HPQ Images

Software

• hpq-fb - display HPQ images on the Linux framebuffer device
• hpq-png - write an HPQ image to a PNG file
• hpq-seq - write some HPQ q-sequences
• hpq.h - library for HPQ images

These programs and libraries are part of Misk. I use hpq-fb to explore the images and hpq-png to save them.

Definition

In an HPQ image, a function p#(q, (x, y)) is applied to every pixel. The #, q, x, y, and output of the function are nonnegative integers. The function is called a painter. The # is called the identifier (ID). The q is called the q-argument. The x and y are the horizontal and vertical coordinates of a pixel, respectively. Pixel (x, y) is white if p#(q, (x, y)) = 0, otherwise it is black.

For pixel (x, y), the pixel to its right is at (x + 1, y), and the pixel below it is at (x, y + 1). The top-left pixel is at (x₀, y₀), where x₀ and y₀ are nonnegative integers called x-offset and y-offset, respectively. Pixel (0, 0) is called the origin.

There are 50 defined painters, 0 through 49, divided into 10 classes, 0 through 9. They are defined in such a way that p#(q, (x, y)) = p#(q, (y, x)). Painters in odd classes divide by q, so q = 0 is undefined for them. The following is a list of all painters, where & denotes bitwise AND, | denotes bitwise inclusive OR, ^ denotes bitwise exclusive OR, * denotes multiplication, which has the highest precedence, and % denotes modulo:

• Class 0
  • p0 = (x & y) & q
  • p1 = (x | y) & q
  • p2 = (x ^ y) & q
  • p3 = (x + y) & q
  • p4 = (x * y) & q
• Class 1
  • p5 = (x & y) % q
  • p6 = (x | y) % q
  • p7 = (x ^ y) % q
  • p8 = (x + y) % q
  • p9 = (x * y) % q
• Class 2
  • p10 = (x * x & y * y) & q
  • p11 = (x * x | y * y) & q
  • p12 = (x * x ^ y * y) & q
  • p13 = (x * x + y * y) & q
  • p14 = (x * x * y * y) & q
• Class 3
  • p15 = (x * x & y * y) % q
  • p16 = (x * x | y * y) % q
  • p17 = (x * x ^ y * y) % q
  • p18 = (x * x + y * y) % q
  • p19 = (x * x * y * y) % q
• Class 4
  • p20 = (q * x & q * y) & q
  • p21 = (q * x | q * y) & q
  • p22 = (q * x ^ q * y) & q
  • p23 = (q * x + q * y) & q
  • p24 = (q * x * q * y) & q
• Class 5
  • p25 = (q * x & q * y) % q
  • p26 = (q * x | q * y) % q
  • p27 = (q * x ^ q * y) % q
  • p28 = (q * x + q * y) % q
  • p29 = (q * x * q * y) % q
• Class 6
  • p30 = ((q & x) * x & (q & y) * y) & q
  • p31 = ((q | x) * x & (q | y) * y) & q
  • p32 = ((q ^ x) * x & (q ^ y) * y) & q
  • p33 = ((q + x) * x & (q + y) * y) & q
  • p34 = ((q * x) * x & (q * y) * y) & q
• Class 7
  • p35 = ((q & x) * x & (q & y) * y) % q
  • p36 = ((q | x) * x & (q | y) * y) % q
  • p37 = ((q ^ x) * x & (q ^ y) * y) % q
  • p38 = ((q + x) * x & (q + y) * y) % q
  • p39 = ((q * x) * x & (q * y) * y) % q
• Class 8
  • p40 = ((q & x) * x + (q & y) * y) & q
  • p41 = ((q | x) * x + (q | y) * y) & q
  • p42 = ((q ^ x) * x + (q ^ y) * y) & q
  • p43 = ((q + x) * x + (q + y) * y) & q
  • p44 = ((q * x) * x + (q * y) * y) & q
• Class 9
  • p45 = ((q & x) * x + (q & y) * y) % q
  • p46 = ((q | x) * x + (q | y) * y) % q
  • p47 = ((q ^ x) * x + (q ^ y) * y) % q
  • p48 = ((q + x) * x + (q + y) * y) % q
  • p49 = ((q * x) * x + (q * y) * y) % q

Example

For an HPQ image with width 3, height 2, x-offset x₀ = 1, and y-offset y₀ = 2, the pixels have the following coordinates:

Painter 8 with q-argument 4 is applied to every pixel as follows:

• p8(4, (1, 2)) = (1 + 2) % 4 = 3
• p8(4, (2, 2)) = (2 + 2) % 4 = 0
• p8(4, (3, 2)) = (3 + 2) % 4 = 1
• p8(4, (1, 3)) = (1 + 3) % 4 = 0
• p8(4, (2, 3)) = (2 + 3) % 4 = 1
• p8(4, (3, 3)) = (3 + 3) % 4 = 2

The pixels are colored as follows:

Pixel (2, 2) is white because p8(4, (2, 2)) = 0. Pixel (1, 2) is black because p8(4, (1, 2)) ≠ 0.

Observations

• HPQ images are symmetric around the line y = x.
• The Sierpinski sieve or variations of it seem to occur in painters 0, 5, 6, 20, 30, 31, 32, 35, and 36.
• Cantor dust or variations of it seem to occur in painters 1, 40, 45, and 46.
• The box fractal or variations of it seem to occur in painters 25, 26, 27, and 35.
• Painter 9 visualizes the factors of its q-argument.

Demo

Here is an interactive demo of HPQ images. It has options that let you select a painter and input its q-argument to paint a canvas. Note that the options are limited in comparison with those of the main program hpq-png.