ModularWord

The unique positive word in $L$ and $R$ of a matrix in $\mathrm{PSL}(2,\mathbb{Z})$ with non-negative entries — which is also its path down the Stern–Brocot tree.

WikipediaModular groupMathWorldModularGroupGammaWikidataQ1834342

ModularWord(matrix)the positive

LR

word

ModularMatrix(a, b, c, d)a matrix; every head here also accepts the word in its place

ModularSTWord(matrix)the alternating

S

/

T

factorisation, as its list of

T

-exponents

Typed signatures

Type	Package	Applies
`(expression<ModularMatrix> \| list \| string) -> string`	`modular`	always

Domain: The modular group

Details

$L = \begin{pmatrix}1&0\\1&1\end{pmatrix}$ and $R = \begin{pmatrix}1&1\\0&1\end{pmatrix}$ generate the positive cone
The peel never searches: $R$ comes off when $a \ge c$ and $b \ge d$ , $L$ when $c \ge a$ and $d \ge b$ , and both at once would force determinant $0$
ModularKind sorts by $|\mathrm{tr}|$ against $2$ — elliptic, parabolic, hyperbolic
A positive word is hyperbolic exactly when it uses both letters; all- $L$ and all- $R$ are parabolic
Rebuilding from an $S$ / $T$ word can return $-M$ : that sign is the $\pm I$ that $\mathrm{PSL}$ quotients out

Examples

Basic #

the shortest geodesic

In

\mathrm{ModularWord}(\mathrm{ModularMatrix}(1, 1, 1, 2))

Out

\text{LR}

a word evaluates as its matrix

In

\mathrm{ModularTrace}(\text{LRLR})

Out

7

Scope #

the $S$ / $T$ exponents are a continued fraction

In

\mathrm{ModularSTWord}(\mathrm{ModularMatrix}(1, 5, 0, 1))

Out

\bigl\lbrack5\bigr\rbrack

the identity matrix

In

\mathrm{ModularKind}(\mathrm{ModularMatrix}(1, 0, 0, 1))

Out

\text{Identity}

$S$ is elliptic

In

\mathrm{ModularKind}(\mathrm{ModularMatrix}(0, -1, 1, 0))

Out

\text{Elliptic}

$T$ is parabolic

In

\mathrm{ModularKind}(\mathrm{ModularMatrix}(1, 1, 0, 1))

Out

\text{Parabolic}

$LR$ is hyperbolic

In

\mathrm{ModularKind}(\mathrm{ModularMatrix}(1, 1, 1, 2))

Out

\text{Hyperbolic}

the trace of a plain matrix, not just a word

In

\mathrm{ModularTrace}(\mathrm{ModularMatrix}(1, 1, 1, 2))

Out

3

the trace of a bare word

In

\mathrm{ModularTrace}(\text{LR})

Out

3

a word wrapped as ModularMatrix reads the same

In

\mathrm{ModularTrace}(\mathrm{ModularMatrix}(\text{LRLR}))

Out

7

a nested List matrix works too

In

\mathrm{ModularTrace}(\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)

Out

3

Properties #

$T^5$ fixes a cusp

In

\mathrm{ModularKind}(\mathrm{ModularMatrix}(1, 5, 0, 1))

Out

\text{Parabolic}

the $S$ / $T$ round trip can return $-M$

In

\mathrm{ModularTrace}(\mathrm{ModularFromSTWord}(\mathrm{ModularSTWord}(\mathrm{ModularMatrix}(3, 2, 4, 3))))

Out

-6

About

Examples	13
Crosswalk	Wikipedia MathWorld Wikidata

See also: ContinuedFraction, ModularClasses, RademacherSymbol