Issues with loss of precision are caused by bad or misapplied numerical algorithms; the choice of units (degrees versus radians) or whether the quantity being approximated is rational or irrational should not make a difference.
As for hyperspheres, there is no reason to stop at any number of dimensions: area and volume are defined in Euclidean n-space and the surface area of a unit sphere is exactly 2π[sup]n/2[/sup]/Γ(n/2). In fact, inspecting this formula shows that as the number of dimensions grows the surface area tends to zero, even though in low dimensions it looks like it’s growing. Also, despite this, the ratio of volume to surface also gets smaller (e.g., the area of a disc is 1/2 of its radius times its circumference, but for a sphere the ratio is only 1/3, and so on.) I suppose that actually using hyperradians to measure things is rare in practice, though I would love to hear about such cases.
I’m an astronomer and I have no idea what you’re talking about. When astronomers use degrees, arcminutes, arcseconds etc, etc, it’s because of tradition, the same reason the US still uses inches. We keep using it because that’s what we learned in college. As far as computational accuracy is concerned , there is no benefit to a full circle being a rational number of degrees. Units are used for measuring many different things, most of which aren’t a rational number of degrees or steradians. And even if one of your numbers is rational, you lose that advantage as soon as you actually use that number in a calculation, because the next intermediate result will not be rational.
And my colleagues and I generally use steradians for radiometric calculations. Though I’ve used square arcseconds in a few situations because that gives a result that’s easier to interpret. (For observing a small astronomical object, “10 photons per second per square arcsecond” is easier to interpret than “2e11 photons per second per steradian”.)
In other words, 360x180=64800? That’s the number of grids on the surface of the Earth where the sides are latitude and longitude lines exactly one degree apart. But these grids vary in size, getting smaller as you leave the equator. At the equator, they resemble squares. But at the poles they are extremely skinny triangles that don’t even resemble squares at all.
Try taking one such square at the equator, one degree of latitude by one degree of longitude. It’s a square 111.3 km on a side. That’s about 12,388 km2. The total area of Earth is 510.1 million km2, so that’s a ratio of about 41,160. If you could come up with a way to tile the surface of the Earth with 41,160 sectors of equal size, each one of them would be the same size as a one-degree-by-one-degree square at the equator. One option would be to adjust their north-south heights as you move away from the equator, while keeping the widths at exactly one degree of longitude. That gives sectors that resemble trapezoids, 360 of them in each row, and we’d need about 114 rows (57 up from the equator to the north pole and 57 down from the equator to the south pole) for a total of 41,040 sectors. This would cause some headaches for navigation because the tops and bottoms aren’t one degree of latitude apart. But each sector would be the same size, approximately 1 square degree.
That’s if you want to try to stick with “degrees”. The logical alternative is to switch to radians.
Imagine a piece of rope the same length as your arm. Now swing your arm in a complete circle. How many ropes would it take the cover the circle traced by your finger tip? Just over six. That’s why there’s 6.283 radians in a circle.
Imagine a large birthday square birthday cake where each side is the same length as your arm. Now swing both arms to trace a sphere. How many birthday cakes would it take to cover the sphere? Just over 12 1/2. That’s why there’s 12.566 steradians in a sphere.
I say the answer is 12.566 steradians in a sphere (or 2 times tau if you want to be precise).
Just throwing in that the concept of square degrees is commonly used in astronomy. You might say, for instance, that the constellation Orion has an area (geometrically called a solid angle, but astronomers often call it area) of 594 square degrees (out of the 129,600/π
I really think you are saying “degrees” in a sphere when you should be saying “points” in a sphere. The answer is, of course, an infinite number because, for any two points you could name, I can name a point that is mathematically between them, and I can do this ad infinitum.
Seems my post somehow got garbled, and it’s too late to edit it. Here is the complete thing:
Just throwing in that the concept of square degrees is commonly used in astronomy. You might say, for instance, that the constellation Orion has an area (geometrically called a solid angle, but astronomers often call it area) of 594 square degrees (out of the 129,600/π of the entire spherical sky, or the 64,800/π square degrees of the hemispherical sky you see above you).
We are very privileged with this. A google search tells me this is a hapex legomenon, so anyone searching in the future will have to come to the Dope to spot it in the wild.