Pythagoras table biography

So, in Fig. In order to get an idea of ​​how the numbers are located in the Pythagorean table, which give the same residues when dividing, for example, we paint numbers, giving residues 0, 1, 2, 3, 4, each with its own color. Surprisingly, the Pythagoras table turns out to be dissected into the squares of Fig. Similar breakdown is obtained by dividing the number of the table into any other natural number K, which is easy to verify by replacing the number 5 with it in the program.

Due to the property of the periodicity of the Pythagoras table, a variety of mosaics occur on the remains on the screen. Obviously, the more K, the more R remains of R, the more colors will be required. So that the patterns are not too motley, we will confine ourselves, for example, with three colors. To do this, we group the remains according to the module 3, that is, the first color is painted over the number of the table with remains of 1, 4, 7, you can divide any of these mosaics into three monocrine, complementing one another to complete mosaic.

Each of them separately is also of interest to Fig. Another version of tricolor mosaics is shown in Fig. Here, for greater symmetry, not only numbers with the same residue R are painted in the same color, but also the number with the remainder complementing R to K. Interesting mosaics also arise when not all numbers are painted, but selectively. For example, a tricolor pattern in Fig.

Circular monochrome pattern Fig. And if you include a random number generator in the program to determine the sizes of the squares of K, lying in the period of numbers of the extended table of Pythagoras and the Nomers C, then using the computer the table will turn into a peculiar kaleidoscope of amazing and unique patterns of RIS. In Fig. Here, each number is depicted with a blue or green cell.

Moreover, the numbers of the first, third, fifth, etc. is clear that if the work N X M is constantly, then there is a reverse proportion between numbers, therefore, alternating blue and green stripes have a hyperbolic shape. With an increase in the work of N X M, the width of the strips decreases, and then the stripes are completely torn and disintegrated into single -color islands, which are grouped with islands of the same color, but from another hundred, forming symmetrical forms of rice.

Here, each number of the X table is depicted by a pixel point.

Mysteriously, the Pythagoras table is not wider into a periodic structure. I wonder how this can be explained? If you carefully and patiently study the properties of the Pythagoras table, then you will undoubtedly find new, no less beautiful patterns based on this ancient numerical scheme.