Using the normal Euclidean geometry we learned in high school, it seems to me that there are an infinite number of lines that pass through the center of any given two-dimensional circle. I would also think that there are an infinite number of lines passing through the center of any given three-dimensional sphere.
Which is greater, the number of lines passing through the center of a circle or the number of lines passing through the center of a sphere or are they the same?
Both describe a point. You’re asking which is greater, the number of lines through a point, or the number of lines through a point.
I probably could have worded it better. In the two-dimensional case I was thinking about lines just in those two dimensions and in the three-dimensional case I was thinking about lines in three dimensions.
In that case, you’re comparing an infinity and an infinity of infinities. If you stipulate lines have to be a degree apart from one another, the sphere wins by a big factor. However, as it is you’re comparing infinities, and I don’t know the rules for that. Let’s wait for a mathematician.
You’re asking if you can map (theta, rho) to (theta). Yes, you interleave the decimal digits. Same cardinality.
Caveat, I’m not a mathematician, but pretty sure.
Perhaps the use of the word “number” in this sentence was a poor choice since infinity is not a number? What would be a better word, cardinality?
My degree was in Applied Mathematics (a long time ago) but I don’t consider myself one. I do seem to recall that there is a difference in infinities, and I would think the infinite lines going through the center of a sphere in 3D is greater than the infinite lines going through the center of a circle in 2D.
There’s no rigidity in my guess, though.
Again not a mathematian but I think it’s analagous to the infinite set of all reals vs all complex numbers. There is a infinite number of lines through the center of a circle, but that just represents a sub set of the lines through the center of a sphere (the ones where z=0). For each line through a circle there are an infinite number of lines through a sphere where the X and Y components are the same but Z is different.
@Snarky_Kong has it right, at least under the usual conventions for talking about infinities. Usually, questions of “which infinity is larger” are answered in terms of cardinality, the number of elements in a set. Two sets have the same cardinality if the elements of one of them can be put into a one-to-one correspondence with the elements of the other.
The set of all lines through a point in 3 dimensions can be labeled with an ordered pair of real numbers, which you can think of as the latitude and longitude of the point on the sphere where the line passes through it. The set of all ordered pairs in 2 dimensions can be labeled with a single real number, the angle of the line. So basically, what you’re asking is if the set of ordered pairs of real numbers is larger than the set of real numbers, or in other words, if it’s possible to assign a different real number to each ordered pair of real numbers.
And the answer is yes. You can construct such a number by, as @Snarky_Kong says, by interleaving the digits. So for instance, if you have the ordered pair (0.31415926…, 0.27182818…), you can construct the number 0.3217411852982168… to correspond to it.
At least, that’s the gist of it. There are a few fine details to work out, like the fact that the poles of a sphere exist at every latitude, and that some real numbers have two different decimal representations (the 1 vs. 0.999… problem), etc., but none of those change the final answer, that the two infinities are the same size.
Note that not all infinities are the same size, though. For instance, the set of all integers is also infinite, but it’s a smaller infinity than the set of all reals. And the set of all functions from real numbers to real numbers is a larger infinity than the set of all reals.
I agree with @Chronos.
The OP’s question is similar to the question of whether there are more points on a sphere than there are on a circle (or, in general, more points on a higher-dimensional object than on a lower-dimensional object). And, counterintuitive as it may be, higher dimension does not imply higher cardinality.
I think George Cantor proved that some infinite sets are larger than others. if so this would imply that the infinite set of points on a sphere is larger than the infinite set of points on a circle.
But I’m no theoretical mathematician
The cardinality of the reals is larger than the cardinality of the integers, but that says nothing about the surface of a sphere (R^2) vs the surface of a circle (R).
I am not a mathematician; I am not even an arithmetician. But if you can pass an infinite number of lines through a point, doesn’t that mean you can always pass another one through? And if you can always pass more lines through, what does it matter if you’re passing them along a single plane or through a sphere?
This is true. Here’s one explanation of how he did so that I like:
This is false.
For one infinite set of points to be shown to be larger than another, in the sense that Cantor was dealing with, one would have to show that it is impossible to match them up one-to-one—that no matter how one tried to match them up, there would always be points from the larger set left over.
Reasoning about infinities can be tricky and counterintuitive, so it’s necessary to make sure your arguments are rigorous. You can’t get consistent results by just hand-waving about “some infinities are larger than others, therefore this infinity must be larger than that”
Cantor defines two sets as having the same cardinality if there is a bijection (one-to-one correspondence) between them. That’s the only rule. So to determine that two sets have the same cardinality (intuitively, are the same size), it’s sufficient to construct a bijection between them. To show that two sets do not have the same cardinality, it’s sufficient to prove that there can be no bijection between them. This is typically harder than constructing a bijection.
I claim that the set of points on a circle has the same cardinality as the set of points on a sphere. Proof: by Chronos’s construction above. A point on a circle can be identified by one real number A between 0 and 1, and a point on a sphere can be represented by two real numbers B and C between 0 and 1. Let 0.a_1 a_2 a_3 \dots be the decimal representation of A, and similarly for B = 0.b_1 b_2 b_3 \dots and C = 0.c_1 c_2 c_3 \dots. Then define the function f(B,C) = 0.b_1 c_1 b_2 c_2 b_3 c_3 \dots. This is almost a bijection; it’s flawed by the fact that some real numbers have two representations, one ending in all zeros and one ending in all nines. With a little fiddling this can be patched up and made into a true bijection.
There are similar arguments to prove that the set of points in a line of any size has the same cardinality as the set of points in the whole real line, which is the same as that of the set of points in the plane, the same as the set of points in the whole 3-D space R^3, and the same as the set of points in a space of any dimension R^n. However this cardinality is different (and larger than) the cardinality of the set of all integers, and is different (and smaller than) the cardinality of the set of all subsets of the points in a line. This can all be rigorously proved by arguments about possible bijections.
When dealing with infinities, you aren’t dealing with things but with carefully defined mathematical objects. Differently defined objects can behave differently.
The positive integers are infinite; you can always add one to whatever your highest number is. But you can’t put things between integers. Nothing is between 3 and 4 or between 1002 and 1003.
If you use different objects, like decimal fractions, you can indeed stuff items between 2 and 3. In fact, you can stuff an infinity of decimals between integers. That results in an infinity of a higher cardinality.
The cardinality of the surface of a 2-sphere turns out to be lower than the cardinality of the surface of a 3-sphere. You might think that the surface of a 4-sphere has an even higher cardinality but it doesn’t. All spheres 3 and higher have the same cardinality. You need to understand more math than I remember to explain why.
ETA: I didn’t see markn_1’s post but I think I’m saying the same thing but using just words.
You can stuff an infinity of rational numbers between two integers, but the set of rational numbers is the same cardinality as the set of integers.
The square root of 2 has a decimal representation but is not rational. Pi has a decimal representation but is not rational. The former is part of a set of algebraic numbers; the latter part of a set of transcendental numbers. I was talking about real numbers, which contain all three sets, and has cardinality Alpha.
I don’t think so? But again, IANAM.
When you wrote “That results in an infinity of a higher cardinality,” I couldn’t tell whether you meant (or whether other people might interpret you to mean) “That necessarily results in (i.e. logically implies) an infinity of higher cardinality.” It does not; and the set of rational numbers between 2 and 3 provides a counterexample.