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a few months Ago, looking for material for my last book ( The secret of the multiplication, Cataract, 2019), I found an interesting article by professor of French algerian Zoheir Barka, who is passionate about mathematics, titled The symmetries hidden in the multiplication table. In this article we are going to start a little walk by some of them.
The idea of Zoheir Barka is to create different patterns geometric color planes on the multiplication table , of varying sizes, associating colors to the multiples of some numbers. Therefore, the starting point is a times table square or rectangular, with a certain number of rows and columns, depending on the aesthetic needs of the employer you want to perform.
The next image is a table of multiply normal, with the products of the first ten numbers, from 1 to 10, to which has been added in addition to the products by zero, that is to say, all zeros, resulting in a square grid with 11 rows and 11 columns.
once we have the multiplication table, the size that is considered appropriate, it is coloring each cell according to whether the number of the cell is, or is not, a multiple of a number or of any of the selected numbers.
The simplest case would be to color the multiples of a number, for example 2, and leave it without color, or use a different one, to those that are not multiples of 2. We thus obtain the following pattern, which is very simple.
it Is evident that, if we take the multiples of a prime number as 2, but also 3, 5 and 7, the patterns are simple lattices, like the one above, but with areas of square, white, or uncolored, larger still. For the 2 the white areas were simple cells, to the 3 would be squares of 2 x 2 cells, for the 5 squares of 4 x 4 cells, and so for the rest of the prime numbers.
that Is, it creates a symmetrical pattern in which they are repeating, horizontal and vertical, basic blocks of size equal to the number whose multiples are being considered. Below, we see the basic building blocks 2, 3 and 5.
But if we consider the multiples of numbers are not cousins, as the number 4, whose divisor is not trivial is 2 (4 = 2 x 2), or the number 6, whose divisors with non-trivial are the 2 and the 3 (6 = 2 x 3), the structure is complicated a little more, as we see below.
For aesthetic reasons we may call the “basic area” of each example to the grid of size (n + 1) x (n + 1) if we are considering the multiples of the number n, which consists of adding to the basic block the following row and column, whose cells have color (as are the first few multiples of the number n) and shut down the basic building blocks.
So the basic zones of the cases in which we considered the multiples of the numbers 4, 6 and 10, which are product of two primes (equal or distinct), are the following.
And if the number considered is divisible by more primes (the same or different), such as 12, which is equal to the product 2 x 2 x 3, it complicates a bit more the fabric. Let’s see it.
Another example is the following, which shows the basic area of number 30, which is equal to the product 2 x 3 x 5.
As we see, the structure is enriched in function of the number of prime numbers to generate the number whose multiples are are coloring.
The next natural step, which is also considered Zoheir Barka in your article, is to consider the multiples of two or more numbers, using as many colors as numbers. Let’s start with the multiples of 2 and 3, the smaller numbers possible for this to make sense, and coloreemos the multiples of 2 green multiples of 3 in blue. Here arises a doubt, what to do with the numbers that are multiples of two, and then multiples of 6. We would have three options, keep the color of the greatest multiple, which in this case is 3 (blue),
keep the color of multiple smaller, which in this example is 2 (green),
or even, use another color for the multiples of 6 = 2 x 3, which in the following picture, we used the yellow.
now Let’s look at an example in which one of the two numbers is not prime, for example, 4 = 2 x 2, but the prime numbers that compose it, 2 (two times), there are the other prime number, 3. In this case, the core areas that are repeated in the three options are the of the following images. In each of the cases we have added to the version with numbers, one without numbers, which allows us to better see the geometric pattern that is generated.
The following is an example of two numbers that share a prime number, such as the numbers 6 and 9, for the that 3 is a divisor of both. We show the basic zones in cases in which premium, in the first, the color of the number 9 and, in the second, the color of the number 6.
To finish, we take an example in which one of the numbers is a multiple of the other. For example, the numbers 6 and 12.
To generate the above symmetrical patterns have been used in small numbers. With larger numbers, with more primes in its decomposition into factors, the structures will be more rich, but also with basic building blocks of larger and need much more space to represent them.
For example, for the numbers 10 and 12, the basic block is a grid of size 60 x 60, since 60 is the least common multiple of 10 and 12. The next twist would be to use more numbers and colors.
The symmetries hidden in the multiplication table are a fun activity and teaching that connects mathematics with the art. Even more, they constitute an interesting tool for artistic creation.
Raúl Ibáñez Torres is professor of Mathematics at the University of the Basque Country / Euskal Herriko Unibertsitatea.
This article has been originally published in “The Conversation”.
A version of this article was originally published in the Notebook of Scientific Culture, a publication of the Chair of Scientific Culture of the UPV/EHU.
