# Can there be a "center" of something infinite?

Imagine a plane that's infinitely large. Would there still be a center of it? I feel like there couldn't be one... But something tells me there should be in some way!

• See Centre (geometry) : if we have no "limits" (boundary, edges, etc) we cannot apply the defining property : equidistant from the boundary. Commented Oct 8, 2016 at 21:29
• It depends on what extra structure you have in your plane. If you have none there is no center, if you have concentric circles drawn centered at some point, or rays emanating from it, etc., then that point will be the center because your extra structure is centrally symmetric with respect to it. Commented Oct 9, 2016 at 0:40
• @MauroALLEGRANZA He did use quotes, there are other definitions of center. You are too hasty to dismiss him. Most simply, there are centers of symmetry as well as centers of measure. Some spaces themselves have limited symmetry, like conic sections, which are all bilaterally symmetric, but not translationally symmetric (like a straight line that you can move any distance in either direction and overlay itself.) So you don't need extra decoration, as noted by Conifold to make this relevant.
– user9166
Commented Oct 9, 2016 at 1:14
• @jobermark In the case of conic sections the extra decorations are the metric and the embedding in 3-space. Without them, topologically or even projectively, they have no centers either. Commented Oct 9, 2016 at 1:56
• the center is everywhere in an infinite boundless universe. A corollary to this question is what is our position in the universe in relation to the point where the big bang occurred? Answer: We are at the point where the big bang occurred. And if you go to a position 100,000,000 light years from here, you will still be at the point where the big bang occurred. Commented Oct 9, 2016 at 10:24

Your title allows for better answers than your example. And I will answer from the title question, rather than limiting myself to the plane.

You know where the centerpoint of a parabola is, even though it extends infinitely far away from there in both directions -- it is the point closest to the focus, the point of maximal curvature. It is a center in a specially defined sense which recognises a sense of the symmetry a center should confer. Similarly a cubic has the opposite kind of center, a uniquely flat point. However you rotate the curves, these central references remain obvious, and they are historically relevant enough that the ordinary sense of center is often extended to include them.

A hyperbolic paraboloid has a saddle point where its tangent space splits it into ascending and descending parts, and that makes it a centerpoint of the space in important ways, again related to symmetry rather than distance.

So in the sense of the point that everything is intuitively situated in terms of, like a city center, certain infinite spaces have centers. If our own universe is "too light", it may have a hyperhyperboloid shape, with the third-dimensional equivalent of the saddle point, and in this sense a relative center.

You can define that center as existing everywhere.

Put an x and y axis on that plane. Those axiis also extend infinitely.

There, now you have the center.

• At least a short explanation for your claim or definition of your proposed operations would be nice. Commented Oct 8, 2016 at 22:33
• You can by why would you? What is to be gained?
– user9166
Commented Oct 9, 2016 at 0:26
– user20153
Commented Oct 15, 2016 at 22:37
• "Everywhere"? Surely you mean "anywhere". The very concept of "a centre" implies the selection of a single point. You may be able to put an x and y axis on the plane at any point and call that point the centre. But then that point is the centre and nowhere else. Every point cannot simultaneously be the centre. So the centre (if there is one) can be anywhere, but not everywhere. Commented Nov 11, 2016 at 1:58

As implied by Conifold and RodolfoAP the answer depends both on the definition of centre and the nature of the space.

If we choose "centre of symmetry" as the definition then (for example) an infinite plane can have either one or an infinite number of centres, depending on its structure. The example of a series of uniform concentric rings implies a single centre. If there are identical structures at every point (m,n) (m and n both integers), there will be a centre of symmetry at every point (r/2, s/2) (r and s both integers). If the plain is continuous and of uniform density, then every (finite) point on the plain will be a centre of symmetry.

Various writers cite the universe. It is generally believed not to be repetitively cyclic, in which case it will be infinite in whatever number of dimensions is needed to allow it to be defined as a stationary entity (this would ignore the possibility of a universe corresponding to every possible outcome in the present one, which would seem to map to an infinite number of dimensions). QM appears to preclude a regularly repetitive universe, which means that our universe would appear to have a single centre, albeit the location could be uncertain to a Planck length or so.

– user2953
Commented Apr 24, 2017 at 5:03

Zero is the center of an number line reaching to infinity in the positive and negative directions (as well as the center of the infinite complex plane). It fits right in with all the other points, but it is still structurally unique.

• I respectfully disagree. The complex number -839+1732i is the centre of lines reaching infinity in all directions, it fits right in, and is unique. So, obviously that must be the plane's centre!
– ngn
Commented Feb 20, 2018 at 2:37

The definition of center is your issue. This is not philosophy or math, this is a rhetoric problem.

If center means the axis of rotation and your plane rotates, there is your center. Centers are equidistant points to sets of systems (borders, lines, weights, moving points, etc.). Then it depends.

This is like asking if you can think of nothing. It all depends on your definition of thinking. There are no absolute truths on language, where concepts are usually subjective. Only on logic and causality (except that causality has been broken on Q. Physics and we're rebuilding our concepts... Again).

the concept of infinity in mathematics, means undefine. can you define something on top of undefine? no, at least mathematically.

however, on top of something which is not only mathematics, and it is a progressing infinite object, like an object growing on the forever looping computer program, yes, you can.

for example: assuming you have defined a point as 0,0, and a program loop forever to extend both of the axis to infinity. you will achieve infinity plane, and 0,0 is your center. the same concept applied for any dimensions and thus explained any objects in the universe.

however, for something already created, and especially you are within its lower dimension, you can never find the center. as you don't even able to find the whole picture of it, for you, as civilized human, inifinity, is just undefine.

• Can you provide references to support "the concept of infinity in mathematics, means undefine [sic]"? Welcome to Philosophy.SE, by the way.
– user2953
Commented Dec 21, 2016 at 19:22
• i don't think it is meaningful to discuss infinity on mathematics. the main concept is without the whole picture you cannot find the center or medium, it is just as trivial as instinct. so reversely, if you are the person who starting an infinite event, is it very simple for your to define the center of an object (you just make sure your program extend both ways equally). Commented Dec 22, 2016 at 2:20
• So I repeat my question: can you provide references to support "the concept of infinity in mathematics, means undefine [sic]"? On this site, you cannot just write anything, it needs to have some basis in literature. Also, what is the program you are referring to and how is it relevant to the question?
– user2953
Commented Dec 22, 2016 at 5:42
• in mathematics infinity is undefine, i don't think it is something that required reference. what do you mean "it needs to have some basis in literature". do you mean philosophy is something that always from the literature? but not i thought from yourself? Commented Dec 22, 2016 at 8:33
• and for your question "what is the program". program is just a simple object. and you are talking infinite thing, its definition doesn't exist in the real world, in your brain, you can imagine you have a pen and a infinite big paper. you start to draw your object from the origin (0,0), and then extend both axis by 1 and -1, when time go to infinite, your object will be infinite, and 0,0 is your center. it is a typical progressive against static model. it is actually straight forward for me ( or you may have other philosophical consideration? but i didn't see it in your question) Commented Dec 22, 2016 at 8:36