Skip to main content# Patterns in deep Mandelbrot zooms

# Abstract

# Patterns in deep Mandelbrot zooms

## The Mandelbrot set

## Sources

## Method

## Analysis

### “Polefcra” by Jonathan Leavitt

### “Old Wood Dish” by JWM

### “Evolution of Trees” by DinkydauSet

### “Mandelbrot Deep Julia Morphing 1” by Microfractal

## Synthesis

### “Polefcra” variations

### “Old Wood Dish” variations

### “Evolution of Trees” variations

## Conclusions

A simple formula leads to emergent complexity.

Published onNov 11, 2023

Patterns in deep Mandelbrot zooms

The Mandelbrot set emerges from iterations of

The Mandelbrot set is a famous fractal (a complicated shape with detail visible at all scales) which emerges from the dynamics under iteration of the simple formula

The fractal contains a cardioid decorated with recursive trees of discs and thin filaments. Smaller copies of the whole ensemble appear in the filaments, interspersed with spirals with varying numbers of arms. Each solid region is associated with a positive whole number, called the period (there are multiple regions for each period in general). A child disk attached at internal angle

For a gentle introduction to the Mandelbrot set see the first half of [2]. The radial edges in binary decomposition colouring are parts of external rays with terminating binary expansion. For a mathematical treatment of external rays, one of the key tools used to analyze locations, see [3].

Periodic (with repeating binary expansion) external rays land in pairs on the cusps of cardioids and bond points to disks (which have the same period as the rays) while pre-periodic (with eventually repeating binary expansion) external rays land on the centres of spirals [4]. Algorithms for converting between representations can be found in [5] [6].

The

It is unclear what license the coordinates are under, or even what kind of legal framework applies to coordinates. However, the artists JWM, DinkydauSet, and Microfractal have granted permission for images and analysis to be included in this paper. FractalMonster and Olbaid-ST also agreed but couldn’t be included due to space constraints. No contact details for Jonathan Leavitt could be found.

When constructing a deep Mandelbrot zoom image, there are generally relatively few decision points, because after each decision, the history repeats before anything new appears, increasing in speed and rotational symmetry by factors of two. If the artist decides not to change the focus of zooming, the repetition of history ends at a baby Mandelbrot set with rings of tiny details surrounding it.

Around half the way through the zoom is the last time a two-fold rotationally symmetric figure appears. This is typically when artists make decisions to vary the focus of the zoom, choosing a focus point not in the centre of the symmetry. Then zooming in to this new focus history repeats again, until the next decision point, which means the exponent in the magnification typically goes up by a factor of ^{80} to 10^{120} to 10^{180}. Magnification depth is a limiting factor because deeper zooms require greater numerical precision which means images take longer to calculate.

By mathematically analyzing the behaviour of the coordinates under iteration, it is possible to deduce the decision points, and discover the patterns of nearby (pre)periodic points and their external rays.

There are several parts of the analysis. First is to find a sequence of increasing periods of nearby regions, at successive minima of |z|. Using Newton’s root finding method, their locations can be found. Their distance from the coordinates of the original image gives the size of the view where a zooming decision was made. Images can then be generated to see what the decisions were.

More precisely, tracing external rays from near each region found determines their external angles. This is only feasible for low periods (below about 100000) as computation time increases rapidly. The periodic blocks of each angle pair can then be expressed as concatenations of the periodic blocks of lower period angle pairs in the sequence.

0 and 1 are the binary digits that comprise external angles. The O and I columns contain P digits each. The corresponding external rays are these columns repeated with period P. O(p) and I(p) denote sequences of binary digits of length p, referring to earlier rows in the same table (different tables have different unrelated sequences). B^{m} denotes m copies of B concatenated together. n denotes the row of the table, so that O(P(n-1)) and I(P(n-1)) denote the sequences from the previous row of the table. X denotes a choice between O and I. Choices may differ between the O and I columns, but the structure of the decomposition in terms of lower periods is the same. Choices are constrained by the row forming a ray pair.

Published 1998-2001, magnification 10^{100} [15].

P (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

2 | 01 | 10 |

3 | 011 | 100 |

124 | O(3) | O(3) |

292 | I(124) O(124) I(3) | I(124) I(124) O(3) |

625 | I(292) O(292) I(3) | I(292) I(292) O(3) |

1294 | I(625) O(625) I(3) | I(625) I(625) O(3) |

2629 | I(1294) O(1294) I(3) | I(1294) I(1294) O(3) |

Pattern could be continued by repeating:

P(n) | O(P(n)) | I(P(n)) |
---|---|---|

P(n-1) + P(n-1) + 44 | I(P(n-1)) O(P(n-1)) I(3) | I(P(n-1)) I(P(n-1)) O(3) |

P(n-1) + P(n-1) + 41 | I(P(n-1)) O(P(n-1)) I(3) |

Published 2010-11-25, magnification 10^{152} [16].

