Wednesday, 27 December 2017

tile styles

When I posted Pavement Mathematics last month, my aim was to analyse the different ways in which a plane could be filled by objects of the same size and shape, in that case by 2×1 rectangles represented by bricks. However, in addition to the ubiquity of brick-covered pavements in Hong Kong, there are many examples of different shapes being used to cover larger areas, and this post discusses some of the many examples I’ve come across in the past two months.

I’ll start with a short section of ‘crazy’ paving on the campus of the Chinese University of Hong Kong (ChineseU). I couldn’t choose between the photos I took, so I’ve included three here:

At first glance, it seems that each shows a network of irregular quadrilaterals, but closer inspection shows that there are a few triangles. Nevertheless, I imagine that it must take considerable ingenuity to make all the shapes fit, although the fit isn’t precise—there is some variation in the spaces between pieces.

The next photo was also taken on the campus of ChineseU:

You might think that it shows a network of white, light grey and dark grey squares, but you would be mistaken. These shapes are actually rhombuses! Incidentally, can you spot the rhombus that is the wrong colour?

The Hong Kong Science Park is located next to ChineseU, and I spotted this in a bicycle-parking area as I rode past:

It isn’t easy to determine what the pattern is here. Obviously, there is a repeating 4×5 rectangle, which in each case consists of a 3×3 square, a 2×2 square, two 2×1 rectangles and three unit squares, one of which is black. At least I thought that this was the case, but the larger rectangle in the upper centre of the photograph has five unit squares and just one 2×1 rectangle. I shall take a closer look the next time I pass, because it is often the case that when a tile needs replacing, any pre-existing pattern is ignored. If I don’t find any other ‘irregular’ arrangements, then this is what must have happened. By the way, I’m now wondering how many ways it is possible to combine a 3×3 square, a 2×2 square, two 2×1 rectangles and three unit squares to form that 4×5 rectangle.

Closer to home, I took the next two photos in the area surrounding Fu Tei Pai village hall:

Compare these with the following photo, which I took in Cornmarket, Penrith, last summer:

These are ‘stone’ rather than ceramic components, but the design is the same.

Even closer to home, there are three high-rise residential estates on the eastern edge of Fanling. The largest is Regentville, and the approach to the main entrance of the estate’s shopping mall looks like this:

The basic pattern here is an 18×18 square of pink and purple unit squares, with an 8×8 square of white, yellow and black occupying one corner. This is a closer look at a single 18×18 square:

I can’t help wondering why the white, yellow and black square isn’t precisely a quarter of the larger square, because the way the larger square has been designed, with strings of three alternating pink and purple unit squares, means that there is a remainder of one in each line.

Inside the mall itself, the floor of polished, square red, green and white tiles follows a repeating pattern of larger squares:

This is the view from the mezzanine floor of the central atrium, which shows the pattern more clearly:

I can imagine a fiendishly complex board game being played on this grid.

Regentville was already there when we moved into the area ten years ago, but a new estate, occupying the corner of Sha Tau Kok Road and Ma Sik Road, was completed at the beginning of last year. The associated shopping mall, Green Code Plaza, features the most unusual floor patterns to be seen anywhere. I’ve included just one photo in this collection, because I plan to devote an entire post to these patterns in the future.

The white is polished marble, and the other colours appear to be various igneous rocks. I can’t help wondering whether there is some kind of mathematical formula for these rounded shapes, which are either grey or brown, except where they overlap, when they are black. I keep thinking that I’ve spotted a rule, but then I notice an obvious exception. For example, almost all these shapes are vaguely three-sided, with rounded sides and corners, but there are a couple of four-sided shapes. And, while most of the shapes curve outwards, I’ve noticed the odd one that curves inwards, like a banana. The orange bars are exactly half the width of the basic unit squares, except for a couple that are the same width. In every case, however, the locations and outlines of these shapes appear to be completely arbitrary. I wonder what the thinking behind them was.

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