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探奇小径
为好奇者准备的可选支路。它们从不计入成绩，也总会带你回到原来的位置。
- 幼儿园至二年级Boxes inside boxesWhat is inside a box that contains only an empty box? Is it empty? And can every description you can think of make a collection?起点 一组对象 →SetsRussell's paradox and why sets need axiomsZFC: axiomatic set theory踏上小径
- 三至五年级Corners, edges and facesCount the corners, edges and faces of a cube, then work out corners − edges + faces. Try a pyramid and a tetrahedron too. Why do you keep getting the same number — and can you find a shape that gives a different one?起点 Shapes →Surfaces, polyhedra and the Euler characteristicAlgebraic topologyRiemannian geometry踏上小径
- 六至八年级How many infinities?There are infinitely many counting numbers and infinitely many infinite 0–1 sequences. Are these two infinities the same size?起点 Sets →Countability and Cantor's diagonal argument踏上小径
- 六至八年级Primes forever2, 3, 5, 7, 11, 13, … The primes get rarer as the numbers grow. Do they ever run out? And how rare do they get?起点 Factors, multiples and primes →Divisibility, Euclid's algorithm and unique factorisationNumber theoryAnalytic number theory踏上小径
- 六至八年级Halfway to the wallWalk halfway to a wall. Then walk half of what is left, then half of that, and so on, forever. Do you ever reach the wall? How can infinitely many steps add up to a finite distance?起点 Fractions and decimals →Sequences and seriesReal analysisMeasure theory and integration踏上小径
- 六至八年级Symmetry makes groupsA square can be turned or flipped and still fit exactly into its outline. How many different moves like that are there? And what do you get if you do one move and then another?起点 Symmetry: slides, turns and flips →GroupsMatrix groups and continuous symmetryLie groups and Lie algebras踏上小径
- 六至八年级Knots and coloursTie a knot in a piece of string and glue the two ends together. How could you prove that it really is knotted — that no amount of wiggling, without cutting, will turn it into a plain loop?起点 Dots and lines: networks and puzzles →Knot theoryLow-dimensional topology: 3- and 4-manifoldsCategorical representation theory踏上小径
- 九至十二年级Can a computer decide?Some programs stop and some loop forever. Could someone write one clever program that looks at any program, together with its input, and always correctly says whether it will stop?起点 Repetition and decisions →ComputabilityGödel's incompleteness theoremsComputational complexity踏上小径
- 六至八年级Clock arithmeticIt's 9 o'clock. What time will it be in 5 hours — and in 29 hours? Can a clock teach us a whole new kind of arithmetic?起点 Integers →Clock arithmetic: remainders and modular arithmeticCongruencesQuotient groups and quotient rings踏上小径
- 六至八年级Inverses and structureEvery integer can be added to another integer to get . So why can almost no integer be multiplied by another integer to get ?起点 Integers →Algebraic structures: monoids, groups and ringsGroupsRings, ideals and fields踏上小径
- 六至八年级Remainder familiesSort every integer — positive, negative and zero — into families by its remainder after dividing by 3. If you add a member of one family to a member of another, does the family of the answer depend on which members you chose?起点 Integers →Relations, equivalence relations and partitionsCongruencesQuotient groups and quotient rings踏上小径
- 九至十二年级The product, by its mapsCan you describe the set of pairs without ever looking inside it — only by how functions go into it?起点 Functions →Universal propertiesLimits and colimits踏上小径
- 六至八年级Life on a torusIn some video games, walking off the right edge of the screen brings you back on the left, and walking off the top brings you back at the bottom. If that world were a real surface, what shape would it be?起点 The coordinate plane →Surfaces, polyhedra and the Euler characteristicHomotopy and the fundamental groupLow-dimensional topology: 3- and 4-manifolds踏上小径
- 六至八年级Triangles on a ballOn flat paper the angles of every triangle add up to 180°, and a² + b² = c² in every right triangle. Draw a triangle on a ball instead. Is either rule still true?起点 The Pythagorean theorem →Non-Euclidean geometry: spherical and hyperbolicDifferential geometry of curves and surfacesRiemannian geometry踏上小径
- 六至八年级Will the walker return?A walker tosses a fair coin every minute: heads, one step east; tails, one step west. Is she certain to come back to where she started? What if she can also step north and south — or up and down in space?起点 Probability →Markov chainsStochastic processesRandom structures: walks, percolation and random geometry踏上小径
- 九至十二年级Why do planets move on ellipses?Planets move around the Sun on ellipses, with the Sun at one focus. Why ellipses — and not circles, egg shapes or spirals?起点 Conic sections →Classical mechanicsThe geometry of spacetime: special and general relativitySymplectic geometry and topology踏上小径
- 九至十二年级Loops around a holeWalk around a tree and come back to where you started. How many times did you go around it? Can you change that number by wandering differently — without ever walking through the tree?起点 Complex numbers →Homotopy and the fundamental groupComplex analysisAlgebraic topology踏上小径
- 九至十二年级How sharply does a road bend?Driving along a winding road, how sharply does it bend at each point? Is there a circle that fits the road best at each point — and what is its radius?起点 Introductory calculus →Differential geometry of curves and surfacesRiemannian geometryThe geometry of spacetime: special and general relativity踏上小径
- 九至十二年级Arrows that composeWhat do functions between sets, maps between groups and continuous maps between shapes have in common? What if we studied only the arrows?起点 Composition →CategoriesFunctors踏上小径