A Math Museum interactive exhibit

Geometry as Algebra

See how equations were once built from areas, how shapes become proofs, and how symmetry turns local rules into infinite structure.

Room 1

When Algebra Was Made of Shapes

Before symbolic algebra became standard, many equations were understood as lengths and areas. Solving an equation could literally mean rearranging pieces.

Ancient Babylon

Quadratic problems in words

Scribes solve area problems that are equivalent to quadratic equations.

c. 300 BCE

Euclid's geometric algebra

Relationships we now write symbolically are proved through constructions and areas.

800s CE

Al-Khwarizmi completes the square

Quadratic equations are explained through geometric rearrangement, centuries before modern notation.

Room 2

Completing the Square, Literally

Start with x² and two rectangular strips. Add one small corner and the pieces become a perfect square.

Room 3

A Proof You Can See

Four identical right triangles can be arranged in two ways. The empty regions reveal why a² + b² = c².

Room 4

One Cone, Four Curves

Tilt a cutting plane through a cone. The same solid produces circles, ellipses, parabolas, and hyperbolas.

Room 5

Geometry Can Move Without Breaking

Translations, rotations, reflections, and scaling change where a shape appears while preserving different kinds of structure.

Room 6

Patterns Without Gaps

A tessellation repeats shapes to cover a surface without overlaps or empty spaces. Symmetry turns a local rule into an infinite pattern.

Final Room

The Idea Continues

The symbol is only the doorway. The real exhibit is the connection it reveals.

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