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The crossmetric tensor and the geometrical meaning of the imaginary numbers

This paper introduces the "crossmetric tensor," a generalization of Hamilton's quaternion matrix to non-Cartesian bases composed of metric and cross tensors, to provide a unified geometric interpretation of imaginary numbers and crystallographic quaternions as pairs of oriented planes that explains the consistency of Hamilton's product rules across different crystal families.

Original authors: Cyril Cayron

Published 2026-08-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Cyril Cayron

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Secret Language of Spinning Things

Imagine you are trying to describe how a top spins, or how a crystal twists and turns in a microscope. For centuries, scientists have used a special kind of math called "quaternions" to do this. Think of quaternions not as numbers you count with, but as a secret code for rotation. In the 1800s, a mathematician named William Rowan Hamilton invented a way to write these codes using a real number and three "imaginary" symbols: i, j, and k.

Now, here is the tricky part that has confused people for a long time: What exactly are these imaginary symbols? Are they just weird numbers? Do they represent a 90-degree turn? For a long time, the answer was a bit fuzzy, and many scientists just used the math because it worked, without really knowing what the symbols meant in the real world. This is like driving a car because the engine works, but not knowing how the pistons move. This paper dives into that mystery, specifically looking at how these spinning codes work not just in perfect, square boxes (like a standard 3D grid), but in the weird, stretched-out shapes found in crystals. The author wants to replace the confusing "imaginary numbers" with something you can actually see and touch: pairs of flat, floating mirrors.

The Paper's Big Idea: Mirrors Instead of Magic Numbers

In this study, the author, Cyril Cayron, takes the old math of quaternions and gives it a brand new, visual makeover. He proposes that we stop thinking of the imaginary symbols i, j, and k as mysterious numbers, and instead think of them as pairs of oriented mirrors.

Imagine you have two flat mirrors floating in space. They aren't just sitting there; they are "oriented," meaning they have a front and a back, like a piece of paper with a front side and a back side. If you place these two mirrors so they intersect (cross each other) at a specific angle, they define a rotation. If the mirrors are perpendicular (at a 90-degree angle) to each other, they represent a "half-turn" or a 180-degree spin.

The paper suggests that the "imaginary" part of the quaternion isn't a number at all; it's the pair of mirrors itself.

The symbol i isn't a number; it's a specific pair of mirrors.
The symbol j is another pair.
The symbol k is a third pair.

When you "multiply" two quaternions (which is how you combine two rotations), you aren't doing complex math with numbers. You are doing a game of "connect the mirrors." If the second mirror of the first pair matches the first mirror of the second pair, you can snap them together. The result is a new pair of mirrors: the first mirror of the first pair and the last mirror of the second pair. The author calls this the "head-tail" rule, similar to how you might connect train cars or link arrows in a chain.

The "Crossmetric Tensor": A New Rulebook for Weird Shapes

The paper does something even more specific. Most math for spinning things assumes everything is a perfect cube (like a standard video game world). But crystals in the real world are often stretched, squashed, or tilted. They don't have right angles.

The author introduces a new tool called the "crossmetric tensor." Think of this as a universal rulebook or a translator.

The old rulebook (Hamilton's original math) only worked for perfect cubes.
This new "crossmetric tensor" is a 4x4 grid of symbols that works for any shape, whether it's a perfect cube, a stretched box, or a weird crystal.

This new grid uses symbols like s (for the scalar part), and a, b, c (which represent the specific axes of the crystal). The paper shows that you can calculate how rotations work in these weird shapes by using this grid, just like you would multiply numbers. The author calculated this grid for all six major families of crystal shapes found in nature.

Why the Old Rules Were "Right" (But for a Different Reason)

One of the most fun parts of the paper is explaining why Hamilton's famous rules (like i² = -1) work.

The Old Confusion: People used to think i was a 90-degree turn, so doing it twice (i²) would be 180 degrees, which is the same as flipping everything upside down (which is -1).
The Paper's Correction: The author argues that i is actually a pair of mirrors at a 90-degree angle. When you combine a pair of mirrors with itself (using the head-tail rule), the middle mirrors cancel out, leaving you with a pair of mirrors that are actually the same mirror flipped over. This represents a full 360-degree spin, which in the world of quaternions is treated as -1 (a "double cover" of the rotation).

