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The Geometry of Space Is Wired in the Brain

Do people have intuitions about spaces with unusual geometry? Probably not.

Key points

  • People have excellent spatial abilities, which helps them navigate the world.
  • Research suggests that the brain has built-in assumptions about the geometry of the world.
  • Those built-in assumptions are not changed by a small amount of experience in a world with weird geometry.

You have lots of ways of thinking about space. Some of these ways help you to judge the relative location of things over very large distances. For example, you can easily judge that New York City is east of Los Angeles just by knowing that New York State is east of California. This global strategy can cause problems with questions like whether Reno, Nevada is east or west of Los Angeles. Many people will say that Reno is east of Los Angeles, because Nevada is generally east of California. Because of the shapes of the two states, though Reno is actually a little west of Los Angeles.

Clearly, then, not all of our ways of thinking about spaces preserve the geometry of real space. But, the way we think about very large distances may be quite different than the way we think about spaces of a size that we could reasonably travel around in a short period of time. What we could think of as navigable spaces.

For example, think of the town you live in now. You’re probably pretty good at walking for a while away from your house and still knowing roughly what direction your house is from where you’re standing at that moment. That ability suggests that you can preserve some of the geometry of the space around you.

Source: From Wikimedia Commons (Artist, Cmglee)
Three different geometries that formed the basis of the worlds used in the studies described here.
Source: From Wikimedia Commons (Artist, Cmglee)

The Earth is a huge ball (more-or-less a sphere), and we live on the outer edge of it. The earth is so large, though, that the spaces we can walk (or bike) around are basically flat. As a result, we can measure spaces using Euclidean geometry (which is basically the geometry most of us learn in grade school and high school). Euclidean geometry is illustrated in the center part of the figure here.

Suppose you lived on a much smaller sphere, perhaps one that you could walk completely around in 30 steps. Then, geometry would work differently. For one thing, you could walk completely around the sphere in 30 steps, so walking straight ahead would eventually lead you back to the starting point. For another, if you walked a bit in one direction and then turned slightly and walked a bit more, pointing back to your starting point would require pointing in a different direction than if you were walking on a flat surface. You can see this in the sphere part of the figure.

A fascinating paper by Christopher Widdowson and Ranxiao Frances Wang in a 2021 issue of the journal Cognition looked at how good people are at deciding where they are in Euclidean spaces as well as two non-Euclidean spaces like the sphere I just described.

The studies were done in a virtual-reality environment in which people wore headsets and were able to traverse through space with planets serving as landmarks in the space. The space people occupied had a Euclidean geometry or one of two different non-Euclidean geometries (the “sphere” and “saddle surface” geometries in the figure).

After having a chance to explore the space, participants were instructed to take a certain number of steps in one direction, turn in a second direction, and take a certain number of additional steps. Then, they were asked to point back to the spot where they started. That is, they were completing a triangle from the starting place, to the place where they turned, and the place where they stopped.

The angle they are supposed to point back depends on the type of space that they are in. However, when the researchers analyzed the angles participants used, they found that people were systematically using angles that would have been correct only in a Euclidean space. That is, even though they spent time getting used to the “weird” non-Euclidean spaces, people’s mental maps of the virtual space obeyed the geometry of their experience on Earth.

These findings suggest that evolution has prepared humans for the environment we live in. We have excellent spatial abilities. In order to make those work effectively, though, our visual and spatial systems have to make some assumptions about the world. One of those assumptions that seems to be wired in is that the world has a Euclidean geometry.

Of course, it is possible that we could learn some new spatial abilities with a lot of practice. The adults in these studies had years of practice living in a world with Euclidean geometry. Perhaps we shouldn’t expect that a few minutes of practice in a weird world would overcome a lifetime of experience. It would be interesting to know whether more practice in a virtual world like this would help people to develop spatial abilities to navigate spaces with a different geometry.


Widdowson, C. & Wang, R.F. (2021). Human navigation in curved spaces. Cognition, 218.