How best to measure the shape of the universe? One crucial insight is that geometry depends on scale. At large scales, the Earth is clearly spherical. At small scales, it is hard to distinguish a region of the Earth's surface from a plane. This is why we can draw faithful scale maps of a city or even a country. Maps of larger regions of the globe, on the other hand, will always be distorted. The same holds true for the cosmos at large. To make sensitive measurements of cosmic geometry, we should consider the largest triangles possible.

How do we find a triangle of sufficient size? When it comes to huge distances, there's no place like outer space. It would be ideal if we could construct the triangle shown on the right for some astronomical object, and if we knew that object's size (at right angles to our line of sight), and its distance from us. The angle α is readily determined by astronomical observation. It corresponds to  the object's apparent size in the night sky.

These angles and lengths, highlighted in green in the image to the right, would be sufficient to test the geometry of space.

It is not easy to find an object or structure in deep space with the required properties — a structure where we know both size and distance. Also, there is a trade-off: typical astronomical objects are very, very small, compared with their distance from us. In other words: while we can make two sides of the triangle very long, the third side will be comparatively small — resulting in very inaccurate tests of cosmic geometry.

In order to find a suitable triangle, we need to go to much greater distances than those of the farthest galaxies or galaxy clusters. The key idea to keep in mind is that, as we look out at distant objects, we look into the past. For instance, you never see the Sun as it is at the very moment of observation. You will always see the Sun (as in the image on the left) as it was eight minutes ago, since that is the time it takes light from the Sun to reach Earth.

Since the mid-1920's, astronomers know that our universe is expanding: as time passes, the average distances between galaxies increase further and further. Conversely, the further we look back into the past, the closer all galaxies will have been closer together. At some point in the pass, all matter will have been in such close proximity that there were no individual stars or galaxy. The cosmos was filled with a hot gas. Go back even further, and the average temperature in the cosmos will have been so great that atoms cannot exist. Instead, the cosmos was filled with a plasma, a swirling mixture of electrons and atomic nuclei.

Here's the catch: plasma is opaque. When we look at the night sky, we cannot look arbitrarily far into the distance or, equivalently, arbitrarily far into the past. We cannot see further than the primordial plasma, just as we cannot see through a wall. By now, the images of this plasma in the night sky — known as the cosmic background radiation — are very faint. But they can be detected with specialized satellite telescopes such as the NASA probes COBE or, more recently WMAP (pictured on the left).

The clue: the cosmic background radiation bears the minute traces of regions of different density in the primordial plasma. And using the basic physics of waves, it is possible to estimate those traces' actual size. (This is a story of its own, which you can follow in our Explorations.)

From other astronomical observations (and some more subtle properties of the cosmic background radiation), cosmologists can also reconstruct how long ago the "echo of the big bang" was emitted — equivalently: how far away it is. And current observations yield the fluctuation's apparent size in the night sky. With the fluctuation's size, apparent size, and distance known, we have our cosmic triangle.