G2 (mathematics)
Based on Wikipedia: G2 (mathematics)
In 1887, a German mathematician named Wilhelm Killing sat down to write a letter that would announce the discovery of something genuinely strange. He had found a mathematical structure that didn't fit the patterns anyone expected—a fourteen-dimensional object with symmetries so unusual that it would take decades for mathematicians to fully understand what he had stumbled upon. He called it G₂, and it remains one of the most exotic creatures in the mathematical bestiary.
This has nothing to do with the "G2" you might encounter in discussions of geopolitics, where the term refers to a proposed partnership between the United States and China as co-managers of global affairs. That G2 is about two nations. The mathematical G₂ is about something far more abstract: a particular way that symmetry can organize itself in higher dimensions.
What Is a Lie Group, and Why Should You Care?
To understand G₂, you first need to understand what mathematicians mean by symmetry. When you rotate a square by ninety degrees, it looks the same. That's a symmetry. The collection of all such rotations—ninety degrees, one hundred eighty degrees, two hundred seventy degrees, and back to the start—forms what mathematicians call a group.
But some symmetries are continuous rather than discrete. Think about rotating a circle. You can rotate it by any angle whatsoever, not just specific increments. The collection of all possible rotations forms a continuous group, and when that group has certain smoothness properties, mathematicians call it a Lie group (pronounced "Lee," after the Norwegian mathematician Sophus Lie).
Lie groups turn out to be everywhere in physics. The standard model of particle physics is built on them. The ways that forces transform, the relationships between different particles, the very structure of spacetime—all of these involve Lie groups. Understanding these mathematical objects isn't just an abstract exercise; it's essential for describing how the universe works.
The Classification Problem
In the late nineteenth century, mathematicians undertook an ambitious project: classify all the simple Lie groups. "Simple" here is a technical term meaning that the group can't be broken down into smaller pieces in certain ways—think of it as being mathematically indivisible, like a prime number.
What they found was surprising. Most simple Lie groups fall into four infinite families, labeled A, B, C, and D. These are like the regular patterns you'd expect—the mathematical equivalent of finding that most crystals have predictable, repeating structures.
But then there were five exceptions.
These five didn't fit any pattern. They just existed, strange and solitary, like platypuses in the animal kingdom. Mathematicians named them G₂, F₄, E₆, E₇, and E₈. They're called the exceptional Lie groups, and G₂ is the smallest of them.
The Octonion Connection
Here's where things get genuinely weird. You probably know about the real numbers—the ordinary numbers we use every day. You might know about complex numbers, which add an "imaginary" unit that, when squared, gives negative one. Complex numbers form a two-dimensional system.
There's another step up: the quaternions, discovered in 1843 by William Rowan Hamilton in a famous flash of insight while walking across a bridge in Dublin. Quaternions are four-dimensional numbers with three different imaginary units. They're strange—multiplication isn't commutative anymore, meaning that multiplying A by B gives a different result than multiplying B by A. But they're incredibly useful for describing rotations in three dimensions, which is why video game engines and aerospace engineers use them constantly.
Then there are the octonions. Eight-dimensional numbers. Four imaginary units weren't enough, so we have seven. And here's the truly strange part: octonion multiplication isn't even associative. That means (A × B) × C can give a different answer than A × (B × C). This breaks one of the most basic rules of arithmetic.
G₂ is the symmetry group of the octonions.
More precisely, G₂ consists of all the ways you can transform the octonions while preserving their multiplicative structure. It's the collection of all the symmetries that these exotic eight-dimensional numbers possess. The fact that such a strange number system has such a specific, fourteen-dimensional symmetry group is not obvious at all—it's something that had to be discovered and proved.
Rolling Balls and Hidden Geometries
In 1893, both Élie Cartan and Friedrich Engel independently noticed something remarkable. They were studying certain geometric structures called distributions—imagine at every point in a five-dimensional space, you mark out a two-dimensional plane passing through that point, and these planes vary smoothly as you move around. The symmetries of a particular distribution turned out to be exactly G₂.
Years later, mathematicians realized this wasn't just abstract geometry. It describes a ball rolling on another ball.
Picture a small ball rolling on the surface of a larger ball, three times its radius. The small ball can roll in any direction, but it must roll without slipping and without twisting—pure rolling contact. The space of all possible positions and orientations of the rolling ball is five-dimensional. The constraints of rolling without slipping give you a two-dimensional distribution at each point. And the symmetries of this mechanical system are exactly G₂.
This connection between an abstract exceptional Lie group and a simple physical setup—two balls rolling on each other—is the kind of unexpected bridge that makes mathematics feel like discovery rather than invention.
The Technical Details
G₂ has dimension fourteen and rank two. Dimension tells you how many parameters you need to specify an element of the group. Rank is more subtle—it's the dimension of the largest commutative subgroup, which determines a lot about the group's structure.
The group comes in three flavors: a complex form, a compact real form, and a split real form. The compact form is the one most directly connected to the octonions. It can also be described as a subgroup of SO(7)—the rotation group in seven dimensions—consisting of those rotations that preserve a particular vector in an eight-dimensional space called the spinor representation.
If you've encountered spinors in physics, you know they're the mathematical objects that describe particles with half-integer spin, like electrons. The fact that G₂ shows up inside the rotation group acting on spinors hints at deeper connections to physics that mathematicians and physicists are still exploring.
