Why is the Hilbert Curve Beautiful?

The Hilbert Curve is one of the results of David Hilbert’s vision of mathematics as a network of symbolic systems. Defined in 1891, this curve is a fractal that fills a two-dimensional plane with a one-dimensional line in the limit. It exhibits self-similarity. It emerged from representing higher-dimensional spaces through continuous, lower-dimensional entities. A curve that grows within a bounded area remains continuous but non-differentiable. Yet it possesses infinite length. Perhaps no one will ever truly see or fully grasp it. Yet it carries the potential to fill a higher-dimensional space through infinite iteration. It is one of the most elegant way of Hilbert’s belief in the power of axiomatic structures.

Using Grasshopper Python, I created a script that generates L-systems with letters and symbols, one that represents the Hilbert Curve here in this post. But is this code the Hilbert Curve?

The script outputs coordinates that Rhino can process. But is that geometric data the Hilbert Curve?

To Hilbert, mathematics was a symbolic construction of human thought. His contemporary, Gottlob Frege, held a different view. Frege, one of the founding figures of the analytic tradition in philosophy, regarded the world and mathematics as within the realm of logic. For him, mathematics was a branch of logic. He searched for the foundations behind everything that could be deemed true. This is the mode of thought I always feel close to Frege. Searching for a deeper truth. Just one year after Hilbert introduced his curve, in 1892, Frege published the famous On Sense and Reference. This text offered a powerful tool for understanding meaning and language as part of his project to reconstruct mathematics through logic. It also planted the early seeds of the “linguistic turn”. This turn would probably reach one of its peaks with Wittgenstein, who is another very intuitive person for me.

The fractal drawing appears on the screen through computed data. So, does that count as the Hilbert Curve?

In the next step, I tried to 3D print an infill with a robot-controller extruder. I developed a Grasshopper script that generated a KRL file to be read by a robot within a predefined setup. Are these codes and robot instructions the Hilbert Curve?

According to Frege, a word’s meaning has two layers. The sense is how the word appears within a particular context. The various things we might refer to when we say “Hilbert Curve”: the code, the image on screen, the CAD file, the KRL file.

Finally, the KRL code moved the robot and printed the curve onto a platform. Is that printed object the Hilbert Curve?

According to Frege, reference is the entity to which we anchor senses. In this case, what underlies all these representations is the mathematical idea in Hilbert’s mind. I think this is why Frege is thought to be a pseudo-Platonic. There is a conceptual essence of infinity through formalism that appears in many different senses. For me, it was interesting to see Hilbert’s mind through the lens of Frege’s method.