Every shape can be cut exactly in half with one straight line
The article explains the mathematical guarantee that any bounded 2D shape can be divided into two equal areas by a single straight line. This concept, rooted in the Intermediate Value Theorem, is explored through the lens of game design and human visual perception.
Why it matters
It highlights the intersection of abstract mathematical proofs and practical application in software development and user experience design.
Take any bounded 2D region with positive area. Make it ugly. Add dents. Stretch it. Punch holes in it.
There is a clean mathematical guarantee: there is always at least one straight line, oriented in that exact direction, that divides your shape into two pieces of identical area.
You don't need to guess if a solution exists; the mathematics guarantees it is there. The interesting part is trying to find it anyway.
I recently built a small browser game around this idea called Bisecto , and while implementing the cutting engine, I stumbled into a fascinating gap between mathematical certainty and human perception.
Why is a perfect cut guaranteed for any direction? The foundation is a basic consequence of continuity: the Intermediate Value Theorem (IVT) .
(Here we're talking about a 2D region and a 1D straight line. In 3D, the same argument works with a plane and volume.)
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