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arxiv: 2604.10574 · v1 · submitted 2026-04-12 · 🧮 math.CO · cond-mat.other· math-ph· math.MP

Tiles from projections of the root and weight lattices of A_n

Nazife Ozdes Koca , Mehmet Koca , Rehab Nasser Al Reasi This is my paper

Pith reviewed 2026-05-10 16:10 UTC · model grok-4.3

classification 🧮 math.CO cond-mat.othermath-phmath.MP
keywords A_n latticesVoronoi tessellationweight latticeroot latticeCoxeter planegolden ratiotilingsprojections
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The pith

Projections of the weight lattice A_4 star produce hexagons and golden-ratio rhombi unlike the root lattice.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a general technique for projecting the Voronoi tessellation of the weight lattice A_n star using a set of k_i vectors. For A_4 star this yields a tiling with two types of hexagons and two types of rhombuses whose edges have lengths in the golden ratio. This scheme differs from the tiling obtained from projecting the Voronoi cell of the root lattice A_4. The method also applies to projections of the root lattice A_n and distinguishes between Voronoi and Delone cell projections.

Core claim

The projection of the Voronoi tessellation of the weight lattice A_4 star produces a totally different tiling scheme than the tiling obtained from the Voronoi cell projection of the lattice A_4. The 2D faces of the Voronoi cell of the lattice A_4 star are of two types: regular hexagons and squares in 4-dimensions but project into two types of hexagons and two types of rhombuses with edges of two lengths in proportion to golden ratio.

What carries the argument

A set of linearly dependent non-orthogonal vectors k_i where the first two components in an orthogonal basis determine the Coxeter plane for the projection.

If this is right

  • The same technique can be used for the projections of the root lattice A_n.
  • Vertices of the Voronoi cell of A_n are the union of orbits of the weight vectors omega_1 to omega_n.
  • The Voronoi cell of A_n star is the permutohedron of order n+1 with regular hexagons and squares as faces.
  • Projections of Delone cells of both A_n and A_n star produce the same type of tiles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The appearance of the golden ratio suggests that the projection preserves certain metric relations from the higher-dimensional lattice.
  • Such projected tilings may offer new ways to construct planar patterns with irrational scaling factors from lattice symmetries.

Load-bearing premise

The specific choice of the first two components of the k_i vectors defines a projection onto the Coxeter plane that faithfully captures the combinatorial and metric structure of the higher-dimensional Voronoi faces.

What would settle it

Measure the edge lengths in the projected 2D figures from the 4D hexagons and squares of the A_4 star Voronoi cell to verify whether they consist of exactly two distinct lengths whose ratio is the golden ratio.

Figures

Figures reproduced from arXiv: 2604.10574 by Mehmet Koca, Nazife Ozdes Koca, Rehab Nasser Al Reasi.

Figure 1
Figure 1. Figure 1: Coxeter-Dynkin diagram an. The root system and certain polytopes of the group W(an ) ≈ Sn+1 has a symmetry called Dynkin diagram symmetry γ which acts on the simple roots and weights as γ : α1 ←→ αn, α2 ←→ αn−1, . . . ; γ : ω1 ←→ ωn, ω2 ←→ ωn−1, . . . . (2) Therefore, the root system and certain polytopes symmetric under the Dynkin diagram symmetry has a larger symmetry W(an ) : C2 of order 2 × (n + 1)!. T… view at source ↗
Figure 2
Figure 2. Figure 2: Thin and thick hexagons projected from the sets A and B. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Thin (red) and thick (blue) rhombuses with respective edges 1 and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tiling of the projected Voronoi cell of A∗ 4 by the four tiles. Two patches of centrally symmetric tilings with the four tiles are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Centrally symmetric uncolored (a) and colored (b) patches obtained by projection of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Main purpose of this work is to introduce a general technique of projection of the Voronoi tessellation of the weight lattice $A_n^\ast$ and apply it for the lattice $A_4^\ast$. The projection of the Voronoi tessellation of the weight lattice $A_4^\ast$ produces a totally different tiling scheme than the tiling obtained from the Voronoi cell projection of the lattice $A_4$. The 2D faces of the Voronoi cell of the lattice $A_4^\ast$ are of two types: regular hexagons and squares in 4-dimensions but project into two types of hexagons and two types of rhombuses with edges of two lengths in proportion to golden ratio. The mathematical technique employed is also useful for the projections of the root lattice $A_n$. A convenient set of linearly dependent and non-orthogonal $\left(n+1\right)$ vectors $k_i$ is introduced. The simple roots and the fundamental weights are defined as $\alpha_i=k_i-k_{i+1},\left(i=1,2,\ldots,n\right) ,\omega_i=k_1+k_2+\ldots+k_i$, respectively. When the vectors $k_i$ are defined in an orthogonal basis, the first two components of $k_i$ determine the Coxeter plane. Projection of the Delone cells of $A_n$ and $A_n^\ast$ on the Coxeter plane displays the same type of tiles and tilings but the Voronoi cell projection of these lattices yields different tiles and tilings. Vertices of the Voronoi cell $V(0)$ of $A_n$ is the union of the orbits of the weight vectors $W(a_n){(\omega}_1)\cup W\left(a_n\right)(\omega_2)\cup\ldots\cup W\left(a_n\right)(\omega_n)$ and the 2D faces are the rhombuses. The Voronoi cell ${V(0)}^\ast$ of $A_n^\ast$ is the permutohedron of order $(n+1)$ and its vertices are the permutations of the vectors ${k}_i$ of the vertex $\frac{1}{n+1}[\left(n+1\right)k_1+nk_2+\ldots+k_{n+1}]$. It has regular hexagons and squares as 2D faces in $n$-dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces a general technique for projecting the Voronoi tessellation of the weight lattice A_n^* onto the Coxeter plane using a set of linearly dependent vectors k_i (with roots α_i = k_i - k_{i+1} and weights ω_i defined as partial sums), and applies it to the case n=4. It claims that the projection of the Voronoi cell of A_4^* (identified as the 4D permutohedron with regular hexagonal and square 2D faces) yields a distinct 2D tiling consisting of two types of hexagons and two types of rhombuses with edge lengths in the golden ratio, in contrast to the rhombus tiling obtained from the Voronoi cell projection of the root lattice A_4. Projections of Delone cells are asserted to produce the same tile types for both lattices.

