Tiling: The mathematical termsNovember 11, 2020
After taking a look at the skies last week , let’s get back to the ground we stepped on, because math can tell us a few things too. Our busy daily life does not allow us to enjoy everything that surrounds us, or at least stop to think at some point how the things we live with are like. With this article I am going to start a series on how we organize and configure something that we cannot do without tiling the floor .
Since ancient times, human beings have used different motifs to decorate the pavement of streets and houses, to decorate walls, ceilings and objects such as boxes, furniture, etc. And he has devised different ways to do it.
That’s what we’re going to talk about: how to coat the plane, said in mathematical terms. In case the language can be an inconvenience (many times this is what happens in mathematics; we say that we do not understand the jargon used, but on many occasions the branches simply do not let us see the forest, and things are simpler than what seem; others do not, of course).
We use the expression “the plane” to designate a surface with two dimensions, length and width. That word tries to respond to the intuitive concept of everything that is like that, flat, that has no height, or is irrelevant to what we are doing.
The walls, the ceilings are flat (even if they are sloping; that does not concern us for what we want to deal with, which is the covering of them), a sheet of paper, and of course, the floor tiling.
We will call a set of pieces (tiles or tesserae) mosaic or tessellation of the plane with which we will try to completely cover the plane, without leaving any gap, and without mounting or overlapping some pieces with others.
Anyone who has tried to tile a bathroom, uneven room, etc. knows from experience that the task is not trivial. It is not trivial if you want to do well, of course. If what we want is to finish quickly, we make small pieces and we fill in any way. But you already know that mathematicians want it to be perfect, symmetrical, that makes us want to look at it, not that it frightens our eyes.
Tiling Regular polygons
Today we are going to start with the simplest: using only regular polygons (you know, triangles, squares, pentagons, hexagons, etc.). We will be more sophisticated later. You already know that in mathematics things are not done by tun-tun (trial-error; sometimes to try to understand a situation, it can be proved, but in matters as simple as this, it is not necessary), so let’s take a small account. We are going to look at a hypothetical vertex of the tiling, that is, at a point where several tiles that we are going to use meet.
If from that point we draw a circle with the compass, we will immediately realize that the plane is completely covered with that circle, since it encloses all points around that vertex. That is to say, the plane is completed with 360º, which is the total amplitude of a circumference. Next, we need to know the degrees of each of the angles of the regular polygons. Being regular (that is, all the sides have the same length), the angles are also equal for each polygon.
At school they taught us at the time that the sum of all the angles of a regular polygon with n sides is given by the expression: 180 (n-2)
Thus, for the triangle (n = 3), the sum of all the angles is 180º; for a square (n = 4), said sum is 360º; for a pentagon (n = 5), it is 540º, and so on. If we want to know the amplitude of each angle, simply divide the previous expression by n (the number of sides):
As we must distribute those values of the angles in 360º, only those that divide at that value serve us. Thus, for example, for n = 3 (triangles), we can fill the floor tiling with six tiles, because 360 is divisible by 60, and the quotient is 6. This is how we see it in the following image: at each vertex (they only appear all in the central one) six triangles are arranged.
The next value that divides 360º would be that of n = 4 (squares), that is, we also regularly fill the plane with four squares, because 90 x 4 = 360.
Here we will find variations in our streets, because such a floor tiling would be monotonous and boring (although there are). If we move the rows a little, we slightly eliminate that feeling. In the following image, which I have taken in a street near my house, we observe both on the floor tiling and on the wall, tessellated with squares (or rectangles; it doesn’t matter because the angles are still 90º).
However, on the floor tiling the arrangement is different from that of the wall, in which both the brick and white rows are counterbalanced. In the case of the wall, this action gives more solidity to the whole (those of us who have played as children to build).
Another variation of the same tessellation is to play with the two directions, horizontal and vertical, look for symmetries, play with different colors, etc., as we see in the following photographs.
Going back to the table, there is only one possibility of tiling the plane with regular and equal polygons: with hexagons (n = 6), because 120 x 3 = 360 (you can check that there are no more).