Mathematics in action: tiling the plane

In the next post of our mathematics in action series, we look at tessellations of the plane. The most familiar of these are the three regular tilings, using tiles that are regular triangles, squares, or (as below) hexagons.

Photo: Claudine Rodriguez

The great Dutch artist M. C. Escher is famous for his distorted versions of such tilings, such as this tiling on the wall of a museum in Leeuwarden:

Photo: Bouwe Brouwer

Alternatively, the regular tilings can be extended by mixing different kinds of regular polygon. Of particular interest are the eight semiregular tilings, in which the tiles all meet edge-to-edge, and each vertex is equivalent to each other vertex (i.e. each vertex can be mapped to each other vertex through rotations, reflections, translations, or glide reflections). Here is one of the eight:

Photo: “AnnekeBart”

Because of the high level of symmetry, an exhaustive list of the 11 regular and semiregular tilings can be made by considering all possible meetings of polygons at a vertex, such as these two:


Penrose tilings, discovered by Roger Penrose in 1974, loosen the regularity and symmetry conditions, while still using a fixed number of kinds of tile, and while still being “almost” symmetrical. In the image below, Penrose is standing on a Penrose tiling. His 1974 discovery goes to show that fairly simple mathematical truths can still be discovered today.

Photo: “Solarflare100”

Infrared spectroscopy

Infrared (IR) spectroscopy exploits the quantum-theoretic fact that the stretching or bending of chemical bonds involves specific amounts of energy, which correspond to specific IR frequencies (lower, microwave, frequencies cause molecules as a whole to move; higher, UV, frequencies can actually break chemical bonds).

The technique of IR spectroscopy was pioneered by William Coblentz at Cornell University during 1903–1905 (using the apparatus shown above), and had become a standard technique of chemistry by the 1950s. Traditionally, a prism or diffraction grating splits infrared light into different frequencies, while a movable mirror bounces specific frequencies of infrared light through a sample, and a detector measures how much of that light is absorbed. Prisms for this purpose cannot be made of glass, which absorbs infrared light, but prisms made of sodium chloride and other salts have been used. Modern devices use Fourier transform techniques, which do not require a prism or diffraction grating.

The image below is the result of using modern IR spectroscopy equipment (like that above). The vertical axis in this plot measures how much infrared light gets through:

This IR spectrum (in the mid-IR range 2.5–17 μm) is for aniline, which has an NH2 group attached to a benzene ring (see molecular structure below). Some of the key absorbance peaks are marked; these correspond to stretching and bending of N–H, C–H, C–N, and C–C bonds. The set of visible peaks form a fingerprint, which immediately identifies the substance aniline. For unknown compounds, the IR spectrum provides valuable clues to any molecular detective trying to determine the structure. Thank you, William Coblentz!

Ground-penetrating radar and archaeology

Ground-penetrating radar (shown in action above) is a useful application of science to archaeology. Exploring the underground with microwaves saves a lot of digging!

The image below (click for details) is of a “slice” though an historic cemetery. The vertical axis shows depth. Yellow arrows mark probable human burials, while dashed blue lines mark probable lines of bedrock. The upper half-metre is a tangle of tree-roots, which it would have been difficult to dig through (had that been permitted, which it was not).

You can imagine how useful this technique would be in searching for a lost and buried city!

Mathematics in action: affine transformations and PostScript

Here’s the next in the mathematics in action series.

An important group of geometric transformations are the affine transformations, such as translations, reflections, and rotations. They are defined by a combination of a matrix multiplication and a translation:

Such transformations are an important part of the PostScript page description language. For example, the PostScript code below, after the initial header, specifies a black letter F, and another in blue, translated by (50, 50):

%%BoundingBox: 0 0 196 118
%%Pages: 0
%%DocumentFonts: Times-Roman
%%DocumentNeededFonts: Times-Roman
%%Orientation: Portrait
gsave newpath 100 dict begin /Times-Roman findfont 50 scalefont setfont

gsave 0 setgray 0 0 moveto (F) show grestore

gsave 0 0 1 setrgbcolor 50 50 translate 0 0 moveto (F) show grestore

The output is included in this picture:

But of course that is using only the (ef) part of the transformation. More interesting is a rotation through an angle θ, where the matrix is:

PostScript allows this to be specified using a translation and a rotation angle in degrees (red F) or as an (abcdef) matrix which combines the rotation matrix with a (180, 0) translation (grey F):

gsave 1 0 0 setrgbcolor 100 0 translate 30 rotate 0 0 moveto (F) show grestore

gsave 0.5 setgray 30 [0.8660 0.5 -0.5 0.8660 180 0] concat 0 0 moveto (F) show grestore

A diagonal matrix provides scaling in the x and y directions, with negative scaling factors giving a reflection. Again, PostScript allows the scaling factors to be provided directly (green F), or as an (abcdef) matrix (brown F):

gsave 0 0.7 0 setrgbcolor 100 100 translate -2 0.5 scale 0 0 moveto (F) show grestore

gsave 0.5 0.3 0 setrgbcolor [-2 0 0 0.5 175 75] concat 0 0 moveto (F) show grestore

end grestore showpage

Copying and pasting the three blocks of PostScript into a text file with a .EPS extension will give the coloured image above, which can be viewed by printing it, or by inserting it into a Microsoft Word document. Experimentation with other, more complex, affine transformations is easy that way.

The Bunsen burner

The Bunsen burner was invented in 1855 by the German chemist Robert Bunsen at the University of Heidelberg, assisted by Peter Desaga, an instrument maker there. Bunsen wanted a device that could produce heat without light, unlike the gas flames used for lighting at the time.

Bunsen was particularly interested in using the burner to identify elements by the colour of the flame they produced (or, more precisely, to identify elements by their emission spectrum). The image above shows the flames produced by placing salts of lithium, sodium, potassium, and copper in the flame of a Bunsen burner, for example. The image below shows the corresponding emission spectra (from top to bottom: Li, Na, K, Cu).


Rainbows are one of the most frequently observed atmospheric phenomena, although double rainbows can still get a strong reaction.

Rainbows form when light is refracted and reflected in droplets of water from rain (or some other source) as shown below. The light emerges at angles of up to 42°, so that the primary rainbow forms a circular halo around the antisolar point, at an angle of 42° from it. For the secondary rainbow, light enters the droplet from below and is internally reflected twice, emerging at angles of 51° or more, thus forming a larger halo (with reversed colours) around the antisolar point.

No light is refracted into the region between the primary and secondary rainbow, and this dark region (shown below in a photo by L.T. Hunter) is called Alexander’s band, after Alexander of Aphrodisias, who first discussed it in around 200 AD, in his commentary on Aristotle’s Meteorology.