Guides

Walkthroughs of the mathematics, each one following a family of objects far enough to see how it is built and what it is good for. Live cells throughout — a claim you can re-evaluate is worth more than one you have to take on faith.

Guides explain. The explorations are for tinkering: fewer words, more dials. Where a guide names an object worth turning over in your hands, it links to the exploration that lets you.

Each guide is a page of the package it introduces, in that package's docs/; the docs list them with the rest of each package's pages.

Counting

• Ranking and unranking — a combinatorial family as an indexed collection: Count and At over subsets, partitions, Dyck paths and permutations, answered without building the list.
• Numeral systems — one base slot, nine systems: factoradic, Zeckendorf, balanced, negative, bijective, mixed, primorial, combinatorial and residue — two of which are the unranking maps above in disguise.
  • Complex bases — a Gaussian or Eisenstein base, one digit per residue class, and the fractal tiles its numerals make.
• Adèles and idèles — numbers known modulo m as values: profinite integers and rationals, the adèle ring and idèle group of ℚ, profinite Fibonacci numbers and strong approximation, after Hertogh's Sage package.
• Quadratic integers — the rings $\mathcal{O}_d$ beyond the Gaussian integers on a lattice: norms, primes, and the irreducibles that are not prime where the class number exceeds 1.

Algebras

Each of these introduces an extension library and then follows where it leads. The heads they declare are catalogued in the symbol reference.

• Hypercomplex algebras — six families of imaginary unit as ordinary subscripted symbols: multicomplex, split, dual, Clifford and Grassmann, with arithmetic, norms and inverses on compute-engine's own operators.
  • Finite: ℤ/m and the places — where these units already live, one CRT channel at a time.
• Diagram algebras — the partition algebra and its subalgebras (Brauer, Temperley–Lieb, Motzkin, rook, symmetric group), whose bases are pictures and whose dimensions are the Bell, Catalan and Motzkin numbers.
• Hecke algebras — $H_n(q)$, the q-deformation of the symmetric group algebra: same basis, deformed multiplication, and the group algebra back at $q = 1$.
• Incidence algebras — the algebra of a poset's intervals, where the zeta function's inverse is the Möbius function, and specialising the poset recovers number theory's $\mu$ and inclusion–exclusion.
• Path algebras — a quiver's paths under concatenation, and the first family here that need not be finite-dimensional: one loop and there is no basis at all.
• Combinatorial Hopf algebras — NSym and QSym on compositions: the first structures here with a coproduct, an antipode, and the compatibility that ties them to the product.
• Group algebras — k[G] for cyclic, dihedral and product groups, where the interesting part is the centre: a commutative subalgebra spanned by class sums, one per conjugacy class.

Groups, words and knots

• The modular group — PSL(2,Z) as a free product, its words as continued fractions, and its closed geodesics as binary necklaces — which, by Ghys's theorem, are knots in the complement of a trefoil.
• Knots and braids — the Artin presentation, closures, the Burau representation and Alexander polynomials, and the Lorenz braids that turn a modular geodesic into an actual knot.

Not here

How an expression is written and what it comes back out as — LaTeX, MathJSON, Wolfram, NumPy, the shader languages — is reference material, not a walkthrough: see formats. The components these pages are built from have their own reference, and the playground exercises each one in isolation.

The analytic special functions — Hurwitz zeta, the Lerch transcendent, the polylog and polygamma of @enumeratio/analytic — are turned over with sliders rather than explained at length, in the explorations. Running any of this outside a browser is the command line's job.