Strange Shapes in Space

Which is a stranger place to find an interesting shape: on the north pole of Saturn or in 248 dimensions? In either case, without 21st century technology, we wouldn't be seeing anything at all.

In the first case, astronomers knew about this strange sight since the 1980s, but didn't get a clear view until Cassini hit the right angle:

A hexagonal form surrounding the north pole of Saturn.

This image reveals atmospheric activity on Saturn at a wavelength typically invisible to the human eye. Here's NASA's description:

In this image, the blue color shows high-altitude emissions from atmospheric molecules excited by charged particles smashing into the atmosphere along Saturn's powerful magnetic field lines, producing the aurora at very high altitudes in Saturn's atmosphere. The red color indicates the amount of 5-micron wavelength radiation, or heat, generated in the depths of the warm interior of Saturn that escapes the planet. Clouds blocking this light are revealed as silhouettes against the background thermal glow of the planet.

Of course, all the fancy camera equipment and spacecraft we can build couldn't give us a better perspective on symmetrical forms which exist in hundreds of dimensions. We can, however, build computers which can map such forms, and give us representations that can be displayed on a plain-old 2-D moniter. Mathematicians managed this feat recently by mapping the most complex Lie group, E₈:

E₈ Root System

So, what is it?

The American Institute of Math explains:

At the most basic level, the E₈ calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.

Lie groups come in families. The classical groups A₁, A₂, A₃, ... B₁, B₂, B₃, ... C₁, C₂, C₃, ... and D₁, D₂, D₃, ... rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G₂(shown on the right--K), F₄, E₆, E₇ and, towering above them all, E₈. E₈ is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E₈ itself is 248-dimensional!

To describe the new result requires one more level of abstraction. The ways that E₈ manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E₈. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E₈, and a precise description of the relations between them, all encoded in a matrix with 205,263,363,600 entries.

Since the mathematicians didn't know how many of those 200 billion entries would have meaningful values-or equal zero-they had no idea how long it would take. They knew it would be a big a project:

The magnitude of the E₈ calculation invites comparison with the Human Genome Project. The human genome, which contains all the genetic information of a cell, is less than a gigabyte in size. The result of the E₈ calculation, which contains all the information about E₈ and its representations, is 60 gigabytes in size. That is enough space to store 45 days of continuous music in MP3 format. While many scientific projects involve processing large amounts of data, the E₈ calculation is very different: the size of the input is comparatively small, but the answer itself is enormous, and very dense.

Sometimes, as I try to picture higher spatial dimensions (like four, rather than three) I feel woefully unequipped. Our brains barely manage to reconstruct 3-D images. Here, a computer "pictures" a 248-D image. The effect is lost on the computer, in a way... it cannot appreciate a complex, symmetrical image the same way the human mind can. Yet, the human mind, even with the aid of the computer, can still only grasp, at best, 3-D representations of the image. Even if we memorized each representation, like pieces of a puzzle, we'd have no way to construct the pieces in our perspective. Still, by studying these representations, we can get a better grasp of the way the universe is put together... even if our perspective is relatively inadequate.

Credits: Image of Saturn's north pole via NASA. Image of E₈ and Jeffery Adams depicting a representation of a G₂-type Lie group via AIM.