Crystal Amplituhedron: Symmetry as Boundary

The Gram Matrix as Truth Object

Every lattice is uniquely characterised by its Gram matrix — the matrix of inner products of basis vectors. For a 3D lattice with basis vectors $\mathbf{a}$ , $\mathbf{b}$ , $\mathbf{c}$ meeting at angles $\alpha$ , $\beta$ , $\gamma$ :

$G = \begin{pmatrix} a^2 & ab\cos\gamma & ac\cos\beta \\ ab\cos\gamma & b^2 & bc\cos\alpha \\ ac\cos\beta & bc\cos\alpha & c^2 \end{pmatrix}$

The metric vector $\mathbf{g} = (g_{11}, g_{22}, g_{33}, g_{23}, g_{13}, g_{12})$ lives in $\mathbb{R}^6$ , where $g_{23} = 2bc\cos\alpha$ , $g_{13} = 2ac\cos\beta$ , $g_{12} = 2ab\cos\gamma$ . The positive-definite condition (Sylvester’s criterion: all three leading minors positive) carves a cone in this 6-dimensional space. Every point inside the cone is a lattice; every point outside is not.

The parameter space is not a list — it is a geometry.

Boundary Stratification Encodes Symmetry

Crystal systems are not an arbitrary taxonomy. They are the codimension- $k$ boundary strata of the positive-definite cone — the faces where equality constraints activate.

System	Constraint	Codimension	Free parameters
Triclinic	$a \neq b \neq c, \, \alpha \neq \beta \neq \gamma$	0	6
Monoclinic	$\alpha = \gamma = 90°$	2	4
Orthorhombic	$\alpha = \beta = \gamma = 90°$	3	3
Tetragonal	$a = b, \, \alpha = \beta = \gamma = 90°$	4	2
Trigonal	$a = b = c, \, \alpha = \beta = \gamma$	4	2
Hexagonal	$a = b, \, \gamma = 120°, \, \alpha = \beta = 90°$	4	2
Cubic	$a = b = c, \, \alpha = \beta = \gamma = 90°$	5	1

Moving toward a boundary face means approaching higher symmetry. Crossing it means a symmetry enhancement — a phase transition in the language of the lattice.

$\lvert\text{Cubic}\rvert \subset \lvert\text{Tetragonal}\rvert \subset \lvert\text{Orthorhombic}\rvert \subset \lvert\text{Monoclinic}\rvert \subset \lvert\text{Triclinic}\rvert$

Triclinic

codimension 0

a1.00

α75.0°

b1.20

β80.0°

c1.40

γ85.0°

RefinementSystems (7)

Polytopehidden

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The 4D Projection: Spatial Computing for Free

Six dimensions do not render to a screen. But there is a principled collapse that preserves the essential structure. The metric vector decomposes into diagonal (shape) and off-diagonal (angular) components:

$\mathbf{g}_{\text{diag}} = (g_{11}, g_{22}, g_{33}), \qquad \mathbf{g}_{\text{off}} = (g_{23}, g_{13}, g_{12})$

The 4D intermediate projection maps:

$\Phi_4(\mathbf{g}) = (g_{11}, \, g_{22}, \, g_{33}, \, \lVert\mathbf{g}_{\text{off}}\rVert)$

In the 3D scene, the first three coordinates become spatial axes. The fourth — the off-diagonal norm — displaces the point along the body diagonal $(1,1,1)/\sqrt{3}$ , exactly like the spin-gap displacement in the periodic topology visualization.

This gives immediate visual semantics:

Orthorhombic lattices (all angles 90°) sit on the base plane where $\lVert\mathbf{g}_{\text{off}}\rVert = 0$
Monoclinic and triclinic lattices are displaced from the base plane, with displacement proportional to angular deviation from orthogonality
System boundaries become visible surfaces: the tetragonal stratum is the $g_{11} = g_{22}$ plane on the base, the cubic stratum is the $(t,t,t)$ ray

Proximity in this projection directly encodes cell shape similarity (diagonal components) and angular orthogonality (displacement from base plane).

Three Charts of the Same Object

The visualization shows three coordinate charts on the same underlying object — the Gram matrix $G$ .

Crystal chart $\Phi_{\mathrm{crystal}}(G)$ : renders $G$ as the parallelepiped unit cell it defines. Three basis vectors ( $\mathbf{a}$ in red, $\mathbf{b}$ in blue, $\mathbf{c}$ in green), six faces, and a surrounding 3D grid of repeated cells. The crystal system appears as a color.

