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Tile Pattern

Tile Pattern
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2020-06-21 00:05
PolygonTransformationGeometric Construction
This example demonstrates how to create a tile pattern using GeoGebra. It involves concepts of geometric transformations and polygons. By using commands like Rotate, Intersect, and Polygon, complex patterns can be created. Suitable for middle school and above students to learn geometric construction.

GeoGebra Commands & Parameters

  1. Rotate(A, (-(90°)), B) → AB
  2. Rotate(B, 90°, A) → BA
  3. Rotate(B, 135°, A) → BAR
  4. Rotate(B, (-(135°)), A) → ABA
  5. Intersect(Ray[ABA, Rotate[A, 45°, ABA]], Ray[B, Rotate[A, (-(45°)), B]]) → ABAB
  6. Intersect(Ray[A, Rotate[B, 45°, A]], Ray[B, Rotate[A, (-(45°)), B]]) → ABH
  7. Intersect(Ray[A, Rotate[B, 67.5°, A]], Ray[B, Rotate[A, (-(67.5°)), B]]) → ABHR
  8. Intersect(Ray[B, Rotate[A, (-(22.5°)), B]], Ray[BA, Rotate[A, 22.5°, BA]]) → ABL
  9. Rotate(A, (-(45°)), B) → ABR
  10. Intersect(OrthogonalLine[BAR, Line[A, B]], OrthogonalLine[B, Line[A, BAR]]) → ABT
  11. Rotate(B, 135°, A) → BBA
  12. Intersect(Ray[B, Rotate[A, (-(67.5°)), B]], Ray[BBA, Rotate[A, 67.5°, BBA]]) → BAJ
  13. Intersect(OrthogonalLine[BAR, Line[A, BAR]], OrthogonalLine[B, Line[A, B]]) → BARB
  14. Intersect(OrthogonalLine[B, Line[A, B]], OrthogonalLine[BBA, Line[BBA, A]]) → BBAB
  15. Polygon(A, B, AB, BA) → Square1, sq12, sq14, sq13, sq11
  16. Segment(A, AB) → sq1pijl1
  17. Segment(B, AB) → sq1pijl2
  18. Segment(BA, AB) → sq1pijl3
  19. Segment(A, B) → sq1vet1
  20. Segment(A, BA) → sq1vet2
  21. Dilate(A, 0.75, Centroid[Square1]) → A'
  22. Dilate(B, 0.75, Centroid[Square1]) → B'
  23. Dilate(AB, 0.75, Centroid[Square1]) → AB'
  24. Dilate(BA, 0.75, Centroid[Square1]) → BA'
  25. Polygon(Dilate[A, 0.75, Centroid[Square1]], Dilate[B, 0.75, Centroid[Square1]], Dilate[AB, 0.75, Centroid[Square1]], Dilate[BA, 0.75, Centroid[Square1]]) → Square1', f, g, i, j
  26. Polygon(A, B, ABR, BAR) → Rhombus2, rh24, rh23, rh22, rh21
  27. Dilate(A, 0.75, Centroid[Rhombus2]) → A'_1
  28. Dilate(B, 0.75, Centroid[Rhombus2]) → B'_1
  29. Dilate(ABR, 0.75, Centroid[Rhombus2]) → ABR'
  30. Dilate(BAR, 0.75, Centroid[Rhombus2]) → BAR'
  31. Polygon(A'_1, B'_1, ABR', BAR') → Rhombus2', a'_1, b'_1, abr', bar'
  32. Segment(A, ABR) → rh2pijl1
  33. Segment(B, ABR) → rh2pijl2
  34. Segment(BAR, ABR) → rh2pijl3
  35. Segment(A, B) → rh2vet1
  36. Segment(BAR, A) → rh2vet2
  37. Polygon(A, B, BBAB, BBA) → Almond3, alm34, alm33, alm32, alm31
  38. Dilate(A, 0.75, Centroid[Almond3]) → A'_2
  39. Dilate(B, 0.75, Centroid[Almond3]) → B'_2
  40. Dilate(BBAB, 0.75, Centroid[Almond3]) → BBAB'
  41. Dilate(BBA, 0.75, Centroid[Almond3]) → BBA'
  42. Polygon(Dilate[A, 0.75, Centroid[Almond3]], Dilate[B, 0.75, Centroid[Almond3]], Dilate[BBAB, 0.75, Centroid[Almond3]], Dilate[BBA, 0.75, Centroid[Almond3]]) → Almond3', k, l, m, n
  43. Segment(A, BBAB) → alm3pijl1
  44. Segment(B, BBAB) → alm3pijl2
  45. Segment(A, B) → alm3vet1
