Figurative numbers
The habit of representing numbers pictorially (they were called "figured numbers") led to further discoveries - https://domyhomework.club . For example, triangular numbers, including the "sacred" tetraktys, and square numbers were known.
From the representation of the nested square numbers - https://domyhomework.club/math-problem-solver/ , one can immediately read that the sum of the first n odd numbers always results in the sum n2.
The following, for example, was derived from other representations:
The sum s of the first n natural numbers is s=12n(n+1).
For the sum of the numbers that leave the remainder 1 when divided by 3, we get s=12n(3n-1).
And further: The angle piece tinted in picture 4 (such a thing was called a "gnomon") consists of nine points, thus itself represents a square number. Thus, from the smaller square, the larger square and the gnomon, one obtains a Pythagorean triple of numbers, namely 4; 5 and 3 and a way to obtain further such triples.
The sign of the Pythagoreans
The Pythagorean League also had its secret sign. This was the (regular) five-pointed star, the "pentagram", which was used as an amulet in the Middle Ages - statistics homework helper . (Goethe's Faust also draws such a figure, called "Drudenfub", on his threshold to keep the devil away).
It happened that once an old, frail and sick wanderer knocked at a house and asked for something to eat and drink. The landlord of the house was a kind-hearted man and gave the wanderer food and a place to sleep for the night and continued to care for him, as the old man was terminally ill. When he was at the end of his life, he asked his landlord for a tablet and a piece of chalk, drew a pentagram on it and asked the landlord to put the tablet in the window, for he had nothing else with which to repay the kindness he had experienced. Then he died. Some time later, a distinguished traveller passed by the house, saw the tablet and asked the innkeeper what the matter was. When the innkeeper told him what had happened, he was rewarded by the traveller with a bag of money.
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