What proportion of the volume of Gabriel's horn could be filled by its surface area?

Question

Space filling curves imply that lower dimensional objects can fill a proportion of higher dimensional objects. Eg. in the case of the Hilbert curve and the 2D space it fills, this proportion is 1.

Gabriel's horn has an infinite surface area, but a finite volume. What proportion of the volume of Gabriel's horn could be filled by its surface area?

Is there a general way of thinking about the size of an object in one dimension compared to its size in the next higher dimension, which would answer this question as a special case?

Apologies for the problems with the question. All input appreciated. Cheers!

Answer 1

tl; dr: The "plane-filling" property of the Hilbert curve (e.g.) is not restricted to raising the dimension by one. For example, there exists a surjection from the number line to Gabriel's horn.

It's a red herring that the surface area of Gabriel's horn is infinite.

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$\newcommand{\Reals}{\mathbf{R}}$Let $h:\Reals \to \Reals^{2}$ be a continuous, surjective mapping, and write $h = (h_{1}, h_{2})$. The mapping $h \times h:\Reals^{2} \to \Reals^{4}$ defined by $$ (h \times h)(s, t) = (h(s), h(t)) = (h_{1}(s), h_{2}(s), h_{1}(t), h_{2}(t)) $$ is also surjective, so the composition $H = (h \times h) \circ h$, defined by $$ H(t) = (h(h_{1}(t)), h(h_{2}(t))), $$ maps $\Reals \to \Reals^{4}$ continuously and surjectively. Iterating this construction gives us a continuous surjection from $\Reals$ to a Cartesian space of arbitrarily high dimension.

To map $\Reals$ continuously onto Gabriel's horn, we can:

1. Start with $H:\Reals \to \Reals^{4}$; then
2. Map $\Reals^{4} \to \Reals^{3}$ continuously and surjectively by a coordinate projection; then
3. Map $\Reals^{3}$ continuously onto a closed cylinder by sending $(x, y, z)$ to $(x, y, z)$ if $x^{2} + y^{2} \leq 1$ and to $(x/\sqrt{x^{2} + y^{2}}, y/\sqrt{x^{2} + y^{2}}, z)$ otherwise; then
4. Map the closed cylinder to a semi-infinite cylinder by sending $(x, y, z)$ to $(x, y, z)$ if $1 < z$ and to $(x, y, 1)$ otherwise; then
5. Map the closed semi-infinite cylinder $\{(x, y, z) : x^{2} + y^{2} \leq 1, 1 \leq z\}$ continuously onto Gabriel's horn by axial scaling, sending $(x, y, z)$ to $(x/\sqrt{z}, y/\sqrt{z}, z)$.