Iterative Tangents

A curiosity journal of math, physics, programming, astronomy, and more.

Torus world

Torus with Earth's surface area and volume

In the previous post I supposed a torus roughly the radius of Earth, but because of my arbitrarily chosen proportions, the torus had only a fraction of Earth's surface area and volume. In this post I'll work out the dimensions that would give a torus a surface area and volume closer to Earth's.

The surface areas and volumes of Earth and a torus are

StartLayout 1st Row 1st Column upper A Subscript e a r t h 2nd Column equals 4 pi upper R Subscript e a r t h Superscript 2 Baseline 2nd Row 1st Column upper V Subscript e a r t h 2nd Column equals four thirds pi upper R Subscript e a r t h Superscript 3 Baseline 3rd Row 1st Column upper A Subscript t o r u s 2nd Column equals 4 pi upper R r 4th Row 1st Column upper V Subscript t o r u s 2nd Column equals 2 pi squared upper R r squared EndLayout

Setting the surface areas equal to each other, and the volumes equal to each other,

StartLayout 1st Row 1st Column 4 pi upper R Subscript e a r t h Superscript 2 2nd Column equals 4 pi upper R r 2nd Row 1st Column four thirds pi upper R Subscript e a r t h Superscript 3 2nd Column equals 2 pi squared upper R r squared EndLayout

gives us two equations with two unknowns, R and r. Let's solve the first equation for R:

StartLayout 1st Row 1st Column upper R 2nd Column equals StartFraction upper R Subscript e a r t h Superscript 2 Baseline Over r EndFraction EndLayout

Plug it into the second and solve for r:

StartLayout 1st Row 1st Column four thirds pi upper R Subscript e a r t h Superscript 3 2nd Column equals 2 pi squared StartFraction upper R Subscript e a r t h Superscript 2 Baseline Over r EndFraction r squared 2nd Row 1st Column StartFraction 2 Over 3 pi EndFraction upper R Subscript e a r t h 2nd Column equals r EndLayout

Now plug that back into the equation for R:

StartLayout 1st Row 1st Column upper R 2nd Column equals StartStartFraction upper R Subscript e a r t h Superscript 2 Baseline OverOver StartFraction 2 Over 3 pi EndFraction upper R Subscript e a r t h Baseline EndEndFraction 2nd Row 1st Column upper R 2nd Column equals StartFraction 3 pi Over 2 EndFraction upper R Subscript e a r t h EndLayout

Our final equations, then, for the torus's major and minor radii are

StartLayout 1st Row 1st Column upper R 2nd Column equals StartFraction 3 pi Over 2 EndFraction upper R Subscript e a r t h Baseline 2nd Row 1st Column r 2nd Column equals StartFraction 2 Over 3 pi EndFraction upper R Subscript e a r t h EndLayout

To have the same surface area and volume as Earth, our planetary toroid would have to have a major radius about five times larger than Earth and a minor radius about a fifth of Earth's. Compared to the drawing in the previous post, such a torus would look more like this (Earth's size shown at the center):

While this could be an interesting scenario to explore, I'd like to stick with a smaller, fatter torus, which I think will provide some more interesting skyscapes.