experimenters finally proved him right. The neutrino, though notoriously one of the most elusive particles in nature, was indeed real.

Einstein went back and forth throughout his career on whether or not extra dimensions were real. This indecision related to his mixed feelings about the role of experimentation in physics. Philosophically, he had one foot firmly in each of two camps. He often argued that experimentation was needed to establish any proposition. That’s why he breathed easier once the Mercury precession and light-bending measurements seemed to confirm general relativity. On the other hand, he spent much of his later years trying to use his own intuition to surmise the deep mathematical principles underlying reality. At least to the outside world, these musings seemed to have little to do with what was experimentally known at the time.

Einstein’s propositions that the universe is shaped like a hypersphere, and that a cosmological constant is needed to bolster it from either expansion or collapse, could not be tested for many years. Only in recent times have astronomers been able to map the likely shape of the cosmos and consider the likelihood of an antigravity term. Nevertheless, just by bringing up these issues Einstein ushered in a new age for cosmology. For the first time, science addressed the possibility that space itself has an overall shape.

A sphere is not the only way a surface can be curved. Saddles, for example, are often curved one way on the bottom, to accommodate the horse, and another way on the top, to provide comfort to the rider. Similarly, three-dimensional spaces can curve several ways into a higher dimension while preserving constant curvature. Besides a hypersphere (known as closed or positively curved), spaces can be saddled-shaped hyperboloids (known as open or negatively curved). The third possibility is for the space to be completely flat (known as zero curvature).

In 1922, Russian mathematician Alexander Friedmann explored each of these geometric possibilities for the universe. In the absence of a cosmological constant, he found that they corresponded to three



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