# The front matter: welcome, the formula, the color key *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#page-book, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Chapter ᚏ(R)](ch01.md) # Teclam Áirme ## Welcome to the Teclam Áirme! I created this Mathematical Compendium for my daughter Rose, but all are welcome. As an English major, my goal was to present it as a light, artistic “performance” designed to engage, rather than as a dry analytical tome designed to prove—I’ll leave the proofs to those more capable, and I am grateful so many have dedicated their time over the centuries to give us such a rich playground. In that spirit, the chapter headings use Pictish Ogham—the Scottish inheritance of an ancient Celtic alphabet, carved on some three dozen stones from Aberdeenshire to Shetland—which, hilariously ... has no numerals at all. For the structure, I built a mathematical formula that basically proves nothing: its answer is settled before you start, and only the reasons keep changing. Rather than the traditional goal of mathematics to reduce something to its simplest form, I want the opposite and tried to create an incredibly diverse formula with intricacies reaching across a broad spectrum. The rich set of mathematical symbols are the fabric, with 0 acting as the threads that weave it together. Bonus: I invented a mathematical symbol ([ᚏ_4.16](ch01-cards.md#g-rose)) in the hope of inspiring anyone (even us English majors) to enjoy and create with math! For my more math-y kin, see the [mathematical and artistic disclaimer](appendices.md). **Teclam Áirme** ᚛ᚈᚓᚉᚂᚐᚋ ᚐᚔᚏᚋᚓ᚜ — “Mathematical Compendium”: a gathering or collection of calculation and numbers. ## Teclam Áirme᚛ᚈᚓᚉᚂᚐᚋ ᚐᚔᚏᚋᚓ᚜ This non-exhaustive compendium exists to document the formula below, which in essence says 0 = 0. Each chapter works in a different mathematical framework, chosen from an abundance of candidates mostly because it brings its own symbols or has an interesting way of arriving at 0. ### Color Key (That’s right, math ... in color lol) Anything on a gray background has a popup: hover over it to read it, or click or tap it to go to its glossary card. If it is a number, a colored border says what kind: #### Type of number Integer or rational · Algebraic irrational · Proven transcendental · Status open · Definable but uncomputable · Undefinable · Non-real object · Infinity or limit device · Function or operator · #### Physical Physics: fixed by definition · Physics: measured · Physics: derived · ## The formula^{[*](front.md)} [ᚏ_4.16](ch01-cards.md#g-rose) [:=](ch01-cards.md#g-defsym) x / 4.16, for one x [∈](ch01-cards.md#g-elem) [ℝ](ch01-cards.md#g-reals) [∖](ch01-cards.md#g-setminus) [D](ch01-cards.md#g-defreals), conceived once and held tightly. [ᚏ(R)](ch01.md)The gateway([ᚏ_4.16](ch01-cards.md#g-rose))/(([𝟘](ch01-cards.md#g-tropical))) = [᚛(Prologue)](ch02.md)0ur starting point( [𒑊](ch02-cards.md#g-babyzero)·[𝋠](ch02-cards.md#g-mayazero)·[ο](ch02-cards.md#g-greekzero)·[nulla](ch02-cards.md#g-dionzero)·[०](ch02-cards.md#g-devzero)·[០](ch02-cards.md#g-khmerzero)·[N](ch02-cards.md#g-bedezero)·[٠](ch02-cards.md#g-arabzero)·[○](ch02-cards.md#g-gwalior)·[•](ch02-cards.md#g-bakhzero)·[0](ch02-cards.md#g-latinzero)·[〇](ch02-cards.md#g-chinazero)·[零](ch02-cards.md#g-jpzero) ) · [ᚁ(B)](ch03.md)The