P (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

2 | 10 | |

34 | O(2) | O(2) |

70 | O(34) O(34) 11 | O(34) I(34) 00 |

142 | O(70) O(70) 11 | O(70) I(70) 00 |

286 | O(142) O(142) 11 | O(142) I(142) 00 |

574 | O(286) O(286) 11 | O(286) I(286) 00 |

862 | O(574) O(286) 11 | O(574) I(286) 00 |

1438 | I(862) O(574) 11 | I(862) I(574) 00 |

2878 | I(1438) O(1438) 11 | I(1438) I(1438) 00 |

5758 | O(2878) O(2878) 11 | O(2878) I(2878) 00 |

The pattern is mostly a repetition of

P(n) | O(P(n)) | I(P(n)) |
---|---|---|

P(n-1) + P(n-1) + 2 | O(P(n-1)) O(P(n-1)) 11 | O(P(n-1)) I(P(n-1)) 00 |

or its swapped (interchange O and I columns) and inverted (interchange 0 with 1 and O with I) version

P(n) | O(P(n)) | I(P(n)) |
---|---|---|

P(n-1) + P(n-1) + 2 | I(P(n-1)) O(P(n-1)) 11 | I(P(n-1)) I(P(n-1)) 00 |

but at periods 862 and 1438 it deviates significantly.

Published 2013-09-25, magnification 10^{227} [17].

P (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

93 | (10) | (10) |

145 | I(93) (10) | I(93) (10) |

3913 | (I(145) O(145)) | (I(145) O(145)) |

4348 | I(3913) O(145) I(145) I(145) | I(3913) I(145) O(145) O(145) |

8841 | I(4348) O(4348) I(145) | I(4348) I(4348) O(145) |

17628 | O(8841) O(4348) I(4348) (10) | O(8841) O(4348) I(4348) (10) |

35455 | O(17628) I(8841) O(8841) I(145) | O(17628) I(8841) I(8841) O(145) |

70910 | I(35455) I(17628) O(8841) O(8841) I(145) | I(35455) I(17628) O(8841) I(8841) O(145) |

141965 | X(70910) X(35455) X(35455) X(145) | note 1 |

283930 | X(141965) X(70910) X(70910) X(145) | note 1 |

567135 | X(283930) X(141965) X(70910) X(35455) X(17628) X(8841) X(4348) X(3913) X(145) | note 2 |

Conjecture based on periods alone.

Conjecture based on periods alone. The step turns a tree into an S-shape.

The repeating part of the pattern seems to be

P(n) | O(P(n)), I(P(n)) |
---|---|

P(n-1)+P(n-2)+P(n-2)+145 | X(P(n-1)) X(P(n-2)) X(P(n-2)) X(145) |

Published 2021-03-14, magnification 10^{4341} [18].

P (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

2 | 01 | 10 |

3 | 011 | 100 |

4 | 0111 | 1000 |

8 | O(4) I(4) | I(4) O(4) |

137 | O(8) | O(8) |

278 | I(137) O(137) I(4) | I(137) I(137) O(4) |

560 | I(278) O(278) I(4) | I(278) I(278) O(4) |

1124 | O(560) O(56O) I(4) | O(560) I(560) O(4) |

2252 | O(1124) O(1124) I(4) | O(1124) I(1124) O(4) |

4508 | I(2252) O(2252) I(4) | O(2252) I(2252) O(4) |

9020 | O(4508) O(4508) I(4) | O(4508) I(4508) O(4) |

18044 | I(9020) O(9020) I(4) | I(9020) I(9020) O(4) |

36052 | note 1 | note 2 |

72144 | X(36052) X(18044) X(18044) X(4) | note 3 |

144248 | X(72144) X(36052) X(36052) | |

288540 | X(144248) X(72144) X(72144) X(4) | |

577036 | X(288540) X(144248) X(144248) | |

1154120 | X(577036) X(288540) X(288540) X(4) | |

2308192 | X(1154120) X(577036) X(577036) | |

4616436 | X(2308192) X(1154120) X(1154120) X(4) |

O(36052) = I(18044) I(9020) O(4508) I(2252) O(1124) O(560) I(278) O(137) O(8)

^{15}I(4) 0 I(4)I(36052) = I(18044) I(9020) O(4508) I(2252) O(1124) O(560) I(278) I(137) O(8)

^{15}I(4) 1 O(4)Conjectures based on periods alone.

