John Willson - Mosaic and Tessellated Patterns: How to Create Them, with 32 Plates to Color

1. A single tessellation which is the reciprocal of the semiregular pattern (). It has a sixfold center of rotational symmetry but no lines of mirror symmetry.
2. A semiregular pattern with two lines of mirror symmetry. It is formed by overlap of lozenge-shaped decagons.
3. A single tessellation which is the reciprocal of Plate 2. It can be viewed as a grid of large distorted hexagons superimposed on an identical grid.
4. A semiregular pattern formed by overlap of dodecagons or expansion of a regular hexagon grid.
5. A double tessellation formed by infilling the hexagons in Plate 4.
6. A triple tessellation formed by one method of infilling the semiregular pattern of dodecagons and triangles in .
7. A double tessellation formed by infilling the hexagons in Plate 6.
8. A triple tessellation, related to Plate 6, formed by the alternative method of infilling the semiregular pattern of dodecagons and triangles in .
9. A double tessellation formed by infilling the hexagons in Plate 8.
10. A semiregular pattern formed by expansion of the tessellation of dodecagons and triangles in .
11. A triple tessellation formed by one method of infilling the dodecagons in Plate 10. The alternative method forms the semiregular pattern in Plate 4.
12. A double tessellation formed by infilling the hexagons in Plate 11.
13. A periodic pattern of decagons, pentagons and distorted hexagons based on the arrangement of ten pentagons surrounding one decagon. At every intersection the angles are 108, 108 and 144.
14. A periodic pattern of pentagons and diamonds; the angles involved are 36, 108 and 144.
15. A nonperiodic arrangement of pentagons and diamonds which has a fivefold center of rotational symmetry and five lines of mirror symmetry.
16. A nonperiodic pattern of stars and diamonds based on Plate 15; the angles involved are all multiples of 36 .
17. A nonperiodic pattern related to Plate 16; the angles involved are all multiples of 36.
18. A periodic tessellation of squares, octagons and eight-pointed stars superimposed on a semiregular grid of octagons and squares.
19. A double tessellation in which no tiles of the same shape share a common edge. It can be viewed as tiles of one shape surrounding holes of the other shape. The shading determines which shape forms the holes.
20. A periodic tessellation based on two triangular grids superimposed on a pattern of overlapping hexagons.
21. A nonperiodic design with a threefold center of rotational symmetry. The eight different shapes are formed by applying a single modification process to some of the lines of a triangular grid.
22. A grid created by adding diagonals to the shapes formed when concentric circles and radiating lines overlap. This is the basis of all the designs in .
23. A grid formed by combination of eight adjacent areas in Plate 22.
24. A deformed square grid in which the intervals along the edges and main axes follow an arithmetic progression.
25. Grid related to , formed by two sets of straight lines which radiate at 10 intervals. The slight curving appearance is an illusion.
26. Grid formed by two sets of lines which radiate to points equally spaced around the perimeter.
27. Grid formed by six sets of lines which radiate at 15 intervals from the corners of a hexagon.
28. Grid formed by four sets of steadily increasing concentric circles and a set of radiating lines which are drawn as tangents to these circles.
29. Grid formed by a set of concentric triangles and their inscribing and circumscribing circles.
30. Grid formed from six sets of parallel lines oriented at 30 intervals.
31. Grid formed by extending all the sides of the shapes in Plate 15. It consists of five sets of parallel lines oriented at 36 intervals; the spacings between the lines are related to the golden ratio 1:1.618.
32. A more complex grid formed by further development of part of Plate 31.

A Roman mosaic consisted of many small stones which were laid edge to edge to cover a surface, such as a wall or a floor. These tiles were called tesserae, and the term tessellation is now used for any arrangement of shapes that completely covers a surface. This technique of space-filling can be seen in paved or parquet flooring, in ceramic tiling and in stained-glass windows. Tessellations even occur naturally on some crystal surfaces and in many cellular structures such as honeycombs. Tessellation is a form of design which has been used for thousands of years. Patterns found in a Roman villa in Greece, an Islamic mosque in Iran and the Alhambra palace in Spain are shown in respectively. Similar designs occur in many other countries which have been influenced by Greek, Roman, Byzantine or Arabic culture.

These patterns have inspired artists and designers throughout the centuries. In the last few decades, they have found a new lease on life through the work of Victor Vasarely, Bridget Riley, Maurits Escher, Ensor Holiday and many others.

In this book the basic principles of tessellation will be studied and various methods of creating new patterns will be explained. Some curious effects of shading will be analyzed, and in the center pages of the book there are many designs to explore, modify and color.

Effects of Color

Many tessellations take on new appearances or produce interesting optical effects when shaded or colored. The effects are often most noticeable when the designs are viewed from a distance or in subdued lighting or with ones eyes partially closed or screwed up. Hence the square pattern in , where the pattern of rectangles involves only straight lines but on shading (7) develops a curving, three-dimensional appearance.

Fig. 7

This effect in some ways resembles the moire effect in which bands of light and shade are produced when two grids or sets of lines overlap. The relationship can be seen in this tessellation of various shapes produced by overlapping two sets of lines radiating from two points (8). On coloring in alternate areas we see bands sweeping over the pattern in a series of horseshoe shapes. A similar set of curves is seen if the original grid is drawn using much thicker lines so as to produce a moire effect (9).