Dot Patterns, Pascal Triangle and Lucas Theorem

In the drawing area of the applet below, we have either rows of digits or circles with colors corresponding to the digits. Patterns in the drawing area are defined row-by-row starting from the upper row which consists of clickable digits (or circles.) The value p of a node is defined by values (q1 and q2) of the two nodes immediately above it according to the following formula:

p = q1 + q2 (mod N),

where N is the modulus of the arithmetic used. Think of the applet as presenting a finite view of an infinite lattice of nodes filling the lower half plane. All omitted nodes are assigned the value of 0. The applet has the following controls:

  1. Every dot in the upper row is clickable. With every click the digit (or the corresponding color) cycles through the sequence of residues 0, 1, 2, 3, ..., N-1.
  2. When creating a new pattern, you can select to get a single nonzero digit in the upper row, or a random pattern, or the whole upper row carrying the same digit (1).
  3. By checking "Multiplies" you request to associate all nonzero digits with a single color. In this case, the pattern consists of two colors only with 0 using the foreground color. So that the colors are in a sense reversed.
  4. You can also chose to see a pure triangle with a single node in the upper row. The apex of the triangle is still clickable.

(Please note that when the number of rows is close to the maximum of 50, the drawing is slow. Be patient.)

As of 2018, Java plugins are not supported by any browsers (find out more). This Wolfram Demonstration, Pascal-like Triangles Mod k, shows an item of the same or similar topic, but is different from the original Java applet, named 'Patterns'. The originally given instructions may no longer correspond precisely.

(image below from deprecated 'Patterns' applet)

Dot Patterns, Pascal Triangle and Lucas Theorem

You are to investigate the distribution of 0s in rows of the pattern emanating from a single dot.

About Fractals

Pascal's Triangle and the Binomial Coefficients


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