Traveling around the world tends to shift my perspective. If for only a moment, I remember our planet is round, not flat. I know it is round, sure, but I often live as if the world is flat. For example, I tend to think of streets as parallel and perpendicular lines. (Surprising, since I live in Austin, where all roads seem to merge into one.)
There is nothing wrong per se with operating as though the world is flat. Mental constructs help us filter out extraneous information. I don’t need to consider the roundness of the Earth while I am walking down the street. In fact, I tend to run into things while I am simultaneously thinking about terrestrial geometry and walking. But, if I am on an international flight, having the roundness of the world loaded into memory can be useful.
The Earth is not a sphere. All the mountains and valleys muck it up, not to mention tidal forces from our moon stretching the globe like a kickball kicked a bit off kilter. Still, a sphere is not a bad construct for this planet of ours.
If the seemingly parallel streets of ours follow great circles, then they are not parallel at all. Think beach ball. The streets would meet. Twice. On opposite ends of the earth. Not necessarily at the north and south poles, but the two intersections would be diametrically opposed, in a geometric sense, not philosophical.
So, here comes the kicker.
- How might you find the surface area of a spherical lune?
- How might you determine whether two lunes are congruent?
- How many tennis balls end up in Shoal Creek each year?