The Formula Containing Every Image
Your name, history, future scenery...
Everything exists within a single formula from the beginning.
1. The Formula
Published by Jeff Tupper in 2001, this inequality is famous for its self-referential property: it graphs its own formula at a specific location on the $(x, y)$ plane.
$$ \frac{1}{2} < \left\lfloor \mod\left(\left\lfloor \frac{y}{17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mod(\lfloor y \rfloor, 17)}, 2\right) \right\rfloor $$
The formula itself is simple, but the magic lies in the "drawing range." When you plot this inequality at a very specific, immense height on the $y$-axis (between $k$ and $k+17$), meaningful shapes emerge.
2. Generator: Exploring the Ocean of Math
By changing the height $k$ on the $y$-axis, this formula can display any bitmap image that fits within a height of 17 pixels. Use the generator below to calculate which "floor" (height $k$) your text resides on.
This is the location where your image is found within the infinite $y$-axis.
3. Why does the "Height" change?
In a standard graph, defining the function $y=f(x)$ determines the shape of the line. However, Tupper's formula takes a different approach.
In this context, the height $k$ (Y-coordinate) is the data itself.
Computer images consist of a sequence of 0s and 1s (a bitstream). If you interpret this long sequence of bits as a single gigantic integer, you get $k$.
- Data for "A": When converted to a number, it might be "Height 500". $\rightarrow$ You find "A" at $y=500$.
- Data for "B": Since the data is different, the number changes to "Height 900". $\rightarrow$ You find "B" at $y=900$.
In other words, we are not "drawing a new picture" but rather "searching for the address (number)" that corresponds to that picture. If the input text changes, the number representing it changes, and thus the display position on the graph moves to a completely different location.
4. Dr.WataWata's Insight
"Creativity is Discovery."
Michelangelo once said, "The sculpture is already complete within the marble block, before I start my work. It is already there, I just have to chisel away the superfluous material." This formula embodies that concept mathematically.
The text you just typed, the photos you took today, and the blueprints for inventions not yet conceived—they all already exist within this formula, provided you find the correct height $k$. We aren't creating something new; perhaps we are simply "discovering" it from an infinite number line.