Geometry: Intermediate

Foundations of differential geometry and manifolds (advanced undergraduate level)

Overview

At the intermediate level you study differential geometry, which uses calculus to analyze how curves and surfaces bend, then move on to the notion of manifolds built by patching together local coordinates, and to the beginnings of topology.

Learning goals

Be able to compute the curvature and torsion of a curve
Understand the first and second fundamental forms of a surface
Understand the notions of a manifold and its tangent space
Understand topological invariants such as the Euler characteristic
Understand geodesics, Riemannian metrics, and the fundamental group

Ch. 1 Curve Theory
Curvature, torsion, and the Frenet-Serret formulas.
Ch. 2 Surface Theory
The first and second fundamental forms, Gaussian curvature, and mean curvature.
Ch. 3 Foundations of Manifolds
A space formed by gluing together local coordinate patches: topological manifolds, differentiable structures, and tangent spaces.
Ch. 4 Introduction to Riemannian Geometry
Riemannian metrics, geodesics, and curvature tensors.
Ch. 5 Topology
The Euler characteristic, the fundamental group, and covering spaces.
Ch. 6 Distance in n-Dimensional Space
Covers Euclidean distance, the distance formulas from a point to a line and to a hyperplane, and the concept of projection.
Appendix Affine Transformation
Definition and classification, homogeneous coordinates, composition and inverse, the affine group, invariants, and the relation to projective transformations.
Appendix Rodrigues' Rotation Formula
Rotation by axis-angle representation, the explicit rotation matrix formula and the exponential map, SO(3), and applications to 3D and robotics.
Helly's Theorem
A theorem on the intersection of families of convex sets in $\mathbb{R}^d$, with a proof via Radon's theorem and the centerpoint theorem.

Topics: Plane Curves, Competition Geometry, and Solids

Geometry topics a step beyond high-school mathematics: curves given by parametric or polar equations, competition-geometry theorems, and polyhedra and spherical geometry.

Plane Curves & Special Figures (93)

Triangle & Circle Theorems (Competition Geometry) (20)

Solids, Polyhedra & Spherical Geometry (23)

