The tetractys and the Cartesian coordinate system
The four basic operations of arithmetic in the four quadrants of a coordinate system. To understand the primes, you have to switch back and forth between calculating and drawing.
A tetractys is also contained in the four basic operations of arithmetic. Draw a Cartesian coordinate system:
- The two axes carry the natural numbers. From the origin they correspond to addition and subtraction.
- Each of the four quadrants is a multiplication table and a division table at once. Division and multiplication depend on each other.
- The cells within the quadrants are coloured by the three kinds of divisibility, exactly as in the division table.
One quadrant: the division table, coloured by divisibility. In the full coordinate system this pattern repeats, mirrored, in all four quadrants.
The number line itself and the three kinds of divisibility together give the four basic figures from The key to the tetractys.
Points instead of digits
The message of this chapter is one of method. To really understand the primes, you have to switch in your mind from arithmetic to geometry and back. Numbers must be replaced by points or other figures that stand in a ratio to one another. Only in this way can you escape the illusions of a counting system. The Pythagoreans, who calculated with pebbles, already knew this. Pure calculation often obscures what a simple drawing reveals.
The coordinate system is reminiscent of Peter Plichta’s prime number cross and deals with the same subject, but is built quite differently.