Positive geometry chart $\Phi_{\mathrm{amp}}(G)$ : renders $G$ as a point in the 4D-projected Niggli cone. Eight system boundary strata become visible geometric loci — the cubic body diagonal, three tetragonal segments, two orthorhombic face anti-diagonals, and two hexagonal displaced lines. Distance-to-boundary lines encode how close the lattice is to each symmetry enhancement. This is the crystal amplituhedron — a positive geometry whose faces mean something.

Seed chart $\Phi_{\mathrm{seed}}(G)$ : renders $G$ projected onto the $E_6$ root system. The 72 roots in $\mathbb{R}^6$ are projected to $\mathbb{R}^3$ via a matrix that separates length asymmetry, tetragonality, and angular deviation. Each root is colored by its associated crystal system and rendered as a dot on the sphere surface. The sphere mesh itself is tessellated into system-colored regions by nearest-root classification.

Morphing between charts changes the lens, not the lattice.

Why “Amplituhedron”?

The amplituhedron of Arkani-Hamed and Trnka (2013) is a positive geometry whose boundary stratification encodes scattering amplitudes in $\mathcal{N}=4$ super-Yang-Mills. In the framework of Arkani-Hamed, Bai, and Lam (2017), a positive geometry $(X, X_{\geq 0}, \Omega)$ is:

A semialgebraic subset $X_{\geq 0}$ of an algebraic variety $X$
Equipped with a canonical rational form $\Omega$ with logarithmic singularities on — and only on — its boundary faces
With the recursive property that the residue of $\Omega$ on each boundary face is the canonical form of that face

The Niggli cone satisfies all three conditions. It is a convex cone (positive geometry in the simplest sense). Its boundary faces are exactly the loci where crystal symmetry enhances. The residues on those faces recover the lower-dimensional parameter spaces of the higher-symmetry systems.

Boundary stratification is combinatorial classification. The principle is the same whether the boundaries encode particle scattering or crystal symmetry.

The $E_6$ Root System

The $E_6$ root system — 72 roots in $\mathbb{R}^6$ — is the natural coordinate system for the 6-dimensional Gram metric space. The normalised metric vector $\hat{\mathbf{g}} = \mathbf{g}/\lVert\mathbf{g}\rVert$ lies on $S^5$ . The Weyl chambers of $W(E_6)$ partition this sphere. The mapping

$\Psi: G \mapsto \hat{\mathbf{g}} \in S^5$

sends each lattice to a point whose Weyl chamber encodes its crystal system. Boundary strata of the Niggli cone align with walls between chambers: approaching a wall means approaching a symmetry enhancement.

The 72 roots decompose into two families: 60 roots with two nonzero components (capturing pairwise metric relationships) and 12 axis-aligned roots (capturing individual metric components). Under the 3x6 projection to $\mathbb{R}^3$ , roots aligned with diagonal directions map to the orthorhombic/tetragonal/cubic family, roots with off-diagonal components map to monoclinic/triclinic, and roots with specific symmetry patterns map to hexagonal/trigonal.

Polytope Growth: The Lattice as Wireframe Geometry

The polytope layer reveals structure that the parallelepiped hides. Where the unit cell shows one period of the lattice, the polytope shows the lattice itself — expanding shells of points connected through all 26 neighbor directions (6 nearest, 12 face-diagonal, 8 body-diagonal), producing a dense internal wireframe that resembles a polytope projected from higher dimensions.

Shell $N$ contains all lattice points $(i,j,k)$ where $\max(|i|,|j|,|k|) = N$ . At 3 shells (343 points, thousands of edges), the connectivity is already rich enough to see the crystal system in the wireframe topology:

Cubic lattices produce cube-symmetric polytopes with visible fourfold axes
Hexagonal lattices produce hex-prism wireframes with sixfold symmetry along $\mathbf{c}$
Triclinic lattices produce asymmetric structures where no two directions look alike

The rendering uses a depth-attenuated glow shader — front edges bright, back edges dim — with additive blending so the wireframe appears luminous against the dark background. The polytope rotates with the crystal view camera and morphs smoothly when switching between presets, making the connection between lattice parameters and polytope shape continuous and immediate.

The classification does not stop at crystal systems. Within each system lie finer distinctions:

Level 0: 7 crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic)
Level 1: 14 Bravais lattices — primitive and centered variants (aP, mP, mS, oP, oS, oI, oF, tP, tI, hR, hP, cP, cI, cF)
Level 2: 32 crystallographic point groups — the discrete rotation/reflection symmetries
Level 3+: Barycentric subdivision of the face complex

At each refinement level, the number of faces grows as $O(k \cdot 6^N)$ . The pattern of boundary-within-boundary repeats: every time a new constraint tightens, the stabiliser group jumps. This is structural self-similarity — not a fractal in the Mandelbrot sense, but the same algebraic pattern of symmetry enhancement recurring at every depth.

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