Tessellations

John Golden

Interesting plane coverings

Table of Contents

  1. Monotile
    1. A Tessellation
    2. Daisy Tessellation
    3. Triangle Tessellation
    4. Isosceles Triangle Tessellation
    5. Equilateral Triangle Tessellation
    6. Isosceles Right Tessellation
    7. Golden Triangle Fractal
    8. Tessellation
    9. Kite Tessellation
    10. H. Weyl "Symmetry" (PRINCETON) via Ijon Ticky
    11. Tessellation
    12. Indiana Puzzle Quilt
    13. Tessellation in a Square
    14. Yemeni Squares
    15. Gábor's Tiling
    16. Gato escheriano
    17. Cat Tiling
    18. Turtle Tiles
    19. Square Double Glide
    20. Parallelogram Tiling
    21. Rectangular Tessellation
    22. Square Duck
    23. Isosceles Trapezoid Tessellation
    24. Double Glide Reflection Tessellation
    25. Double Glide Reflection Tessellation
    26. An Alhambra Tessellation
    27. pentagonal tiling from squares and parallelograms
    28. Escherized Pentagon Tessellation
    29. Pentagon Spiral Tiling
    30. Escherized Pentagons
    31. Pentagon Rotation
    32. Escherized Pentagonal Tiling
    33. Hexagon Tessellation
    34. Decorated Polygons
    35. Socolar Taylor Tiles
    36. Hexágonos simetría central teselan
    37. Lagarto escheriano
    38. Hexagon Tiling
    39. Hexagon Spiral Tessellation
    40. Escherized Hexagon 2
    41. Hexagonal Slide
    42. Tessellated Polyline
    43. Hexagonal Fish
    44. Two Tiles Glide Reflection
    45. Hexagonal Translation Tiling
    46. Escherized Concave Cairo Pentagons
    47. Escherized Cairo Pentagons
  2. Multitile
    1. Pythagorean Tiling
    2. Napoleon Tiling
    3. Semiregular tessellation
    4. Triangle Tessellation with Hexagonal Gap
    5. Generalized García Octagonal Patterm
    6. Hexagonal Star
    7. Kite Tessellation
    8. Lee Sallows' Self Tilers
    9. Dual Dilations
    10. Tessellation kite & hexagons
    11. Stars and Hexes
    12. Hex, Rhomb and Isosceles
    13. Triangle Tessellation with Gaps
    14. Three Piece Hexagonal Tessellation
    15. Escherized Two Tile: Octagon and Square
    16. Two Rhomb Tiling after Escher
    17. Kites and Rhombs
    18. Two Hexagon Tiling
    19. Tiles
  3. Nonregular
    1. Fathauer Kites
    2. Fractal Tiling
    3. Dilated Tessellation
    4. Similar Quadrilaterals
    5. Similar Quadrilaterals
    6. Construction of non periodic tiling (extended Version)
    7. Penrose Tiling
    8. Penrose Tiles with Matching
    9. Penrose Tiles without Matching
    10. Duck Tiles
    11. an 'einstein' tile - een 'einstein' tegel...

1.

Monotile

  • 1. A Tessellation
  • 2. Daisy Tessellation
  • 3. Triangle Tessellation
  • 4. Isosceles Triangle Tessellation
  • 5. Equilateral Triangle Tessellation
  • 6. Isosceles Right Tessellation
  • 7. Golden Triangle Fractal
  • 8. Tessellation
  • 9. Kite Tessellation
  • 10. H. Weyl "Symmetry" (PRINCETON) via Ijon Ticky
  • 11. Tessellation
  • 12. Indiana Puzzle Quilt
  • 13. Tessellation in a Square
  • 14. Yemeni Squares
  • 15. Gábor's Tiling
  • 16. Gato escheriano
  • 17. Cat Tiling
  • 18. Turtle Tiles
  • 19. Square Double Glide
  • 20. Parallelogram Tiling
  • 21. Rectangular Tessellation
  • 22. Square Duck
  • 23. Isosceles Trapezoid Tessellation
  • 24. Double Glide Reflection Tessellation
  • 25. Double Glide Reflection Tessellation
  • 26. An Alhambra Tessellation
  • 27. pentagonal tiling from squares and parallelograms
  • 28. Escherized Pentagon Tessellation
  • 29. Pentagon Spiral Tiling
  • 30. Escherized Pentagons
  • 31. Pentagon Rotation
  • 32. Escherized Pentagonal Tiling
  • 33. Hexagon Tessellation
  • 34. Decorated Polygons
  • 35. Socolar Taylor Tiles
  • 36. Hexágonos simetría central teselan
  • 37. Lagarto escheriano
  • 38. Hexagon Tiling
  • 39. Hexagon Spiral Tessellation
  • 40. Escherized Hexagon 2
  • 41. Hexagonal Slide
  • 42. Tessellated Polyline
  • 43. Hexagonal Fish
  • 44. Two Tiles Glide Reflection
  • 45. Hexagonal Translation Tiling
  • 46. Escherized Concave Cairo Pentagons
  • 47. Escherized Cairo Pentagons

