## The Unreasonable Effectiveness of Quasirandom Sequences

I present a new low discrepancy quasirandom sequence that offers many substantial improvements over other popular sequences such as the Sobol and Halton sequences. Figure 1a. Comparison of the various low discrepancy quasirandom sequences. Note that the newly proposed $R$-sequence produces more evenly spaced points than any of the other methods. Furthermore, all other current methods require careful selection of basis parameters, and if not chosen carefully can lead to degeneracy (eg top right).

## Evenly Distributing Points in a Triangle.

Most  two dimensional quasirandom methods focus on sampling over a unit square. However, sampling evenly over the triangle is also very important in computer graphics.  Therefore,  I describe a simple and direct construction method for a point sequence to evenly cover an arbitrary shaped triangle. Figure 1. A new direct method for constructing an open (infinite) low discrepancy quasirandom sequence over an triangle of arbitrary shape and size. Shown are the point distributions for twelve random triangles for the first 150 points.

## Going beyond the Golden Ratio.

I show that for the same reason that the golden ratio, $\phi=1.6180334..$, can be considered the most irrational number, that $1+\sqrt{2}$ can be considered the 2nd most irrational number, and indeed why $(9+\sqrt{221})/10$ can be considered the 3rd most irrational number.