Morley's Trisector Theorem — Morley's trisector theorem: the three intersection points of adjacent trisectors of the interior angles of any triangle form an equilateral triangle. The theorem statement, an SVG diagram, a proof outline, and numerical examples are explained.
Desargues' Theorem — Desargues' theorem: if two triangles are in perspective from a point, then the lines through corresponding vertices meet at a single point and the intersections of corresponding sides are collinear. This fundamental theorem of projective geometry is explained.
Pappus's Hexagon Theorem — Pappus's hexagon theorem: given two lines each containing three points, the three intersection points of the three pairs of opposite sides of the resulting hexagon are collinear. This fundamental theorem of projective geometry, a proof outline, and numerical examples are explained.
Pascal's Theorem — Pascal's theorem: the three intersection points of opposite sides of a hexagon inscribed in a conic section are collinear (the Pascal line). The dual relationship with Brianchon's theorem, a proof outline, and numerical examples are explained.
Brianchon's Theorem — Brianchon's theorem: the three main diagonals of a hexagon circumscribed about a conic section meet at a single point (the Brianchon point). The dual relationship with Pascal's theorem, the proof, and numerical examples are explained.
Feuerbach's Theorem — Feuerbach's theorem: the nine-point circle of a triangle is internally tangent to the incircle and externally tangent to each of the three excircles. The theorem statement, the nine-point circle, a proof outline, and numerical examples are explained.
Isogonal Conjugate — The isogonal conjugate: the point where the lines symmetric to the angle bisectors at each vertex of a triangle meet at a single point.
Pedal Triangle — Defines the pedal triangle, describes its construction and area formula, its limiting relationship to the Simson line, and its connections to the Feuerbach point and nine-point circle at an intermediate level.
Medial Triangle — Defines the medial triangle, explains its similarity ratio 1:2 to the original triangle, its area equal to 1/4 of the original, its relationship to the nine-point circle, and its connections to the centroid and triangle centers at an intermediate level.
Orthic Triangle — Defines the orthic triangle, explains the angle formulas (e.g., 180° − 2A), its perimeter-minimizing property inside acute triangles (Fagnano's problem), and the analogy with light reflection at an intermediate level.
Brocard Points — Defines the Brocard points, explains the Brocard angle formula cot ω = cot A + cot B + cot C, their existence and uniqueness, the special case of the equilateral triangle, and their relationship to isogonal conjugates at an intermediate level.
Isoperimetric Inequality — Isoperimetric inequality L^2 >= 4πA: among all curves of a given perimeter, the circle encloses the maximum area. Concrete examples for quadrilaterals and triangles, proof strategies (Fourier analysis, Steiner symmetrization), and the equality condition are explained at an intermediate level.
Monge's Theorem — Monge's theorem: the three external centers of similitude of each pair of three circles in the plane are collinear. The projective-geometry intuition, a proof via three-dimensional interpretation, and numerical examples are explained at an intermediate level.
Steiner Inellipse — Defines the Steiner inellipse (the ellipse tangent to each side of a triangle at its midpoint), explains that its center coincides with the centroid, gives the area formula π/(3√3)×Δ, and shows that it is the largest ellipse inscribed in the triangle, at an intermediate level.
Fagnano's Problem — Fagnano's Problem: among all triangles inscribed in an acute triangle, the orthic triangle has the minimum perimeter. The proof via Schwarz's reflection method and concrete examples are explained at an intermediate level.
Truncated Tetrahedron — Truncated Tetrahedron: an Archimedean solid obtained by cutting the corners of a regular tetrahedron. Its face, edge, and vertex counts, verification via Euler's polyhedron formula, the net, and the volume and surface area formulas are explained at an intermediate level.
Truncated Octahedron — Truncated Octahedron: an Archimedean solid obtained by truncating a regular octahedron. Its face, edge, and vertex counts, its property as a space-filling polyhedron, and the volume and surface area formulas are explained at an intermediate level.
Rhombicuboctahedron — Rhombicuboctahedron: an Archimedean solid with square and triangular faces. Its face, edge, and vertex counts, volume and surface area formulas, and its relation to the Ruffini lantern are explained at an intermediate level.
Truncated Cuboctahedron — Truncated Cuboctahedron: an Archimedean solid and the largest in the cubic family. Its face, edge, and vertex counts, its status as the largest Archimedean solid in the cubic group, and its volume and surface area formulas are explained at an intermediate level.
Snub Cube — Snub Cube: an Archimedean solid distinguished by its chirality (it exists in two non-superimposable mirror forms). Its face, edge, and vertex counts and its volume and surface area formulas are explained at an intermediate level.
Icosidodecahedron — Icosidodecahedron: a quasiregular Archimedean solid combining icosahedral and dodecahedral symmetry. Its face, edge, and vertex counts and its volume and surface area formulas are explained at an intermediate level.
Cuboctahedron — Explains the cuboctahedron: its definition, the counts of faces, edges, and vertices, volume and surface area formulas, and its properties as a quasiregular polyhedron at an intermediate level.
Truncated Cube — Explains the definition of the truncated cube, its numbers of faces, edges, and vertices, and its volume and surface area formulas, as an Archimedean solid, at an intermediate level.
