Fractal models: Standard-precision implementations

Standard precision implementations are limited by float64 format precision.

class fractalshades.models.Mandelbrot(directory: str)[source]

__init__(directory: str)[source]

A standard power-2 Mandelbrot Fractal.

\[\begin{split}z_0 &= 0 \\ z_{n+1} &= z_{n}^2 + c\end{split}\]

Parameters:

directorystr: Path for the working base directory

calc_std_div(*, calc_name: str = 'base_calc', subset: Fractal_array | None = None, max_iter: int = 10000, M_divergence: float = 1000.0, epsilon_stationnary: float = 0.01, calc_d2zndc2: bool = False, calc_orbit: bool = False, backshift: int = 0)[source]

Basic iterations for Mandelbrot standard set (power 2).

Parameters:

calc_namestr: The string identifier for this calculation
subsetOptional fractalshades.postproc.Fractal_array: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
max_iterint: the maximum iteration number. If reached, the loop is exited with exit code “max_iter”.
M_divergencefloat: The diverging radius. If reached, the loop is exited with exit code “divergence”
epsilon_stationnaryfloat: A small criteria (typical range 0.01 to 0.001) used to detect earlier points belonging to a minibrot, based on dzndz1 value. If reached, the loop is exited with exit code “stationnary”
calc_d2zndc2:: If True, activates the additional computation of d2zndc2, needed only for the alternative normal map shading ‘Milnor’.
calc_orbit: bool: If True, stores the value of an orbit point @ exit - orbit_shift
backshift: int (> 0): The number of iteration backward for the stored orbit starting point

Notes

The following complex fields will be calculated: zn and its derivatives (dzndc, d2zndc2*[optionnal], *dzndz [optionnal]). If calc_orbit is activated, zn_orbit will also be stored Exit codes are 0: max_iter, 1: divergence, 2: stationnary.

newton_calc(*, calc_name: str = 'newton_calc', subset: Fractal_array | None = None, known_orders: int | None = None, max_order: int = 500, max_newton: int = 20, eps_newton_cv: float = 1e-12)[source]

Newton iterations for Mandelbrot standard set interior (power 2).

Parameters:

calc_namestr: The string identifier for this calculation
subsetOptional fractalshades.postproc.Fractal_array: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
known_ordersNone | int list: If not None, only the integer listed of their multiples will be candidates for the cycle order, the other rwill be disregarded. If None all order between 1 and max_order will be considered.
max_orderint: The maximum value tested for cycle order
eps_newton_cvfloat: A small float to qualify the convergence of the Newton iteration Usually a fraction of a view pixel.

Notes

Note

The following complex fields will be calculated:

zr: A (any) point belonging to the attracting cycle
attractivity: The cycle attractivity (for a convergent cycle it is a complex of norm < 1.)
dzrdc: Derivative of zr
dattrdc: Derivative of attractivity

The following integer field will be calculated:

order: The cycle order

Exit codes are 0: max_order, 1: order_confirmed.

References

[1]

<https://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html>

[2]

<https://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html>

class fractalshades.models.Mandelbrot_N(directory: str, exponent: int)[source]

__init__(directory: str, exponent: int)[source]

A standard power-N Mandelbrot Fractal set implementation.

\[\begin{split}z_0 &= 0 \\ z_{n+1} &= {z_{n}}^N + c\end{split}\]

Parameters:

directorystr: Path for the working base directory

calc_std_div(*, calc_name: str, subset: Fractal_array | None = None, max_iter: int, M_divergence: float, epsilon_stationnary: float, calc_d2zndc2: bool = False, calc_orbit: bool = False, backshift: int = 0)[source]

Basic iterations for the Mandelbrot standard set (power n).

Parameters:

calc_namestr: The string identifier for this calculation
subsetOptional fractalshades.postproc.Fractal_array: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
max_iterint: the maximum iteration number. If reached, the loop is exited with exit code “max_iter”.
M_divergencefloat: The diverging radius. If reached, the loop is exited with exit code “divergence”
epsilon_stationnaryfloat: A small float to early exit non-divergent cycles (based on cumulated dzndz product). If reached, the loop is exited with exit code “stationnary” (Those points should belong to Mandelbrot set interior). A typical value is 1.e-3
calc_d2zndc2: bool: If True, activates the additional computation of d2zndz2, needed only for the alternative normal map shading ‘Milnor’.
calc_orbit: bool: If True, stores the value of an orbit point @ n of exit - backshift
backshift: (> 0): The number of iteration backward for the stored orbit starting point

Notes

The following complex fields will be calculated: zn and its derivatives (dzndz, dzndc). If calc_orbit is activated, zn_orbit will also be stored Exit codes are max_iter, divergence, stationnary.

newton_calc(*, calc_name: str = 'newton_calc', subset: Fractal_array | None = None, known_orders: int | None = None, max_order: int = 500, max_newton: int = 20, eps_newton_cv: float = 1e-12)[source]

Newton iterations for Mandelbrot standard set interior (power N).

Parameters:

calc_namestr: The string identifier for this calculation
subsetOptional fractalshades.postproc.Fractal_array: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
known_ordersNone | int list: If not None, only the integer listed of their multiples will be candidates for the cycle order, the other rwill be disregarded. If None all order between 1 and max_order will be considered.
max_orderint: The maximum value tested for cycle order
eps_newton_cvfloat: A small float to qualify the convergence of the Newton iteration Usually a fraction of a view pixel.

