I01Available
The Law of Cosines
Move through the moduli space of triangle shapes, then translate a square grid across the two-colour tessellation.
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Active now2 interactives
Tessellations
Interactive investigations in which periodic patterns turn area identities into movable geometric proofs.
Open project in the LabThe investigation
How much geometry can be made visible by superimposing periodic tilings and moving one layer across another?
Tessellations turn a single area diagram into a family. A periodic pattern can be clipped by a movable square grid, and every grid position produces another dissection of the same identity.
For right triangles, two square tilings express Pythagoras. For a general triangle, the same architecture reveals cosine parallelograms as overlaps, gaps, or a right-angle transition.
GeometryTessellationsArea identities
Try it yourself
Interactives in this project
I02Available
Infinitely many “proofs” of Pythagoras
Slide one square tiling over another and visit the medieval, Perigal, and Ferrarese dissection phases.
Method & scope
How this project works
Periodic tilings are treated as families over triangle shape and grid phase, with signed areas turning the experiment into a proof.
Interactives last tested 29 July 2026.