Thought Experiments


Dante’s Universe, and Ours

When a team of astronomers in 1992 released the first full-sky map of the cosmic microwave background—also known as the afterglow of the big bang—George Smoot, one of the group’s leaders and later a Nobel laureate, said, “If you’re religious, this is like looking at God.”

Mystical undertones stir passions and risk muddying our understanding of science. But whatever one’s views, it is an intriguing coincidence that a possible key to reading Smoot’s words comes to us from none other than Dante Alighieri’s “Paradiso,” written in the early years of the 14th century. The cosmic microwave background, or CMB, shows us a slice of the universe as it looked more than 13.7 billion years ago, and the structure of that universe bears a striking resemblance to that of Dante’s heaven—at least according to some commentators. It is as if the poet had presaged some of the most striking developments of modern mathematics and cosmology six centuries before they emerged.

“Paradiso,” the third and final part of the “Divine Comedy,” narrates an allegorical journey in which Dante ascends from Earth, visits heaven, and eventually gets to behold the creator himself. First, Dante crosses a series of concentric spheres, all centered at Earth, which hold the planets, Sun, moon, and the stars. The next sphere he reaches is one that encloses the entire physical universe. As he crosses it, he steps into the spiritual realm.

The otherworld however also has a geometric structure, and it is completely symmetrical to that of the physical world, with nine concentric spheres, which are inhabited by angels and the souls of the most virtuous dead. But instead of growing ever larger, these spheres grow ever smaller. And at the center, Dante says, sits God, occupying a single point and emanating a blinding light.

Thus Dante’s entire universe—both physical and spiritual—consists of two sets of concentric spheres, one centered at Earth, the other at God. If you were to point a laser vertically up toward the sky from any point on Earth, you’d be pointing it straight at that single point where Dante places God.

In a sense, then, the successive spheres of the spiritual world enclose all of the physical spheres, Dante seems to imply, even though they get smaller and smaller as you move farther away from Earth and closer to God. Such a geometry seems impossible, and the passage has mystified commentators for centuries. In fact, these bizarrely nested spheres are both mathematically and physically possible. To discover why, we have to turn to mathematics that wouldn’t be discovered until centuries after Dante’s death.

In the geometry of our everyday experience, also known as Euclidean geometry, if we draw a sphere around us, the larger the sphere’s radius, the larger its circumference; more precisely, doubling the radius of a sphere doubles its circumference. But this is an empirical fact and not a logical necessity: there is such a thing as non-Euclidean geometry, in which it is perfectly allowable for a sphere to have a circumference that is not proportional to its radius.

Moreover, non-Euclidean geometry is not just a bizarre, abstract invention of mathematicians. In fact, Einstein showed in his theory of general relativity that the geometry of the universe itself is fundamentally non-Euclidean. This is what allows space to twist and bend like a cosmic contortionist.

The discrepancy between the real world and Euclidean geometry is tiny in ordinary situations—a satellite’s orbit around Earth, for example, may be a few inches shorter compared to what you would expect from Euclidean geometry—but becomes substantial in extreme situations such as around black holes.

Dante’s universe, then, can be interpreted as an extreme case of non-Euclidean geometry, one in which concentric spheres don’t just grow at a different pace than their diameters, but at some point they actually stop growing altogether and start shrinking instead. That’s crazy, you say. And yet, modern cosmology tells us that that’s the structure of the cosmos we actually see in our telescopes.

We can think of the observable universe as being made of concentric spheres, just like Dante’s universe. Because light travels at a finite speed, we see distant galaxies as they were in the past, at the time when they emitted the light that we now receive from them. By definition, light covers one light-year of distance every year. Thus, for example, we can picture all galaxies that we see as they were one billion years ago as residing on a sphere centered at our position and of radius one billion light-years. (These spheres are of course not solid objects, and they not absolute but relative to the observer, contrary to those in Dante’s 14th-century cosmology.)

Now, the universe we see all sprang up from a very small region of space, and has been expanding ever since. Cosmologists have placed the beginning of time at about 13.7 billion years ago. That means that our game of drawing concentric spheres cannot be pushed to an arbitrary distance. But it also has another consequence. As the radius of the spheres pushes close to that magic number of 13.7 billion and change, we are looking at smaller and smaller regions of space, despite the fact that those regions still span our entire field of view, in all directions of the sky.

In fact, when astronomers map the CMB, they are mapping a sphere that surrounds us and that is very close to that initial moment—at roughly 400,000 years after the big bang—and thus has a “radius” of around 13.7 billion light-years. But its circumference is a lot smaller than what you would expect from Euclidean geometry—more than a thousand times smaller. Spheres that are even closer to the big bang are even smaller, until our field of view converges to that single point we call the big bang. Theoretically, we could cast a laser in any direction and still aim at that single point.

One very bizarre consequence of the non-Euclidean nature of the observable universe is that distant objects appear larger than their true size. For the first 10 billion light years or so, galaxies look smaller if they are farther away, but beyond that distance they instead start taking up a larger and larger field of view in the sky, as if space itself acted like a magnifying lens. In practice, the effect is exceedingly difficult to actually observe in our telescopes, because at those distances galaxies look extremely faint. But in recent years astronomers have begun several projects to detect the magnification effect in their observations, not by looking at the apparent size of galaxies but at their spacing. To do so, they map hundreds of thousands of galaxies over a range of distances spanning many billions of light-years. “You look at where the galaxies formed, not at how big they are,” explains astronomer Tamara Davis of the University of Queensland, who participates in one such mapping effort called WiggleZ.

Of course, Dante lived five centuries before any mathematicians ever dreamed of notions of curved geometries. We may never know if his strange spheres were a mathematical premonition or esoteric symbolism or simply a colorful literary device.

Go Deeper
Editor’s picks for further reading

Non-Euclidean Geometry Online: a Guide to Resources
Mircea Pitici’s brief introduction to non-Euclidean geometry.

The Poetry of the Universe
Mathematician Robert Osserman’s volume of “math for poets.”

The World of Dante
Explore Dante’s writing with interactive maps, images, music, and more.