The paper also points out a cool secret about cubes. In a perfect cube, the "elementary" quaternions (the ones based on the axes) and the "complementary" quaternions (the ones based on the faces) turn out to be the exact same thing. This is why we only need three symbols (i, j, k) for cubes. But in a weird, stretched crystal, these two types of quaternions are different. You would need six symbols to describe the rotations perfectly. The fact that cubes are so symmetrical is what made the math look simpler than it really is.

The Bottom Line

This paper doesn't just say "here is a new formula." It offers a new way to see the math. It suggests that the "imaginary numbers" Hamilton invented are actually just pairs of floating, oriented planes.

What it proves: The author shows mathematically that this "mirror pair" idea perfectly matches the existing rules for how crystals rotate. They even used computer software (Mathematica) to calculate these new grids for all crystal types and confirmed they work.
What it changes: It changes the story of what these symbols are. Instead of being abstract, confusing numbers, they are geometric objects you can visualize.
The "Aha!" Moment: The paper suggests that the confusion around imaginary numbers comes from trying to force 2D ideas (like a 90-degree turn on a flat piece of paper) onto 3D space. In 3D, the "imaginary" part is really about the relationship between two planes, not a single number.

So, the next time you see a spinning top or a crystal, imagine it not as a collection of numbers, but as a dance of floating mirrors snapping together, head to tail, creating the beautiful, complex rotations we see in nature.

Technical Summary: The Crossmetric Tensor and the Geometrical Meaning of Imaginary Numbers

Problem Statement
The paper addresses two interconnected issues in the theory of quaternions. First, while Hamilton's quaternions provide a compact algebraic framework for 3D rotations in Cartesian bases, their generalization to non-Cartesian (crystallographic) bases has historically relied on separating scalar and cross products. Second, the geometric interpretation of the imaginary units ( $i, j, k$ ) in Hamilton's algebra remains ambiguous and often lacks a coherent physical representation, leading to historical resistance from physicists like Gibbs and Heaviside who preferred vector analysis. The author notes that existing explanations of imaginary numbers as "90-degree rotation operators" or abstract basis elements fail to provide a consistent geometric intuition, particularly regarding the factor of two difference between 2D complex rotations and 3D quaternion rotations.

Methodology
The study proceeds in two main phases: algebraic generalization and geometric interpretation.

Algebraic Generalization: Building on previous work (Cayron, 2026), the author generalizes the composition of rotational quaternions to non-Cartesian crystallographic bases. Instead of treating the scalar product (governed by the metric tensor $\mathcal{M}$ ) and the cross product (governed by the cross matrix $\mathcal{X}$ ) as separate operations, the author combines them into a single 4x4 symbolic matrix. This matrix, termed the crossmetric tensor ( $\mathcal{Q}_c$ ), allows the product of two crystallographic quaternions to be written as a single quadratic form: $\mathbf{q}_3 = \mathbf{q}_1^T \mathcal{Q}_c \mathbf{q}_2$ . The symbols used in this tensor are $\mathbf{s}$ (scalar axis) and the crystallographic axes $\mathbf{a}, \mathbf{b}, \mathbf{c}$ .
Geometric Interpretation: The author proposes a new geometric model where a unit quaternion is not an operation itself, but a set of pairs of oriented planes (mirror planes). A quaternion representing a rotation of angle $\alpha$ around axis $\mathbf{u}$ is defined as the infinite set of pairs of planes $(\mathbf{m}_1, \mathbf{m}_2)$ intersecting along $\mathbf{u}$ with an angle of $\alpha/2$ between them. The composition of quaternions is then modeled using a groupoid head-tail rule: the target plane of the first quaternion must coincide with the source plane of the second.

