Cut-and-project schemes and model sets

The cut-and-project recipe

Start with a lattice ℒ in a product of physical space and internal space. Each lattice point has a physical projection and an internal projection, often called its star map. Retain a lattice point when its internal image lies in an acceptance window; the retained physical projections form a model set. Irrational orientation prevents an ordinary physical-space period while the parent lattice preserves long-range order.^[31]

Duneau and Katz established this projection viewpoint for quasiperiodic patterns.^[75] De Bruijn’s Penrose pentagrid supplies a worked algebraic example: five indexed line families assign integer coordinates, and dualizing grid intersections produces rhombi. Singular offsets require explicit boundary conventions; merely counting visible directions is not a cut-and-project proof.^[76]

Windows, regularity, and diffraction

The window controls allowed local configurations and Fourier amplitudes. A regular model set uses a relatively compact window whose boundary has measure zero under the usual hypotheses, giving pure-point diffraction. Different windows on the same lattice can produce different point sets, and different weights can create extinctions.^[31][32][35]

CAP is a self-similar Hat-family representative with a 4:2 cut-and-project description. CASPr gives the Spectre-family analogue using five Rauzy-fractal windows. These are theorem-backed model-set constructions, not generic consequences of every drawing of a Hat or Spectre patch.^[31][32]

CAP versus Nan Ma’s lift

Nan Ma’s coherent ℝ⁴ edge lift splits two edge-direction classes into two coordinate planes and integrates them across a simply connected tiling.^[54] It elegantly unifies Tile(a,b) projections, but it does not by itself identify a lattice, star map, or acceptance window. CAP/CASPr instead use return modules, algebraic conjugation, and explicit windows to prove model-set and spectral statements.

Van Dongen’s “lift” is different again: a three-dimensional architectural construction made by replacing double-kites with polyhedral surface modules. It can produce continuous non-periodically textured walls, but it is neither Ma’s ℝ⁴ height function nor a cut-and-project theorem.^[69]

Checklist for a Spectre claim

A proposed Spectre model set should state the ambient lattice or module, physical and internal projections, injectivity/density conditions, star map, window, treatment of boundary points, and a proof that the selected projections reproduce the intended control-point hull. A Fourier module of a certain rank or a visually convincing high-dimensional projection is evidence to investigate, not a substitute for those data.

See also

Four-dimensional lift, Diffraction and dynamical spectrum, Sturmian lattices and aperiodic tile sets

Categories: Mathematics