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帕斯卡三角形冰球杆模式演示

帕斯卡三角形冰球杆模式演示
几何画板及绘图内容加载中，请稍等……
2019-02-12 01:25
向量数列几何作图
①本范例演示帕斯卡三角形中冰球杆模式的图形现象；②涉及核心数学概念包括数列、组合数、向量与几何作图；③构造要点提炼自GeoGebra命令：UnicodeToLetter, Polygon, Vector, nCr, Midpoint, Text, Sequence, Random, Join, RemoveUndefined, LaTeX, If；④适合高中数学教学与学习，用于理解组合数学与几何模式。

GeoGebra 命令及参数

  1. UnicodeToLetter(10007) → xmark
  2. UnicodeToLetter(10003) → cmark
  3. Polygon(BLviewbox, (x(BLviewbox), y(TRviewbox)), TRviewbox, (x(TRviewbox), y(BLviewbox))) → viewbox, vb1, vb4, vb2, vb3
  4. Polygon(A, B, 6) → hexagon1, Aside_2, F, Aside_1, Aside_3, Aside_4, Aside_5, Aside_6, C, D, E
  5. Vector(A, E) → v
  6. Vector(F, B) → u
  7. Polygon(A + (B2 * u) + (C2 * v), B + (B2 * u) + (C2 * v), 6) → A2
  8. Polygon(A + (B3 * u) + (C3 * v), B + (B3 * u) + (C3 * v), 6) → A3
  9. Polygon(A + (B4 * u) + (C4 * v), B + (B4 * u) + (C4 * v), 6) → A4
  10. Polygon(A + (B5 * u) + (C5 * v), B + (B5 * u) + (C5 * v), 6) → A5
  11. Polygon(A + (B6 * u) + (C6 * v), B + (B6 * u) + (C6 * v), 6) → A6
  12. Polygon(A + (B7 * u) + (C7 * v), B + (B7 * u) + (C7 * v), 6) → A7
  13. Polygon(A + (B8 * u) + (C8 * v), B + (B8 * u) + (C8 * v), 6) → A8
  14. Polygon(A + (B9 * u) + (C9 * v), B + (B9 * u) + (C9 * v), 6) → A9
  15. Polygon(A + (B10 * u) + (C10 * v), B + (B10 * u) + (C10 * v), 6) → A10
  16. Polygon(A + (B11 * u) + (C11 * v), B + (B11 * u) + (C11 * v), 6) → A11
  17. Polygon(A + (B12 * u) + (C12 * v), B + (B12 * u) + (C12 * v), 6) → A12
  18. Polygon(A + (B13 * u) + (C13 * v), B + (B13 * u) + (C13 * v), 6) → A13
  19. Polygon(A + (B14 * u) + (C14 * v), B + (B14 * u) + (C14 * v), 6) → A14
  20. Polygon(A + (B15 * u) + (C15 * v), B + (B15 * u) + (C15 * v), 6) → A15
  21. Polygon(A + (B16 * u) + (C16 * v), B + (B16 * u) + (C16 * v), 6) → A16
  22. Polygon(A + (B17 * u) + (C17 * v), B + (B17 * u) + (C17 * v), 6) → A17
  23. Polygon(A + (B18 * u) + (C18 * v), B + (B18 * u) + (C18 * v), 6) → A18
  24. Polygon(A + (B19 * u) + (C19 * v), B + (B19 * u) + (C19 * v), 6) → A19
  25. Polygon(A + (B20 * u) + (C20 * v), B + (B20 * u) + (C20 * v), 6) → A20
  26. Polygon(A + (B21 * u) + (C21 * v), B + (B21 * u) + (C21 * v), 6) → A21
  27. Polygon(A + (B22 * u) + (C22 * v), B + (B22 * u) + (C22 * v), 6) → A22
  28. Polygon(A + (B23 * u) + (C23 * v), B + (B23 * u) + (C23 * v), 6) → A23
  29. Polygon(A + (B24 * u) + (C24 * v), B + (B24 * u) + (C24 * v), 6) → A24
  30. Polygon(A + (B25 * u) + (C25 * v), B + (B25 * u) + (C25 * v), 6) → A25
  31. Polygon(A + (B26 * u) + (C26 * v), B + (B26 * u) + (C26 * v), 6) → A26
  32. Polygon(A + (B27 * u) + (C27 * v), B + (B27 * u) + (C27 * v), 6) → A27
  33. Polygon(A + (B28 * u) + (C28 * v), B + (B28 * u) + (C28 * v), 6) → A28
  34. Polygon(A + (B29 * u) + (C29 * v), B + (B29 * u) + (C29 * v), 6) → A29
