Tessellations: Techniques and Teaching

Greg Petrics, Mar 2, 2020

Middlebury Math Teachers' Circle -- March 2020

Table of Contents

  1. Euclidean Tessellations
    1. Introduction
    2. Rule for Euclidean Tessellations by Regular Polygons
    3. M.C. Escher's Triangle Trick
    4. M.C. Escher's Hexagon Trick
    5. M.C. Escher's Hexagon Trick Part 2
    6. Teaching Tessellations?
  2. Hyperbolic Tessellations
    1. Circle Limit
    2. Tessellating with Hyperbolic Triangles?
    3. Tessellation Failure
    4. The Poincaré Disk
    5. Polygon Angle Sums in Hyperbolic Geometry
    6. Similar Triangles Are Congruent
    7. Regular Polygons in Hyperbolic Geometry
    8. Rule for Hyperbolic Tessellations by Regular Polygons
    9. The Hyperbolic (5,4) Tessellation
    10. The Hyperbolic (3,7) Tessellation
    11. Make Your Own Hyperbolic Tessellation
    12. Conclusion
    13. BONUS: Quasi-Regular Hyperbolic Tessellations

1.

Euclidean Tessellations

In the first half of tonight's session we'll study tessellations in the Euclidean (standard) plane, paying special attention to tessellations by regular polygons (and their cousins).

  • 1. Introduction
  • 2. Rule for Euclidean Tessellations by Regular Polygons
  • 3. M.C. Escher's Triangle Trick
  • 4. M.C. Escher's Hexagon Trick
  • 5. M.C. Escher's Hexagon Trick Part 2
  • 6. Teaching Tessellations?

1.1.

Introduction

[b]Definition[/b]: A shape P [b]tessellates [/b]if it is possible to fill the entire plane with rotations and or translations of P with no overlapping area. [br][br]Three regular polygons tesselate the Euclidean plane. Which ones?[br][br]Adjust "Sides" and "Copies" to explore.
What are the three regular polygons that tessellate the plane? [br][br]If p = the number of sides of the polygon and q = the minimum number of copies at a vertex, list all pairs (p,q) that tessellate the plane.[br][br]Why does the regular pentagon not tessellate the plane?[br][br]Why are there no tessellations by regular polygons with 7 or more sides?
– Greg Petrics

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2.

Hyperbolic Tessellations

In the second half of tonight's session we'll explore Hyperbolic Geometry by way of the Poincaré Disk, and learn some techniques for constructing Hyperbolic tessellations.

  • 1. Circle Limit
  • 2. Tessellating with Hyperbolic Triangles?
  • 3. Tessellation Failure
  • 4. The PoincarĂ© Disk
  • 5. Polygon Angle Sums in Hyperbolic Geometry
  • 6. Similar Triangles Are Congruent
  • 7. Regular Polygons in Hyperbolic Geometry
  • 8. Rule for Hyperbolic Tessellations by Regular Polygons
  • 9. The Hyperbolic (5,4) Tessellation
  • 10. The Hyperbolic (3,7) Tessellation
  • 11. Make Your Own Hyperbolic Tessellation
  • 12. Conclusion
  • 13. BONUS: Quasi-Regular Hyperbolic Tessellations

2.1.

Circle Limit

M.C. Escher utilized Hyperbolic Geometry (specifically: the Poincaré Disk model) to create interesting tessellations in his "Circle Limit" series.[br][br][br][center][/center][img]https://upload.wikimedia.org/wikipedia/en/5/55/Escher_Circle_Limit_III.jpg[/img][center][/center][br][br]
– Greg Petrics

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