Tessellation Morph 2D: Polar-Equation Regular-Polygon Tiling

Morph

Morph speed 1.0x

Pulse amplitude (apothem) 1.0x

Rotation speed 1.0x

Lattice density 9 × 7

Live Readout

Sides n(t)

3.00

Nearest tiling

{3,6}

Interior angle

60.0°

Cell area

0.00

Cells drawn

Total tiled area

0.00

How it works

Every cell is one closed curve traced from the exact polar equation of a regular polygon of apothem a (its inradius, the flat-to-flat half-width):

r(θ) = a / cos( ((θ mod 2π/n) − π/n) )

Unlike drawing a fixed vertex list, this formula accepts any real n ≥ 3 — so animating n continuously from 3 to 4 to 6 sweeps the shape smoothly through a true triangle, square and hexagon (and every irregular polygon in between), with vertices appearing and merging exactly where the formula predicts. The three regular tessellations of the plane use these Schläfli symbols:

{3,6} — 6 triangles meet at each vertex, interior angle 60°
{4,4} — 4 squares meet at each vertex, interior angle 90°
{6,3} — 3 hexagons meet at each vertex, interior angle 120°

Interior angle is computed from the live n via (n−2)·180/n, and cell area from n·a²·tan(π/n) — both update in the readout as the shape morphs, so you can verify the geometry is genuine rather than a texture swap. Cells sit on a triangular (honeycomb-offset) lattice sized to the current apothem so neighbouring cells stay edge-adjacent at the three exact tessellation points of the cycle.

Cell outline (r(θ) formula) Fill (palette by lattice parity)