Rhombic Dodecahedron — Defines the rhombic dodecahedron and explains its properties, the formulas for its volume and surface area, and its properties as the dual of the cuboctahedron and as a space-filling polyhedron, at an intermediate level.
Zonohedron — Explains the definition, properties, and construction of the zonohedron, along with its characteristics as a convex polyhedron with centrally symmetric faces, at an intermediate level.
Truncated Dodecahedron — Truncated Dodecahedron: an Archimedean solid whose faces consist of regular decagons and equilateral triangles. Its face, edge, and vertex counts, its relation to the golden ratio, and its volume and surface area formulas are explained at an intermediate level.
Truncated Icosahedron — Truncated Icosahedron: an Archimedean solid whose faces consist of regular pentagons and hexagons. Its face composition, vertex and edge counts, volume and surface area formulas, its relation to the soccer ball and fullerene C60, and numerical examples are explained at an intermediate level.
Rhombicosidodecahedron — Rhombicosidodecahedron: an Archimedean solid with vertex configuration 3.4.5.4. Its face composition, vertex and edge counts, volume and surface area formulas, and its position among the Archimedean solids are explained at an intermediate level.
Snub Dodecahedron — Snub Dodecahedron: a chiral Archimedean solid with vertex configuration 3.3.3.3.5. Its face composition, vertex and edge counts, volume and surface area formulas, and its chirality (non-superimposable on its mirror image) are explained at an intermediate level.
Triakis Tetrahedron — Triakis Tetrahedron: a Catalan solid and the dual of the truncated tetrahedron. Its face composition, vertex and edge counts, and volume and surface area formulas are explained at an intermediate level.
Tetrakis Hexahedron — Tetrakis Hexahedron: a Catalan solid and the dual of the truncated octahedron. Its face composition, vertex and edge counts, and volume and surface area formulas are explained at an intermediate level.
Rhombic Triacontahedron — Rhombic Triacontahedron: a Catalan solid and the dual of the icosidodecahedron, whose faces are golden rhombi. Its face composition, vertex and edge counts, volume and surface area formulas, and its relation to the golden ratio are explained at an intermediate level.
Pentakis Dodecahedron — Pentakis Dodecahedron: a Catalan solid and the dual of the truncated icosahedron. Its face composition, vertex and edge counts, and volume and surface area formulas are explained at an intermediate level.
Deltoidal Icositetrahedron — Deltoidal Icositetrahedron: a Catalan solid with 24 kite-shaped faces. Its face shape, vertex configuration, dual polyhedron, and key properties are explained at an intermediate level.
Disdyakis Dodecahedron — Disdyakis Dodecahedron: a Catalan solid and the dual of the truncated cuboctahedron. Its face shape, vertex configuration, its relationship to the truncated cuboctahedron, and its properties are explained at an intermediate level.
Triakis Octahedron — Triakis Octahedron: a Catalan solid and the dual of the truncated cube. Its face shape, vertex configuration, its dual relationship with the truncated cube, and its properties are explained at an intermediate level.
Small Stellated Dodecahedron — Small Stellated Dodecahedron: one of the four Kepler–Poinsot polyhedra. The concept of star polyhedra, the face shape (pentagrammic faces), and its properties as a Kepler–Poinsot solid are explained at an intermediate level.
Great Stellated Dodecahedron — Great Stellated Dodecahedron: a Kepler–Poinsot polyhedron with Schläfli symbol {5/2,3}. Its face shape, its properties as a Kepler–Poinsot solid, and its relation to the golden ratio are explained at an intermediate level.
Great Dodecahedron — Great Dodecahedron: a Kepler–Poinsot polyhedron with Schläfli symbol {5,5/2}. Its face shape, its position among the Kepler–Poinsot polyhedra, and its relationship to its dual polyhedron are explained at an intermediate level.
Great Icosahedron — Great Icosahedron: a Kepler–Poinsot polyhedron with Schläfli symbol {3,5/2}. Its face shape, its dual relationship with the great stellated dodecahedron, and its vertex coordinates are explained at an intermediate level.
Stella Octangula — Stella Octangula: a star polyhedron formed as the compound of two regular tetrahedra. Its vertex, edge, and face counts, volume and surface area formulas, and its relationship to dual polyhedra are explained.
Deltahedron — Deltahedron: a polyhedron whose faces are all equilateral triangles. The eight convex deltahedra — tetrahedron, triangular bipyramid, triangular antiprism, dipyramid variants, snub disphenoid, and icosahedron — are classified with their edges, faces, and vertices described.
Snub Disphenoid — Snub disphenoid (Johnson solid J84): a convex deltahedron with 12 equilateral triangular faces, 8 vertices, and 18 edges. Its vertex coordinates, classification as a Johnson solid, and its place among the eight convex deltahedra are explained.
Szilassi Polyhedron — Szilassi polyhedron: a toroidal polyhedron with 7 hexagonal faces, 14 vertices, and 21 edges in which every pair of faces shares an edge. Its connection to the seven-color theorem for the torus and its dual relationship with the Csaszar polyhedron are explained.
Csaszar Polyhedron — Csaszar polyhedron: a toroidal polyhedron with 7 vertices, 21 edges, and 14 triangular faces in which every pair of vertices is connected by an edge. Its toroidal topology, dual relationship with the Szilassi polyhedron, and vertex coordinates are explained.
