Tiles from projections of the root and weight lattices of $A_n$

This paper introduces a general projection technique for the Voronoi tessellation of the weight lattice $A_n^*$, demonstrating that projecting the Voronoi cell of $A_4^*$ onto the Coxeter plane yields a unique tiling scheme composed of golden-ratio-related hexagons and rhombuses that differs significantly from the tiling obtained from the root lattice $A_4$.

The Gist

Imagine you have a giant, invisible 4-dimensional crystal ball. Inside this ball, there are two different ways to arrange tiny, invisible marbles (mathematical points called "lattices"). One arrangement is called the Root Lattice ($A_n$), and the other is its mirror image, the Weight Lattice ($A^*_n$).

This paper is about what happens when you shine a light through these 4D crystal balls and look at their shadows on a 2D wall (our flat world).

The Main Idea: Two Different Shadows

Usually, when mathematicians look at these 4D shapes, they focus on the "Delone cells." Think of these as the individual rooms in a 4D house. When you project the shadows of these rooms, both the Root Lattice and the Weight Lattice look exactly the same: they make a famous pattern called the Penrose Tiling. This pattern uses two shapes (kites and darts, or thick and thin rhombuses) to cover a floor without ever repeating the pattern. It's like a puzzle that goes on forever without a repeating design.

But here is the twist: This paper asks, "What if we don't look at the rooms, but instead look at the walls of the house?"

In math terms, the authors looked at the Voronoi cells. If the Delone cells are the rooms, the Voronoi cells are the "territories" or "neighborhoods" surrounding each marble. The paper discovers that when you project the shadows of these neighborhoods from the Weight Lattice, you get something completely new and different.

The New Shapes: A Golden Ratio Party

When the authors projected the 4D "neighborhoods" of the Weight Lattice onto the 2D wall, they didn't just get the usual rhombuses. They found a mix of four distinct shapes:

1. Two types of Hexagons: One is "thin" and one is "thick."
2. Two types of Rhombuses (Diamonds): One is "thin" and one is "thick."

The secret sauce here is the Golden Ratio ($\tau$ or $\phi \approx 1.618$). This is the number that appears in sunflowers, nautilus shells, and ancient Greek art.

• The edges of these new shapes are either length 1 or length $\tau$.
• It's like the shapes are made of two different sizes of Lego bricks, where the big brick is exactly 1.618 times bigger than the small one.

The Analogy: The 4D Dice

To understand how this works, imagine a 4-sided die (a tetrahedron) in 3D space. Now, imagine a 5-sided die in 4D space.

• The Root Lattice is like arranging these dice so they fit together perfectly in a specific way.
• The Weight Lattice is like arranging them in a slightly different, "dual" way.

The authors used a special mathematical "camera" (called the Coxeter plane) to take a picture of these 4D dice.

• When they photographed the Root Lattice, they got the classic Penrose pattern (rhombuses).
• When they photographed the Weight Lattice, the 4D "faces" of the dice (which were perfect hexagons and squares in 4D) got squashed and stretched by the camera lens.
  • The 4D squares turned into rhombuses.
  • The 4D hexagons turned into weird, irregular hexagons.

Why Does This Matter?

You might ask, "Who cares about 4D shadows?"

1. Quasicrystals: In the real world, scientists have found materials called quasicrystals. These are metals that have an ordered structure but no repeating pattern (like the Penrose tiling). This paper helps explain how these materials might be built. The "Weight Lattice" projection offers a new blueprint for how atoms could arrange themselves in these strange materials.
2. New Patterns: Before this, we mostly knew about the Penrose tiling. This paper says, "Hey, there's a whole other family of patterns waiting to be discovered if you look at the 'neighborhoods' instead of the 'rooms'."
3. Future Designs: The authors suggest that if we look at even higher dimensions (like 6D or 8D), we might find patterns with 8-fold or 12-fold symmetry. This could help design new materials or even new types of art and architecture that use these golden-ratio patterns.

The Takeaway

Think of the universe as a giant, multi-dimensional puzzle.

• Most people have been looking at the puzzle pieces (Delone cells) and seeing a familiar pattern (Penrose).
• This paper says, "Let's look at the gaps between the pieces (Voronoi cells)."
• When they did, they found a new, richer set of shapes (hexagons and rhombuses of two sizes) that fit together in a beautiful, non-repeating way, governed by the Golden Ratio.

It's a bit like realizing that while you've been admiring the bricks of a wall, the mortar between them actually holds the secret to a completely different, more complex, and beautiful design.

Technical Summary

1. Problem Statement

The paper addresses the geometric classification and projection of high-dimensional lattices, specifically the root lattice $A_n$ and its dual, the weight lattice $A^*_n$, onto the 2D Coxeter plane. While the projection of root lattices ($A_n$) and their Delone cells is well-established (yielding Penrose tilings for $n=4$), the projection of the Voronoi tessellation of the weight lattice ($A^*_n$) remains less explored.