  35. Polygon(A + (B30 * u) + (C30 * v), B + (B30 * u) + (C30 * v), 6) → A30
  36. Polygon(A + (B31 * u) + (C31 * v), B + (B31 * u) + (C31 * v), 6) → A31
  37. Polygon(A + (B32 * u) + (C32 * v), B + (B32 * u) + (C32 * v), 6) → A32
  38. Polygon(A + (B33 * u) + (C33 * v), B + (B33 * u) + (C33 * v), 6) → A33
  39. Polygon(A + (B34 * u) + (C34 * v), B + (B34 * u) + (C34 * v), 6) → A34
  40. Polygon(A + (B35 * u) + (C35 * v), B + (B35 * u) + (C35 * v), 6) → A35
  41. Polygon(A + (B36 * u) + (C36 * v), B + (B36 * u) + (C36 * v), 6) → A36
  42. nCr(B2 - 1, C2 - 1) → F2
  43. nCr(B3 - 1, C3 - 1) → F3
  44. nCr(B4 - 1, C4 - 1) → F4
  45. nCr(B5 - 1, C5 - 1) → F5
  46. nCr(B6 - 1, C6 - 1) → F6
  47. nCr(B7 - 1, C7 - 1) → F7
  48. nCr(B8 - 1, C8 - 1) → F8
  49. nCr(B9 - 1, C9 - 1) → F9
  50. nCr(B10 - 1, C10 - 1) → F10
  51. nCr(B11 - 1, C11 - 1) → F11
  52. nCr(B12 - 1, C12 - 1) → F12
  53. nCr(B13 - 1, C13 - 1) → F13
  54. nCr(B14 - 1, C14 - 1) → F14
  55. nCr(B15 - 1, C15 - 1) → F15
  56. nCr(B16 - 1, C16 - 1) → F16
  57. nCr(B17 - 1, C17 - 1) → F17
  58. nCr(B18 - 1, C18 - 1) → F18
  59. nCr(B19 - 1, C19 - 1) → F19
  60. nCr(B20 - 1, C20 - 1) → F20
  61. nCr(B21 - 1, C21 - 1) → F21
  62. nCr(B22 - 1, C22 - 1) → F22
  63. nCr(B23 - 1, C23 - 1) → F23
  64. nCr(B24 - 1, C24 - 1) → F24
  65. nCr(B25 - 1, C25 - 1) → F25
  66. nCr(B26 - 1, C26 - 1) → F26
  67. nCr(B27 - 1, C27 - 1) → F27
  68. nCr(B28 - 1, C28 - 1) → F28
  69. nCr(B29 - 1, C29 - 1) → F29
  70. nCr(B30 - 1, C30 - 1) → F30
  71. nCr(B31 - 1, C31 - 1) → F31
  72. nCr(B32 - 1, C32 - 1) → F32
  73. nCr(B33 - 1, C33 - 1) → F33
  74. nCr(B34 - 1, C34 - 1) → F34
  75. nCr(B35 - 1, C35 - 1) → F35
  76. nCr(B36 - 1, C36 - 1) → F36
  77. nCr(B37 - 1, C37 - 1) → F37
  78. Midpoint(F, C) → Center
  79. Text(F2, Center + (B2 * u) + (C2 * v) - (0.1 + (Length[Text[F2]] * 0.1), 0.2)) → H2
  80. Text(F3, Center + (B3 * u) + (C3 * v) - (0.1 + (Length[Text[F3]] * 0.1), 0.2)) → H3
  81. Text(F4, Center + (B4 * u) + (C4 * v) - (0.1 + (Length[Text[F4]] * 0.1), 0.2)) → H4
  82. Text(F5, Center + (B5 * u) + (C5 * v) - (0.1 + (Length[Text[F5]] * 0.1), 0.2)) → H5
  83. Text(F6, Center + (B6 * u) + (C6 * v) - (0.1 + (Length[Text[F6]] * 0.1), 0.2)) → H6
  84. Text(F7, Center + (B7 * u) + (C7 * v) - (0.1 + (Length[Text[F7]] * 0.1), 0.2)) → H7
  85. Text(F8, Center + (B8 * u) + (C8 * v) - (0.1 + (Length[Text[F8]] * 0.1), 0.2)) → H8
  86. Text(F9, Center + (B9 * u) + (C9 * v) - (0.1 + (Length[Text[F9]] * 0.1), 0.2)) → H9
  87. Text(F10, Center + (B10 * u) + (C10 * v) - (0.1 + (Length[Text[F10]] * 0.1), 0.2)) → H10
  88. Text(F11, Center + (B11 * u) + (C11 * v) - (0.1 + (Length[Text[F11]] * 0.1), 0.2)) → H11
  89. Text(F12, Center + (B12 * u) + (C12 * v) - (0.1 + (Length[Text[F12]] * 0.1), 0.2)) → H12
  90. Text(F13, Center + (B13 * u) + (C13 * v) - (0.1 + (Length[Text[F13]] * 0.1), 0.2)) → H13
