Sawuare's Site

HXY Images

HXY images are mathematical images. I discovered them around 2020.


This program is part of Misk.


Let ℕ be the set of all nonnegative integers. For an HXY image with width w and height h, in pixels, w, h ∈ ℕ ∖ {0}, let X = {x ∈ ℕ ∣ x < w}, Y = {y ∈ ℕ ∣ y < h}, and P = {x ⋅ y ∣ x ∈ X, y ∈ Y}. Number the pixels 0, 1, … from top to bottom and left to right. A pixel is white if its number is in P, otherwise it is black.


For the HXY image with width w = 3 and height h = 2, the sets defined above become:

The pixels are numbered as follows:


And colored as follows:


Pixel 0 is white because 0 is in P. Pixel 3 is black because 3 is not in P.



Here are some images written using hxy.

Figure 1. The square HXY images with side lengths 256 and 251.
Figure 2. The special shapes of Figure 1.
Figure 3. The horizontal projections of Figure 1.
Figure 4. The vertical projections of Figure 1.

To see the relationship between Figure 1 and Figure 2, do the following in GIMP (or do its equivalent in another graphics program):

  1. Copy and paste (or save and open) an image from Figure 1.
  2. Use ToolsPaint ToolsBucket Fill to flood-fill the top row with a color other than black or white.
  3. Use SelectBy Color to select all white pixels, then use ColorsColorize to color them black.