Scutoid — Scutoid: a solid discovered in 2018 with pentagonal and hexagonal parallel faces joined by a triangular lateral face. Its geometric properties, its role in the packing of epithelial cells in living tissue, and its vertex, edge, and face counts are explained.
Geodesic Polyhedron — Geodesic polyhedron: a sphere-approximating polyhedron formed by triangulating a sphere, classified by frequency and class (I, II, III). Formulas for vertex, edge, and face counts, the connection to Fuller's geodesic dome, and the dual Goldberg polyhedron are explained.
Square Pyramid — Square pyramid: a pyramid with a square base, 5 vertices, 8 edges, and 5 faces. The volume formula V=sh/3, surface area, slant height, verification of Euler's polyhedron formula, and its classification as a Johnson solid are explained.
Triangular Cupola — Triangular cupola (Johnson solid J3): a cupola with a triangular top, 9 vertices, 15 edges, and 8 faces. Its volume formula, surface area, symmetry group, and relationship to Archimedean solids are explained.
Koch Snowflake — Koch snowflake: a fractal built by repeatedly replacing each side of an equilateral triangle with four segments. The divergence of its perimeter, convergence of its area, Hausdorff dimension log4/log3, self-similarity, and relationship to the Koch curve are explained.
Sierpinski Triangle — Sierpinski triangle: a fractal formed by iteratively removing the central triangle from an equilateral triangle. The convergence of area to 0, Hausdorff dimension log3/log2, construction via the chaos game, and its connection to binomial coefficients are explained.
Sierpinski Carpet — Sierpinski carpet: a planar fractal formed by iteratively removing the central ninth of a square. The convergence of area to zero, Hausdorff dimension log8/log3, and its property as a universal plane curve are explained.
Menger Sponge — Menger sponge: a three-dimensional fractal formed by iteratively removing the central seventh of each face of a cube. The convergence of volume to zero, divergence of surface area to infinity, Hausdorff dimension log20/log3, and its property as a universal curve are explained.
Dragon Curve — Dragon curve (Heighway–Harter curve): a fractal generated by repeatedly folding a strip of paper in half. Its L-system description, fractal dimension of 2, plane-tiling property, and other mathematical properties are explained.
Hilbert Curve — Hilbert curve: a space-filling curve constructed recursively so that it passes through every point of the unit square in the limit. Its fractal dimension of 2, construction procedure, numerical properties, and applications in image processing and spatial indexing are explained.
Peano Curve — Peano curve: the first known space-filling curve, constructed via a ternary recursive subdivision to cover the unit square. Its fractal dimension of 2, historical significance as a continuous surjection from an interval to a square, and modern applications are explained.
Levy C Curve — Levy C curve: a self-similar fractal generated by repeatedly replacing each segment with two sides of an isosceles right triangle. Its fractal dimension, self-similar tiling property, IFS description, and numerical examples are explained.
Gosper Curve — Gosper curve (flowsnake or hexagonal Peano curve): a space-filling fractal generated by an L-system substitution on a triangular grid. Its fractal dimension, L-system construction, and the self-similar hexagonal Gosper island tiling are explained.
Apollonian Gasket — Apollonian gasket: a fractal formed by repeatedly filling the interstices between mutually tangent circles with new tangent circles. Descartes' circle theorem (Soddy's formula), recursive curvature computation, fractal dimension, integer-curvature gaskets, and comparison with the Sierpinski gasket are explained.
Pythagoras Tree — Pythagoras tree: a fractal tree constructed by attaching squares to each leg of right triangles derived from the Pythagorean theorem. Its fractal dimension, symmetric and asymmetric variations, numerical examples, and applications are explained.
Vicsek Fractal — Vicsek fractal: a fractal formed by iteratively replacing a square with a plus-sign or diagonal-cross pattern of five smaller squares. Its fractal dimension log5/log3, IFS description, comparison with the Sierpinski carpet, and applications in physics are explained.
Penrose Tiling — Penrose tiling: an aperiodic tiling of the plane using two tile sets (P2 kite-and-dart, P3 thick-and-thin rhombus). Inflation rules, fivefold symmetry, and the connection to quasicrystals are explained.
Wallpaper Group — Defines the wallpaper groups and explains the classification into 17 types, the classification by rotations, reflections, and glide reflections, the crystallographic restriction theorem, and concrete pattern examples.
Frieze Group — Frieze group: one of the seven symmetry groups of infinite strip patterns. All seven classes, their international symbols, and their symmetry elements — translation, reflection, glide reflection, and 180-degree rotation — are explained with concrete pattern examples.
Pinwheel Tiling — Explains the definition of the pinwheel tiling, its 1:2:√5 right-triangle tile, the property that edge orientations rotate densely, its substitution rule, and statistical isotropy.
Pentagonal Pyramid — Pentagonal pyramid: a pyramid with a regular pentagonal base, 6 vertices, 10 edges, and 6 faces. Verification of Euler's formula, volume and surface area formulas, classification as a Johnson solid, and inscribed and circumscribed spheres are explained.
Triangular Bipyramid — Triangular bipyramid: a solid with 5 vertices, 9 edges, and 6 faces. Its Euler characteristic, volume and surface area formulas, classification as Johnson solid J12, and its status as a deltahedron are explained.
Pentagonal Bipyramid — Pentagonal bipyramid: a solid with 7 vertices, 15 edges, and 10 faces. Its volume and surface area formulas, classification as Johnson solid J13, its status as a deltahedron, and its relationship to the regular icosahedron are explained.