  46. Segment(A, BBA) → alm3vet2
  47. Segment(BBA, BBAB) → alm3pijl3
  48. Polygon(A, B, BAJ, BBA) → Jug4, jug44, jug43, jug42, jug41
  49. Segment(A, BAJ) → jug4pijl1
  50. Segment(BBA, BAJ) → jug4pijl2
  51. Segment(B, BAJ) → jug4pijl3
  52. Segment(A, B) → jugvet1
  53. Segment(BBA, A) → jugvet2
  54. Dilate(A, 0.75, Centroid[Jug4]) → A'_3
  55. Dilate(B, 0.75, Centroid[Jug4]) → B'_3
  56. Dilate(BAJ, 0.75, Centroid[Jug4]) → BAJ'
  57. Dilate(BBA, 0.75, Centroid[Jug4]) → BBA'_1
  58. Polygon(A'_3, B'_3, BAJ', BBA'_1) → Jug4', a'_3, b'_3, baj', bba'_1
  59. Polygon(A, B, ABH) → HSquare5, hsq53, hsq52, hsq51
  60. Dilate(A, 0.75, Centroid[HSquare5]) → A'_4
  61. Dilate(B, 0.75, Centroid[HSquare5]) → B'_4
  62. Dilate(ABH, 0.75, Centroid[HSquare5]) → ABH'
  63. Polygon(A'_4, B'_4, ABH') → HSquare5', abh', a'_4, b'_4
  64. Segment(Midpoint[A, B], ABH) → hsq5pijl1
  65. Segment(A, ABH) → hsq5pijl2
  66. Segment(B, ABH) → hsq5pijl3
  67. Segment(A, B) → hsq5vet1
  68. Polygon(A, B, ABHR) → HRhombus6, hrh63, hrh62, hrh61
  69. Segment(Midpoint[A, B], ABHR) → hrh6pijl1
  70. Segment(A, ABHR) → hrh6pijl2
  71. Segment(B, ABHR) → hrh6pijl3
  72. Segment(A, B) → hrh6vet1
  73. Dilate(A, 0.75, Centroid[HRhombus6]) → A'_5
  74. Dilate(B, 0.75, Centroid[HRhombus6]) → B'_5
  75. Dilate(ABHR, 0.75, Centroid[HRhombus6]) → ABHR'
  76. Polygon(A'_5, B'_5, ABHR') → HRhombus6', abhr', a'_5, b'_5
  77. Segment(A, BA) → sq7pijl2
  78. Segment(A, B) → sq7pijl3
  79. Segment(A, B) → rh8pijl2
  80. Segment(A, BAR) → rh8pijl3
  81. Polygon(A, B, ABT, BAR) → SmBiped9, smb94, smb93, smb92, smb91
  82. Polygon(A, B, ABL, BA) → LBiped10, lb104, lb103, lb102, lb101
  83. Segment(A, ABL) → lbp10pijl1
  84. Segment(A, B) → lbp10pijl2
  85. Segment(A, BA) → lbp10pijl3
  86. Polygon(A, B, BA) → HSquare11, hsq113, hsq112, hsq111
  87. Dilate(A, 0.75, Centroid[HSquare11]) → A'_6
  88. Dilate(B, 0.75, Centroid[HSquare11]) → B'_6
  89. Dilate(BA, 0.75, Centroid[HSquare11]) → BA'_1
  90. Polygon(A'_6, B'_6, BA'_1) → HSquare11', ba'_1, a'_6, b'_6
  91. Segment(A, Midpoint[BA, B]) → hsq11pijl1
  92. Segment(A, BA) → hsq11pijl2
  93. Segment(A, B) → hsq11pijl3
  94. Polygon(A, B, BAR) → HRhombus12, hrh123, hrh122, hrh121
  95. Dilate(A, 0.75, Centroid[HRhombus12]) → A'_7
  96. Dilate(B, 0.75, Centroid[HRhombus12]) → B'_7
  97. Dilate(BAR, 0.75, Centroid[HRhombus12]) → BAR'_1
  98. Polygon(A'_7, B'_7, BAR'_1) → HRhombus12', bar'_1, a'_7, b'_7
  99. Segment(A, Midpoint[BAR, B]) → hrh12pijl1
  100. Segment(A, BAR) → hrh12pijl2
  101. Segment(A, B) → hrh12pijl3
  102. Polygon(ABA, A, B, ABAB) → SmBiped13, smbp134, smbp133, smbp132, smbp131
  103. Dilate(ABA, 0.75, Centroid[SmBiped13]) → ABA'
  104. Dilate(A, 0.75, Centroid[SmBiped13]) → A'_8
  105. Dilate(B, 0.75, Centroid[SmBiped13]) → B'_8
  106. Dilate(ABAB, 0.75, Centroid[SmBiped13]) → ABAB'
  107. Polygon(Dilate[ABA, 0.75, Centroid[SmBiped13]], Dilate[A, 0.75, Centroid[SmBiped13]], Dilate[B, 0.75, Centroid[SmBiped13]], Dilate[ABAB, 0.75, Centroid[SmBiped13]]) → SmBiped13', p, q, r, s
