This is the visual side of mathematics — the study of shape, space and the relationships between them. Geometry measures the rigid world of lengths and angles; topology throws the ruler away and keeps only what survives stretching and bending. Together they run from the triangle on a page to the curvature of spacetime itself.

The areas to explore

Start with geometry — points, lines, angles, circles and the proofs that tie them together. Add the mathematics of angles and rotation in trigonometry. Then loosen the rules: topology studies the properties of shapes that don't care about distance at all — what's connected, what has holes, what a coffee cup and a doughnut secretly share. Finally, bring calculus to curved spaces in differential geometry, the study of curves, surfaces and the manifolds that underpin general relativity.