This site contains few information about chryzodes themselves, and it only covers very few aspects of chryzodes, since I myself know only a small part of the very wide field of chryzodes.
In order to get more precise and more complete information about chryzodes, please go to the official chryzode site. Try it, it is really worth a click : The chryzode website
Also, one must keep in mind that the very first works on chryzodes were only started 25 years ago, by Jean-Paul Sonntag.
Jean-Paul Sonntag started to work alone by drawing sketches by hand on small notebooks, even before having technological means
like computers to draw complex chryzodes. Then he carried out research on chryzodes nearly alone, with a very small
team around him.
That's why chryzodes are not well known and why there is still so much to discover about them.
Chryzodes ( name created from the greek words chryzos and zoide, and thus meaning "golden writing on a circle ) are geometrical and graphical representations of numbers and mathematical operations by the mean of a circle. Chryzodes are a way to show various properties of numbers and mathematical operations, and to show the relationship between numbers. They also show some aspects of undulatory phenomena and thus may give us another understanding of these phenomena.
The general method to draw a chryzode is yet quite simple. Everything begins by taking a circle an integer m, called coding modulus
of the chryzode. Then m equidistant points are put on the circle, and these points are ordered and numbered from 0 to m-1.
Then we have series of numbers, generally mathematically defined and representing a precise operation. Then for each of the m points
on the circle, we draw a line from this point to the point it is linked to according the series of numbers.
Thus we end up with a set of lines drawn on the initial circle, of which the chryzode is made.
Then the chryzode can be represented by actually drawing the lines, or ( when the number of line is big )
by plotting with a computer only the points of intersections between the lines ( That is what the Chryzodus program does ).
Then we have what we call a simple chryzode, but we can then have more complex things by, for example, doing superpositions of chryzodes drawn
with different modulus m.
The previous short general explanation might not be clear enough. Therefore, we will, as an example, draw a simple chryzode.
Let's draw the chryzode representing the multipication by 2 in the modulus 29.
So, We draw a circle with 29 equidistant points on it, numbered from 0 to 28.
We take the series of numbers, where each number is the previous number doubled :
When a number is greater ( or equal ) than the modulus 29, we take the remainder of the division by 29 : ( since we work on a circle, everything is periodic : when it reaches 29, it starts back at 0 )
Then we have do draw a line from each number to his double : a line from 1 to 2, from 2 to 4, from 4 to 8, from 8 to 16, from 16 to 3, from 3 to 6, etc...
The result, which can be easily drawn by hand since there are not too many lines, is shown below :
Chryzode in lines |
Points of intersection of the lines |