We saw last week that the eight types of semi-regular tessellations respond to the notation:
22.214.171.124.6, 126.96.36.199.4, 188.8.131.52.4, 184.108.40.206, 220.127.116.11, 3.12.12 , 4.6.12, 4.8.8
The numbers separated by points indicate the number of sides of each one of the regular polygons that converge in a vertex, beginning with those with less sides and in a rotary direction. Thus, 18.104.22.168 means that at each vertex of the tessellation an equilateral triangle, two squares and a regular hexagon converge, in the order triangle-square-hexagon square, as seen in the figure
The notation does not indicate the relative size of the regular polygons involved; but it is that for each combination there is only one possible correlation of sizes.
Regarding the possibility of tessellation of the plane with any quadrilateral and with some irregular hexagons, our regular commentator Luca Tanganelli has sent the following reflection (illustrated):
“Los Quadrilaterals are divided into triangles that tile the plane into "strips" (dashed lines). In addition, a second tessellation is revealed with hexagons that are obtained by duplicating each quadrilateral (in red) by rotational symmetry. Therefore, that a hexagon can be divided into two equal quadrilaterals is a sufficient condition to tile the plane (a concave hexagon can also tile); but if it is necessary, I don't know yet. ”
Is it a necessary condition, for a hexagon to be able to tessellate the plane, that it can be divided into two equal quadrilaterals? Other yes: How else can the necessary and sufficient condition be formulated?
Convex monohedral pentagonal tessellations
It is evident that the plane cannot be tessellated with regular pentagons, since the interior angle of a regular pentagon measures 108º, which is not a divisor 360º (yes, regular pentagons can, however, cover a spherical or hyperbolic surface).
With some irregular pentagons it is possible to tile the plane, as we saw when talking about the Cairo mosaic, the best known example of covering a flat surface using convex pentagonal pieces of the same shape and size: what is technically called a convex monohedral pentagonal tessellation
Throughout the 20th century and so far into the 21st, 15 different types of these tessellations have been discovered (the last as recently as 2015), and it appears that there are no more possibilities (there is a proof of impossibility pending verification, although it is likely that in the time of writing these lines has already been checked). Not surprisingly, Martin Gardner and his Scientific American math games section played an important role in this investigation; But it is true that an amateur mathematician without any university training discovered 4 of the 15 possible convex monohedral pentagonal tessellations: this is the extraordinary case of the recently deceased Marjorie Rice (1923-2017), who even created her own form of operative notation and its personal line of work (developed, as she herself said, in her home kitchen), and whose Intriguing Tessellations website deserves a visit from my astute readers.
In the image, the 4 types of pentagonal tessellation discovered by Rice, which showed that an amateur mathematician – and an amateur mathematician – can still make contributions at the highest level.
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including 'Damn physics', 'Damn maths' or 'The great game'. He was a screenwriter for 'La bola de cristal'.
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