OTHER TESSELLATIONS

Our main interest is in tessellations which involve the regular polygons, but there are several other types of tessellation which either commonly occur, or which have great intrinsic interest. We look at these briefly.

  Brick paving patterns

A common paving brick forms a 2 x 1 rectangle.

• (a) Investigate various types of brick paving in your neighbourhood, and draw / describe the recurring patterns.
   (b) In what ways do these tessellations fail to be regular or semiregular?

Now check your answers ...

OTHER TESSELLATIONS

Our main interest is in tessellations which involve the regular polygons, but there are several other types of tessellation which either commonly occur, or which have great intrinsic interest. We look at these briefly.

  Brick paving patterns

A common paving brick forms a 2 x 1 rectangle.

(a) Investigate various types of brick paving in your neighbourhood, and draw / describe the recurring patterns.
    (b) In what ways do these tessellations fail to be regular or semiregular?

Several types of common brick paving patterns are shown below. Of course, as a tessellation, there is no difference between the first two, but the angle of the second gives the paving a different appearance.

Herringbone Herringbone (45°) Running bond Basket weave

Since each of these tessellations only involves one basic shape, they can’t be classed as semiregular, but they also fail to be regular as the rectangle is not a regular polygon. Notice too that in each case, each brick has at least two vertices at the midpoints of neighbouring edges – a behaviour disallowed in our previous definitions.
Kites and darts

In 1974, while Roger Penrose was a graduate student at Cambridge, he became interested in non-periodic tilings. By non-periodic we mean that there is no translation which maps the tiling onto itself. Penrose found a wonderful tiling of this type which only involves two tiles: a kite and a dart.

• The figure at right shows the construction and shape of the kite and dart components. Think about where you might have seen this rhombus before. Then calculate the lengths of the remaining segments in the figure.

Hint.

We know that the ratio of the diagonal of a regular pentagon to the side length is 1 : , where  =  (1 + 5)/2    1.618. From the hint, we deduce that the length of the long diagonal of the rhombus is . Now using the similar isosceles triangles in the figure, we deduce that the three equal shorter segments have length 1/.

It is clear that the kite and dart can be used to form a periodic tessellation: we simply tessellate the given rhombus in the obvious way. But the kite and dart fit together in other more interesting ways. For example:

Here are some properties of the Penrose tilings.

•• Every finite portion of any tiling is contained infinitely often in every other tiling
•• In any finite tiled region, only one tiling is possible.
•• No finite patch of tiles can determine the rest of the tiling.
•• It is impossible to tell from any patch of tile which tiling it is on.
•• In an infinite tiling of the plane, any tiling of a region that occurs is repeated infinitely often elsewhere in the plane and must recur within twice the diameter of the region from where you first found it.
•• Only at their infinite limits are the different patterns distinguishable.

It is fun to explore these tilings on your own. You can buy a commercial set of tiles, or print off several copies of this linked page onto light card, cut out the tiles, and then play!

The Cairo tessellation

This is a tessellation which appears in the streets of Cairo, and in Islamic art. We can think of the tessellation as being made up of two overlapping tessellations of congruent hexagons, intersecting each other at right angles.

• If you measure the angles in the above tessellation you will find that each hexagon has angles 90°, 144°, 90°, 108° and 108°.
Given the two right angles as fixed, are there any constraints on the remaining angles? Are all the angle completely determined?

Check your answer ...




























The Cairo tessellation

This is a tessellation which appears in the street pavements of Cairo, and in Islamic art. We can think of the tessellation as being made up of two overlapping tessellations of congruent hexagons, intersecting each other at right angles.

If you measure the angles in the above tessellation you will find that each hexagon has angles 90°, 144°, 90°, 108° and 108°.
Given the two right angles as fixed, are there any constraints on the remaining angles? Are all the angles completely determined?


In fact, the remaining angles are not completely determined. We would like our pentagons to retain an axis of symmetry, meaning the two base angles should be equal, and the three non-right angles must add to 360°, but other than that, they can vary. Here is a variation with the non-right angles each 120°.

The Cairo tessellation occurs as a projection of the regular dodecahedron (later) and as the dual of the semiregular tessellation of squares and equilateral triangles, 3.3.4.3.4.

References

Brick paving patterns

This Quikcrete page has a good selection : http://www.quikrete.com/diy/PatiosWalkwaysAndDrives.html


Kites and darts

This advertisement show some of the possibilities : http://www.pburch.net/toys/B1915068958/C522364530/E1218854721/

A wonderful gallery of Penrose tilings : http://www.josleys.com/creatures34.htm


The Cairo tessellation

This page has a resizable Cairo tessellation http://descartes.cnice.mecd.es/ingles/3rd_year_secondary_educ/Tessellation/Otras_teselaciones.htm