Modulorama

R = X modulo N is a mathematical operation whose result R is the remainder of the division of X by N. For example: 127 modulo 10 = 7. Although the modulo is a non-reversible function, it is very practical in the analysis of cyclical phenomena, such as clocks.
 see https://en.wikipedia.org/wiki/Modulo

X Modulorama N
 A modulorama is an image showing a circular rather than a Cartesian vision.
 This image illustrates the modulo N of the multiplication of X by all the integers smaller than N. The goal is to visualize this multiplication table.
The value N indicates the quantity of numbers that have been placed on the perimeter of a circle or a plane figure. For example, on the clock like the drawing on the right, the classic case of modulo 12, there are the 12 integers from 0 to 11 distributed regularly around the circle. Each straight line refers to the multiplication of each of these integers i by the value X in order to illustrate this table of Xi modulo N in a circular array.
Here, for X = 4, is represented the 4 Modulorama 12 with all its lines originating from each integer. For example, the blue line represents the union of the integer 5 with the modulo 12 of the multiplication of X by this integer 5; ie 20. Since 20 is not part of this circle, we calculate 20 modulo 12 = 8 and this 8 finally locates the other end of the blue line. Just like 20 o'clock is 8 o'clock. So the blue straight line connects 5 to 8. Thus each of the other lines is also calculated.
Quite simple, even banal. But the image becomes more and more complex if the modulo grows by increasing the number N of lines and thus of harmonic crossings.
And if we gradually modify X, then the cinema of Modulorama unfolds with a surprising mathematical beauty that only waits for our imagination to brew these parameters (on yellow background ) to explore what can be hidden in all these Moduloramas.
The parameter dX indicates the variation to give to X between each image.
The Period of frame is the duration of each image animated with the buttons "Animate", "Stop" or "Inverse". Also, typing Esc or clicking on the image Modulorama can stop, move forward or back the animation to locate the exact quantities of X and N from which gushes of sublime mandalas.
Good excursion in the resonance of Moduloramas!
Note that straight lines become animated by rotating at different speeds, slower for smaller integers. This augurs subtle phenomena of synchronization, harmonic and resonance in a jumble of modulated multiplications with singular beats to the rhythm of arithmetic.
And, an extra charm is available besides black and white.
 by clicking the button "Black and white or 3 colors";
 red, green or blue determined by i modulo 3.
Also, the projection can be done on a circle, a triangle, a square, 2 parallels or a crossing of 2 perpendiculars in X.
Finally the blue buttons are some suggestions of remarkable moduloramas to watch ...

X Modulorama N

MicMath Wiki: Français English Mathologer Arithmétique modulaire de Gérard Villemin

MicMath
Wiki: Français English
Mathologer
Arithmétique modulaire de Gérard Villemin