exupero's blog

# An Earth-sized torus

In the previous post I speculated about some characteristics of the horizon on the outer equator of a toroidal world. To make the discussion a bit more concrete, let's pick a size for our hypothetical toroid. Personally, I like the look of a torus with a 4-to-1 ratio of major radius to minor radius:

Let's give our torus a major radius equal to Earth's radius. How does the surface area compare to Earth's?

Solving for Rearth:

The surface area of a torus is

We've chosen r = R/4 and R = Rearth, so

Our toroidal world's surface area is only a quarter Earth's, which makes sense given how much top and bottom have to be cut off a sphere to make a torus like the one above.

How does volume compare?

Our toroid's volume would be less than a third Earth's.

Part of the reason to calculate volume is so we can calculate mass and gravitational pull. Of course, talking about the gravity of a toroidal world is an exercise in absurdity, since on the inner equator gravitational attraction is toward the center of mass in the (empty) middle of the toroid, meaning objects would be pulled off the inner surface. In a more full-fledged hypothetical, we'd have to imagine a countering force that keeps objects on the inner surface, but that force would also potentially lessen gravitational pull on the outer equator. For now I'll ignore all that and only calculate the gravitational pull on the outer equator.

Giving our torus an Earth-like density, its mass is

Gravitational acceleration on Earth's surface is

So gravitational acceleration on our torus's outer equator is

Due to the smaller mass and further distance from the center of gravity, downward pull on the outer equator would be less than 20% of Earth's. If the residents of this world are familiar with Newton's law of gravitation (perhaps from observations of neighboring spherical planets and their moons), and they have some expectation of their planet's Earth-like density, the lack of gravity would be very surprising. Most likely residents would speculate that they lived on a hollow sphere, in this case not an unreasonable conclusion.

However, such a theory would be quickly undermined by expeditions off the outer equator, which we'll discuss in a subsequent post.