  91. Text(F14, Center + (B14 * u) + (C14 * v) - (0.1 + (Length[Text[F14]] * 0.1), 0.2)) → H14
  92. Text(F15, Center + (B15 * u) + (C15 * v) - (0.1 + (Length[Text[F15]] * 0.1), 0.2)) → H15
  93. Text(F16, Center + (B16 * u) + (C16 * v) - (0.1 + (Length[Text[F16]] * 0.1), 0.2)) → H16
  94. Text(F18, Center + (B18 * u) + (C18 * v) - (0.1 + (Length[Text[F18]] * 0.1), 0.2)) → H18
  95. Text(F19, Center + (B19 * u) + (C19 * v) - (0.1 + (Length[Text[F19]] * 0.1), 0.2)) → H19
  96. Text(F20, Center + (B20 * u) + (C20 * v) - (0.1 + (Length[Text[F20]] * 0.1), 0.2)) → H20
  97. Text(F21, Center + (B21 * u) + (C21 * v) - (0.1 + (Length[Text[F21]] * 0.1), 0.2)) → H21
  98. Text(F22, Center + (B22 * u) + (C22 * v) - (0.1 + (Length[Text[F22]] * 0.1), 0.2)) → H22
  99. Text(F23, Center + (B23 * u) + (C23 * v) - (0.1 + (Length[Text[F23]] * 0.1), 0.2)) → H23
  100. Text(F24, Center + (B24 * u) + (C24 * v) - (0.1 + (Length[Text[F24]] * 0.1), 0.2)) → H24
  101. Text(F25, Center + (B25 * u) + (C25 * v) - (0.1 + (Length[Text[F25]] * 0.1), 0.2)) → H25
  102. Text(F26, Center + (B26 * u) + (C26 * v) - (0.1 + (Length[Text[F26]] * 0.1), 0.2)) → H26
  103. Text(F27, Center + (B27 * u) + (C27 * v) - (0.1 + (Length[Text[F27]] * 0.1), 0.2)) → H27
  104. Text(F28, Center + (B28 * u) + (C28 * v) - (0.1 + (Length[Text[F28]] * 0.1), 0.2)) → H28
  105. Text(F29, Center + (B29 * u) + (C29 * v) - (0.1 + (Length[Text[F29]] * 0.1), 0.2)) → H29
  106. Text(F30, Center + (B30 * u) + (C30 * v) - (0.1 + (Length[Text[F30]] * 0.1), 0.2)) → H30
  107. Text(F31, Center + (B31 * u) + (C31 * v) - (0.1 + (Length[Text[F31]] * 0.1), 0.2)) → H31
  108. Text(F32, Center + (B32 * u) + (C32 * v) - (0.1 + (Length[Text[F32]] * 0.1), 0.2)) → H32
  109. Text(F33, Center + (B33 * u) + (C33 * v) - (0.1 + (Length[Text[F33]] * 0.1), 0.2)) → H33
  110. Text(F34, Center + (B34 * u) + (C34 * v) - (0.1 + (Length[Text[F34]] * 0.1), 0.2)) → H34
  111. Text(F35, Center + (B35 * u) + (C35 * v) - (0.1 + (Length[Text[F35]] * 0.1), 0.2)) → H35
  112. Text(F36, Center + (B36 * u) + (C36 * v) - (0.1 + (Length[Text[F36]] * 0.1), 0.2)) → H36
  113. Text(F37, Center + (B37 * u) + (C37 * v) - (0.1 + (Length[Text[F37]] * 0.1), 0.2)) → H37
  114. Sequence(Sequence[Polygon[A + (k * u) + (j * v), B + (k * u) + (j * v), 6], j, 1, k], k, 1, 8) → pascalOutline
  115. Sequence(RemoveUndefined[Sequence[If[Element[cols, j - 1] ≟ i, j], j, 2, 37]], i, 1, 8) → AllCols
  116. Sequence(RemoveUndefined[Sequence[If[Element[rows, j - 1] ≟ i, ("A" * j)], j, 2, 37]], i, 1, 8) → AllRows
  117. Random(1, 5) → QuestionStickCol
  118. Random(3, 8 - QuestionStickCol) → QuestionStickSize
  119. Sequence((("SetValue(D" * i) * ",false)"), i, 2, 37) → cmdTurnOffAll
  120. Sequence((("RunClickScript(A" * Element[Element[AllCols, QuestionStickCol], i]) * ")"), i, 1, QuestionStickSize) → cmdLevel1Setup
