What’s the Radius of a Day?

I never know which cartoons will strike a chord.

But judging by the bizarre and amusing commentary that flooded my Twitter mentions after posting this one, it struck a real chord. Among badly drawn math cartoons, it was like the chord at the start of “A Hard Day’s Night.”

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This cartoon’s strange Sean Carroll meets Lewis Carroll mash-up (heady astrophysics + total illogic) prompted so many questions that I feel compelled to provide a series of FAQs and responses.

What, exactly, are you smoking?

Only the bargain-priced hallucinogen of pure mathematics.

You’re wrong! You’re wrong on the internet! The day isn’t a circle. It’s a spiral, ending in a different place than it began.

First: not a question.

Second: Sure, let’s go with that. After all, you begin at 12am on one date, and end at 12am on a different date. In terms of polar geometry, you’re at same angle, but you’re more distant from the origin. So it’s not crazy to say that our days are a tremendous spiral, with the Big Bang at the center.

(Okay, it is crazy, but it’s not crazy crazy.)

Still, this isn’t a solution; it’s a greater paradox! After all, the radius of a spiral is continuously growing. This suggests that each day is longer than the previous one!

Why, then, do we experience each day as identically 24 hours? Are we traversing the spiral at a faster and faster rate? If so, does this explain our occasional feelings of nausea as we rocket through time?

(This is what happens when you ask a non-question. Ask a question, get an answer; ask an answer, get a boatload of questions.)

If you’re imagining the day as 2D, why not go further? Why not a 3D sphere?

Aha! Perhaps the full day is measured not in square hours, but cubic hours, because the 24 hours that we traverse are merely a great circle on the exterior of a sphere. Trippy thought, my friend.

But why not go even further? Couldn’t the day be a hypersphere, or a 4-sphere in 5-space, or a 100-dimensional sphere in 110-dimensional space?

To explain the circularity of the day, we need to posit a second dimension. But once you arbitrarily posit a third, it seems there’s no reason to stop. Which is perhaps a reason for stopping at 2D.

Also: who says the 24 hours must be a great circle? If the day is spherical, perhaps we are circling near the pole. It’s like we’re flying around the earth at the latitude of Greenland. If so, we could experience a longer day merely by moving towards the equator.

What exactly is a square hour?

Dude, I wish I knew.

Perhaps a hint lies in our measure of acceleration. We measure speed in miles per hour, and so we measure acceleration miles per hour per hour, i.e., “miles per square hour.”

Thus, the day’s “area” can be thought of as a distance, divided by an acceleration. But after that, I’m stumped.

Further Carroll-ian insights (of either kind) will be appreciated.

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12 thoughts on “What’s the Radius of a Day?

  1. Clock time is clearly a helix, with time progressing linearly in one dimension while the clock hand circles in the perpendicular plane. There are occasional glitches (daylight savings time and leap seconds, for example).

    Twelve-hour and 24-hour clocks are helices of different pitch and radius, but the same slope. One can envision a minute hand and even a second hand spinning much faster but with smaller radii, also having the same slope.

  2. If a butterfly measures its life not in hours, but moments, then you can map their day in square moments. This way we can measure the area of time humans spend with butterflies in the untapped unit of hour-moments.

  3. I have no insights but I have to say your original post did make me think. I just wrote it off as something I couldn’t wrap my head around and I was OK with that. You are right. The post struck a chord and made me think far longer than normal. In other words, I traveled far further around that circumference of that circle than normal in deep thought!

  4. I seem to recall being told that each successive day IS longer than the previous one. In the early Carboniferous period, according to Sci Amer, an Earth year was around 385 days, or less than 23 hours per day.
    On that basis, the day is indeed a spiral, IMHO, albeit a wobbly one (day lengths vary within each year).
    Dev-Ilsa D. Vocate

  5. The 24 hour day is a lie!

    The sidereal day, one Earth rotation period, is about 23 minutes 56 seconds: it varies because the Earth is a crappy clock, hence leap seconds. It’s less than the solar day because as the Earth turns it revolves around the Sun as well.

    The solar day itself, the time from midnight to midnight, is 24 hours only on average. The apparent motion of the physical Sun deviates from the so-called “mean Sun” by up to ~15 minutes in either direction as specified by the equation of time: the shortest apparent day is around 3 November, the longest around 12 February. The graph expressing variation against time is called an analemma, and can be seen at the above page or on most globes and world maps.

    The local civil day is 23 hours on Spring Ahead day, 25 hours on Fall Back day, and can vary secularly in all sorts of ways due to political decisions right down to 0 hours, as on December 31, 1845 in the Philippines, when the islands switched from Mexico City time all the way across the Pacific to local time and a whole day was skipped.

    1. > The 24 hour day is a lie!

      “Earth has 4 days simultaneously each rotation. You erroneously measure time from 1 corner. Earth body 4 corner time equals 4 leg mobility. Your ignorance of Harmonic Cube is demonic.”

      R.I.P Gene Ray (Time Cube)

  6. Isn’t 24 hours just an arbitrary division in an infinite sequence? If we were living on Mars, or a moon of Jupiter, we would very likely select a different frame of reference.

  7. This reminds me of some clever student at the school dance.

    The chaperone says to keep at least 6″ between himself and his dance partner.

    “But there are at least 6 cubic inches between us.”

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