"If you look for truth, you may find comfort in the end; if you look for comfort you will not get either comfort or truth only soft soap and wishful thinking to begin, and in the end, despair." -C. S. Lewis

And yet, when you search for the truth, you often find answers that butt up against your sensibilities, your preconceptions, and even your very notions of common sense. Such is the case in this week's Ask Ethan, where longtime reader and commenter MandoZink asks:

I have a question that has perplexed me for most of my life. Recently I sought out and re-read more expert explanations of the non-existence of a "center-of-the-universe". [...] But as a person with a very firm grasp of geometry (my career was drafting) I cannot conceive of a volume that would lack a central geometric point. Even if space is constantly expanding, a snapshot of the universe at any one instant would contain a center of volume. [...] I have not accepted this "no center" concept since the original and inadequate "stretching balloon surface" analogy. I would like a competent explanation that i can grasp. I do not care how in-depth it is, as long as it is not simply a "it doesn't exist" explanation.

So let's get to it: what's the deal with the geometry of the Universe, and whether (or not) there's a center?

First off, it's important to draw a distinction between what we know as "our Universe" -- or what I'll call the observable Universe -- and "the Universe", which includes everything that's ever been connected to us since the Big Bang and a lot more. When we look out into the abyss of a dark, night sky, past the Solar System, beyond the stars, gas and dust in our own galaxy, out beyond our local galactic neighbors and into the vast void of what seems like eternity, that's us peering at our observable Universe.

Image credit: NASA, European Space Agency / Wikimedia Commons user Noodle snacks.

We can look beyond the farthest observable galaxy, back to before the first stars formed, all the way back to the Universe's dark ages. Before even that, some 13.8 billion years ago, when the Universe was smaller, hotter and denser, it was too hot to form neutral atoms, and so it's this location that we see the relic cosmic microwave background (CMB) radiation. It's just a short 380,000 years before that to the moment of the Big Bang, or the creation of the hot, dense, expanding and cooling state that describes our Universe and all the matter-and-radiation in it.

Image credit: NASA, ESA and A. Felid (STScI).

That's our observable Universe, as visible from our location in space. The part that we can see is -- to an excellent approximation -- spherical, and about 46 billion light-years in radius, centered on us.

But that doesn't mean we're at the center of the Universe. Quite to the contrary, when we talk about the Universe, we have to talk about it in the context of the physical theory that describes it on its largest scales: general relativity. And relativity tells us that space isn't flat, like a 3D grid, but rather curved by the presence of matter and energy.

Image credit: Christopher Vitale of Networkologies and the Pratt Institute.

On the largest scales -- as in, on a global scale -- there's also the opportunity for the Universe to have curvature, depending on all the relative components of the different types of matter and energy and the Universe's expansion rate. It could either be negatively curved -- where the angles connecting any three points add up to less than 180 degrees -- positively curved -- where they add up to more than 180 degrees -- or flat, like the 3D analogue of a piece of paper, where the angles add up to exactly 180 degrees.

These curvatures also cause two lines to either converge, diverge, or remain at the same relative angle to one another as they proceed ever farther away.

Image credit: Shashi M. Kanbur at SUNY Oswego.

Based on the best observations of our Universe coupled with our understanding of the physics governing it, we've not only been able to conclude that the Universe is flat to within our best error bars (and they're only on the order of 1%), but that the size of the unobservable Universe is somewhere on the order of at least 150 times bigger than the part we can observe. The best observations leading us to this conclusion come from the CMB.

Image credit: Smoot Cosmology Group / Lawrence Berkeley Labs.

The way we see the Universe at our location in space, assuming our physical understanding of the Universe is valid, is no different on the largest scales than any other observer at any other location would see it.

Yes, there'd be differences on small scales, as far as how grouped together galaxies were and local distributions and flows of matter. Some directions will have great clusters of galaxies while others will have great voids; some spots in the CMB will be hotter or colder in some directions than average. And these particulars will be inconsistent for observers located at different points in space.

Image credit: The Hercules Galaxy Cluster, by Russ Croman / RC Optical Systems.

But on the largest scales, any observer would find a Universe that looked roughly like the map you see, below. While the portion that any observer can see is always going to be a sphere (of radius defined by the light-travel-distance since the Big Bang) centered on that point, space -- in all three dimensions -- just appears to go on and on, with more Universe in whatever direction you try and perceive.

Image credit: V. Springel et al., MPA Garching, and the Millenium Simulation.

But what about the whole "center" thing?

That depends on how speculative you're willing to get. On the one hand, the structure of space does have an overall shape on the largest scales, dependent on its curvature. If it's positively curved, it's shaped like a sphere, and will close in on itself. If it's completely flat, it could still be shaped like a torus, or a cylinder that loops around and reconnects with itself, like a donut. Or if it's negatively curved, it could be shaped like a saddle on small scales, but even these could close in on themselves in a complicated way.

Image credit: NASA / WMAP science team / Gary Hinshaw.

In other words, regardless of the Universe's curvature, it could be either finite or infinite. And if it is finite, you'd be tempted to ask about the center. After all, if you imagine spacetime is a sphere, you'd say that the center of the sphere is the center. If spacetime is a donut, you'd say the center of the hole is the center.

But there's a problem with your reasoning, tempting though it may be.

Image credit: Wikimedia Commons user Jhausauer.

You are thinking in three dimensions. And who can blame you, really? The problem is, all of these surfaces -- the surface of a sphere; the surface of a donut; the surface of a saddle -- are all two dimensional. And you are finding the center by going into the third dimension.

But space is three dimensional. Rather than the Universe expanding like the surface a rubber sheet (or balloon) getting stretched, it's expanding like a loaf of baking bread (in a zero-gravity oven), expanding in all three dimensions. When you're thinking of a sphere, a donut, or a saddle, you're thinking of two-dimensional objects, taking one dimension of space away. I can't blame you; visualizing an extra dimension is pretty hard! If the Universe actually turns out to be finite, and actually has a center, that center would exist in higher dimensional space!

The surface-of-the-circle on the left is a one-dimensional line. It has a center in two dimensions, but if all you can do is perceive and walk along the circumference of the circle, asking "where is its center?" is a question that doesn't make sense.

The surface-of-the-sphere on the right is a two-dimensional plane. It has a center in three dimensions, but if all you can do is perceive and walk along the surface of the sphere, asking "where is its center?" is a question that doesn't make sense.

And then there's what we know as space. Our three-dimensional space.

It's totally plausible that this is the surface of some higher-dimensional structure, and that structure actually has a center. But if it does, it's completely beyond our ability to observe it. Moreover, asking where is the center of our Universe is a question that doesn't make sense, the same way asking where is the center of a circle's circumference doesn't make sense.

And that is why the Universe doesn't have a center!

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