Consider the guitar leaning against the wall over there—you got it because you wanted to look cool, sure, but notice that it is unlike a piano (where you press one key and you get one note) and also unlike a violin (where you can press anywhere, and you look less cool). The guitar is somewhere in between because its frets are positioned where they are for a reason, not arbitrarily, but you can still make tiny changes to notes, as tiny as you like. It’s kinda digital at the same time it’s kinda analog.
Mathematics has a similar area of study, coolly named Wonderland Spectra that is handy for describing singular-continuous phenomena like certain kinds of waves, or energy levels in atoms. They are not discrete, like a piano note, but they aren’t continuous either, like a violin string. They are something in between – scattered but in a very specific, structured way, kind of like frets on a guitar or the pattern of a fractal. This type of math draws on several specialties, including functional analysis, harmonic analysis, spectral theory, probability theory, and fractal geometry. Wonderland Spectra can help scientists understand the neither-here-nor-there aspects of quantum physics that everyone is chasing after because they might have potential applications in quantum computing or other technologies that exploit quantum coherence and entanglement.