The Dynkin Diagram
Mathematicians have developed a beautiful visual language for describing simple Lie groups: Dynkin diagrams. Each simple Lie group corresponds to a specific diagram made of dots and lines. The four infinite families have predictable patterns. The five exceptional groups have their own quirky diagrams.
The diagram for G₂ consists of just two dots connected by a triple line with an arrow. That triple line indicates a specific ratio between the lengths of certain vectors called roots, which encode the group's structure. It's the only simple Lie group whose Dynkin diagram has a triple line.
The Weyl group of G₂—a discrete group that captures some of the structure—is the dihedral group of order twelve, which is the same as the symmetry group of a regular hexagon. This hexagonal symmetry shows up repeatedly in the theory of G₂.
G₂ in Modern Mathematics and Physics
G₂ isn't just a curiosity. It plays important roles in contemporary mathematics and theoretical physics.
In differential geometry, there's a concept called holonomy—roughly, what happens to a vector when you parallel transport it around a closed loop on a curved surface. The possible holonomy groups are constrained, and G₂ is one of the special possibilities. Manifolds with G₂ holonomy, called G₂-manifolds, have exactly the right properties to serve as extra dimensions in certain versions of string theory and M-theory.
Finding explicit examples of G₂-manifolds is hard. The first complete examples were constructed by Robert Bryant in 1987, a century after Killing's initial discovery. These seven-dimensional spaces have special geometric properties that make them important in theoretical physics, where extra dimensions beyond our familiar four (three of space, one of time) are needed to make the mathematics work out.
Finite Groups of Type G₂
There's another direction to take this: instead of working over the real or complex numbers, you can ask what happens if you use finite fields—number systems with only finitely many elements.
The mathematician Leonard Eugene Dickson did exactly this in the early 1900s. The resulting finite groups, denoted G₂(q) where q is the size of the field, form an infinite family of finite simple groups. These are important building blocks in the classification of finite simple groups, one of the major achievements of twentieth-century mathematics.
The order—meaning the number of elements—of G₂(q) is q⁶(q⁶ − 1)(q² − 1). For most values of q, this is a simple group. There's also a twisted version, discovered by Rimhak Ree in 1960, with a different but related structure.
These finite versions of G₂ have connections to other exceptional structures. The Janko group J₁, one of the sporadic simple groups (isolated exceptions in the classification of finite simple groups, analogous to how G₂ is exceptional among Lie groups), was first discovered as a subgroup of G₂(11).
Representations
When mathematicians want to understand an abstract group, they often study its representations—ways of realizing the group as concrete matrices acting on vector spaces. The representations of G₂ have dimensions that form a specific sequence: 1, 7, 14, 27, 64, 77, 77, 182, and so on.
The seven-dimensional representation is the action of G₂ on the imaginary octonions—remember, octonions are eight-dimensional, but one dimension is the real part, leaving seven imaginary dimensions. The fourteen-dimensional representation is called the adjoint representation, where the group acts on its own Lie algebra.
Interestingly, there are two different fourteen-dimensional representations, two different seventy-seven-dimensional representations, and so on. These pairs are not equivalent to each other, reflecting subtle asymmetries in the group's structure.
The Historical Thread
The story of G₂ winds through some of the greatest names in mathematics. Wilhelm Killing found it while working on the classification problem in 1887. Élie Cartan, the French geometer who would become one of the most important mathematicians of the twentieth century, clarified its structure and discovered its connection to the octonions. Friedrich Engel, Sophus Lie's collaborator, independently found the same distribution-theoretic characterization as Cartan.
In older texts, you'll sometimes see G₂ referred to as E₂, part of an alternative naming convention that didn't catch on. The modern notation uses G₂ through G₂, F₄, and E₆ through E₈ for the five exceptional groups, with the number indicating something about the group's structure (specifically, its rank plus certain adjustments).
Why Does This Matter?
You might reasonably ask: why should anyone care about a fourteen-dimensional symmetry group discovered in the nineteenth century?
One answer is aesthetic. Mathematics is full of patterns, and the exceptional Lie groups are genuine anomalies—structures that exist for no apparent reason, following no general pattern, yet possessing their own intricate beauty. G₂ is the smallest of these anomalies, simple enough to study in detail yet strange enough to remain surprising.
Another answer is practical. The mathematics of Lie groups underlies particle physics, and while G₂ itself doesn't appear directly in the standard model, its relatives do. Understanding exceptional structures helps physicists explore what kinds of theories are mathematically possible. G₂-manifolds appear in string theory and M-theory as candidates for the extra dimensions those theories require.
A third answer is foundational. The classification of simple Lie groups is one of mathematics' great organizing achievements, reducing an apparent wilderness of possibilities to a small, comprehensible list. The five exceptional groups are the parts of that list that refuse to fit the patterns, and understanding them deepens our understanding of symmetry itself.
Wilhelm Killing, writing his letter in 1887, couldn't have known any of this. He simply found a fourteen-dimensional mathematical structure that behaved in unexpected ways. Nearly a century and a half later, mathematicians and physicists are still discovering what that structure means.
``` The essay transforms the dense Wikipedia article into an accessible narrative that: - Opens with the historical hook of Killing's 1887 discovery - Immediately clarifies this is not the geopolitical "G2" mentioned in the related Substack article - Builds from basic concepts (symmetry, groups) to more advanced ones (Lie groups, octonions) - Uses the memorable "rolling ball" analogy to ground abstract mathematics in physical intuition - Explains technical terms like "rank," "holonomy," and "representation" in plain language - Provides context about why this matters for physics and mathematics