This page is a small collection of complex numbers and the..

Mandelbrot Sequences and Orbits
...which correspond to them. Almost all the entries are focused on points near a bulb of period 14, attached to the main cardiod. Send entry suggestions to: Entry Editor



Each entry of this encyclopedia is a complex number c = x+ yi. The sequence associated to this number is found by iterating z_(n+1) = (z_n)^2 + c.

Pictured are the plotted terms in the sequence, graphed using javascript.
Cut and paste the value into interactive animated javascript, with suggested zoom, to see the sequence animated.

Below each picture is a designation of whether the original number is in the Mandelbrot set M or not; that is, whether its sequence is bounded or not.

The link in each entry to the wolfram alpha plot will give an approximate location, and an approximate Julia set.
SEQUENCES
Density

(Is in M)

c = -0.260143 + 0.6337i

wolfram alpha plot.
interactive animated javascript: use zoom = 900.
Some that got away:
-0.26077 + 0.6337i
-0.26071 + 0.6337i
-0.2608 + 0.6337i
top    index

White hole sun

(Is not in M)

c = -0.26036 + 0.6337i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.

interactive animated javascript with pause and hold: use zoom = 1000.
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Fractal Periodic (period of 7*95)

(Is in M)

c = -0.260355 + 0.6337i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
95 tiny spirals: -0.2603471 + 0.6337i
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Star: 14 bent spiral arms

(Is in M)

c = -0.257 + 0.6337i

wolfram alpha plot.
interactive animated javascript: use zoom = 900.
top    index

Counter Star: counterclockwise

(Is in M)

c = -0.255 + 0.63345i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Sunflower

(Is in M)

c = -0.24 + 0.635i

wolfram alpha plot.
interactive animated javascript: use zoom = 1200.
top    index

Space

(Is in M)

c = -0.26 + 0.6332i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Counter Space

(Is in M)

c = -0.25 + 0.6355558i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Crossed Scallop

(Is in M)

c = -0.26 + 0.6337i

wolfram alpha plot.
interactive animated javascript: use zoom = 900.
2-tone version: see 204 arms
Even more arms: -0.26012 + 0.6337i
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Whirled

(Is in M)

c = -0.257 + 0.63432i

wolfram alpha plot.
interactive animated javascript: use zoom = 1200.
top    index

Counter Whirled

(Is in M)

c -0.2548 + 0.6345i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Whirlier

(Is not in M)

c = -0.257 + 0.634456i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Whirliest

(Is not in M)

c = -0.25733999 + 0.634456i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Center cannot hold

(Is not in M)

c = -0.2573399923 + 0.63445602i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Period of 81 points

(Is in M)

c = -0.261 + 0.6337i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
95 tiny spirals: -0.2603471 + 0.6337i
Period 3x81 ? -0.261189 + 0.6337i
Totoro ? -0.2612 + 0.6337i
Screensaver: -0.2612 + 0.63373i
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Lesser Chaos

(Is in M)

c = -0.256625459 + 0.6345309i

wolfram alpha plot.
interactive animated javascript: use zoom = 875.
Worth a try:
-0.256625459 + 0.6345312i (gappy chaos)
-0.256625459 + 0.634501i (super whirly)
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Not Whirly

(Is not in M)

c = -0.257341 + 0.634456i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Party Time

(Is not in M)

c = -0.256625459 + 0.634532i

wolfram alpha plot.
interactive animated javascript: use zoom = 860.
top    index

Frayed Knot

(Is not in M)

c = -0.2566254588 + 0.634532i

wolfram alpha plot.
interactive animated javascript: use zoom = 860.
top    index

Semi Chaos

(Is not in M)

c = -0.2625107 + .64i

wolfram alpha plot.
interactive animated javascript: use zoom = 1000.
top    index

Repelled from center four times

(Is not in M)

c = -0.2599966743201 + 0.6350001i

wolfram alpha plot.
interactive animated javascript: use zoom = 900.
top    index

14 plus one repelling points

(Is not in M)

c = -0.2601112 + 0.63537781i

wolfram alpha plot.
interactive animated javascript: use zoom = 900.
top    index

14 mini 22-arm spirals

(Is in M)

c = -0.26 + 0.635375i

wolfram alpha plot.
interactive animated javascript: use zoom = 900.
top    index

3 armed Spiral

(Is in M)

c = -0.14832 + 0.646999i

wolfram alpha plot.
interactive animated javascript: use zoom = 1200.
top    index

Three 1-armed spirals

(Is in M)

c = -0.145 + 0.654i

wolfram alpha plot.
interactive animated javascript: use zoom = 1200.
top    index

Chaos

(Is in M)

c = -0.01183223669 + 0.63816572702i

wolfram alpha plot.
interactive animated javascript: use zoom = 860.
top    index

Misiurewicz point

(Is in M)

c = -0.77568377 + 0.13646737i

wolfram alpha plot.
wiki
interactive animated javascript: use zoom = 920.
top    index

[ slides]    [ Plot Mandlebrot sequences in Excel ]
...or even better in javascript.

Here's an excellent tutorial on the relationship between Mandelbrot and Julia sets, and on how the Fibonacci numbers appear.
Lots more in Mu-Ency... an encyclopedia of features of the Mandelbrot set.
Here's the Wolfram Mathworld article stating that it is unknown whether the Mandlebrot set is path connected.

A drag-and-drop demonstration of the orbits on a mandelbrot set.
...and a very nice mouse-over julia set generator.

Beautiful collection of lots more orbits by Stefan Bion; including composite pictures of orbits.
INDEX
Density
Star: 14 bent spiral arms
White hole sun
Counter Star: counterclockwise
Fractal Periodic
Sunflower
Space
Counter Space
Crossed Scallop
Whirled
Counter Whirled
Whirlier
Whirliest
Center cannot hold
Period of 81 points
Lesser Chaos
Not Whirly
Party Time
Frayed Knot
Semi Chaos
Repelled from center four times
14 plus one repelling points
14 mini 22-arm spirals
3 armed Spiral
Three 1-armed spirals
Chaos
Misiurewicz point

 Back to research page. 

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