Notes

Note

The following complex fields will be calculated:

zr: A (any) point belonging to the attracting cycle
attractivity: The cycle attractivity (for a convergent cycle it is a complex of norm < 1.)
dzrdc: Derivative of zr
dattrdc: Derivative of attractivity

The following integer field will be calculated:

order: The cycle order

Exit codes are 0: max_order, 1: order_confirmed.

References

[1]

<https://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html>

[2]

<https://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html>

class fractalshades.models.Burning_ship(directory: str, flavor: ~typing.Literal[<enum 'BS_flavor_enum'>] = 'Burning ship')[source]

__init__(directory: str, flavor: ~typing.Literal[<enum 'BS_flavor_enum'>] = 'Burning ship')[source]

A Burning Ship Fractal (power 2). The basic equation a detailed below:

\[\begin{split}x_0 &= 0 \\ y_0 &= 0 \\ x_{n+1} &= x_n^2 - y_n^2 + a \\ y_{n+1} &= 2 |x_n y_n| - b\end{split}\]

where:

\[\begin{split}z_n &= x_n + i y_n \\ c &= a + i b\end{split}\]

Parameters:

directorystr: Path for the working base directory
flavorstr: The variant of Burning Ship detailed implementation, defaults to “Burning Ship”. Acceptable values are listed below in notes.

Notes

Note

Several variants (flavor parameter) are implemented with the following iteration formula:

“Perpendicular burning ship” variant of the Burning Ship Fractal.

\[\begin{split}x_{n+1} &= x_n^2 - y_n^2 + a \\ y_{n+1} &= 2 x_n |y_n| - b\end{split}\]

“Shark fin” variant

\[\begin{split}x_{n+1} &= x_n^2 - y_n |y_n| + a \\ y_{n+1} &= 2 x_n y_n - b\end{split}\]

“Celtic” variant

\[\begin{split}x_{n+1} &= |x_n^2 - y_n^2| + a \\ y_{n+1} &= 2 x_n y_n - b\end{split}\]

“Buffalo” variant

\[\begin{split}x_{n+1} &= |x_n^2 - y_n^2| + a \\ y_{n+1} &= 2 |x_n y_n| - b\end{split}\]

calc_std_div(*, calc_name: str, subset, max_iter: int, M_divergence: float, calc_orbit: bool = False, backshift: int = 0)[source]

Basic iterations for Burning ship set.

Parameters:

calc_namestr: The string identifier for this calculation
subsetfractalshades.postproc.Fractal_array: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
max_iterint: the maximum iteration number. If reached, the loop is exited with exit code “max_iter”.
M_divergencefloat: The diverging radius. If reached, the loop is exited with exit code “divergence”
calc_orbit: bool: If True, stores the value of an orbit point @ exit - orbit_shift
backshift: int (> 0): The count of iterations backward for the stored orbit starting point

Notes

The following complex fields will be calculated: xn yn and its derivatives (dxnda, dxndb, dynda, dyndb). If calc_orbit is activated, zn_orbit will also be stored Exit codes are max_iter, divergence.

class fractalshades.models.Power_tower(directory)[source]

__init__(directory)[source]

The tetration fractal - standard precision implementation

\[\begin{split}z_0 &= 1 \\ z_{n+1} &= c^{z_n} \\ &= \exp(z_n \log(c))\end{split}\]

This class implements limit cycle calculation through Newton search

Parameters:

directorystr: Path for the working base directory

newton_calc(*, calc_name: str, subset=None, compute_order=True, max_order, max_newton, eps_newton_cv)[source]

Newton iterations for the tetration fractal

Parameters:

calc_namestr: The string identifier for this calculation
subset: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
compute_orderbool: If True, the period of the limit cycle will be computed If None all order between 1 and max_order will be considered.
max_orderint: The maximum value for cycle order
eps_newton_cvfloat: A small float to qualify the convergence of the Newton iteration Usually a fraction of a view pixel.

Notes

Note

The following complex fields will be calculated:

zr: A (any) point belonging to the attracting cycle
dzrdz: The cycle attractivity (for a convergent cycle it is a complex of norm < 1.)

The following integer field will be calculated:

order: The cycle order

Exit codes are max_order, order_confirmed.

class fractalshades.models.Collatz(directory)[source]

__init__(directory)[source]

The Collatz fracal:

\[\begin{split}z_0 &= c \\ z_{n+1} &= 0.25 (2 + 7 z_n - (2+5z_n) \cos(\pi z_n))\end{split}\]

This fractal is linked to the Syracuse conjecture.

Parameters:

directorystr: Path for the working base directory

base_calc(*, calc_name: str, subset, max_iter: int, M_divergence: float, epsilon_stationnary: float)[source]

Basic iterations for Mandelbrot standard set (power 2).

Parameters:

calc_namestr: The string identifier for this calculation
subsetfractalshades.postproc.Fractal_array: A boolean array-like, where False no calculation is performed If None, all points are calculated. Defaults to None.
max_iterint: the maximum iteration number. If reached, the loop is exited with exit code “max_iter”.
M_divergencefloat: The diverging radius. If reached, the loop is exited with exit code “divergence”
epsilon_stationnaryfloat: A small float to early exit non-divergent cycles (based on cumulated dzndz product). If reached, the loop is exited with exit code “stationnary” (Those points should belong to Mandelbrot set interior). A typical value is 1.e-3

Notes

The following complex fields will be calculated: zn and its derivatives (dzndz, dzndc, d2zndc2). Exit codes are max_iter, divergence, stationnary.