actual world( (([Λ](ch03-cards.md#g-Lambda)·[G](ch03-cards.md#g-G)·[ℏ](ch03-cards.md#g-hbar))/([c](ch03-cards.md#g-c)³))·([Ω_Λ](ch03-cards.md#g-OmegaL) + [Ω_m](ch03-cards.md#g-Omegam) + [Ω_r](ch03-cards.md#g-Omegar) + [Ω_k](ch03-cards.md#g-Omegak)) {brace: gravity} · (([k_B](ch03-cards.md#g-kB)·[T_CMB](ch03-cards.md#g-Tcmb))/([m_e](ch03-cards.md#g-me)·[c](ch03-cards.md#g-c)²))·(([b](ch03-cards.md#g-wien)·[x_W](ch03-cards.md#g-wienx)·[k_B](ch03-cards.md#g-kB))/([h](ch03-cards.md#g-h)·[c](ch03-cards.md#g-c))) {brace: heat} · ([N_A](ch03-cards.md#g-NA)·[1 mol](ch03-cards.md#g-mol))·(([R](ch03-cards.md#g-gasR))/([N_A](ch03-cards.md#g-NA)·[k_B](ch03-cards.md#g-kB)))·(([F](ch03-cards.md#g-faraday))/([N_A](ch03-cards.md#g-NA)·[e](ch03-cards.md#g-echarge))) {brace: amount} · (([σ](ch03-cards.md#g-sigma)·[ℏ](ch03-cards.md#g-hbar)³·[c](ch03-cards.md#g-c)²)/([k_B](ch03-cards.md#g-kB)⁴)) {brace: light} · ((([e](ch03-cards.md#g-echarge)²)/(4π·[ε₀](ch03-cards.md#g-eps0)·[ℏ](ch03-cards.md#g-hbar)·[c](ch03-cards.md#g-c))))/([α](ch03-cards.md#g-alpha))·([k_e](ch03-cards.md#g-coulombk)·4π·[ε₀](ch03-cards.md#g-eps0))·((2·[h](ch03-cards.md#g-h)·[R_∞](ch03-cards.md#g-rydberg))/([α](ch03-cards.md#g-alpha)²·[m_e](ch03-cards.md#g-me)·[c](ch03-cards.md#g-c))) {brace: electricity} · ([μ₀](ch03-cards.md#g-mu0)·[ε₀](ch03-cards.md#g-eps0)·[c](ch03-cards.md#g-c)²)·(([e](ch03-cards.md#g-echarge)²·[R_K](ch03-cards.md#g-RK))/([h](ch03-cards.md#g-h)))·(([Z₀](ch03-cards.md#g-Z0))/([μ₀](ch03-cards.md#g-mu0)·[c](ch03-cards.md#g-c)))·([Φ₀](ch03-cards.md#g-fluxq)·[K_J](ch03-cards.md#g-josephson)) {brace: magnetism} · ([m_p/m_e](ch03-cards.md#g-mpme)·[α_s](ch03-cards.md#g-alphas)·[sin²θ_W](ch03-cards.md#g-weakmix)·[a_e](ch03-cards.md#g-ae)·[α_G](ch03-cards.md#g-alphaG)) {brace: particle physics} {brace: [physical constants](ch03.md#ch-T3-physical-constants)} + [ds²_light](ch03-cards.md#g-dslight) + [∇·𝐁](ch03-cards.md#g-divB) + [dQ/dt](ch03-cards.md#g-noether) + [(E − mc²)](ch03-cards.md#g-emc2) + [∇_μG^μν](ch03-cards.md#g-bianchi) + [∂_μj^μ](ch03-cards.md#g-continuity) {brace: [physical laws](ch03.md#ch-T3-physical-laws)} ) · [ᚂ(L)](ch04.md)Garden variety( (([(9³ + 10³) − (1³ + 12³)](ch04-cards.md#g-taxicab)))/(([0!](ch04-cards.md#g-zerofact) + [4 + 9](ch04-cards.md#g-squares) + [20](ch04-cards.md#g-twenty) + [42](ch04-cards.md#g-fortytwo) + [60](ch04-cards.md#g-sixty) + [6 + 28 + 496 + 8128](ch04-cards.md#g-perfect) + [142857](ch04-cards.md#g-cyclic) + [6174](ch04-cards.md#g-kaprekar) + [Belphegor’s](ch04-cards.md#g-belphegor) + [220 + 284](ch04-cards.md#g-amicable) + [561](ch04-cards.md#g-carmichael) + [F₁₂](ch04-cards.md#g-fib144) + [7!](ch04-cards.md#g-fact7) + [5#](ch04-cards.md#g-primorial5) + [65537](ch04-cards.md#g-fermat4)) {brace: small} · ([M₃₁](ch04-cards.md#g-m31)·[3↑↑3](ch04-cards.md#g-knuth3)·[10⁶³](ch04-cards.md#g-sand)·[無量大数](ch04-cards.md#g-muryo)·[136·2²⁵⁶](ch04-cards.md#g-eddington)·[10¹⁰⁰](ch04-cards.md#g-googol)·[10¹²⁰](ch04-cards.md#g-shannon)·[M_136279841](ch04-cards.md#g-lkp)·[10¹⁰¹⁰⁰](ch04-cards.md#g-googolplex)·[Skewes’](ch04-cards.md#g-skewes)·[Moser’s](ch04-cards.md#g-moser)·[G₆₄](ch04-cards.md#g-g64)·[TREE(3)](ch04-cards.md#g-tree3)·[SSCG(3)](ch04-cards.md#g-sscg3)) {brace: