Portability	portable
Stability	unstable
Maintainer	claudiusmaximus@goto10.org

Fractal.RUFF.Mandelbrot.Address

Description

External angles give rise to kneading sequences under the angle doubling map. Internal addresses encode kneading sequences in human-readable form, when extended to angled internal addresses they distinguish hyperbolic components in a concise and meaningful way.

The algorithms are mostly based on Dierk Schleicher's paper Internal Addresses Of The Mandelbrot Set And Galois Groups Of Polynomials (version of February 5, 2008) http://arxiv.org/abs/math/9411238v2.

Synopsis

Documentation

type Angle = Rational Source

Angle as a fraction of a turn, usually in [0, 1).

double :: Angle -> Angle Source

Angle doubling map.

wrap :: Angle -> Angle Source

Wrap an angle into [0, 1).

data Knead Source

Elements of kneading sequences.

Constructors

Zero
One
Star

Instances

Bounded Knead
Enum Knead
Eq Knead
Data Knead
Ord Knead
Read Knead
Show Knead
Typeable Knead
Pretty Knead

data Kneading Source

Kneading sequences. Note that the Aperiodic case has an infinite list, which the Pretty instance truncates arbitrarily.

Constructors

Aperiodic [Knead]
PrePeriodic [Knead] [Knead]
StarPeriodic [Knead]
Periodic [Knead]

Instances

Eq Kneading
Data Kneading
Ord Kneading
Read Kneading
Show Kneading
Typeable Kneading
Pretty Kneading

kneading :: Angle -> Kneading Source

The kneading sequence for an external angle.

period :: Kneading -> Maybe Integer Source

The period of a kneading sequence, or Nothing when it isn't periodic.

unwrap :: Kneading -> [Knead]Source

Unwrap a kneading sequence to an infinite list.

data InternalAddress Source

Internal addresses are a non-empty sequence of strictly increasing integers beginning with '1'.

Constructors

InternalAddress [Integer]

Instances

Eq InternalAddress
Data InternalAddress
Ord InternalAddress
Read InternalAddress
Show InternalAddress
Typeable InternalAddress
Pretty InternalAddress

internalAddress :: Kneading -> Maybe InternalAddress Source

Construct an InternalAddress from a kneading sequence.

associated :: Kneading -> Maybe (Kneading, Kneading)Source

A star-periodic kneading sequence's upper and lower associated kneading sequences.

upper :: Kneading -> Maybe Kneading Source

The upper associated kneading sequence.

lower :: Kneading -> Maybe Kneading Source

The lower associated kneading sequence.

internalFromList :: [Integer] -> Maybe InternalAddress Source

Construct a valid InternalAddress, checking the precondition.

internalToList :: InternalAddress -> [Integer]Source

Extract the sequence of integers.

data AngledInternalAddress Source

Angled internal addresses have angles between each integer in an internal address.

Constructors

Unangled Integer
Angled Integer Angle AngledInternalAddress

Instances

Eq AngledInternalAddress
Data AngledInternalAddress
Ord AngledInternalAddress
Read AngledInternalAddress
Show AngledInternalAddress
Typeable AngledInternalAddress
Pretty AngledInternalAddress

angledInternalAddress :: Angle -> Maybe AngledInternalAddress Source

The angled internal address corresponding to an external angle.

angledFromList :: [(Integer, Maybe Angle)] -> Maybe AngledInternalAddress Source

Builds a valid AngledInternalAddress from a list, checking the precondition that only the last 'Maybe Angle' should be Nothing, and the Integer must be strictly increasing.

angledToList :: AngledInternalAddress -> [(Integer, Maybe Angle)]Source

Convert an AngledInternalAddress to a list.

externalAngles :: AngledInternalAddress -> Maybe (Rational, Rational)Source

The pair of external angles whose rays land at the root of the hyperbolic component described by the angled internal address.

stripAngles :: AngledInternalAddress -> InternalAddress Source

Discard angle information from an internal address.

parse :: String -> Maybe AngledInternalAddress Source

Parse an angled internal address, accepting some unambiguous abbreviations.