exupero's blog

Toroidal poles

In a previous post I worked out some math and physics regarding a planet-sized torus. Now I'd like to venture off the outer equator and head for the north polar rim. This is when I think resident scientists would start to identify major differences between their expectations and experimental results.

Heading north along the meridian, the noon sun would be unusually lower day after day. Given an equatorial circumference of 31,092 miles, one degree of longitude is 86 miles, meaning an expedition traveling east or west would see the sun would change its east-west position in the sky 1° for every 86 miles traveled. Anyone expecting to see the same traveling north would be surprised. Based on our chosen size and proportions, our torus has a meridional circumference of 6,218 miles, making 1° of latitude only 17 miles. A north-bound expedition would see the noon sun drop in the southern sky five times faster than expected. At a walking speed of 20 miles per day, it would drop by the width of a fist at arm's length in a little more than a week. That might not be particularly noticeable at first, but after five weeks an expedition would be around the 45th parallel and the noon sun would peak at only half its starting height.

I tend to imagine theoreticians resisting the implausible explanation that their world is toroidal and instead reaching for more spherical likelihoods, such as their world being an extremely oblate spheroid or having tremendous ice caps whose weight deforms the planet and causes it to bulge at the ice-free equator. That sort of "muffin-top hypothesis" would have support not only from the greater north-south curvature but also from temperatures dropping rapidly as one approached polar latitudes.

The illusion would be broken, however, as explorers approach 90°N. Just after the polar circle appears on the horizon, a second horizon appears behind it: the other side of the torus.

The latitude at which the second horizon appears depends on the observer's height:

Assuming a six-foot explorer and a minor radius of 990 miles, that's 89.91°N, or 1.5 miles from the polar rim. If our torus does have lower gravity than Earth, human residents might be taller, but not much. Even were they 10 feet tall, the double horizon would only appear half a mile sooner.

However, as Anders mentions in his analysis, if there are active plate tectonics on our toroidal world, plates moving toward the poles are likely to fold and produce mountains. On a world with lower gravity, they could be significantly higher than the tallest mountains on Earth. At an Appalachian peak of 6,700 feet, the double horizon would appear at 87.1°N, 50 miles from the pole. A Rocky Mountain peak of 14,400 feet makes it visible at 85.8°N, 73 miles from the pole. And Mount Everest, at 29,000 feet, would expose it at 84°N, 104 miles from the pole. On Earth, those latitudes are all north of Greenland, so beyond arctic cold, any explorers to get such a view would also have to contend with difficult terrain and a thinner atmosphere. Probably it's easier to reach 90°N at zero elevation.

How much of the far side of the torus appears from a polar rim depends on its apparent size above the horizon:

Here, H is the height of the object beyond the horizon, α is the angle between the observer and the object's base, and d is the distance from the observer to the top of the object.

(Technically, observers on one rim would see a tiny bit beyond 90°N on the opposite rim, but at planetary scales the distance is small enough to ignore, so I'll use the north rim as the point of visual contact to keep the math simple.)

We can find the "height" of the far side of the torus using the Pythagorean theorem and subtracting the torus's minor radius:

To find α, use basic trigonometry:

And to find d, use the Pythagorean theorem again:

Calculating these values using the dimensions of our torus and a six-foot observer produces an apparent size above the horizon of less than a tenth of a degree, just a sliver along the horizon. Possibly our explorers would just think it was a distant, very smooth, range of mountains.

One aspect that's hard to calculate is the view east and west on the polar rim. The rim itself is flat, so in the absence of obstructions I'd guess objects in the distance shrink toward vanishing points, but limited atmospheric visibility could mean they fade out before they shrink to nothing. Beyond a certain distance, however, land might reappear. The curve of the planet could bend it around such that line of sight passed through less air, in essence looking vertically out of the atmosphere rather than horizontally through it, similar to the way a planetary ring could be seen beyond the atmosphere. Whether line of sight across the meridional horizon passes through little enough air to see the far side probably depends on atmospheric composition and lighting. If you have thoughts on this, feel free to email me.

While only a sliver of the far side might be visible from 90°N, its size will increase as our explorers continue toward the inner equator, which we'll imagine in the next post.