GARGANTUAN} ) {brace: [integers](ch04.md#ch-T4-integers)} · ((3·[⅓](ch04-cards.md#g-third) − 1) + ([0.999…](ch04-cards.md#g-nines) − 1))/(([22/7 + 355/113](ch04-cards.md#g-piapprox) + [B₂](ch04-cards.md#g-bern2) + [H₄](ch04-cards.md#g-h4))) {brace: [rationals](ch04.md#ch-T4-rationals)} · (([φ² − φ − 1](ch04-cards.md#g-phipoly)))/(([φ](ch04-cards.md#g-phi) + [√2](ch04-cards.md#g-sqrt2) + [δ_S](ch04-cards.md#g-silver) + [cos(2π/17)](ch04-cards.md#g-cos17)) {brace: [constructible](ch04-cards.md#g-constructible)} · ([ρ](ch04-cards.md#g-plastic) + [∛2](ch04-cards.md#g-cbrt2) + [2cos(2π/7)](ch04-cards.md#g-cos7) + [τ₃](ch04-cards.md#g-tribonacci) + [λ_L](ch04-cards.md#g-lehmer) + [Conway’s](ch04-cards.md#g-conway)) {brace: [not constructible](ch04-cards.md#g-nonconstructible)} ) {brace: [algebraic](ch04.md#ch-T4-algebraic)} · ([sin π](ch04-cards.md#g-sinpi)·([√2](ch04-cards.md#g-sqrt2)·[2^√2](ch04-cards.md#g-gelfond)·([Champernowne](ch04-cards.md#g-champ))/([Copeland–Erdős](ch05-cards.md#g-cope)))^([∛2](ch04-cards.md#g-cbrt2)))/(([e](ch04-cards.md#g-e) + [π](ch04-cards.md#g-pi) + [2/π](ch04-cards.md#g-twoverpi) + [ln 2](ch04-cards.md#g-ln2) + [sin 1 + cos 1](ch04-cards.md#g-sincos1) + [Laplace limit](ch04-cards.md#g-laplacelim)) {brace: [Hermite, Lindemann](ch04-cards.md#g-lindweier)} · ([2^√2](ch04-cards.md#g-gelfond) + [e^π](ch04-cards.md#g-epi) + [iⁱ](ch04-cards.md#g-ii) + [e^π√163](ch04-cards.md#g-ram) + [log₂ 3](ch04-cards.md#g-log23) + [√2^√2](ch04-cards.md#g-sqrt2sqrt2)) {brace: [Gelfond, Schneider](ch04-cards.md#g-gelfschn)} · ([Liouville’s](ch04-cards.md#g-liou) + [Champernowne](ch04-cards.md#g-champ) + [Thue–Morse](ch04-cards.md#g-tm) + [q_KL](ch04-cards.md#g-komornik) + [P_fold](ch04-cards.md#g-paperfold)) {brace: built by hand} · ([ϖ](ch04-cards.md#g-lemn) + [Dottie](ch04-cards.md#g-dottie) + [Gauss’s](ch04-cards.md#g-gaussc) + [parabolic P](ch04-cards.md#g-parab) + [Cahen’s](ch04-cards.md#g-cahen) + [Γ(¼)](ch04-cards.md#g-gamma14)) {brace: other theorems} ) {brace: [transcendental](ch04.md#ch-T4-transcendental)} ) · [ᚃ(F)](ch05.md)The local park( (([Legendre’s](ch05-cards.md#g-legendre) − 1))/(([E_EB](ch05-cards.md#g-eb) + [ψ](ch05-cards.md#g-recipfib) + [Copeland–Erdős](ch05-cards.md#g-cope) + [ζ(3)](ch05-cards.md#g-apery) + [Mills’](ch05-cards.md#g-mills)) {brace: irrational, transcendence open} · ([γ](ch05-cards.md#g-gamma) + [δ](ch05-cards.md#g-feig) + |[α_F](ch05-cards.md#g-feigA)| + [G](ch05-cards.md#g-catalan) + [K](ch05-cards.md#g-khin) + [A](ch05-cards.md#g-glaisher) + [B₂](ch05-cards.md#g-brun) + [Gompertz](ch05-cards.md#g-gompertz) + [C₂](ch05-cards.md#g-twinprime) + [Meissel–Mertens](ch05-cards.md#g-mm) + [Golomb–Dickman](ch05-cards.md#g-gd) + [Landau–Ramanujan](ch05-cards.md#g-lr) + [ζ(5)](ch05-cards.md#g-zeta5) + [γ_L](ch05-cards.md#g-levy) + [F](ch05-cards.md#g-fransen) + [β](ch05-cards.md#g-bernstein) + [1.1319…](ch05-cards.md#g-viswanath) + [c_N](ch05-cards.md#g-niven) + [C_Artin](ch05-cards.md#g-artin) + [K_S](ch05-cards.md#g-sierpinski) + [MRB](ch05-cards.md#g-mrb) + [σ](ch05-cards.md#g-somos) + [ρ_KB](ch05-cards.md#g-keplerb) + [B_N](ch05-cards.md#g-backhouse) + [π^π + e^e](ch05-cards.md#g-pipi) + [(π + e) + π·e + π^e](ch05-cards.md#g-pie)) {brace: not even known irrational} ) {brace: [unresolved constants](ch05.md#ch-T5-unresolved-constants)} · ([lim](ch05-cards.md#g-limop)_{n→[∞](ch05-cards.md#g-inf)}(([TREE(3)](ch04-cards.md#g-tree3))/([Rayo’s](ch05-cards.md#g-rayo)))^(n))/(([Ω](ch05-cards.md#g-omega)·[BB(6)](ch05-cards.md#g-bb6))) {brace: [uncomputable and infeasible](ch05.md#ch-T5-uncomputable-and-infeasible)} ) · [ᚄ(S)](ch06.md)The city streets( ([(12! + 1) mod 13](ch06-cards.md#g-wilson))/([φ(13)](ch06-cards.md#g-totient) + [2¹² mod 13](ch06-cards.md#g-fermatlittle) + [13 mod 4](ch06-cards.md#g-mod4) + [(3/13)](ch06-cards.md#g-legsym) + [ord₁₃(2)](ch06-cards.md#g-ord13)) {brace: [modular arithmetic](ch06.md#ch-T6-modular-arithmetic)} · ([μ(6174)](ch06-cards.md#g-mobius) + ([gcd(12, 18)·lcm(12, 18) − 12·18](ch06-cards.md#g-gcdlcm)))/([gcd(12, 18)](ch06-cards.md#g-gcd1218) + [lcm(12, 18)](ch06-cards.md#g-lcm1218)) {brace: [divisibility](ch06.md#ch-T6-divisibility)} · ([p(9) mod 5](ch06-cards.md#g-partition5))/([p(4)](ch06-cards.md#g-p4)) {brace: [partitions](ch06.md#ch-T6-partitions)} · ([C(2, 5)](ch06-cards.md#g-binomial))/([R(3, 3)](ch06-cards.md#g-ramsey33) + [C₃](ch06-cards.md#g-catalan3) + [B₃](ch06-cards.md#g-bell3) + [!4](ch06-cards.md#g-derange4)) {brace: [combinatorics](ch06.md#ch-T6-combinatorics)} · ([P(K₃, 2)](ch06-cards.md#g-chrom) + ([10·3 − 2·15](ch06-cards.md#g-handshake)))/([P(K₃, 3)](ch06-cards.md#g-chromK33) + [|E(Petersen)|](ch06-cards.md#g-petersenE)) {brace: [graph theory](ch06.md#ch-T6-graph-theory)} ) · [ᚅ(N)](ch07.md)The stadium( ([3 ⊕ 5 ⊕ 6](ch07-cards.md#g-nim356))/([1 ⊕ 2 ⊕ 4](ch07-cards.md#g-nim124) + [2 ⊗ 2](ch07-cards.md#g-nimprod)) {brace: [combinatorial game theory](ch07.md#ch-T7-combinatorial-game-theory)} · ([v(matching pennies)](ch07-cards.md#g-mpvalue))/([½](ch07-cards.md#g-mixhalf) + [u(D, D)](ch07-cards.md#g-pdpayoff)) {brace: [classical game theory](ch07.md#ch-T7-classical-game-theory)} ) · [ᚆ(H)](ch08.md)The observation car( ([[∃ n, x, y, z ∈ ℤ⁺, n > 2 : xⁿ + yⁿ = zⁿ]](ch08-cards.md#g-fermat) + ([x ∧ ¬x](ch08-cards.md#g-boolean)))/([[1+1=2] + [√2 ∉ ℚ] + [∞ many primes] + [ℵ₀ < 𝔠]](ch08-cards.md#g-truths) + ([x ∨ ¬x](ch08-cards.md#g-excluded))) ) · [ᚇ(D)](ch09.md)The locomotive( (([(λf.λx.x) S 0](ch09-cards.md#g-church)) + [[∃ a halting decider]](ch09-cards.md#g-halting))/([BB(5)](ch09-cards.md#g-bb5) + [A(3, 3)](ch09-cards.md#g-ack) + [BB(1) + BB(2) + BB(3) + BB(4)](ch09-cards.md#g-bbsmall)) {brace: [computability](ch09.md#ch-T9-computability)} · [[P ⊆ NP]](ch09-cards.md#g-pnpsub) {brace: [complexity](ch09.md#ch-T9-complexity)} · ([fl₆₄(1+2⁻⁵³)−1](ch09-cards.md#g-float64zero))/([2⁻⁵²](ch09-cards.md#g-eps64)) {brace: [numerical analysis](ch09.md#ch-T9-numerical-analysis)} ) · [ᚈ(T)](ch10.md)The caboose( ([det [[1, 2], [2, 4]]](ch10-cards.md#g-detA) + [δ₁₂](ch10-cards.md#g-kronecker) + [(1, 2) · (2, −1)](ch10-cards.md#g-dot0) + [N²](ch10-cards.md#g-nilp) + [A² − 5A](ch10-cards.md#g-cayham))/([tr A](ch10-cards.md#g-trA) + [‖A‖_F](ch10-cards.md#g-frob) + [λ_max(A)](ch10-cards.md#g-lmax) + [rank A + null