  108. Segment(A, ABAB) → smbp13pijl1
  109. Segment(ABA, ABAB) → smbp13pijl2
  110. Segment(B, ABAB) → smb13pijl3
  111. Segment(ABA, A) → smbp13vet1
  112. Segment(A, B) → smbp13vet2
  113. Polygon(A, B, BARB, BAR) → Almond14, alm144, alm143, alm142, alm141
  114. Segment(A, BARB) → alm14pijl1
  115. Segment(A, B) → alm14pijl2
  116. Segment(A, BAR) → alm14pijl3
  117. Dilate(A, 0.75, Centroid[SmBiped9]) → A'_9
  118. Dilate(B, 0.75, Centroid[SmBiped9]) → B'_9
  119. Dilate(ABT, 0.75, Centroid[SmBiped9]) → ABT'
  120. Dilate(BAR, 0.75, Centroid[SmBiped9]) → BAR'_2
  121. Polygon(A'_9, B'_9, ABT', BAR'_2) → SmBiped9', a'_9, b'_9, abt', bar'_2
  122. Segment(A, ABT) → smb9pijl1
  123. Segment(A, B) → smb9pijl2
  124. Segment(A, BAR) → smb9pijl3
  125. Dilate(A, 0.75, Centroid[LBiped10]) → A'_{10}
  126. Dilate(B, 0.75, Centroid[LBiped10]) → B'_{10}
  127. Dilate(ABL, 0.75, Centroid[LBiped10]) → ABL'
  128. Dilate(BA, 0.75, Centroid[LBiped10]) → BA'_2
  129. Polygon(A'_{10}, B'_{10}, ABL', BA'_2) → LBiped10', a'_{10}, b'_{10}, abl', ba'_2
  130. Dilate(A, 0.75, Centroid[Almond14]) → A'_{11}
  131. Dilate(B, 0.75, Centroid[Almond14]) → B'_{11}
  132. Dilate(BARB, 0.75, Centroid[Almond14]) → BARB'
  133. Dilate(BAR, 0.75, Centroid[Almond14]) → BAR'_3
  134. Polygon(A'_{11}, B'_{11}, BARB', BAR'_3) → Almond14', a'_{11}, b'_{11}, barb', bar'_3
  135. Intersect(Line[C, Line[D, E]], Line[E, Line[D, C]]) → CE
  136. Intersect(Line[C, Rotate[D, 45°, Line[C, C + (0, 0, 1)]]], Line[E, Rotate[D, (-(45°)), Line[E, E + (0, 0, 1)]]]) → CESB
  137. Intersect(Line[C, Rotate[D, 22.5°, Line[C, C + (0, 0, 1)]]], Line[E, Rotate[D, (-(22.5°)), Line[E, E + (0, 0, 1)]]]) → CELB
  138. Intersect(Line[C, Rotate[D, 90°, Line[C, C + (0, 0, 1)]]], Line[E, Rotate[D, (-(90°)), Line[E, E + (0, 0, 1)]]]) → CEA
  139. Polygon(D, E, EE, DD) → achter1, d, e, ee, dd
  140. Polygon(C, CC, DD, D) → achter2, c, cc, t, d_1
  141. Polygon(EE, EC, DD) → ESquare1boven1, a, b, ec
  142. Polygon(CC, EC, DD) → ESquare1boven2, f_1, g_1, h_1
  143. Polygon(EE, ECA, DD) → EAlmond3boven1, i_1, j_1, eca
  144. Polygon(CC, ECA, DD) → EAlmond3boven2, k_1, l_1, m_1
  145. Polygon(EE, ECJ, DD) → EJug4boven1, n_1, p_1, ecj
  146. Polygon(CC, ECJ, DD) → EJug4boven2, q_1, r_1, s_1
  147. Polygon(CC, DD, CDHS) → EHSquare5boven, cdhs, t_1, a_1
  148. Polygon(CC, DD, CDHR) → EHRhombus6boven, cdhr, b_1, c_1
  149. Polygon(EE, ECSB, DD) → ESmBiped13boven1, e_1, f_2, ecsb
  150. Polygon(DD, ECSB, CC) → ESmBiped13boven2, g_2, h_2, i_2
  151. Polygon(DU, E, CESB) → ESmBiped9boven1, cesb, du, e_2
  152. Polygon(DU, CESB, C) → ESmBiped9boven2, c_2, j_2, k_2
  153. Polygon(DU, E, CE) → ESquare7boven1, ce, l_2, e_3
  154. Polygon(DU, CE, C) → ESquare7boven2, c_3, m_2, n_2
  155. Polygon(DU, E, CELB) → ELBiped10boven1, celb, p_2, e_4
  156. Polygon(DU, CELB, C) → ELBiped10boven2, c_4, q_2, r_2
  157. Polygon(C, DU, E) → EHSquare11boven, e_5, c_5, s_2
  158. Polygon(DU, E, CEA) → EAmond14onder1, cea, t_2, e_6
  159. Polygon(DU, CEA, C) → EAmond14onder2, c_6, a_2, b_2