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Square Tiling Puzzle

Square Tiling Puzzle
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2018-01-06 05:33
PolygonVectorTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the construction of complex tiling patterns using square grids, involving core mathematical concepts like plane geometry, 2D vector operations, trigonometric relationships, and coordinate geometry. Key GeoGebra commands used include Point, Polygon, Sort, Sequence, Join, If, and Rotate for creating and manipulating geometric shapes and transformations. Suitable for high school mathematics education to explore transformations and combinations of 2D shapes.

GeoGebra Commands & Parameters

  1. Point(Circle[H1, 1]) → H2
  2. Point(Circle[I31, 1]) → I32
  3. Point(Circle[S1, 1]) → S2
  4. Point(Circle[L1, 1]) → L2
  5. Point(Circle[I21, 1]) → I22
  6. Point(Circle[I41, 1]) → I42
  7. Point(Circle[L21, 1]) → L22
  8. Point(Circle[V221, 1]) → V222
  9. Point(Circle[V211, 1]) → V212
  10. Polygon(A1, A2, A5, A3, A4, A1, A5, A6, A7, A8, A9) → Rahmen, f, p, q, g, h, i, j, k, l, m, n
  11. Sort({V11, V12, V13, V14, V211, V212, V221, V222}) → Schwarz
  12. Sort(Union[Union[Union[Union[Union[Union[Union[{S1, S2, S4, S3}, {L1, L2, L3}], {I21, I22}], {I31, I32, I33}], {I41, I42, I43, I44}], {Q1, Q2, Q3, Q4}], {L21, L22, L23, L24}], {H1, H2, H4, H3}]) → Weiss
  13. Sequence(Segment[(0.5, i), (6.5, i)], i, 1.5, 5.5, 1) → XStrecken
  14. Sequence(Segment[(i, 0.5), (i, 6.5)], i, 1.5, 5.5, 1) → YStrecken
  15. Join(Sequence[Sequence[(i, j), j, 1, 6], i, 1, 6]) → Alles
  16. Sort(Union[Weiss, Schwarz]) → Zusammen
  17. If(Zusammen ≟ Alles, 1, 0) → korrekt
  18. Polygon(V12 + (0.5, 0.5), V12 + (-0.5, 0.5), V12 + (-0.5, -0.5), V12 + (0.5, -0.5)) → VielV12, r, s, t, a
  19. Polygon(V11 + (0.5, 0.5), V11 + (-0.5, 0.5), V11 + (-0.5, -0.5), V11 + (0.5, -0.5)) → VielV11, b, c, d, e
  20. Polygon(V13 + (0.5, 0.5), V13 + (-0.5, 0.5), V13 + (-0.5, -0.5), V13 + (0.5, -0.5)) → VielV13, f_1, g_1, h_1, i_1
  21. Rotate(Polygon[V211 + (-0.5, -0.5), V211 + (1.5, -0.5), V211 + (1.5, 0.5), V211 + (-0.5, 0.5)], Angle[V212 - V211], V211) → VielV21
  22. Polygon(V14 + (0.5, 0.5), V14 + (-0.5, 0.5), V14 + (-0.5, -0.5), V14 + (0.5, -0.5)) → VielV14, j_1, k_1, l_1, m_1
  23. Rotate(Polygon[V221 + (-0.5, -0.5), V221 + (1.5, -0.5), V221 + (1.5, 0.5), V221 + (-0.5, 0.5)], Angle[V222 - V221], V221) → VielV22
  24. Rotate(Polygon[I21 + (-0.5, -0.5), I21 + (1.5, -0.5), I21 + (1.5, 0.5), I21 + (-0.5, 0.5)], Angle[I22 - I21], I21) → VielI2
  25. Rotate(Polygon[I31 + (-1.5, -0.5), I31 + (1.5, -0.5), I31 + (1.5, 0.5), I31 + (-1.5, 0.5)], Angle[I32 - I31], I31) → VielI3
  26. Rotate(Polygon[I41 + (-1.5, -0.5), I41 + (2.5, -0.5), I41 + (2.5, 0.5), I41 + (-1.5, 0.5)], Angle[I42 - I41], I41) → VielI4
  27. Polygon(Q1 + (-0.5, -0.5), Q1 + (1.5, -0.5), Q1 + (1.5, 1.5), Q1 + (-0.5, 1.5)) → VielQ, n_1, p_1, q_1, r_1
  28. Rotate(Polygon[L21 + (-0.5, (-0.5 * L2x)), L21 + (1.5, (-0.5 * L2x)), L21 + (1.5, (0.5 * L2x)), L21 + (0.5, (0.5 * L2x)), L21 + (0.5, (2.5 * L2x)), L21 + (-0.5, (2.5 * L2x)), L21 + (-0.5, (0.5 * L2x))], Angle[L22 - L21], L21) → VielL2
  29. Rotate(Polygon[L1 + (-0.5, (-0.5 * Lx)), L1 + (1.5, (-0.5 * Lx)), L1 + (1.5, (0.5 * Lx)), L1 + (0.5, (0.5 * Lx)), L1 + (0.5, (1.5 * Lx)), L1 + (-0.5, (1.5 * Lx)), L1 + (-0.5, (0.5 * Lx))], Angle[L2 - L1], L1) → VielL1
  30. Rotate(Polygon[S1 + (-0.5, (-0.5 * Sx)), S1 + (0.5, (-0.5 * Sx)), S1 + (0.5, (-1.5 * Sx)), S1 + (1.5, (-1.5 * Sx)), S1 + (1.5, (-0.5 * Sx)), S1 + (1.5, (0.5 * Sx)), S1 + (0.5, (0.5 * Sx)), S1 + (0.5, (1.5 * Sx)), S1 + (-0.5, (1.5 * Sx)), S1 + (-0.5, (0.5 * Sx))], Angle[S2 - S1], S1) → VielS
  31. Rotate(Polygon[H1 + (-1.5, (-0.5 * Hx)), H1 + (1.5, (-0.5 * Hx)), H1 + (1.5, (0.5 * Hx)), H1 + (0.5, (0.5 * Hx)), H1 + (0.5, (1.5 * Hx)), H1 + (-0.5, (1.5 * Hx)), H1 + (-0.5, (0.5 * Hx)), H1 + (-1.5, (0.5 * Hx))], Angle[H2 - H1], H1) → VielH