Space, shape and the invariants that survive deformation.
11 topics · 16 curated works
No prior grounding assumed.
Flatland: A Romance of Many Dimensions
Edwin A. Abbott · 1884
A two-dimensional narrator's encounter with a sphere argues, through satire, that a being confined to one dimension cannot conceive of the next by…
+2 more at this level
Assumes you know the vocabulary.
Elements
Euclid · 300 BCE
Derives the whole of classical geometry from five postulates, and in doing so invents the axiomatic method every later mathematics is written in.
+3 more at this level
Primary sources and full treatments.
General Investigations of Curved Surfaces
Carl Friedrich Gauss · 1828
Proves the Theorema Egregium: a surface's curvature can be measured from within the surface itself, without reference to the space it sits in — which…
+8 more at this level
12 of 16 works
Edwin A. Abbott
A two-dimensional narrator's encounter with a sphere argues, through satire, that a being confined to one dimension cannot conceive of the next by reason alone.
about 2.5 hours to readlink checked 17 Sept 2026Derrick Norman Lehmer
Builds the projective plane from incidence and cross-ratio alone, with no coordinates, arguing that Desargues's and Pappus's theorems and the principle of duality are more fundamental than any metric structure laid on top of them.
link checked 17 Sept 2026Notes on this copyGrant Sanderson (3Blue1Brown)
Shows that applying Newton's root-finding method across the complex plane produces a fractal boundary between basins of attraction, arguing the method's local reliability hides genuinely chaotic global behaviour.
link checked 17 Sept 2026Notes on this copyEuclid
Derives the whole of classical geometry from five postulates, and in doing so invents the axiomatic method every later mathematics is written in.
Assumes nothing but a straight edge, a compass, and the willingness to be shown rather than told
link checked 17 Sept 2026Notes on this copyNikolai Lobachevsky
Develops a self-consistent geometry in which Euclid's parallel postulate fails, giving the first worked demonstration that a non-Euclidean geometry is logically possible.
50 pageslink checked 17 Sept 2026Notes on this copyRoberto Bonola
Traces two thousand years of failed attempts to prove the parallel postulate and shows that the failures were the discovery: the postulate is independent, so consistent geometries exist without it.
link checked 17 Sept 2026Notes on this copyMiles Reid
Introduces algebraic geometry through explicit plane curves and projective varieties before any scheme-theoretic machinery, arguing the classical objects are worth understanding concretely first.
link failing as of 17 Sept 2026Notes on this copyCarl Friedrich Gauss
Proves the Theorema Egregium: a surface's curvature can be measured from within the surface itself, without reference to the space it sits in — which is what makes intrinsic geometry possible.
link checked 17 Sept 2026Notes on this copyHermann Minkowski
Introduces convex bodies and lattice points as tools for number theory, proving that any sufficiently large symmetric convex region must contain a nonzero lattice point, the founding theorem of convex geometry.
link checked 17 Sept 2026Luther Pfahler Eisenhart
Develops the geometry of a surface entirely from its first and second fundamental forms, treating Gauss's Theorema Egregium as the fact that curvature is intrinsic and geodesics as the resulting generalisation of a straight line.
link checked 17 Sept 2026Notes on this copyOswald Veblen and John Wesley Young
Rebuilds projective geometry from explicit axioms of incidence alone, demonstrating that a geometry can be developed with no notion of distance or angle anywhere in it.
link failing as of 17 Sept 2026Notes on this copyFelix Hausdorff
Defines a dimension that can take any non-negative real value, not just an integer, giving fractal geometry the measurement tool it would need decades later to make fractional dimension precise.
link checked 17 Sept 2026Notes on this copy