HXY Images

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

Software

hxy - write an HXY image to a PNG file

This program is part of Misk.

Definition

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.

Example

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

X = {0, 1, 2}
Y = {0, 1}
P = {0 ⋅ 0, 0 ⋅ 1, 1 ⋅ 0, 1 ⋅ 1, 2 ⋅ 0, 2 ⋅ 1} = {0, 1, 2}

The pixels are numbered as follows:

0	1	2
3	4	5

And colored as follows:

White	White	White
Black	Black	Black

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

Observations

The top row of an HXY image is white if
- the width and height are equal to 1 or
- the height is greater than 1.
The bottom row is black if the height is greater than 1.
Flood-filling the top row reveals a special shape.

Gallery

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):

Copy and paste (or save and open) an image from Figure 1.
Use Tools → Paint Tools → Bucket Fill to flood-fill the top row with a color other than black or white.
Use Select → By Color to select all white pixels, then use Colors → Colorize to color them black.