I have a consistent attraction to the weird and strange when it comes to math and data. And I'm not talking just "interesting" analytics or talking points like you'd find in most of my other blog posts. I'm talking things like Tupper's Self-Referential Formula. That weird.
Jeff Tupper published a SIGGRAPH paper in 2001 that discusses the (at the time) short-comings with graphical representation of 2-d formulas. I won't get into the weeds, but his methodology takes pixels and translates them to a bitmap that can be bound by some range (he uses k in his paper) which can then be used to translate an image to a formula. The method of finding this formula ultimately relies on the k values where the binary 1's and 0's of the originating image reside. To put things plainly, Tupper came up with a way to graph every single possible combination of pixels and related binary values based on a specific window k to k+n. This is the only formula that can graph itself.
The uses are ultimately endless, because later, Tupper published a series of additional papers that discussed taking images and essentially reverse-finding their necessary formulas. You could find the formula to graph your very own name or face! The trick is finding the necessary window of k.
In my research, I found a great github that allows you to (in Python) use draw any 2-d image (like the card symbols above) and the values of k will be returned.
The downside to this finding (and other weird science) is it's mostly just a curiosity and not a scientific breakthrough. However, it did in some ways help highlight the capabilities of mathematical visualizations and computer graphics.
Beyond the aesthetics, Tupper’s formula also invites philosophical questions. Is there something inherently self-referential about all mathematical systems, or is this a rare anomaly? It evokes a kinship with the famous paradoxes of self-reference, like Gödel’s incompleteness theorems, where the system itself becomes the subject of scrutiny. Can other formulas “describe” themselves, or is this an isolated case? Tupper's formula might seem like a fun math finding, but it hints at deeper concepts of recursion, infinity, and the limits of formal systems.
Some additional readings on this corner of math include:
“Gödel, Escher, Bach: An Eternal Golden Braid” by Douglas Hofstadter
“The Joy of x: A Guided Tour of Math, from One to Infinity” by Steven Strogatz
“The Art of Mathematics” by Béla Bollobás
“Visual Complex Analysis” by Tristan Needham
“Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise” by Manfred Schroeder
Also check out Mathologer on YouTube for interesting math discussions.
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