Geometry
Geometry studies shapes, spaces, transformations, curvature, and the invariants preserved by geometric equivalence.
Discrete geometry, the largest field with about a third of the subject's articles, studies point configurations, lattice points, polyhedra and extremal shapes. Its largest topic, continuous Turán numbers, concerns the supremum of the areas of open subsets of \([0, n]^2\) that avoid a planar pattern \(P\), which for finite \(P\) agrees up to constant factors with the 0-1 matrix extremal function of \(P\). The field also has lattice points in dilated regions, the moving sofa constant, between \(2.2195\) and \(2.37\), the Dehn invariant, distinct and unit distances, Monsky's theorem and the Sylvester–Gallai theorem.
Packings and coverings, a fifth of the subject, has the Kepler conjecture that the supremum of the densities of unit sphere packings in \(\mathbb{R}^3\) is \(\pi/\sqrt{18}\), and covering problems such as Moser's worm problem, Lebesgue's universal covering problem and Hadwiger's covering problem, with every convex body in \(\mathbb{R}^3\) covered by \(14\) translates of its interior. Convex geometry includes the Brunn–Minkowski and Prékopa–Leindler inequalities and Hadwiger's theorem that the continuous isometry-invariant valuations on convex bodies in \(\mathbb{R}^n\) form a space of dimension \(n + 1\).
Euclidean geometry runs from triangle geometry to isometries and crystallographic groups, with the 17 wallpaper groups and the 230 space groups. On the smooth side, Riemannian and Finsler geometry has the Hopf–Rinow theorem, the sphere theorem for compact simply connected strictly quarter-pinched manifolds and Mostow rigidity for lattices in \(\mathrm{PO}(n, 1)\) with \(n \ge 3\), and curves and surfaces include the Fáry–Milnor theorem that a knot of total curvature at most \(4\pi\) is unknotted.
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