Rotate(N', 180°, I) → N''
  38. Rotate(M', 180°, I) → M''
  39. Rotate(A, 180°, I) → A'
  40. Rotate(O, 180°, I) → O'
  41. Rotate(Q, 180°, I) → Q'
  42. Polygon(B', G'_1, F', M'_1, N'_1, N'', M'', A', O', Q', H'_1, H'', G'', C', D', K', J', E'_1) → poly1', b', e'_1, f', m'_1, n'_1, n'', m'', a', o', q', g'_1, h'_1, h'', g'', c', d', k', j'
  43. Rotate(B, 180°, L) → B'_1
  44. Rotate(G, 180°, L) → G'_2
  45. Rotate(H, 180°, L) → H'_2
  46. Rotate(H', 180°, L) → H''_1
  47. Rotate(G', 180°, L) → G''_1
  48. Rotate(C, 180°, L) → C'_1
  49. Rotate(D, 180°, L) → D'_1
  50. Rotate(K, 180°, L) → K'_1
  51. Rotate(J, 180°, L) → J'_1
  52. Rotate(E, 180°, L) → E'_2
  53. Rotate(F, 180°, L) → F'_1
  54. Rotate(M, 180°, L) → M'_2
  55. Rotate(N, 180°, L) → N'_2
  56. Rotate(N', 180°, L) → N''_1
  57. Rotate(M', 180°, L) → M''_1
  58. Rotate(A, 180°, L) → A'_1
  59. Rotate(O, 180°, L) → O'_1
  60. Rotate(Q, 180°, L) → Q'_1
  61. Polygon(B'_1, G'_2, F'_1, M'_2, N'_2, N''_1, M''_1, A'_1, O'_1, Q'_1, H'_2, H''_1, G''_1, C'_1, D'_1, K'_1, J'_1, E'_2) → poly1'_1, b'_1, e'_2, f'_1, m'_2, n'_2, n''_1, m''_1, a'_1, o'_1, q'_1, g'_2, h'_2, h''_1, g''_1, c'_1, d'_1, k'_1, j'_1
  62. Mirror(Polygon[Translate[B, u], Translate[G, u], Translate[H, u], Translate[H', u], Translate[G', u], Translate[C, u], Translate[D, u], Translate[K, u], Translate[J, u], Translate[E, u], Translate[F, u], Translate[M, u], Translate[N, u], Translate[N', u], Translate[M', u], Translate[A, u], Translate[O, u], Translate[Q, u]], g) → poly2
  63. Translate(Polygon[Mirror[B, g], Mirror[G, g], Mirror[H, g], Mirror[H', g], Mirror[G', g], Mirror[C, g], Mirror[D, g], Mirror[K, g], Mirror[J, g], Mirror[E, g], Mirror[F, g], Mirror[M, g], Mirror[N, g], Mirror[N', g], Mirror[M', g], Mirror[A, g], Mirror[O, g], Mirror[Q, g]], Vector[(-u)]) → poly3
  64. Mirror(Translate[I, u], g) → P
  65. Mirror(Translate[L, u], g) → R
  66. Rotate(poly2, 180°, P) → poly2'
  67. Rotate(poly2, 180°, R) → poly2'_1
  68. Translate(Mirror[I, g], Vector[(-u)]) → S
  69. Translate(Mirror[L, g], Vector[(-u)]) → T
  70. Rotate(poly3, 180°, S) → poly3'
  71. Rotate(poly3, 180°, T) → poly3'_1
  72. Rotate(L, 180°, I) → L'
  73. Rotate(B', 180°, L') → B''
  74. Rotate(G'_1, 180°, L') → U
  75. Rotate(H'_1, 180°, L') → V
  76. Rotate(H'', 180°, L') → H'''
  77. Rotate(G'', 180°, L') → G'''
  78. Rotate(C', 180°, L') → C''
  79. Rotate(D', 180°, L') → D''
  80. Rotate(K', 180°, L') → K''
  81. Rotate(J', 180°, L') → J''
  82. Rotate(E'_1, 180°, L') → W
  83. Rotate(F', 180°, L') → F''
  84. Rotate(M'_1, 180°, L') → Z
  85. Rotate(N'_1, 180°, L') → A_1
  86. Rotate(N'', 180°, L') → N'''
  87. Rotate(M'', 180°, L') → M'''
  88. Rotate(A', 180°, L') → A''
  89. Rotate(O', 180°, L') → O''
  90. Rotate(Q', 180°, L') → Q''
  91. Polygon(B'', U, F'', Z, A_1, N''', M''', A'', O'', Q'', V, H''', G''', C'', D'', K'', J'', W) → poly1'', b'', w, f'', z_1, a_1, n''', m''', a'', o'', q'', u_1, v, h''', g''', c'', d'', k'', j''
  92. Rotate(S, 180°, T) → S'
  93. Rotate(poly3'_1, 180°, S') → poly4
  94. Rotate(P, 180°, R) → P'
  95. Rotate(poly2'_1, 180°, P') → poly5
  96. glrefl(D, E, B, A) → i, GR_1, GR_2, l
  97. Mirror(Translate[poly2, l], i) → poly6
  98. Mirror(Translate[P, l], i) → D_1
  99. Mirror(Translate[R, l], i) → E_1
  100. Rotate(poly6, 180°, E_1) → poly6'
  101. Rotate(poly6, 180°, D_1) → poly6'_1
  102. Rotate(E_1, 180°, D_1) → B_1
  103. Rotate(poly6'_1, 180°, B_1) → poly7