  121. Sequence((("SetValue(E" * Element[Element[AllCols, QuestionStickCol], i]) * ",true)"), i, 1, QuestionStickSize) → cmdLevel1Lock
  122. Join({Take[Element[AllCols, QuestionStickCol], 1, QuestionStickSize], {Element[Element[AllCols, QuestionStickCol + 1], QuestionStickSize]}}) → answerCells
  123. Sequence((("RunClickScript(A" * Element[Element[AllCols, QuestionStickCol + 1], QuestionStickSize]) * ")"), i, 1, 1) → cmdLevel2Setup
  124. Sequence((("SetValue(E" * Element[Element[AllCols, QuestionStickCol + 1], QuestionStickSize]) * ",true)"), i, 1, 1) → cmdLevel2Lock
  125. Polygon(A + ((QuestionStickSize + QuestionStickCol) * u) + ((QuestionStickCol + 1) * v), B + ((QuestionStickSize + QuestionStickCol) * u) + ((QuestionStickCol + 1) * v), 6) → answerBlade
  126. Sequence(Polygon[A + (k * u) + (QuestionStickCol * v), B + (k * u) + (QuestionStickCol * v), 6], k, QuestionStickCol, QuestionStickSize + QuestionStickCol - 1) → answerStick
  127. Join({answerStick, {answerBlade}}) → answerDisplay
  128. RemoveUndefined(Sequence[If[Element[selectableObjects, i], i + 1], i, 1, 36]) → cellsClicked
  129. RemoveUndefined(Sequence[If[Element[selectableObjects, i], Element[F2:F37, i]], i, 1, 36]) → cellClickedValues
  130. Polygon(A + (B37 * u) + (C37 * v), B + (B37 * u) + (C37 * v), 6) → A37
  131. Text(F17, Center + (B17 * u) + (C17 * v) - (0.1 + (Length[Text[F17]] * 0.1), 0.2)) → H17
  132. LaTeX(If[(¬correctSum) ∧ (¬noCellsClicked), (("The cells you chose do not add to the sum of " * sumAnswer / 2) * ".\\"), ""]) → fbWrongSum
  133. LaTeX(If[noCellsClicked, "You did not select any cells.\\", ""]) → fbNoSelection
  134. LaTeX(If[correctNumberCells ∨ noCellsClicked, "", "You have not selected the correct number of cells.\\"]) → fbWrongNumCells
  135. LaTeX(("\text{" * If[level ≟ 1, "Click on Level 2 or 3 for a more difficult challenge.", (If[level ≟ 2, "Click on Level 1 for an easier challenge, or Level 3 for a more difficult challenge.", "Click on Level 1 or 2 for an easier challenge."] * "}")])) → instrucLevel
  136. LaTeX((If[level ≟ 1, (instruc1 * "\\"), (instruc2 * "\\")] * instrucLevel)) → SpecInstruc
  137. LaTeX(If[isCorrect, "also ", ""]) → alsoWord
  138. LaTeX(If[¬correctCells, "Do you notice a pattern made by the cells shown\\in this solution?", "Do you notice a pattern made by the cells in your\\solution?"]) → fbWrongCells
  139. If(isHockeyStick, "", If[isCorrect, "Consider also this solution:\\", "Consider the solution shown:\\"]) → fbConsider
  140. Join({answerStick, {answerBlade}}) → answerOutline
  141. LaTeX(((("\text{" * fbConsider) * fbWrongCells) * "}")) → fbComplete

AI 绘图提示语参考

绘制帕斯卡三角形的冰球杆形模式，首先画一个视图框，然后画一个六边形。接着，以六边形的两个顶点为起点，分别绘制两个向量。利用这两个向量，从六边形的每个顶点出发，绘制一系列新的六边形，形成冰球杆形模式。每个新六边形的边长是前一个六边形边长的两倍。最后，计算并标注每个六边形的组合数。
以上提示语为根据原始绘图命令整理而来，仅供参考。