Math
Turn the parallel calipers around a Reuleaux triangle, circle and ellipse. The first two keep their width; the ellipse does not. The displayed perimeter uses a fine polygonal approximation.
Equations w(φ) = const; L = πw
Math
Fit a fourth disc between three touching discs until it touches all of them, with the narrowest gap magnified; compare its curvature with Descartes' relation, then find the second solution - an enclosing circle, a line or an outside circle - and grow the Apollonian packing.
Math
One cord of fixed length on pins: add sides and watch the enclosed area grow toward the circle's while a ruler shows the cord is still the same length; then predict which of two shapes from the same cord holds more, and check.
Math
Compare square and staggered lattices in finite squares. Counts include only discs entirely inside the boundary. The infinite-lattice density is a reference, not a claim of optimal finite packing.