Square Cupola — Explains the definition of the square cupola, its numbers of faces, edges, and vertices, its symmetry, its net, its classification as a Johnson solid, its dual polyhedron, and its applications.
Pentagonal Cupola — Pentagonal cupola: a Johnson solid J5 with its face, edge, and vertex counts, coordinate and volume formulas, symmetry properties, and relationship to the rhombicosidodecahedron are explained.
Pentagonal Rotunda — Pentagonal rotunda: a Johnson solid J6 with its face, edge, and vertex counts, volume formula, symmetry properties, and relationships to the rhombicosidodecahedron and the icosahedron are explained.
Gyrobifastigium — Gyrobifastigium: a Johnson solid J26 with its face, edge, and vertex counts, volume formula, symmetry properties, and its remarkable status as a space-filling polyhedron are explained.
Goldberg Polyhedron — Goldberg polyhedron: the definition, GP(h,k) notation, face/edge/vertex counting, relationship to the soccer ball, and applications to viral capsids and fullerenes are explained.
Compound of Five Cubes — Explains the definition of the compound of five cubes, its relationship to the dodecahedron, the icosahedral symmetry group I_h, the number of its faces, edges, and vertices, its convex hull, and its dual polyhedron.
Compound of Five Tetrahedra — Compound of five tetrahedra: a chiral compound related to the regular dodecahedron and icosahedron, with icosahedral rotation symmetry group I, its face/edge/vertex counts, and its dual (the compound of five inverted tetrahedra) are explained.
Uniform Polyhedron — Uniform polyhedron: the definition (vertex-transitivity and regular polygon faces), classification into Platonic solids, Archimedean solids, prisms, and antiprisms, Schläfli symbols, and relationships to dual polyhedra are explained.
Space-Filling Polyhedron — Space-filling polyhedron: the definition, filling conditions (face-to-face contact, no gaps, no overlaps), major examples including the cube, truncated octahedron, and rhombic dodecahedron, and the Kelvin problem with the Weaire–Phelan structure are explained.
Barnsley Fern — Barnsley fern: the definition, construction via an iterated function system (IFS) of four affine contractions, self-similarity of the attractor, the chaos game algorithm for rendering, and fractal dimension are explained.
Koch Curve — Koch curve: the construction procedure, self-similarity, fractal dimension (log4/log3), the reason its length diverges to infinity, and its relationship to the Koch snowflake are explained.
T-Square Fractal — T-Square fractal: the construction procedure, self-similarity, why its fractal dimension equals 2, the calculation of area convergence, and comparisons with related fractals are explained.
H Tree — H Tree: the construction procedure, the self-similar branching structure formed by repeated H-shapes, why its fractal dimension equals 2, and applications in integrated circuit and VLSI design are explained.
Box-Counting Dimension — Box-counting dimension (Minkowski dimension): the definition, the grid-covering computation procedure, concrete applications to the Koch curve and Sierpinski triangle, and its relationship to the Hausdorff dimension are explained.
L-System — L-system (Lindenmayer system): the definition, alphabet, axiom, and rewriting rules, geometric figure generation via turtle graphics, and applications to the Koch curve, Sierpinski triangle, and plant models are explained.
Iterated Function System — Iterated function system (IFS): the definition, existence of attractors via contractive mappings and Banach's fixed-point theorem, the collage theorem, applications to the Barnsley fern and Sierpinski triangle, and the connection to fractal compression are explained.
Chaos Game — Chaos game: the definition, the probabilistic algorithm, applications to the Sierpinski triangle and pentagon fractal, the relationship to iterated function systems, and a proof of convergence are explained.
Aperiodic Tiling — Aperiodic tiling: the definition, quasi-periodicity, representative examples including Penrose tiles and the einstein tile, the concept of forcing, and the connection to quasicrystals are explained.
Truchet Tiles — Truchet tiles: the definition, connection-pattern generation from two orientations of square tiles, extension to Smith arc tiles, and applications in random generation, percolation, and computer graphics are explained.
Einstein Tile — Einstein tile: the definition, the hat and turtle tiles discovered in 2023, the spectre tile that achieves aperiodic tiling without reflections, and the distinction from aperiodic tile sets are explained.
Schwarz Triangle — Schwarz triangle: the definition, triangulations of the sphere, plane, and hyperbolic plane, the Schwarz triangle symbol (p q r), correspondence with reflection groups, and applications to the Schwarz–Christoffel mapping are explained.
Pentagram — Defines the pentagram and explains its exact construction, the sum of its interior angles, its relation to the golden ratio, its classification as a star polygon, and its historical applications.
Hexagram — Explains the definition of the hexagram, its Schläfli symbol, the sum of its interior angles, the superposition of two equilateral triangles, its area ratio, and its geometric properties.
Star Polygon — Star polygon: the definition, classification by Schläfli symbol {n/k}, the formula for the point angle, regularity conditions, and relationships to the pentagram and Star of David are explained.
Trapezohedron — Trapezohedron: the definition, its relationship to the antiprism as a dual polyhedron, face shape, vertex count, edge count, face count (Euler's formula), and concrete examples are explained.

