Autumn equinox. Photo : Reuters
Autumn equinox. Photo : Reuters

The maximum, minimum, equinox and sine curve

  • A parallel can be drawn between a sine curve and the variation in daylight hours through the year, which helps define our planet’s four broad seasons
  • Summer solstice is when the number of daylight hours reaches a maximum (on the sine curve). Winter solstice is when they reach a minimum

Just a few days ago was the vernal, or spring equinox. On that day in March, of course, the night is exactly as long as the day. Six months from now is the autumn equinox, when, once again, the night will be exactly as long as the day.

What happens between now and then? That’s simple. The days will get longer—and the nights correspondingly shorter—over the next three months, until the summer solstice in June, the longest day of the year. After that, the days start getting shorter again, and the nights correspondingly longer. After the autumn equinox in September, the days keep getting shorter for three more months, till December’s winter solstice, the shortest day and longest night of the year. Then the days start lengthening once more…and round we go again. Just so, do the years pass. (Though if you’re in the Southern Hemisphere, please make some appropriate substitutions in this explanation).

You know all this, no doubt. It’s so familiar that you’re probably already wondering why this article even mentions it. The thing is, mathematicians consider this phenomenon—hours of daylight increasing, reaching a maximum, then decreasing, reaching a minimum, and repeat, year after year, essentially the same maximum and minimum—and recognize the similarity to another phenomenon they know well. It’s called the sine curve.

I mean, it’s also similar to the arrival of waves at the seashore—crest to trough to crest, over and over. Or to a range of hills: Climb to the top of one, descend to a valley, climb to the top of the next one, over and over. Or, think of a hotel room: It’s pristine when you enter, it becomes steadily untidier through your stay, then you leave and the cleaning staff bring it back to pristine, ready for the next guest.

You can imagine other such phenomena, I’m sure. In their regular variation between minimum and maximum, trough and crest, pristine and untidy, peak and valley—they are all reminders of the sine curve.

Perhaps you remember the word from your school days. In any right angled triangle, the sine of an angle is the ratio between the lengths of two of the triangle’s sides: The one opposite the angle in question, and the hypotenuse—the longest side of the triangle, opposite the right angle.

This ratio — and its cousin the cosine — is useful in plenty of ways. But that apart, think of how it varies as the angle itself varies. When the angle is 0 degrees, there is no triangle, and thus the ratio—the sine of 0—is also 0. When it is 90 degrees, the opposite side is itself the hypotenuse; thus the sine of 90 is 1. For angles greater than 90 degrees, we can no longer draw right-angled triangles that include them, but we have ways to calculate their sine.

And if you plot a graph of angle against its sine, you’ll get a familiar and pleasantly curvaceous picture: The sine curve. It rises smoothly from a trough of -1, through 0, all the way to a crest of 1, and back down again. Then it repeats. Just like the variation in the length of the day. That is, if you drew a graph that plots the number of daylight hours for each day of the year, and repeat for a few years, you’d get something broadly similar to the sine curve. Which is why mathematicians recognize the parallel.

So, now if you examine the graph you’ve just drawn, you’ll find the December solstice easily. It’s where the graph bottoms out. From there, it rises smoothly till it reaches a maximum, the solstice in June. Question: How do you locate, on this curve, the year’s two equinoxes? Of course, you know they are halfway between the solstices, thus March and September. But is there a mathematical way to find them on the graph, like we find the solstices by looking for the yearly minima and maxima?

Answer: There is. Consider how the number of hours of daylight changes, through the year.

Let’s start with the summer solstice. In the days just before 22 June, the amount of daylight increases steadily, though by smaller amounts every day. Similarly, in the days just after the solstice, the hours of daylight decrease steadily, though initially in small increments that get larger every day. That is: Before the solstice, the change in daylight hours is positive, because there are more every day; but after the solstice, the change is negative, because there’s less daylight every day. If you think about it, this means that right around the solstice, the number of daylight hours is changing hardly at all, though it is going from positive growth to negative growth. Put it this way: On the day of the solstice the rate of change of daylight hours is zero.

With me so far? We can reason in much the same way for the winter solstice on 22 December. That day, the rate of change of daylight hours is again zero, though the change now is from negative growth before 22 December to positive growth afterwards.

Aside: Remember that we could have explained all this in terms of hours of darkness, i.e., nighttime. Except that those would be increasing when daylight is decreasing, and vice versa.

So the rate of change of daylight hours decreases from zero on 22 June to some presumably minimum negative value and then starts increasing. It is zero again on 22 December. But it keeps increasing, till some presumably maximum (positive) value and then starts decreasing again. And, so it goes.

But hold on, when do those maximum and minimum values occur? Yes indeed, the equinoxes. The spring equinox is when the rate of increase of daylight hours reaches a peak. The autumn equinox is the opposite case. That’s when the rate of decrease of daylight hours reaches a peak —or, which is the same thing, its rate of increase reaches a trough.

Summing up: The summer solstice is when the number of daylight hours reaches a maximum. The winter solstice is when they reach a minimum. The spring equinox is when the rate of change of the number of daylight hours reaches a maximum. The autumn equinox is when the same rate of change reaches a minimum. Those four maxima and minima give us the four important seasonal dates in the year and, in fact, they essentially help define our planet’s four broad seasons.

If you’re a little underwhelmed so far, what if I told you that this discussion is something of an entry point into calculus? The rate of change of something—hours of daylight, here—is called, mathematically, its derivative. That’s the fundamental idea of differential calculus. We find maxima and minima of something—daylight, still—by asking when its derivative, the rate of change in the amount of daylight, becomes zero. Those are the two solstices.

But what’s more, we can apply the same idea to the rate of change, too. It reaches its maxima or minima when its own rate of change—its own derivative, or the second derivative of daylight itself—becomes zero. Those are the two equinoxes.

There’s some more to explore here. You may have noticed above that the way the amount of daylight changes is similar to the way the rate of change of daylight changes. In both cases, there’s a maximum, then a decrease to a minimum, then an increase to a maximum, and repeat. There you have it—yet another reminder of the sine curve. If you plot the rate of change of the ratio we call sine, you’ll get an identical sine curve, only shifted over by 90 degrees. (You can think of that shift as corresponding to the three months that separate a solstice and the following equinox). In fact, that’s the curve for the sine ratio’s cousin, cosine. Sure enough, the derivative of the sine curve is, indeed, exactly the cosine curve.

So what does this all mean? To me, it’s the charm of explaining natural phenomena in mathematical terms. It’s how such explanation then makes mathematics itself come alive, leap out of the abstract. It’s the way drawing parallels between two things can improve your understanding of both. Now of course, I leave it to you to find suitable parallels to untidying and then tidying a hotel room.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance. His Twitter handle is @DeathEndsFun

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