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Can a Regular Pentagon Tessellate?

Can a Regular Pentagon Tessellate?
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2014-04-09 15:47
PolygonAngleGeometric Construction
In collections Tessellations of Common Figures
This example demonstrates attempting to tessellate a regular pentagon using the mid-side tessellation rule, which works for triangles, quadrilaterals, and hexagons. It involves core math concepts like polygon construction, midpoints, rotation, and angles. The construction uses GeoGebra commands such as Polygon, Midpoint, Rotate, and Angle. Suitable for middle school students and math enthusiasts learning about geometric transformations and tiling theories.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 5) → poly1, a, b, c, d, e, C, D, E
  2. Midpoint(A, B) → F
  3. Midpoint(B, C) → G
  4. Midpoint(D, C) → H
  5. Midpoint(D, E) → I
  6. Midpoint(E, A) → J
  7. Polygon(A, B, C, D, E) → poly2, a_1, b_1, c_1, d_1, e_1
  8. Rotate(A, 180*°, I) → A'
  9. Rotate(B, 180*°, I) → B'
  10. Rotate(C, 180*°, I) → C'
  11. Rotate(D, 180*°, I) → D'
  12. Rotate(E, 180*°, I) → E'
  13. Polygon(A', B', C', D', E') → poly2', a', b', c', d', e'
  14. Rotate(A, 180*°, H) → A'_1
  15. Rotate(B, 180*°, H) → B'_1
  16. Rotate(C, 180*°, H) → C'_1
  17. Rotate(D, 180*°, H) → D'_1
  18. Rotate(E, 180*°, H) → E'_1
  19. Polygon(A'_1, B'_1, C'_1, D'_1, E'_1) → poly2'_1, a'_1, b'_1, c'_1, d'_1, e'_1
  20. Rotate(A, 180*°, G) → A'_2
  21. Rotate(B, 180*°, G) → B'_2
  22. Rotate(C, 180*°, G) → C'_2
  23. Rotate(D, 180*°, G) → D'_2
  24. Rotate(E, 180*°, G) → E'_2
  25. Polygon(A'_2, B'_2, C'_2, D'_2, E'_2) → poly2'_2, a'_2, b'_2, c'_2, d'_2, e'_2
  26. Rotate(A, 180*°, F) → A'_3
  27. Rotate(B, 180*°, F) → B'_3
  28. Rotate(C, 180*°, F) → C'_3
  29. Rotate(D, 180*°, F) → D'_3
  30. Rotate(E, 180*°, F) → E'_3
  31. Polygon(A'_3, B'_3, C'_3, D'_3, E'_3) → poly2'_3, a'_3, b'_3, c'_3, d'_3, e'_3
  32. Rotate(A, 180*°, J) → A'_4
  33. Rotate(B, 180*°, J) → B'_4
  34. Rotate(C, 180*°, J) → C'_4
  35. Rotate(D, 180*°, J) → D'_4
  36. Rotate(E, 180*°, J) → E'_4
  37. Polygon(A'_4, B'_4, C'_4, D'_4, E'_4) → poly2'_4, a'_4, b'_4, c'_4, d'_4, e'_4
  38. Angle(d'_1, c) → α
  39. Angle(D, C, B) → β
  40. Angle(B, C, A'_2) → γ
  41. Polygon(A_1, B_1, 5) → poly1_1, a_3, b_3, c_3, d_3, e_3, C_1, D_1, E_1
  42. Midpoint(A_1, B_1) → F_1
  43. Midpoint(B_1, C_1) → G_1
  44. Midpoint(D_1, C_1) → H_1
  45. Midpoint(D_1, E_1) → I_1
  46. Midpoint(E_1, A_1) → J_1
  47. Polygon(A_1, B_1, C_1, D_1, E_1) → poly2_1, a_2, b_2, c_2, d_2, e_2
  48. Angle(D_1, C_1, B_1) → β_1
  49. Rotate(A_1, δ, F) → K
  50. Rotate(B_1, δ, F) → L
  51. Rotate(C_1, δ, F) → M
  52. Rotate(D_1, δ, F) → N
  53. Rotate(E_1, δ, F) → O
  54. Polygon(K, L, M, N, O) → poly3, k, l, m, n, o

AI Drawing Prompt Reference

Draw a regular pentagon and rotate it and its midpoints around the center point to generate a series of new pentagons. Observe if these pentagons can tessellate the plane without gaps. First, draw a regular pentagon, then find the midpoints of each side, and rotate the pentagon and its sides around these midpoints. Repeat this process multiple times, and finally measure the angle changes of the rotated pentagons.
The prompt above was compiled from the original drawing commands and is for reference only.