Tessellations

Interactive investigations in which periodic patterns turn area identities into movable geometric proofs.

STATUS · ACTIVE NOW

The question

How much geometry can be made visible by superimposing periodic tilings and moving one layer across another?

Current focus
Understanding periodic area proofs as families over triangle shape and grid phase.

Latest update
The Pythagorean tiling sandbox now includes a movable anchor, mirror orientation and historically distinguished phases.

Explore infinitely many Pythagorean proofs

Open the Law of Cosines interactive

Why this project

Tessellations turn a single area diagram into a family. A periodic pattern can be clipped by a movable square grid, and every position of that grid produces another dissection of the same identity.

For right triangles, squares of side lengths \(a\) and \(b\) fill a square period of side \(c\). For a general triangle, the same architecture reveals two cosine parallelograms. They appear as overlaps for acute triangles, gaps for obtuse triangles, and collapse at a right angle.

Project strands

Shape

Treat triangles up to similarity as a two-dimensional moduli space rather than a list of special cases.

Tile

Build periodic square patterns whose overlaps and gaps encode exact area corrections.

Explain

Turn the experiment into a proof by identifying one fundamental period and computing its signed areas.

Related material

• Infinitely many “proofs” of Pythagoras’ theorem — the common-lattice proof, historical phases and embedded sandbox.
• Pythagorean tiling proofs — the full-screen right-triangle laboratory.
• The Law of Cosines — the complete exposition and embedded interactive.
• Law of Cosines tessellation — the full-screen laboratory.

Next steps

• catalogue distinguished grid phases and the pieces they produce;
• develop classroom prompts that move from experiment to vector proof;
• investigate further identities that admit periodic signed-area interpretations.