Gauss-Bonnet theorem

The curvature of the entire surface and the number of holes are strictly linked.

Differential geometry and topology

Localized curvature ⟷ Number of holes in the entire curved surface

Gauss–Bonnet Architecture — Total Curvature 

TOTAL CURVATURE HALL

Positive curvature encloses the hall.

Negative curvature opens it to the city.

Geometry changes. Topology remains.

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The building's overall topological structure can be felt even from the inside.

Beyond its interesting shape, Gauss-Bonnet's "relationship between total curvature and phase" is most easily understood.

Mathematical Architecture

space