June 21st: Solstice Geometry and Uncle Woody

Today is the summer solstice. I am in Gloucester, Massachusetts, north of Boston, the part of the world from which my family harkens. We had a lovely dinner tonight overlooking Ipswich Bay, where the sunset is spectacular. It is one of few places on the east coast where you can see the sun set over the land and the water. One of the family members who came for dinner is my mother’s brother, my Uncle Woody. Uncle Woody is an economist who earned his Ph.D. from Princeton University in mathematical economics and political philosophy. He studied under Kenneth J. Arrow and John C. Harsanyi, both of whom won the Nobel Prize in Economics. He is impressive on that front, but what always strikes me is how magical he finds mathematics. The beauty and rationality of algebra and geometry are enough to bring him to tears. Seriously.

Tonight he spoke about how the rules of Euclidean geometry (flat planes, parallel lines that never cross) are thrown totally out of whack by the concept of non-Euclidean geometry that brings the curved line into the picture. On a sphere, straight lines are actually circles, parallel lines eventually meet, and a triangle’s angles add up to more than 180°.

As a teacher of younger children who don’t necessarily interact with these ideas yet… I often try to think of the seed that is planted in the elementary years that sprouts into big ideas. This one brings me to the definition of a polygon as a closed shape with straight lines and angles. Many of the non-examples we work have curves….

Later tonight, mulling over the longest day of the year and thinking about the math involved, as I do sometimes, it occurred to me that the solstice itself is a real world example of non-Euclidean geometry. We live on a great sphere, in fact. So I did a little digging into the mathematics of astronomy and came across this explanation of the calculations behind the solstice by Joe Antognini.

The basic idea is this: The solstice is like the point in the trajectory of a swing at a playground when you feel somewhat suspended in the air. The point at which forth turns to back or back turns to forth. The daylight lasts the longest, and the change in daylight is very gradual over the time around the solstice (which is why summer light lingers even though the solstice is right at the beginning). How can you get specific about the solstice where you are? Well let’s take where I am as an example. Gloucester, MA. What I need is my latitude and the sun’s “tilt” on the specific day in question.

Gloucester, MA is at 42.6° latitude.

The earth is tilted on its axis always, and that tilt is always 23.5°. That is why, as the earth travels around the sun over the course of the year, the Sun’s tilt appears to change. Your latitude and the Sun’s tilt work together to stretch or shrink the daylight. Think of 12 hours as a starting point, and the combination of latitude and tilt as the thing that pushes daylight above or below it. The equinox (Fall and Spring) sees the tilt number at 0. When the tilt is 0, there’s no push in either direction so you get a solid 12 hours of daylight, the same everywhere on Earth. When the Sun’s tilt number is positive (moving from spring into summer), the Northern Hemisphere (where I am) tilts toward the Sun, so days grow longer than those 12 hours. When the Sun’s number is negative (moving from fall into winter), we tilt away, so days become shorter than 12 hours.

Where does latitude come into play? Let’s go back to that swing analogy:

The kid on the swing at the equator is barely moving. The forward and backward are so close together that it hardly matters where the swing is, it’s always seemingly in the middle.

The kid at a northern latitude has been given a real heave. The swing arcs out wide, high at the front, low at the back, so summer days run long and winter days run short.

The kid above the Arctic Circle is the one whose swing is looping all the way over the top bar of the swing set. The swing has gone so far that at the high end it never comes back down at all. That’s the midnight sun: the sun’s daily circle clears the horizon and just stays up. The discrepancy between summer and winter daylight is huge (midnight sun, anyone? it is more than a song by Zara Larsson).

Where I am, 42.6°, the latitude is sort of middling (the highest possible latitude is 90° up at the North Pole). My swing is neither boring nor dangerous.

The actual math is far more complex, but fascinating indeed. For all of you who are mulling this over: consider why the opposite occurs in the southern hemisphere… what changes about latitude numbers? Think integers, y’all!