1.1.

A Tessellation

Can you figure out how this tessellation is made? What is the basic shape? What are the motions? What restrictions did the author make on the shape?
A Tessellation
More GeoGebra at [url]mathhombre.blogspot.com[/url]
– John Golden

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

1.10.

–

1.11.

–

1.12.

–

1.13.

–

1.14.

–

1.15.

–

1.16.

–

1.17.

–

1.18.

–

1.19.

–

1.20.

–

1.21.

–

1.22.

–

1.23.

–

1.24.

–

1.25.

–

1.26.

–

1.27.

–

1.28.

–

1.29.

–

1.30.

–

1.31.

–

1.32.

–

1.33.

–

1.34.

–

1.35.

–

1.36.

–

1.37.

–

1.38.

–

1.39.

–

1.40.

–

1.41.

–

1.42.

–

1.43.

–

1.44.

–

1.45.

–

1.46.

–

1.47.

–

2.

Multitile

  • 1. Pythagorean Tiling
  • 2. Napoleon Tiling
  • 3. Semiregular tessellation
  • 4. Triangle Tessellation with Hexagonal Gap
  • 5. Generalized García Octagonal Patterm
  • 6. Hexagonal Star
  • 7. Kite Tessellation
  • 8. Lee Sallows' Self Tilers
  • 9. Dual Dilations
  • 10. Tessellation kite & hexagons
  • 11. Stars and Hexes
  • 12. Hex, Rhomb and Isosceles
  • 13. Triangle Tessellation with Gaps
  • 14. Three Piece Hexagonal Tessellation
  • 15. Escherized Two Tile: Octagon and Square
  • 16. Two Rhomb Tiling after Escher
  • 17. Kites and Rhombs
  • 18. Two Hexagon Tiling
  • 19. Tiles

2.1.

Pythagorean Tiling

Since this nice tumblr post [url]http://bit.ly/1irdco1[/url] I was interested in making a sketch of this important tiling. It's the origin of many proofs of the Pythagorean theorem as it admits many different dissections of the squares of the legs to make the square of the hypotenuse.
Pythagorean Tiling
More GeoGebra at [url]mathhombre.blogspot.com[/url]
– John Golden

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

2.8.

–

2.9.

–

2.10.

–

2.11.

–

2.12.

–

2.13.

–

2.14.

–

2.15.

–

2.16.

–

2.17.

–

2.18.

–

2.19.

–

3.

Nonregular

  • 1. Fathauer Kites
  • 2. Fractal Tiling
  • 3. Dilated Tessellation
  • 4. Similar Quadrilaterals
  • 5. Similar Quadrilaterals
  • 6. Construction of non periodic tiling (extended Version)
  • 7. Penrose Tiling
  • 8. Penrose Tiles with Matching
  • 9. Penrose Tiles without Matching
  • 10. Duck Tiles
  • 11. an 'einstein' tile - een 'einstein' tegel...

3.1.

Fathauer Kites

In Robert Fathauer's excellent book Tessellations, https://www.mathartfun.com/TessBooks.html, he covers kite spirals, which is a special case of my favorite symmetry of similar quadrilaterals. Daniel Mentrard recently shared his excellent technique for coloring series... so here you go![br][br]Raised the question for me: which kites have perfect filling the plane spirals? I left the tools on so you can measure, poke around and investigate.
– John Golden

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

3.9.

–

3.10.

–

3.11.

–

Information