Significance. If the central geometric claims hold, the work supplies an explicit, parameter-free construction for deriving 2D tilings from higher-dimensional root and weight lattices, with the golden-ratio proportions in the A_4^* case offering a concrete link to pentagonal symmetry relevant for quasicrystal models. The paper's strengths include the direct vector definitions that avoid fitted parameters and the clear distinction drawn between root-lattice and weight-lattice Voronoi projections.

major comments (2)
  1. [Definition of the projection via k_i vectors] The assertion (in the paragraph beginning 'When the vectors k_i are defined in an orthogonal basis...') that the first two components of the k_i vectors determine the Coxeter plane requires an explicit check that this coordinate choice coincides with the invariant plane of a Coxeter element (eigenvalue e^{2πi/5} for A_4). Without such verification, the reported edge-length ratios and combinatorial tile types could depend on an arbitrary orthogonal basis rather than intrinsic lattice geometry, undermining the claim of distinct, golden-ratio-based tiles.
  2. [Voronoi cell V(0)^* of A_n^* and its projection] The central claim that the projected 2D faces of V(0)^* consist of 'two types of hexagons and two types of rhombuses' (abstract and Voronoi-cell section) is load-bearing for the distinction from the A_4 tiling; it must be supported by explicit post-projection coordinates or a table of edge vectors and angles, rather than asserted from the 4D face types alone.
minor comments (3)
  1. Notation: the abstract uses lowercase 'a_n' in W(a_n) while the title and body use A_n; standardize to A_n throughout.
  2. Add a short table or explicit list of the projected vertex coordinates for the A_4^* case to allow readers to reproduce the golden-ratio ratios and tile types.
  3. The manuscript would benefit from one or two references to prior literature on Coxeter-plane projections of A_n lattices or permutohedra to situate the new tiling scheme.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested verifications and explicit data.

read point-by-point responses

  1. Referee: The assertion (in the paragraph beginning 'When the vectors k_i are defined in an orthogonal basis...') that the first two components of the k_i vectors determine the Coxeter plane requires an explicit check that this coordinate choice coincides with the invariant plane of a Coxeter element (eigenvalue e^{2πi/5} for A_4). Without such verification, the reported edge-length ratios and combinatorial tile types could depend on an arbitrary orthogonal basis rather than intrinsic lattice geometry, undermining the claim of distinct, golden-ratio-based tiles.

    Authors: We agree that an explicit verification is required to confirm the alignment with the intrinsic Coxeter plane. In the revised manuscript we will add a calculation of the Coxeter element for A_4 and demonstrate that the plane defined by the first two coordinates of the k_i vectors is invariant under this element with eigenvalue e^{2πi/5}. This will establish that the projection and resulting golden-ratio proportions are independent of the choice of orthogonal basis. revision: yes

  2. Referee: The central claim that the projected 2D faces of V(0)^* consist of 'two types of hexagons and two types of rhombuses' (abstract and Voronoi-cell section) is load-bearing for the distinction from the A_4 tiling; it must be supported by explicit post-projection coordinates or a table of edge vectors and angles, rather than asserted from the 4D face types alone.

    Authors: We acknowledge that explicit post-projection data would strengthen the central claim. In the revised version we will add a table of the projected 2D coordinates of the vertices of the relevant faces of the 4D permutohedron, together with the resulting edge vectors, lengths (confirming the golden-ratio ratio), and interior angles. This will directly verify the two distinct hexagon types and two rhombus types. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructive projection from lattice definitions

full rationale

The paper defines k_i vectors explicitly, expresses roots/weights in terms of them, states that an orthogonal basis choice makes the first two components span the Coxeter plane, and then computes the projected Voronoi faces and their metric properties (including golden-ratio edge ratios) directly from the coordinate geometry of A_4*. No parameters are fitted to data, no predictions are made from subsets of results, and no load-bearing steps reduce to self-citations or prior ansatzes by the same authors. The claimed distinction between A_4 and A_4* tilings follows from the differing vertex sets and face types of the two Voronoi cells under the same projection map. This is a standard constructive derivation in lattice geometry with no reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard theory of root systems and weight lattices of type A_n, the geometric definitions of Voronoi and Delone cells, and the choice of the Coxeter plane via the first two components of the k_i basis; no numerical free parameters are introduced.

axioms (2)
  • standard math Standard properties of the root system and weight lattice of type A_n, including the definitions of simple roots α_i = k_i - k_{i+1} and fundamental weights ω_i as partial sums of the k vectors.
    Invoked throughout the definitions and the description of vertices and faces of the Voronoi cells.
  • standard math The Voronoi cell V(0) of A_n has vertices given by the union of Weyl-group orbits of the fundamental weights.
    Stated directly as background for the projection construction.

pith-pipeline@v0.9.0 · 5769 in / 1682 out tokens · 42617 ms · 2026-05-10T16:10:22.184614+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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