Well, it’s Wednesday, which follows Tuesday, which follows Monday, which follows Sunday, which follows Saturday, which follows Friday, which follows Thursday, which follows…
…that’s right, it brings us right back to Wednesday again. It’s a little bit reminiscent of the Doe, a Deer song from The Sound of Music. Or, maybe you could say it’s a little like Rock Around the Clock. All these cyclical things are amusing in that each member of the closed, recursive group both precedes and follows itself.
That gives me an interesting idea. At least, it’s interesting to me. I’ve seen clocks (dial style clocks obviously, not digital ones) that designate the hours with radians, such that in place of the 12, for instance, there would be π/2, since the whole circumference of any circle is 2 π radians.
Six o’clock is then 3 π/2, nine o’clock is then π, and three o’clock is just 0 (or 2π). The order of these numbers uses the tradition of beginning counting radians (or degrees) at the 3 o’clock position as it were. The other numbers (1, 2, 4, 5, etc.) are various ratios of π, using the trig functions that give us the results for what we would think of (typically) as 30 degrees, 60 degrees, and so on.
Well, why not do a novelty clock with the hour markings represented by complex numbers? In engineering and in physics, they are very frequently used because they are excellent ways of calculating cyclical things, using a unit circle (in this case, a circle with radius 1, centered on the origin of the complex plane). Using such an arrangement, multiplying any complex number by i results in a 90 degree rotation (counterclockwise).
In the case of a complex number clock, the 3 o’clock position is just 1 + 0i, 6 o’clock is 0 – i, 9 o’clock is -1 + 0i, and 12 o’clock is 0 + i. I’ll work out the specific complex numbers for the other hours, and I’ll put them in a footnote* below, but I don’t think I can do it off the top of my head while commuting. For one thing, I don’t recall the sine and cosine of 30 degrees…
Wait! The sine of 30 degrees (aka pi over 6 radians) is just ½. And since we can take that as a leg of a right triangle with hypotenuse length 1 (since it’s a unit circle) then the Pythagorean theorem tells us 1^2 = ½^2 plus the cosine^2 of 30 degrees, so the cosine equals the square root of 1 – ¼ or the square root of ¾, in other words √3/2. So 2 o’clock would be √3/2 + (½)i.
One o’clock (or 60 degrees) just reverses the ordering of the parts of the complex number: ½ + (√3/2)i. Then 4 o’clock is √3/2 – i, five o’clock is ½ – (√3/2)i, 7 o’clock is -½ – (√3/2)i, 8 o’clock is -√3/2 – (½)i, 10 o’clock is then -√3/2 + (½)i, and and 11 o’clock is -½ + (√3/2)i. Neat and tidy!
I’ll leave it as an exercise for the reader to work out the complex numbers for the various minutes of the clock face**. And, of course, if I’ve made any errors above, please let me know.
But anyway, I think you can see that this might make for a novelty clock every bit as exciting as the ones that just show you the hours as radians. It might be slightly more rarefied than the other, but then again, engineers might prefer it (though they tend to use j rather than i to represent √-1, because i already stands for electrical current****, if memory serves.
Wow, how’s that been for some weird digressing?
Is “digressing” a word? Google Docs seems to “think” it is.
Hopefully those quotation marks don’t offend Google Docs if it really does actually think. If it’s any consolation (and even if it isn’t), I would be inclined to use such “scare” quotes when referring to the hypothetical thought processes of many a human.
On to another digression: I had a weird thought just now, while my mind briefly wandered to imagining a potential crime thriller, maybe about a serial killer. What if the killer, anticipating the potential for DNA evidence to be used to try to catch him*****, makes it a habit to pick up strands of hair, discarded tissues, used Band-aids, and various other things from various places such as public (pubic?) restrooms, barber shops, and so on. He then leaves them randomly chosen and scattered at his various crime scenes to throw off investigators.
Of course, he’d want to be careful to get his DNA samples from widely chosen and semi-random places, such as maybe rest areas on major highways. Otherwise the very pattern of those sources might give investigators something with which to work. The FBI, for instance, can be impressively clever about such things. At least, well, they were under previous administrations.
Jeez Louise, I’m really losing my mind, am I not? Certainly it seems to have slipped its leash. My apologies for this chaotic brain sneezing; I hope you don’t catch any mental viruses from me.
With that, I’ll bring this to a close. I hope I end up feeling better today than I did yesterday, though so far it doesn’t seem promising. More importantly, as always, I hope all of you have very good days.
*Well, there’s no point in doing this in a footnote now; I worked it out in the main body of the post. See above.
**Just kidding. Seriously, please don’t feel obligated to do this***. I’m not going to do it (or might I say, “i is not going to do it”?)
***On the other hand, if one of you feels compelled to do the figuring, please share your results with me.
****That could almost be like someone’s protest sign, perhaps one that is held up and shaken outside a courthouse:

*****The vast majority of serial killers are men. It’s a travesty, this wild gender/sex disparity; it’s the last bastion of unrepentant, unapologetic misogyny! We need social programs to try to get more women out there serial killing! Maybe we should provide subsidized child care services, so single moms won’t have as hard a time getting into the field…or wherever they might want to dump the bodies of their victims.



