How would you go about mapping the inside surface of a torus (a doughnut shape as seen in the thumbnail image)? The question comes up now and then in forums but I’ve never seen an answer so I decided to take a crack at it.

If the radius from the center of the torus to the center of the ring portion is enormous compared to the radius of the ring portion itself, an immensely long thin rectangle would suffice. Imagine that the earth, traveling along its orbit, were describing the inside of a circular tube. That tube would be the ring section of the torus. With those ratios (4,000 mile radius of earth vs. 92 million mile radius of orbit), the difference between a true representation and a simple rectangle would be negligible.

### A Simpler Model

I want to look at a more typical torus, with much more “doughnut-like” relative dimensions. I’ve created an image of an unfolded ring section. Imagine taking a doughnut, cutting it into eight equal sections, removing the insides, and unrolling the “skin”. If you did that, it would look something like this.

Actually, instead of perfectly straight linear changes (from top to middle and from middle to bottom), there would be a very slight sinusoidal component. It’s so small that I just didn’t think it was worth the effort to reproduce.

This section makes up one-eighth of the ring. Imagine rolling the image, so that the top and bottom come together and meet at the outside edge of the ring (2,000 miles wide). The middle of the image is the inside edge of the ring (1,000 miles wide). The distance from the top to the bottom is the circumference of a section of the ring (shown as the red line in the image below) and is 4,000 miles around.

If you start at the outside edge of the ring, walk along its surface to the inside edge of the ring, and continue on until you reach the outside edge of the ring again (i.e.– follow the red line), you will have walked 4,000 miles. If you instead walk the length of the ring (one possible path is shown by the pink line in the image below), the distance you walk will depend on your path. If you walk along the inside edge of the ring, you will travel 8,000 miles before you return to your starting point (each section’s inside edge is 1,000 miles wide and there are eight sections). If you walk along the outside edge of the ring, you will travel twice as far to reach your starting point (2,000 miles per section times 8 sections).

### The Completed Map

If you take the section shown earlier (which is one-eighth of the ring) and combine it with seven more sections, you will get this resulting map of the entire inside surface of the torus. The surface area of this model is 1,500 * 8 * 4,000 = 48,000,000 square miles. The surface area of the earth is roughly 200 million square miles. Since 4/5 of that is water, the earth has roughly 40 million square miles of land.

Just to be clear, the white areas are not gaps in the map. The map edges on one side connect to the map edges on the other side. It just appears that there are gaps due to laying it out flat.

### Won’t You Fall Off?

When I was in college, I asked my physics professor whether or not it was possible to live on the inside surface of a dyson sphere, whose radius was the distance of the earth from the sun. Being a great guy, he stuck around and walked me through the problem. As it turns out, it is not only possible but it only requires that the sphere possess a very modest mass (on the order of a thickness of a few meters of steel)! Given that, I am willing to accept that the same is true of a torus without actually going through the math. If you want to verify it though, look into point charges. I’ll warn you though that it will get fairly involved.

### Alternate Projections

The problem that I have with this type of projection is that, when laid out flat like this, north-south lines are skewed near the “blank” areas (those angled lines all run precisely north-south). There is no way around this if distances are to remain consistent (where a measured area represents the same distance at all points on the map). Increasing the number of sections (from 8 to 16 or more) would reduce the north-south skewing but, imo, dividing the map into more sections reduces the readability.

If a simple rectangle is used instead, there will be no north-south skewing. However, the entire map will be severely distorted, starting at the outside edge of the ring and becoming most pronounced at the inside edge (a distance on the map that represents 100 miles at the outside edge of the ring would represent 50 miles at the inside edge).

Unfortunately, that is the nature of displaying curved surfaces on a plane. I greatly prefer the accuracy of the method I described above. But the rectangular map, although distorted, has the benefit of greater readability and simplicity.

### Where Would You Use This?

I’m not suggesting that there is a giant torus floating in space. Clearly, this is grounded in the world of fantasy. Even there, it’s pretty fantastic. However, I think this is a perfectly reasonable geometry for a demi-plane, pocket universe, or whatever you wish to call it. I think it might be fun, if you care about such things, to work out how lighting and weather work in such a place.

Edit:

### Larry Niven’s Ringworld

Torus-shaped ringworlds have been written about for quite a few decades. More recently (in 1970), Larry Niven wrote a book called Ringworld. I have not read it but all sources report that it is amazing. His ringworld was not a torus. Instead it was a flat surface surrounding the sun (Dyson strip perhaps?). In any case, I have received numerous emails informing me that a ringworld is a flat strip. I’m sure Mr. Niven would be pleased to see all the support, but he is, in fact, a latecomer to the term. The most readily available example, although on a much smaller scale, is most any space station from early science fiction. The most common geometry for these stations was a great wheel (a torus with spokes).

For those that are interested, mapping Larry Niven’s Ringworld would be far more straightforward. If my understanding is correct, his ringworld was a perfectly flat strip, facing directly inward toward the sun. As such, it would be best demonstrated as an enormously long thin rectangle. There would be no significant distortion of any sort. If this strip were located 1AU from the sun, it would be roughly 1.95 billion miles long. That’s gonna take a lot of mapping!