Key Contributions and Results

The Crossmetric Tensor ( $\mathcal{Q}_c$ ): The paper derives the explicit form of $\mathcal{Q}_c$ for all six crystal families. This tensor unifies the metric tensor (for scalar products) and the cross tensor (for cross products) into a single symbolic structure. The components of $\mathcal{Q}_c$ are shown to be deducible solely from the metric tensor $\mathcal{M}$ .
Geometric Meaning of Imaginary Units: The paper redefines the imaginary units $\mathbf{i}, \mathbf{j}, \mathbf{k}$ $i, j, k$ (and their crystallographic counterparts $\tilde{\mathbf{a}}, \tilde{\mathbf{b}}, \tilde{\mathbf{c}}$ $\tilde{a}, \tilde{b}, \tilde{c}$ ) not as numbers, but as elementary quaternions representing half-turn rotations ( $180^\circ$ $18 0^{\circ}$ ). Geometrically, these are pairs of perpendicular, oriented planes intersecting along the respective axes.
- For example, $\mathbf{i}$ is represented as a pair of planes $(\mathbf{m}_y, -\mathbf{m}_z)$ .
- The rule $\mathbf{i}^2 = -1$ is derived geometrically: composing a pair of planes with itself (after accounting for orientation reversal) results in a $360^\circ$ rotation, which corresponds to the quaternion $-\mathbf{s}$ .
Elementary vs. Complementary Quaternions: The study distinguishes between "elementary" quaternions (based on the crystallographic axes $\mathbf{a}, \mathbf{b}, \mathbf{c}$ $a, b, c$ ) and "complementary" quaternions (based on the faces of the unit cell, i.e., reciprocal vectors).
- In general crystallographic bases, these are distinct sets. The elementary quaternions satisfy the square rule ( $\mathbf{a}^2 = -1$ ), while the complementary quaternions satisfy the bi-product rules (e.g., $\mathbf{a}'\mathbf{b}' = \mathbf{c}'$ ).
- Cartesian Special Case: In a cubic (Cartesian) system, the elementary and complementary quaternions coincide ( $\mathbf{i} = \mathbf{i}'$ , etc.). This coincidence explains why Hamilton's original system requires only three imaginary units to satisfy both square and bi-product rules simultaneously, whereas a general crystallographic system would require six distinct units to fully describe the algebra.
Reinterpretation of 2D Imaginary Numbers: The author argues that the 2D imaginary number $\mathbf{i}$ should be understood as a pair of perpendicular lines (planes in 2D) representing a $180^\circ$ rotation when squared. While the 2D imaginary number $\mathbf{i}$ is associated with a $90^\circ$ rotation (quarter turn), the paper emphasizes that the geometric object $\mathbf{i}$ itself corresponds to the set of pairs of perpendicular oriented planes. Consequently, the rotation associated with the object $\mathbf{i}$ is a rotation of twice $90^\circ$ , i.e., $180^\circ$ . This perspective clarifies that the complex number $\cos\alpha + \sin\alpha \mathbf{i}$ represents two planes forming an angle $\alpha$ between them, resulting in a rotation of angle $2\alpha$ , rather than a direct representation of a rotation of angle $\alpha$ .

Significance and Claims
The paper claims that the crossmetric tensor provides a rigorous algebraic framework for crystallographic rotations that is consistent with the underlying geometry of the lattice. More importantly, the proposed geometric interpretation of imaginary numbers as "pairs of floating oriented planes" offers a coherent explanation for Hamilton's rules without relying on abstract algebraic conventions.

The author suggests that the ambiguity of the quaternion units ( $+1$ and $-1$ representing the same rotation) is naturally explained by the groupoid structure of these plane pairs. The study posits that the "yoke of quaternion" criticized by Gibbs was due to the lack of a clear geometric representation for the imaginary symbols, a gap this work attempts to fill by linking the algebraic symbols directly to the geometry of mirror planes and the composition of rotations. The paper concludes that the Cartesian quaternions are merely a specific, highly symmetric case of a broader crystallographic quaternion system.

The Secret Language of Spinning Things

The Paper's Big Idea: Mirrors Instead of Magic Numbers

The "Crossmetric Tensor": A New Rulebook for Weird Shapes

Why the Old Rules Were "Right" (But for a Different Reason)

The Bottom Line

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