A](ch10-cards.md#g-ranknull)) ) · [ᚉ(C)](ch11.md)The gallery of motion( ([∫₀¹2x dx − 1](ch11-cards.md#g-ftc))/([(x²)′(1)](ch11-cards.md#g-dsq1)) {brace: [calculus](ch11.md#ch-T11-calculus)} · ([(∂_t−∂_xx)[e^−π²tsin(πx)]](ch11-cards.md#g-heatzero))/([π²](ch11-cards.md#g-decayrate)) {brace: [differential equations](ch11.md#ch-T11-differential-equations)} · ([min_x∈ℝx²](ch11-cards.md#g-minimumzero))/([min_x∈ℝ(x² + 1)](ch11-cards.md#g-minplus1)) {brace: [optimization](ch11.md#ch-T11-optimization)} · ([∇·(∇×𝐯)](ch11-cards.md#g-divcurl) + [∇×(∇f)](ch11-cards.md#g-curlgrad) + [∇²(1/r), r ≠ 0](ch11-cards.md#g-laplace1r) + [d(dω)](ch11-cards.md#g-ddzero))/(([∇·𝐫](ch11-cards.md#g-divr) + [∇²(r²)](ch11-cards.md#g-laprsq)) {brace: [divergence](ch11-cards.md#g-nabla)} · ([∮_S² (𝐫̂/r²)·d𝐀](ch11-cards.md#g-flux) + [½∮_S¹(x dy − y dx)](ch11-cards.md#g-greenarea)) {brace: [flux and circulation](ch11-cards.md#g-stokes)} ) {brace: [vector calculus](ch11.md#ch-T11-vector-calculus)} ) · [ᚊ(Q)](ch12.md)The gallery of shape( ([χ(T²)](ch12-cards.md#g-chiT) + [χ(Klein bottle)](ch12-cards.md#g-chiK) + [b₁(S²)](ch12-cards.md#g-b1S2) + ([∂(∂M)](ch12-cards.md#g-bdbd)))/(([χ(S²)](ch12-cards.md#g-chiS) + [χ(ℝP²)](ch12-cards.md#g-chiRP) + |[χ(Σ₂)](ch12-cards.md#g-chiS2g)| + [χ(•)](ch12-cards.md#g-chipt)) {brace: [Euler characteristics](ch12-cards.md#g-eulerchar)} · ([b₀(S²) + b₂(S²)](ch12-cards.md#g-b02S2) + [b₁(T²)](ch12-cards.md#g-b1T2) + [b₁(Σ₂)](ch12-cards.md#g-b1S2g)) {brace: [Betti numbers](ch12-cards.md#g-homology)} · ([g(T²)](ch12-cards.md#g-genusT)) {brace: [genus](ch12-cards.md#g-classsurf)} ) {brace: [topology](ch12.md#ch-T12-topology)} · ([c(○)](ch12-cards.md#g-unknot))/([c(3₁)](ch12-cards.md#g-trefoil) + [V_○(t)](ch12-cards.md#g-jonesunknot)) {brace: [knot theory](ch12.md#ch-T12-knot-theory)} · ([K(cylinder)](ch12-cards.md#g-cylinder))/([∫_S² K dA](ch12-cards.md#g-gaussbonnet)) {brace: [differential geometry](ch12.md#ch-T12-differential-geometry)} · ([3²+4²−5²](ch12-cards.md#g-pythag345))/([∠A + ∠B + ∠C](ch12-cards.md#g-anglesum) + [5](ch12-cards.md#g-platonic)) {brace: [Euclidean geometry](ch12.md#ch-T12-euclidean-geometry)} ) · [ᚋ(M)](ch13.md)The gallery of chance( ([P(X = ᚏ_4.16)](ch13-cards.md#g-probR) + [P(∅)](ch13-cards.md#g-pempty) + [Var(c)](ch13-cards.md#g-var0) + [Cov(X, Y), X ⟂ Y](ch13-cards.md#g-cov0) + [E[X − E[X]]](ch13-cards.md#g-meandev))/(([√(2π)](ch13-cards.md#g-sqrt2pi) + [1/e](ch13-cards.md#g-secretary) + [2/π](ch04-cards.md#g-twoverpi)) {brace: constants} · ([P(Ω)](ch13-cards.md#g-pomega) + [23](ch13-cards.md#g-birthday) + [2/3](ch13-cards.md#g-montyhall) + [E[fair die]](ch13-cards.md#g-diemean)) {brace: famous answers} ) {brace: [probability](ch13.md#ch-T13-probability)} · ([H(certain)](ch13-cards.md#g-entropy0) + [I(X; Y), X ⟂ Y](ch13-cards.md#g-mutinfo0) + [D_KL(P ‖ P)](ch13-cards.md#g-kl0))/([1 bit](ch13-cards.md#g-onebit) + [ln 2](ch04-cards.md#g-ln2)) {brace: [information theory](ch13.md#ch-T13-information-theory)} · ([E[X̄] − μ](ch13-cards.md#g-xbarunbiased) + [Σᵢ(yᵢ − ŷᵢ)](ch13-cards.md#g-olsresid))/([0.05](ch13-cards.md#g-fisher005) + [1.96](ch13-cards.md#g-z196)) {brace: [statistics](ch13.md#ch-T13-statistics)} · ([‖πP−π‖₁](ch13-cards.md#g-markovstationary))/([p₃](ch13-cards.md#g-polya3)) {brace: [stochastic processes](ch13.md#ch-T13-stochastic-processes)} ) · [ᚌ(G)](ch14.md)The hall of mirrors( (([[x, [y, z]] + [y, [z, x]] + [z, [x, y]]](ch14-cards.md#g-jacobi)) + ([[x, x]](ch14-cards.md#g-liealt)))/(([dim 𝔰𝔲(2)](ch14-cards.md#g-su2) + [dim G₂](ch14-cards.md#g-g2) + [dim F₄](ch14-cards.md#g-f4) + [dim E₆](ch14-cards.md#g-e6) + [dim E₇](ch14-cards.md#g-e7) + [dim E₈](ch14-cards.md#g-e8))) {brace: [Lie theory](ch14.md#ch-T14-lie-theory)} · (([e₁ ∧ e₁](ch14-cards.md#g-wedge)) + ([e₁e₂ + e₂e₁](ch14-cards.md#g-cliffanti)))/([dim Cl(3)](ch14-cards.md#g-cl3dim)) {brace: [geometric algebra](ch14.md#ch-T14-geometric-algebra)} · (([1 + ζ₅ + ζ₅² + ζ₅³ + ζ₅⁴](ch14-cards.md#g-rootsum)) + ([Σ_g∈G χ(g), χ irreducible, χ ≠ 1](ch14-cards.md#g-charsum)))/(([|A₅|](ch14-cards.md#g-a5) + [|S₅|](ch14-cards.md#g-s5) + [26](ch14-cards.md#g-sporadic26) + [196883](ch14-cards.md#g-moonshine) + [|𝕄|](ch14-cards.md#g-monster))) {brace: [group theory](ch14.md#ch-T14-group-theory)} · (([disc((x − 1)²)](ch14-cards.md#g-disc)) + [[the general quintic is solvable in radicals]](ch14-cards.md#g-abelbracket))/([[ℚ(√2) : ℚ]](ch14-cards.md#g-degsqrt2) + [|Gal(ℚ(ζ₅)/ℚ)|](ch14-cards.md#g-galzeta5)) {brace: [Galois theory](ch14.md#ch-T14-galois-theory)} ) · [ᚍ(K)](ch15.md)The listening room( ([⟨0|1⟩](ch15-cards.md#g-braket) + ([θ₁θ₂ + θ₂θ₁](ch15-cards.md#g-grass)) + [[x̂, p̂_y]](ch15-cards.md#g-commxpy) + [⟨+|σ_z|+⟩](ch15-cards.md#g-sigzplus))/(([⟨0|0⟩](ch15-cards.md#g-norm00) + [⟨+|0⟩](ch15-cards.md#g-plus0) + [⟨1|σ_x|0⟩](ch15-cards.md#g-sigx)) {brace: [overlaps](ch15-cards.md#g-hilbert)} · ([[x̂, p̂]/(iℏ)](ch15-cards.md#g-commxp) + [tr(σ_x²)](ch15-cards.md#g-trsx2) + [min ΔxΔp / ℏ](ch15-cards.md#g-kennard) + [2√2](ch15-cards.md#g-tsirelson)) {brace: bounds} ) {brace: [quantum mechanics](ch15.md#ch-T15-quantum-mechanics)} · ([⟨sin|cos⟩](ch15-cards.md#g-fourier) + [‖f̂‖₂²−‖f‖₂²](ch15-cards.md#g-plancherel))/([Si(π)/π − ½](ch15-cards.md#g-gibbs)) {brace: [Fourier analysis](ch15.md#ch-T15-fourier-analysis)} ) · [ᚎ(Z)](ch16.md)The roundabout( ([∮_|z|=1](ch16-cards.md#g-contour) [e^z dz](ch16-cards.md#g-cauchy) + [∂(e^z)/∂z̄](ch16-cards.md#g-wirtinger) + [Res₀ e^z](ch16-cards.md#g-resentire))/([∮_|z|=1 dz/z](ch16-cards.md#g-dzz) + [Res₀ e^1/z](ch16-cards.md#g-res) + [n(γ, 0)](ch16-cards.md#g-winding) + [#{z : z² + 1 = 0, |z| < 2}](ch16-cards.md#g-argprin)) ) · [ᚐ(A)](ch17.md)The mechanic’s workshop( ([li](ch17-cards.md#g-li)([μ](ch17-cards.md#g-mu)) + [1/Γ(0)](ch17-cards.md#g-recipgamma) + [sinc(π)](ch17-cards.md#g-sincpi) + [J₁(0)](ch17-cards.md#g-j1zero))/(([Γ(½)](ch17-cards.md#g-gammahalf) + [Γ(¼)](ch04-cards.md#g-gamma14) + [B(½, ½)](ch17-cards.md#g-betahalf) + [|ψ(1)|](ch17-cards.md#g-digamma1)) {brace: [the gamma family](ch17-cards.md#g-gammafn)} + ([W(1)](ch17-cards.md#g-W1) + [|W₋₁(−1/e)|](ch17-cards.md#g-wm1) + [j_0,1](ch17-cards.md#g-j01) + [erf(1)](ch17-cards.md#g-erf1) + [Ai(0)](ch17-cards.md#g-ai0) + [K(0)](ch17-cards.md#g-K0) + [S(∞)](ch17-cards.md#g-fresnelS)) {brace: the others} )) · [ᚑ(O)](ch18.md)The prime mileposts( ([ζ(−2)](ch18-cards.md#g-zetam2) + [ζ(½ + i·t₁)](ch18-cards.md#g-zetarho) + [1/ζ(1)](ch18-cards.md#g-recipzeta1) + [ζ(2)∏_{p