I've occasionally entertained myself pondering the phenomena one would experience living on a toroidal worldâ€”not a ringworld encompassing a star so much as a donut in Earthlike orbit. This has been explored before, but rather than viewing the situation as a whole, I'd like to look more from ground level, at aspects closer to daily life. In this series of posts I'll walk through the various aspects of toroidal life I've thought about, and I invite you to email me with additional thoughts.

As worked out in a previous post, the distance to the horizon depends on a planetary body's radius. More precisely, it depends on the surface's local curvature. On a sphere, local curvature is the same everywhere; on a torus, it depends where you're standing and what direction you're facing. On the outer equator of a torus, the curvature facing east or west is much less than the curvature facing north or south, meaning the horizon along a meridian is much closer than the horizon along the equator. I've often thought of this as an elliptical horizon, but that's only an approximation at low observation angles, since as an observer's altitude approaches infinity the horizon looks increasingly like the profile of a torus, that is, a rectangle with semicircular ends.

What effect an unusually shaped horizon would have on daily life, I'm not sure. Most of the time we don't see the horizon due to trees, buildings, and other obstructions. One place we do see the horizon is on large bodies of water, so perhaps a toroidal planet would generate some interesting naval strategies, such as pursuers staying east or west of their targets to keep them in sight, while pursuees try to dodge north or south to disappear over the closer horizon.

Given that the ancient Greeks calculated the size of the Earth with primitive technology and basic math, how might a Bronze Age civilization discover they lived on a torus? Calculating the radius of a meridian could be done as Eratosthenes did it, by comparing shadows at different latitudes at the same time of day, but finding the radius of the equator would be more challenging. When measuring shadows along the meridian, the height of the sun in the sky synchronizes measurements. To synchronize measurements along a line of latitude requires either an accurate timepiece or a way to communicate instantly over long distances. While less accurate, probably the easier approach would be to measure the distance between a very tall object, such as a great pyramid, and where its top disappears over the horizon. Such a monument would also make a toroidal world's unusual curvature apparent to laymen, who could test the claim of differing radii with a straight walk north and a straight walk east, seeing how long they had to walk before the peak disappeared over the horizon, no math (and possible sophistry) needed.

In the next post we'll pick some arbitrary parameters for a toroidal world.