The core problem is to determine how the Voronoi cell of the weight lattice $A^*_4$ (a 4-dimensional permutohedron) projects onto the Coxeter plane. The authors aim to:

• Contrast the tiling schemes obtained from the Voronoi projection of $A^*_4$ against those from $A_4$.
• Identify the specific 2D prototiles generated by this projection.
• Establish a general mathematical framework for analyzing these projections using Coxeter-Weyl group theory.

2. Methodology

The authors employ a combination of Lie algebra theory, group theory, and geometric projection techniques:

• Lattice Definition: They define the weight lattice $A^*_n$ using fundamental weight vectors $\omega_i$ derived from a set of $(n+1)$ linearly dependent, non-orthogonal vectors $k_i$. These vectors are expressed in an orthogonal basis to facilitate projection.
• Group Theory: The analysis relies on the Coxeter-Weyl group $W(a_n) \cong S_{n+1}$ (the symmetric group). The symmetries of the polytopes are analyzed through coset decompositions of this group under specific subgroups (e.g., $\langle r_1, r_2 \rangle$ for hexagonal symmetry).
• Polytope Identification:
  • The Delone cell of $A^*_n$ is identified as the fundamental simplex.
  • The Voronoi cell of $A^*_n$ is identified as the permutohedron of order $(n+1)$.
• Projection Technique: The authors project the vertices and 2D faces of the 4D permutohedron ($A^*_4$) onto the Coxeter plane. This plane is defined by the first two components of the vectors $k_i$ in an orthogonal basis.
• Classification: They classify the 2D faces (hexagons and squares) of the 4D permutohedron based on their edge vectors and group orbits, then calculate their 2D projections to determine edge lengths and angles.

3. Key Contributions

• Distinction between Root and Weight Lattice Projections: The paper demonstrates that while the Delone cell projections of $A_n$ and $A^*_n$ yield similar triangular tilings (Robinson triangles for $n=4$), their Voronoi cell projections yield fundamentally different tiling schemes.
• Characterization of the $A^*_4$ Voronoi Cell: The authors provide a detailed structural analysis of the 4D permutohedron, identifying it as having 120 vertices, 240 edges, 150 faces (60 hexagons and 90 squares), and 30 3D facets (10 truncated octahedra and 20 hexagonal prisms).
• Discovery of Four Distinct Prototiles: A primary contribution is the proof that the projection of the Voronoi cell of $A^*_4$ generates four distinct types of 2D tiles, rather than the two types (thick and thin rhombuses) found in the $A_4$ projection.
• Golden Ratio Proportions: The study explicitly links the edge lengths of these tiles to the golden ratio $\tau = \frac{1+\sqrt{5}}{2}$, showing that the tiles possess edges in proportions of $1$ and $\tau$.

4. Key Results

The projection of the Voronoi cell of the weight lattice $A^*_4$ onto the Coxeter plane results in a tiling composed of the following four tile types:

1. Thin Hexagon: Derived from the projection of specific hexagonal faces (Set A). It has edge lengths proportional to $(1, \tau, 1, 1, \tau, 1)$.
2. Thick Hexagon: Derived from the projection of other hexagonal faces (Set B). It has edge lengths proportional to $(1, \tau, \tau, 1, \tau, \tau)$.
3. Thin Rhombus: Derived from the projection of square faces (Set C). It has interior angles of $36^\circ$ and $144^\circ$ with edge lengths proportional to $1$.
4. Thick Rhombus: Derived from the projection of square faces (Set D). It has interior angles of $72^\circ$ and $108^\circ$ with edge lengths proportional to $\tau$.

Note: The projection of the square faces in Set E results in a line segment and does not form a 2D tile.

The authors present a centrally symmetric tessellation of the Coxeter plane using these four tiles (five of each type per projected Voronoi cell), as illustrated in their figures.

5. Significance

• Quasicrystallography: The work provides a unified framework for understanding aperiodic tilings with $(n+1)$-fold symmetry. Specifically, it offers a new perspective on 5-fold symmetric quasicrystals (via $A^*_4$) distinct from the standard Penrose tiling derived from $A_4$.
• Generalization: The methodology is generalizable to $n \geq 5$. The authors suggest that projections of $A^*_n$ for $n=7$ and $n=11$ could lead to 8-fold and 12-fold symmetric quasicrystallographic tilings, respectively.
• Mathematical Rigor: By utilizing the Dynkin diagram symmetry and coset decomposition, the paper offers a rigorous group-theoretical explanation for the emergence of specific geometric tiles, bridging the gap between abstract algebra and physical tiling patterns.
• Dual Lattice Insight: The study highlights the geometric duality between root and weight lattices, showing that while they share symmetry groups, their Voronoi tessellations produce distinct physical structures when projected, expanding the toolkit for modeling complex aperiodic materials.