DRAW · CUT · REASSEMBLE

From cubes to a long rectangle.

Draw a relative cycle. Choose a filling. Watch the paper’s cut-and-stack map assemble it into a new cycle.

Rectangle length N

Number of slabs K

2–20

01 / THE LONG RECTANGLE

The assembled relative cycle

Ψ (z) = \partial_{rel} [⋃_{ℓ, i} E_{ℓ}^{i}]

Shaded fillings Slab & cube guides Highlight new seamsLIVE

Slab widths Select a slab to edit its cubes below.

\sum_{ℓ} t_{ℓ} = 3

Changing one width redistributes the others proportionally. A width of zero collapses that slab; its drawings are retained.

02 / THE INPUT CUBES

Draw the inputs for slab 1

Drag inside any square. Open curves extend to the boundary; near-closed curves become loops.

Each slab has a separate set of inputs. Draw again to add a component modulo 2. The pale region is the filling; the dark curve is its relative boundary.

03 / LOOK INSIDE THE MAP

The four cut-and-translate steps

Input cube

Slab

Intersection width

The unit square first scales uniformly by $ρ = \sqrt{ν}$ . Each subsequent operation only translates pieces. Panels show the fillings, including their outer edges; dashed lines show the final piece partition at every stage. At $ν = 1$ , the translations cancel in the final panel, giving the identity map.

Drawing, fillings, and the construction

Drawing a relative cycle

Open strokes are extended to the nearest edges of the square. The two ends are joined along the shorter boundary arc to select a filling. This boundary arc is not part of the relative cycle. Close a stroke near its starting point to make a loop.

Multiple strokes combine by symmetric difference (coefficients in $ℤ_{2}$ ). Use Flip filling to choose the complementary filling. Presets provide a keyboard-accessible alternative to freehand drawing.

The map used here

For each slab–cube intersection, $ν$ is its width. We apply the paper’s planar map $T_{ν}$ to the filling and translate the result into that intersection. The final curve is the relative boundary of all assembled fillings, with matching edges cancelled modulo 2.

The outer boundary of $Q$ is omitted. Simultaneously complementing every input changes the shading but leaves $Ψ$ unchanged. At $ν = 0$ the image is empty, and at $ν = 1$ the map is the identity.

A model of the parameter construction

This page shows the $d = 2$ version of “An equivariant cube to slab map” and “Competitor sweepouts in a long rectangle” in The anisotropic Weyl law and Grassmannian equidistribution, by Antonio De Rosa and Aria Halavati.

Construction in the paper (PDF, §4.2)arXiv cvGMT

The controls explore fillings and join weights; an arbitrary collection of hand-drawn cycles is not itself a verified $p$ -sweepout. The displayed geometry uses polygonal approximations of your strokes.