The following is not true mathematics, but a little imaginary story about mathematical symbols. I wonder if there is - in parts - a true (etymological) story behind it.
Once there was a symbol $\mathsf{O}$ for "nothing", and there was a symbol $\mathsf{I}$ for "something".
(Switching to present tense:) Topologically, the symbol $\mathsf{O}$ is a circle (without ends), and the symbol $\mathsf{I}$ is a line segment (with two ends). Topologically, these two symbols cannot be continuously transformed into each other.
But the symbol $\mathsf{I}$ can be continuously transformed into a figure looking like $\omega$. This figure $\omega$ can be flipped vertically, and appropriately joint with its original looks like $\infty$.
On the other hand, the symbol $\mathsf{O}$ can be flipped horizontally, and appropriately joint with its original looks like $\infty$, too.
Finally, bending the ends of $\omega$ on and on, the figure $\infty$ is a limit of continuous transformations of $\mathsf{I}$.
My question is:
(How) are these - at first sight only somehow related - mathematical symbols for
"nothing-ness" ($\mathsf{O}$),
"some-ness" ($\mathsf{I}$),
countable infinity ($\omega$), and
arbitrary infinity ($\infty$)
truly - e.g. historically and/or etymologically - related?
There is no such relation between these symbols, but if you enjoy messing with such things, here is a challenge:
Find the smallest set of symbols that can generate the entire English alphabet (where you are allowed to rotate and flip).
Here is a somewhat reasonable answer with $13$ symbols:
b, c, e, f, h, i, k, L, m, o, r, s, x.
For completeness, the equivalence classes are:
{a, e, g}, {b, d, p, q}, {c, n, u}, {f, t}, {h, y}, {i}, {J, r}, {k}, {L, v}, {m, w}, {o}, {s, z}, {x}.
Of course, the font needs to be rendered properly for this to work.