Reading

The Hidden Symmetry of Tilings — Between Wallpaper and Quasicrystals [Essay]
Why are there only 17 kinds of symmetry for floor tilings? A relaxed tour from Penrose tiles, which broke the ban on five-fold symmetry, all the way to quasicrystals.
Conics That Draw the Heavens — How a Cross-Section Became a Planet's Orbit [Essay]
Slice a cone at an angle and ellipses, parabolas, and hyperbolas all appear. A relaxed story of how an ancient Greek geometric pastime came to match the orbits of planets and comets exactly.
Why You Cannot Flatten an Orange Peel — Curvature Seen from Within [Essay]
Why won't an orange peel lie flat, and why must every world map distort? Curvature can be known from inside a surface without looking from outside — a relaxed take on Gauss's Theorema Egregium.

Prerequisites

The content of Geometry (Basic)
Multivariable calculus (partial derivatives, multiple integrals, vector calculus)
Linear algebra (eigenvalues, bilinear forms)
Basic concepts of topological spaces

The Concept of Curvature

Quantifying how curves and surfaces bend.

Frequently Asked Questions

What do you learn in intermediate geometry?

The intermediate level covers curve theory (Frenet formulas, curvature, torsion), surface theory (first and second fundamental forms, Gaussian curvature, the Gauss–Bonnet theorem), the foundations of manifolds (tangent spaces, tangent bundles, orientation), an introduction to Riemannian geometry (geodesics, connections, the curvature tensor), topology (fundamental group, Euler characteristic), and distance in n-dimensional space.

What is the difference between differential geometry and topology?

Differential geometry assumes a smooth structure on a manifold (differentiability and a metric) and studies notions involving derivatives, such as curvature, geodesics, and connections. Topology assumes only continuity and studies properties invariant under continuous deformation (the fundamental group, homology groups, the Euler characteristic). Differential geometry deals with finer structure, topology with coarser, more global structure.

What does the Gaussian curvature of a surface mean?

The Gaussian curvature $K=\kappa_1\kappa_2$ (the product of the principal curvatures) expresses the intrinsic bending of a surface. When $K>0$ (spherical) parallel lines converge, when $K<0$ (saddle) they diverge, and when $K=0$ (flat) they stay parallel. By the Gauss–Bonnet theorem, the integral of $K$ equals the Euler characteristic, a topological invariant.