Mapping from n to n-1 Spaces

To the Lumo chatbot:

May a mapping from n to n-1 spaces be attributed to David Hilbert?

Lumo responded:

Yes, the concept of mapping an $n$-dimensional space onto an $(n-1)$-dimensional space (or more generally, filling a higher-dimensional space with a lower-dimensional curve) is famously attributed to David Hilbert, though he was building on the work of Giuseppe Peano.

Here is the breakdown of the attribution:

• Giuseppe Peano (1890): Peano was the first to discover a continuous surjective curve that fills a square (mapping the unit interval $[0,1]$ onto the unit square $[0,1{]}^2$). This is known as the Peano curve. It demonstrated that dimension is not necessarily preserved under continuous mappings, shocking mathematicians at the time who believed a line could not fill a plane.
• David Hilbert (1891): One year later, Hilbert published a paper titled "Ueber die stetige Abbildung einer Linie auf ein Flächenstück" (On the continuous mapping of a line onto a surface piece). In this work, he provided a simpler, more geometric construction of such a curve, now known as the Hilbert curve. He also generalized the construction to map a line onto an $n$-dimensional hypercube.

While Peano discovered the phenomenon first, Hilbert's construction became the standard example used in textbooks and is often the one people refer to when discussing space-filling curves in the context of mapping $n$ to $n-1$ (or specifically $1$ to $n$) dimensions.

So, while Peano holds the priority of discovery, Hilbert is certainly attributed with the specific, elegant construction that popularized the idea and generalized it to higher dimensions. The mapping itself is a counter-intuitive result in topology showing that continuous bijections do not preserve dimension, though the mapping from a line to a plane is not a homeomorphism (it is not invertible continuously).

Would you like to explore the mathematical properties of the Hilbert curve or how it is constructed?