One hundred years ago, Einstein re-envisioned space and time as a rippling, twisting, flexible fabric called spacetime. His theory of general relativity showed how matter and energy change the shape of this fabric. One might expect, therefore, that the fabric of the universe, strewn with stars, galaxies, and clouds of particles, would be like a college student’s dorm room: a mess of rumpled, crumpled garments. Indeed, if you look at the universe on the scale of stars, galaxies, and even galaxy clusters, you’ll find it puckered and furrowed by the gravity of massive objects. But take the wider view—the cosmologists’ view, which encompasses the entire visible universe—and the fabric of the universe is remarkably smooth and even, no matter which direction you turn. Look up, down, left, or right and count up the galaxies you see: you’ll find it’s roughly the same from every angle. The cosmic microwave background, the cooled-down relic of radiation from the early universe, demonstrates the same remarkable evenness on the very largest scale.

Physicists call a universe that appears roughly similar in all directions “isotropic.” Because the geometry of spacetime is shaped by the distribution of matter and energy, an isotropic universe must posses a geometric structure that looks the same in all directions as well. The only three such possibilities for three-dimensional spaces are positively curved (the surface of a hypersphere, like a beach ball but in a higher dimension), negatively curved (the surface of a hyperboloid, shaped like a saddle or potato chip), or flat. Russian physicist Alexander Friedmann, Belgian cleric and mathematician Georges Lemaître and others incorporated these three geometries into some of the first cosmological solutions of Einstein’s equations. (By solutions, we mean mathematical descriptions of how the three spatial dimensions of the universe behave over time, given the type of geometry and the distribution of matter and energy.) Supplemented by the work of American physicist Howard Robertson and British mathematician Arthur Walker, this class of isotropic solutions has become the standard for descriptions of the universe in the Big Bang theory. However, in 1921 Edward Kasner—best known for his coining of the term “Googol” for the number 1 followed by 100 zeroes—demonstrated that there was another class of solutions to Einstein’s equations: anisotropic, or “lopsided,” solutions. Known as the Kasner solutions, these cosmic models describe a universe that expands in two directions while contracting in the third. That is clearly not the case with the actual universe, which has grown over time in all three directions. But the Kasner solutions become more intriguing when you apply them to a kind of theory called a Kaluza-Klein model, in which there are unseen extra dimensions beyond space and time. Thus space could theoretically have three expanding dimensions and a fourth, hidden, contracting dimension. Physicists Alan Chodos and Steven Detweiler explored this concept in their paper “ Where has the fifth dimension gone? ” Kasner’s is far from the only anisotropic model of the universe. In 1951, physicist Abraham Taub applied the shape-shifting mathematics of Italian mathematician Luigi Bianchi to general relativity and revealed even more baroque classes of anisotropic solutions that expand, contract or pulsate differently in various directions. The most complex of these, categorized as Bianchi type-IX, turned out to have chaotic properties and was dubbed by physicist Charles Misner the “Mixmaster Universe” for its resemblance to the whirling, twirling kitchen appliance.

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Like a cake rising in a tray, while bubbling and quivering on the sides, the Mixmaster Universe expands and contracts, first in one dimension and then in another, while a third dimension just keeps expanding. Each oscillation is called a Kasner epoch. But then, after a certain number of pulses, the direction of pure expansion abruptly switches. The formerly uniformly expanding dimension starts pulsating, and one of those formerly pulsating starts uniformly expanding. It is as if the rising cake were suddenly turned on its side and another direction started rising instead, while the other directions, including the one that was previously rising, just bubbled. One of the weird things about the Mixmaster Universe is that if you tabulate the number of Kasner epochs in each era, before the behavior switches, it appears as random as a dice roll. For example, the universe might oscillate in two directions five times, switch, oscillate in two other directions 17 times, switch again, pulsate another way twice, and so forth—without a clear pattern. While the solution stems from deterministic general relativity, it seems unpredictable. This is called deterministic chaos. Could the early moments of the universe have been chaotic, and then somehow regularized over time, like a smoothed-out pudding? Misner initially thought so, until he realized that the Mixmaster Universe couldn’t smooth out on its own. However, it could have started out “lopsided,” then been stretched out during an era of ultra-rapid expansion called inflation until its irregularities were lost from sight. As cosmologists have collected data from instruments such as the Hubble Space Telescope, Planck Satellite, and WMAP satellite (now retired), the bulk of the evidence supports the idea that our universe is indeed isotropic. But a minority of researchers have used measurements of the velocities of galaxies and other observations, such as an odd line up of temperature fluctuations in the cosmic microwave background dubbed the “ Axil of Evil ” to assert that the universe could be slightly irregular after all. For example, starting in 2008, Alexander Kashlinsky, a researcher at NASA’s Goddard Space Flight Center, and his colleagues have statistically analyzed cosmic microwave background data gathered by first the WMAP satellite and the Planck satellite to show that, in addition to their motion due to cosmic expansion, many galaxy clusters seem to be heading toward a particular direction on the sky. He dubbed this phenomenon “dark flow,” and suggested that it is evidence of a previously-unseen cosmic anisotropy known as a “tilt.” Although the mainstream astronomical community has disputed Kashlinsky’s conclusion, he has continued to gather statistical evidence for dark flow and the idea of tilted universes. Whether or not the universe really is “lopsided,” it is intriguing to study the rich range of solutions of Einstein’s general theory of relativity. Even if the preponderance of evidence today points to cosmic regularity, who knows when a new discovery might call that into question, and compel cosmologists to dust off alternative ideas. Such is the extraordinary flexibility of Einstein’s masterful theory: a century after its publication, physicists are still exploring its possibilities. Go Deeper

Editor’s picks for further reading Ars Technica: Is the lopsided Universe telling us we need new theories?

Science writer Matthew Francis asks whether the “axis of evil” reveals anything meaningful about the fundamental laws of physics. Ask an Astronomer: What do “homogeneity” and “isotropy” mean?

A brief introduction to what astronomers mean when they talk about homogeneity and isotropy. New Scientist: Planck shows almost perfect cosmos – plus axis of evil

A summary of conclusions from the Planck mission, which created the highest resolution map of the entire cosmic microwave background radiation.

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