pavement mathematics

Most of the pavements (Americans read: ‘sidewalks’) in Hong Kong have been laid down using interlocking bricks, and I’m intrigued by the different patterns that can be seen. I often wonder, for example, how many ways an area can be covered using shapes that are twice as long as they are wide, which is the standard ratio for paving bricks, although the precise size does vary from location to location.

I have my own terms for the various possible layouts, although I have checked out what might be called the ‘official’ terminology for different ways of arranging bricks. Where appropriate, these appear in what follows as italic phrases enclosed in parentheses. This is probably the most common layout:

This is the zig-zag (herringbone bond) layout. In this photo, the L-shaped pairs of pale bricks complicate matters, but the basic principle of this layout is clear.

The second most common layout is what I call the three-brick rectangle (single basket-weave bond):

In my part of Fanling, this mixture of reddish, greyish and yellowish bricks is especially common, and I often think that there must be some kind of pattern involving the three colours. While this is clearly incorrect, the distribution is almost certainly not entirely random.

The two-brick square (double basket-weave bond) is much less common. This photo was taken on the forecourt of University station, where the alternating lines of light and dark grey bricks are a complication on the basic pattern:

The least common method for covering large areas is what I call a linear pattern. There are two different types of linear pattern, distinguished by whether the bricks are laid end to end (stretcher course) or side by side (header course). This photo illustrates a rare example of the two types being combined:

Note the two parallel stretcher courses just right of centre, which appear to indicate a (slight) change of colour in the bricks. The half-brick offset of the right-hand course is an anomaly.

Linear brick courses are often used to delineate the edge of a paved area, as in the next photo, which shows a header course bordering a path with a zig-zag arrangement. Note that it has been necessary to trim the bricks adjoining the header course to fit, given that the orientation is effectively a mismatch—any area paved in a zig-zag pattern cannot be enclosed by straight lines without some such adjustment.

These may be the basic patterns, but there is huge scope for variation. For example, the next photo illustrates why I call the herringbone bond a zig-zag pattern:

The next two photos are of arrangements involving bricks of more than one colour and are superficially similar, although a closer examination reveals the dark grey bricks in the first image to be part of a stretcher-course linear arrangement, while the alternating red and grey bricks in the second image are part of a zig-zag pattern.

The use of different colours provides some spectacular possibilities, none more so than the following three-colour linear arrangement, which I recorded on the campus of the Chinese University of Hong Kong:

Other opportunities for creativity are afforded by what I describe as ‘wavy’ bricks. The first photo here shows a basic two-brick-square pattern using such bricks, while the second shows what can be done with three colours in a zig-zag layout:

There is another complication in the way bricks have been arranged on the pavements around these parts: what I call ‘separators’. For what I assume are purely decorative reasons, a brick-paved sidewalk is often broken up by these separators, as shown in the following photo:

The separator here is effectively five parallel stretcher courses, but the use of coloured bricks creates a line of what I call ‘nine-brick squares’. Mind you, a moment’s reflection should be enough to realize that it is, in fact, impossible to arrange nine 2×1 bricks in a square, but what I mean here is the polygon that encloses nine bricks arranged in a 1–2–3–2–1 formation, as in the following photo:

Viewed from an angle rather than from directly above, it is deceptively easy to imagine that this enclosing polygon is in fact a rhombus, but it is straightforward to establish not only that the sides of the polygon are of equal length but also that the four apices are all right angles.

Although the nine-brick-square separator is extremely common, it is invariably used to separate areas with the same type of pattern. In the transition from, say, a three-brick rectangle to a zig-zag arrangement, the usual separator is a single stretcher course in a distinctively different colour:

The next photo shows how separators are usually inserted at regular intervals along a stretch of pavement:

And the separators here—alternating three-brick rectangles (dark grey) and two-brick squares (light grey)—show just how much scope there is for inventiveness using these features. Note the parallel stretcher and header courses on the side further from the road.

I’ve come across quite a few other types of separator in addition to the ubiquitous nine-brick square. Here is a selection:

Finally, you might think that all these pavement patterns are to be found only in urban areas, but my final photo was taken in Ta Kwu Ling, a village close to the Chinese border that was part of the ‘frontier closed area’ until the beginning of last year.

It is the most extensive example of alternating stretcher and header courses that I’ve come across to date.