Merry Christmas!

The Christmas fresco above, by Giotto, shows the Star of Bethlehem as a comet (top centre). It is likely that this fresco depicts Halley’s Comet, which Giotto saw in 1301, about two years before he began the series of frescos of which this is part. This work by Giotto was celebrated in the name of the Giotto spacecraft, which observed the comet in 1986.

Let me take this opportunity to wish all my readers a very happy Christmas (and apologies for duplicating last Christmas’s post).

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!