Mandelbrot convergence counts

Usage

mandelbrot(...)

# S3 method for class 'matrix'
mandelbrot(Z, maxIter = 200L, tau = 2, ...)

# S3 method for class 'numeric'
mandelbrot(
  xmid = -0.75,
  ymid = 0,
  side = 3,
  resolution = 400L,
  maxIter = 200L,
  tau = 2,
  ...
)

mandelbrot_tiles(
  xmid = -0.75,
  ymid = 0,
  side = 3,
  nrow = 2L,
  ncol = nrow,
  resolution = 400L,
  truncate = TRUE
)

Arguments

Z: A complex matrix (\(Z\)) for which convergence counts (\(C\)) should be calculated.
maxIter: Maximum number of iterations (\(c^{max}\)) per bin.
tau: A threshold (\(\tau\)); the radius when calling divergence (\(|z_{i}| > \tau\)).
xmid, ymid, side, resolution: Alternative specification of the complex plane Z, where mean(Re(Z)) == xmid, mean(Im(Z)) == ymid, diff(range(Re(Z))) == side, diff(range(Im(Z))) == side, and dim(Z) == c(resolution, resolution).
nrow, ncol: Number of rows and columns of tiles to partition the complex plane Z into.
truncate: If TRUE, all tiles are truncated to have identical dimensions, otherwise the last row and column of tiles may be smaller.

Value

Returns an integer matrix (of class Mandelbrot) with non-negative counts (\(C\)).

Escape time algorithm

The convergence counts are calculated using the standard escape-time algorithm for the Mandelbrot set: for each complex number \(c\), the sequence \(z_{i+1} \leftarrow z_{i}^2 + c\) (starting at \(z_{0} = c\)) is iterated until either its modulus (\(|z_{i}|\)) exceeds the escape radius \(\tau\) (tau) or the maximum number of iterations (\(c^{max}\)) has been reached. The count (\(c_{i}\)) recorded is the iteration at which the sequence escaped, or \(c^{max}\) for points that never escaped.

References

Mandelbrot set, https://en.wikipedia.org/wiki/Mandelbrot_set, 2026

Examples

counts <- mandelbrot(xmid = -0.75, ymid = 0, side = 3)
str(counts)
#>  'Mandelbrot' int [1:400, 1:400] 1 1 1 1 1 1 1 1 1 1 ...
#>  - attr(*, "params")=List of 3
#>   ..$ Z      : cplx [1:400, 1:400] -2.25-1.5i -2.25-1.49i -2.25-1.48i ...
#>   ..$ maxIter: int 200
#>   ..$ tau    : num 2
if (FALSE) { # \dontrun{
plot(counts)
} # }

if (FALSE) { # \dontrun{
demo("mandelbrot", package = "future", ask = FALSE)
} # }