prime}(1−p⁻²)−1](ch18-cards.md#g-eulerproduct2))/(([ζ(2)](ch18-cards.md#g-zeta2)·[ζ(3)](ch05-cards.md#g-apery)·[ζ(4)](ch18-cards.md#g-zeta4)·[ζ(6)](ch18-cards.md#g-zeta6)·|[ζ(0)](ch18-cards.md#g-zeta0)|·|[ζ(−1)](ch18-cards.md#g-zetam1)|·[|ζ(½)|](ch18-cards.md#g-zetahalf)) {brace: values} · ([t₁](ch18-cards.md#g-t1)·[t₂](ch18-cards.md#g-t2)) {brace: [zeros](ch28-cards.md#g-critline)} · ([(1 − 2⁻²)⁻¹](ch18-cards.md#g-eulerfactor2)) {brace: [the Euler product](ch18-cards.md#g-eulerprod)} ) {brace: [analytic number theory](ch18.md#ch-T18-analytic-number-theory)} ) · [ᚒ(U)](ch19.md)The crossroads( ([X(𝒪)](ch19-cards.md#g-ptinf) + [rank(y² = x³ − x)](ch19-cards.md#g-rank32))/(([Δ(y² = x³ − x)](ch19-cards.md#g-discE) + [j(y² = x³ − x)](ch19-cards.md#g-j1728) + [|E(ℚ)_tors|](ch19-cards.md#g-tors4) + [#E(𝔽₅)](ch19-cards.md#g-pts5)) {brace: [y² = x³ − x](ch19-cards.md#g-weierstrass)} + ([N(E₃₇)](ch19-cards.md#g-cond37)) {brace: [conductor 37](ch28-cards.md#g-E37)} + ([16](ch19-cards.md#g-mazur16) + [5](ch19-cards.md#g-congruent5)) {brace: a bound and a smallest} )) · [ᚓ(E)](ch20.md)The cutting( ([λ(𝒞)](ch20-cards.md#g-lamC) + [λ(ℚ ∩ [0, 1])](ch20-cards.md#g-lamQ) + [c′(½)](ch20-cards.md#g-cantorderiv))/(([dim_H 𝒞](ch20-cards.md#g-dimH) + [Σ 2ⁿ⁻¹/3ⁿ](ch20-cards.md#g-removed) + [c(½)](ch20-cards.md#g-cantorfn) + [λ(SVC)](ch20-cards.md#g-fatcantor)) {brace: [Cantor](ch20-cards.md#g-cantor)} + ([dim_H Koch](ch20-cards.md#g-koch) + [dim_H Sierpiński](ch20-cards.md#g-sierp) + [dim_H Menger](ch20-cards.md#g-menger)) {brace: other dimensions} + ([5](ch20-cards.md#g-banachtarski)) {brace: a paradox} ) {brace: [measure theory](ch20.md#ch-T20-measure-theory)} · ([f₂(½) − ½](ch20-cards.md#g-logisticfixed))/(2·[δ](ch05-cards.md#g-feig)·|[α_F](ch05-cards.md#g-feigA)|) {brace: [dynamical systems](ch20.md#ch-T20-dynamical-systems)} ) · [ᚔ(I)](ch21.md)The first landing( ([e^iπ + 1](ch21-cards.md#g-euler) + ([ε²](ch21-cards.md#g-dual)) + ([Re(i)](ch21-cards.md#g-re_i)))/([i·ī](ch21-cards.md#g-ibar) + [(1+i)(1−i)](ch21-cards.md#g-gaussint) + [|e^iθ|](ch21-cards.md#g-unitcirc) + [j²](ch21-cards.md#g-splitj) + [(cos π/3 + i sin π/3)⁶](ch21-cards.md#g-demoivre) + [#{z : z² + 1 = 0}](ch21-cards.md#g-fta) + [arg i](ch21-cards.md#g-argi)) ) · [ᚕ(J)](ch22.md)The second landing( ([ijk + 1](ch22-cards.md#g-ijk) + ([q i q⁻¹ − j, q = (1 + k)/√2](ch22-cards.md#g-qrot)))/([‖1+i+j+k‖](ch22-cards.md#g-qnorm)·([ij − ji](ch22-cards.md#g-comm))·([(1 + i + j + k)(1 − i − j − k)](ch22-cards.md#g-qconj))·([|Q₈|](ch22-cards.md#g-q8))) ) · [ᚖ(W)](ch23.md)The third landing( ([(e₁e₁)e₂ − e₁(e₁e₂)](ch23-cards.md#g-assoc0) + ([(e₄e₁)(e₂e₄) − e₄((e₁e₂)e₄)](ch23-cards.md#g-moufang)))/([((e₁e₂)e₄ − e₁(e₂e₄))](ch23-cards.md#g-assoc)·([‖1 + e₁ + … + e₇‖²](ch23-cards.md#g-onorm))·([dim G₂](ch14-cards.md#g-g2))) ) · [ᚗ(V)](ch24.md)The top landing( ([(e₃+e₁₀)(e₆−e₁₅)](ch24-cards.md#g-zd) + ([(aa)a − a(aa), a = e₃ + e₁₀](ch24-cards.md#g-powassoc)))/([‖e₃+e₁₀‖·‖e₆−e₁₅‖](ch24-cards.md#g-zdnorm)·([84](ch24-cards.md#g-zd84))) ) · [ᚘ(Y)](ch25.md)Pre-flight controls( ([…222₃ + 1](ch25-cards.md#g-adicid) + [|27|₃ · |27|_∞ − 1](ch25-cards.md#g-prodformula))/(([…890625 + …109376](ch25-cards.md#g-tenadic)) {brace: [10-adic, composite