The first pattern building a tree is similar to “Old Wood Dish” by JWM:

P(n) | O(P(n)) | I(P(n)) |
---|---|---|

P(n-1)+P(n-1)+4 | O(P(n-1)) O(P(n-1)) I(4) | O(P(n-1)) I(P(n-1)) O(4) |

At period 36052 the tree is turned into an S. The second pattern is similar to “Evolution Of Trees” by DinkydauSet:

P(n) | Notes |
---|---|

P(n-1)+P(n-2)+P(n-2)+4 | Building a tree. |

P(n-1)+P(n-2)+P(n-2) | Making an S. |

P(n) (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

2 | 01 | 10 |

3 | 011 | 100 |

P(n-1) + 1 | O(P(n-1)) 1 | I(P(n-1)) 0 |

m | 011…11 | 100…00 |

40 × m + 4 | O(m) | O(m) |

2 × P(n-1) + 14 × m + 2 | I(P(n-1)) O(P(n-1)) I(m) | I(P(n-1)) I(P(n-1)) O(m) |

2 × P(n-1) + 13 × m + 2 | I(P(n-1)) O(P(n-1)) I(m) | I(P(n-1)) I(P(n-1)) O(m) |

Repeating the two last lines k times increases the number of rings of reverse bifurcation. Increasing m increases the spacing between the rings of reverse bifurcation. The original “Polefcra” by Jonathan Leavitt corresponds to k = 4 and m = 3. Images drawn with k = 4 and m = 3 except for the variable under consideration.

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

k | ||||||

m-2 |

P(n) (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

2 | 01 | 10 |

k × 2 + 2 + m | O(2) | O(2) |

2 × P(n-1) + m | O(P(n-1)) O(P(n-1)) 1 | O(P(n-1)) I(P(n-1)) 0 |

where the last line is repeated to refine the structure. Increasing m lengthens the branches of the tree: Increasing k tightens the spirals the tree is made of: The original “Old Wood Dish” by JWM corresponds to m = 2, k = 15. Images drawn with m = 2, k = 15 except for the variable under consideration.

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

m | ||||||

k/5 |

Combining the previous two synthesized variations to simplify, and reduce the periods to make ray tracing feasible:

P(n) (Period) | O(P) (Lower External Angle) | I(P) (Upper External Angle) |
---|---|---|

1 | 0 | 1 |

i | 0 1 | 1 0 |

((j+1)×2+k)×i+1 | (O(i) I(i)) | (O(i) I(i)) |

P(n-1)+l×i+1 | O(n-1) O(i) I(i) | O(n-1) I(i) O(i) |

2×P(n-1)+2+(m+1)×i | O(n-1) O(n-1) 0 I(i) | I(n-1) I(n-1) 1 O(i) |

P(n-1)+2×P(n-2)+2+(m+1)×i | O(n-1) O(n-2) I(n-2) 11 O(i) | O(n-1) I(n-2) O(n-2) 00 I(i) |

P(n-1)+2×P(n-2)+m×i | O(n-1) O(n-2) I(n-2) O(i) | I(n-1) O(n-2) O(n-2) I(i) |

P(n-1)+2×P(n-2)+m×i | O(n-1) I(n-2) O(n-2) O(i) | I(n-1) O(n-2) I(n-2) I(i) |

Increasing i lengthens and straightens the filaments, by choosing an island further along the needle. Increasing j increases the spiral count, by choosing a bulb deeper in Seahorse Valley. Increasing k tightens the spirals, by choosing an embedded Julia set closer to the root of the hair. Increasing l makes the trees bigger and closer together. Increasing m makes the branches of the trees longer, by choosing nodes further from the centre. Repeating the final step an increasing number d times makes the overall structure more refined. Images drawn with i = 5, j = 3, k = 3, l = 3, m = 3, d = 5 except for the variable under consideration.

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

i-2 | ||||||

j | ||||||

k | ||||||

l | ||||||

m | ||||||

d+1 |

Patterns in deep Mandelbrot set zoom artworks by several artists were analyzed and compared. Different variations on these patterns were synthesized. Assisted zooming in existing fractal software (principally period detection, root finding, and target zoom level estimation) supports this line of artistic enquiry. Full analysis and synthesis automation via external ray tracing currently remains out of reach for high period examples due to computational constraints.