The skill of tile-laying has roots dating back to at least 4000BC. Remains of Mesopotamian buildings present evidence that the Sumerians used to decorate the walls with patterns of clay tiles, generally referred to as tesserae.
The term ‘tessellation’ thus refers to the covering of a surface using one or more geometrical figures in such a way that there are no overlaps and no gaps. Restricted to two dimensions, tessellations of the plane using regular, semi-regular or irregular figures is an art that has attracted the ingenuity and creativity of many artists (see Did you know?). However, this skill is not restricted to humans. Honeybees are considered the masters of tessellations.
A polygon is regular if it has equal sides and equal angles. Only some basic mathematical knowledge is required to understand what happens if the tessellation is restricted to one where the tiles used are identical regular polygons. Combining the two well-known facts that the sum of the interior angles of a polygon with n sides is given by 2n - 4 right angles and that a whole revolution is made up of four right angles, we immediately get that the only possible values that n can take are 3, 4 and 6. Therefore, if the plane is tessellated with identical regular polygons, then the equilateral triangle, the square and the regular hexagon are the only three possible shapes that can be availed of. But why did the honeybees decide to use regular hexagons when the other two shapes are clearly easier to construct?
Pappus, a geometer of ancient Greece who lived in the fourth century, conjectured that, apart from being aesthetically pleasing, hexagons making up the honeycomb divide a plane into shapes of equal area with the smallest total perimeter. It is thought that this conjecture actually originated in 36BC and is due to the Roman scholar Marcus Terentius Varro.
As easy as it may sound, the ‘Honeycomb Conjecture’ stood for almost two millennia before some progress was made in 1943 by László Fejes Tóth and then finally proved in 2001 by Thomas C. Hales. In other words, mathematicians proved that bees opt to build a house with no empty gaps between rooms using the tessellation requiring the least amount of wax possible among all possible tessellations of the plane (not necessarily using only one tile which is regular)!
This seems to indicate that bees solved an optimisation problem way before mathematicians did. So, can bees do maths? (See Myth Debunked.)
Did you know?
• The bottom end of each cell in a honeycomb is not flat, but has the form of a pyramid made of three rhombi. Thus, each honeycomb cell is not exactly a hexagonal prism. Honeycombs are composed of two back-to-back layers of cells, and although this configuration is more efficient than if each cell in the two layers is made up of a hexagonal prism, it is approximately two per cent less efficient than Kelvin’s structure (see Sound Bites).
• An infinite number of tiles can be devised to tessellate the plane, for instance, by starting with the square, removing a section from one side and adding it to the opposite side. Maurits Cornelis Escher (1898-1972) is one of the most well-known artists whose work features tessellations with animals such as birds, reptiles and fish. His 1958 book Regular Division of the Plane included several woodcut prints illustrating tessellations that use a combination of irregular shapes (see Photo of the Week).
• A Penrose tiling, named after Sir Roger Penrose (b.1931) (see Photo of the Week), is a tessellation of the plane using a set of tiles in such a way that the design obtained is non-periodic, that is, a shifted copy of the tiling never matches the original (it lacks translational symmetry).
For more trivia, see www.um.edu.mt/think
Sound bites
• In 2001, the American mathematician Thomas Callister Hales (b.1958) showed that the best way to divide a plane into figures of equal area with the least total perimeter is the regular hexagonal tiling (or honeycomb). He improved a result of the Hungarian mathematician László Fejes Tóth (1915-2005), who had shown that the regular hexagonal tiling was the best way to tessellate the plane with least total perimeter under the assumption that all the shapes were convex polygons (that is, every interior angle of the polygon is at most two right angles). For more details, refer to the main article above and to: Hales, T. C. (2001) ‘The Honeycomb Conjecture’, Discrete and Computational Geometry, 25 (1), 1-22. In 1998, Hales also solved the sphere packing conjecture made by Johannes Kepler in 1611, following the approach suggested by Fejes Tóth in 1953 (see the Sound Bites section of page 41 of The Sunday Times of Malta of January 15, 2017).
• The ‘Kelvin Conjecture’, due to Lord Kelvin (1824-1907), is the three-dimensional version of the honeycomb conjecture. It claimed that the most efficient way to partition the three-dimensional space into cells of equal volume with the least total surface area is to use the bitruncated cubic honeycomb (a tessellation of space made up of truncated octahedra). This conjecture stood for almost a century, until it was disproved in 1993 by two physicists from Trinity College Dublin, Denis Weaire and Robert Phelan, who used computers to devise a slightly more efficient solution (0.3 per cent less than that of Kelvin’s structure). However, they did not prove that their structure was optimal. For more details: Weaire, D. & Phelan, R. (1994) ‘A counter-example to Kelvin’s conjecture on minimal surfaces’, Phil. Mag. Lett., 69, 107-110.
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