base](ch25-cards.md#g-ostrowski)} + ([|27|₃⁻¹](ch25-cards.md#g-adicabs) + [v₃(27)](ch25-cards.md#g-val3_27)) {brace: [3-adic](ch25-cards.md#g-Qp)} + ([α², α = √2 in ℤ₇](ch25-cards.md#g-sqrt2_7)) {brace: [7-adic](ch25-cards.md#g-Qp)} )) · [ᚙ(X)](ch26.md)The bridge( ([{ | }](ch26-cards.md#g-surzero) + [ε·ω − 1](ch26-cards.md#g-epsomega))/([ω − 1](ch26-cards.md#g-omm1) + [√ω](ch26-cards.md#g-sqrtomega) + [ω^ω](ch26-cards.md#g-omom) + [ω/2](ch26-cards.md#g-omhalf) + [{0 | 1}](ch26-cards.md#g-halfS)) {brace: [surreal numbers](ch26.md#ch-T26-surreal-numbers)} · ([st(1/H)](ch26-cards.md#g-hyper))/([H](ch26-cards.md#g-hyperH)) {brace: [nonstandard analysis](ch26.md#ch-T26-nonstandard-analysis)} ) · [ᚚ(P)](ch27.md)The possible journeys( ([|∅|](ch27-cards.md#g-empty)·[ℵ₀](ch27-cards.md#g-aleph0)·[ℵ₁](ch27-cards.md#g-aleph1)·[ℵ_ω](ch27-cards.md#g-alephomega)·[𝔠](ch27-cards.md#g-cont)·[ℶ_ω](ch27-cards.md#g-bethomega)) {brace: [cardinal arithmetic](ch27.md#ch-T27-cardinal-arithmetic)} + ([ω·0](ch27-cards.md#g-ordzero) + [∅](ch27-cards.md#g-vnzero)) {brace: [ordinal arithmetic](ch27.md#ch-T27-ordinal-arithmetic)} + ([|𝒫(∅)| − 1](ch27-cards.md#g-pset0)) {brace: [power sets](ch27.md#ch-T27-power-sets)} + ([|Hom_Set(∅, X)|−1](ch27-cards.md#g-initialhom)) {brace: [category theory](ch27.md#ch-T27-category-theory)} ) · [᚜(Epilogue)](ch28.md)The homeward view( [Λ_dBN](ch28-cards.md#g-dbn) {brace: [Riemann](ch28-cards.md#g-critline)} · [L(E₃₇, 1)](ch28-cards.md#g-lE37) {brace: [Birch, Swinnerton-Dyer](ch28-cards.md#g-rankE)} · [[every simply connected closed 3-manifold is S³]](ch28-cards.md#g-poincare) {brace: [Poincaré](ch28-cards.md#g-ricci)} · [[P = NP]](ch28-cards.md#g-pnp) {brace: [P versus NP](ch28-cards.md#g-npcomplete)} · [[every Hodge class is algebraic]](ch28-cards.md#g-hodge) {brace: [Hodge](ch28-cards.md#g-hodgeclass)} · [1/T_∗](ch28-cards.md#g-nsrate) {brace: [Navier–Stokes](ch28-cards.md#g-weaksol)} · [[a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has Δ>0]](ch28-cards.md#g-ymbit) {brace: [Yang–Mills](ch28-cards.md#g-ymgap)} ) * The declaration just above the equation is part of its mathematical setup: it fixes one choice of x for this book and defines ᚏ_4.16 in terms of it. Every later appearance of [Rose’s number](ch01-cards.md#g-rose) uses that same choice, and so does every other ᚏ in this book. One more rule of the piece: every expression on the right is evaluated in the setting it belongs to, and only the plain number it hands back joins the product. See Chapter ᚏ(R) for more on the notation. ## Chapters Symbols from many cultures and ages invite us to look; the structure gives us as much to explore as we wish. At its core, this whole work is one simple equation, and it says 0 = 0. Everything right of the equals sign is something humanity worked out: the constants of nature, the famous numbers, the giants, and two dozen ways that two dozen branches of mathematics say nothing. There are infinitely many ways to write zero; the chapters that follow are made of them. And to balance a right side that rich, built almost entirely from symbols with established histories, I wanted a left side of as few symbols as possible, each of them either arcane or never written before.