2019.12.31 Chart of clock hand positions

I will not let 2019 slide away completely without a single POIB post — didn't make it within the UT year, but I'll be darned if I don't post this before midnight local time! Anyway, appropriately for having time on one's mind: below is a chart I drew 11+ years ago (which coincidentally jibes with the subject matter, as you'll see), in October 2008. The overall area of the chart is the space of all clock hand positions, whether they can actually occur or not; the minute hand's position increases in the horizontal direction, and the hour hand's position increases in the vertical direction. The left and right edges can be considered equal to each other, since at both of them the minute hand is at the :00 position; likewise the top and bottom edges for the hour hand — this is indicated by the gray arrows at the chart edges, matching at opposite edges.

The shallow-sloped, heavy lines are the trace of the positions that the hands of a (12-hour) clock take over the course of a 12-hour period. The steep-sloped light diagonal lines, meanwhile, are the positions that the hands would take over the 12-hour period if they were switched with one another. The particularly interesting parts, to me, are where the actual trace and the inverse trace intersect. Here are the notes I wrote accompanying the chart originally:

Where the clock hand relation intersects the self-crossing line, the clock hands are superimposed. The intersections of the relation and inverse relation indicate positions where a switching of the hands yields another real clock position.

Starting with 12:00 and moving forward by intervals of 1/11 of the 12-hour clock cycle (note that 11 = 12 − 1) gives the times when the clock hands are superimposed, while starting with 12:00 and moving forward by intervals of 1/143 of the 12-hour cycle (note that 143 = 11 × 13 = (12 − 1)(12 + 1) = 12² − 1) gives the times when the hands can be switched to give another actual clock position. Every 13th one of these positions is a self-crossing position.

Note that the domain of these relations is a torus (the product of two circles — the hour cycle and the minute cycle).

Note about the above note in case you're unfamiliar with topology/geometry jargon: a torus is a donut-shaped surface. Imagine printing out the chart, cutting out just the internal square area of the chart, and rolling it into a horizontal tube so that the bottom (0-hour) and top (12-hour) edges meet up. The tube will have open ends at left and right. Then imagine that this paper is stretchy in a miraculous way, so that you can bend the tube around horizontally in a circular shape and make the open ends meet up with one another — this matches the :00-minute ends together.

Now you have a torus, a continuous surface where the cyclicality of both hours and minutes is demonstrated by exactly that continuousness: the cyclicality of hours by cross-section circles of the torus, and that of minutes by circles ringing around the torus. The clock hand relation lines would then appear as different sorts of closed spirals around the torus: the inverse relation would look like a spring with its ends pulled around to meet one another, while the actual relation would loop a bunch of times horizontally around the torus. Wild!