Rational Coordinates
Robert P. Munafo, 2012 Apr 22.
See also Exact Coordinates.
There are a great many "special" points in the Mandelbrot Set which have exact rational coordinates. Of particular interest are those which can be expressed in base 10 without a repeating decimal.
The first set of points that merits discussion are the ones with very simple rational fractions, most of which have an exact multiple of ^{1}/_{4} in either the real or imaginary component:
coordinates | description |
-2 + 0i | utter west, R2F(1/2B1)t |
-1^{3}/_{4} + 0i | cusp of largest island, R2F(1/2B1)S.C(0) |
-1^{1}/_{4} + 0i | seahorse valley west, R2.1/2.C(1/2) |
-1 + ^{1}/_{4}i | bond point between R2.1/2a and R2.1/2.1/4a |
-^{3}/_{4} + 0i | seahorse valley, R2.C(1/2) |
0 + i | A terminal point of period 2 and external argument ^{1}/_{6} |
^{1}/_{4} + ^{1}/_{2}i | quad spiral valley, R2.C(1/4) |
^{1}/_{4} + 0i | elephant valley, R2.C(0) |
List by Jay Hill
The following is a 1993 article to sci.fractals by Jay Hill. Near the bottom of the article is a list of points on the boundary of the main cardioid whose coordinates can be expressed exactly in base 10.
Newsgroups: sci.fractals Path: ...!unogate!mvb.saic!ast.saic!jupiter!hilljr From: hilljr (at) jupiter (dot) saic (Jay R. Hill) Subject: Periodicity checking and chaotic orbits Message-ID: <1993Mar26.235613.13190 (at) ast (dot) saic> Followup-To: sci.fractals Lines: 104 Sender: news (at) ast (dot) saic Organization: SAIC Date: Fri, 26 Mar 1993 23:56:13 GMT Hi pixel counters, On 13 Mar 93 01:10:49 GMT tef@acsu.buffalo.edu (T. E. Frisinger) posted questions about periodicity checking on Mandelbrot set boundary. He had developed a fast boundary tracing algorithm. He asked if periodicity checking is hurting? Are their periods very large? The answer is their periods are usually large. There are five classes of points, c, (z <- z*z + c), as I see it. a) outside M set, the points diverge rather quickly to infinity (at least by order n=1/d, d = distance to nearest part of M. b) inside a bud or cardioid, converge to points rather quickly (at least by order n=1/d, d = distance to nearest exit from M, the number of points is the period of the bud. c) points exactly on the edge of a bud or cardioid where two of these are touching (bud to bud or bud to cardioid). Here the orbit slowly converges to points, the number of convergent points is equal to the smaller period of the two. The convergence is very slow, forming spike starlike patterns when plotted. d) points exactly on the edge of a bud or cardioid where no other bud is touching. Here the orbit time is infinite, its path is a possible multiple fractal like pattern resembling the shore of Austrailia :-) The path goes through (0,0) and c. The path is sort of similar at all scales, eventually filling in one or more wiggly closed curves. If c is on the Cardioid, period=1, there is only one wiggly closed curve. If c is on the period 2 bud, there are two simular wiggly closed curves. One containing (0,0) and the other containing c. e) points on filaments. These orbits never converge, they are like case d). Along the real line these points are what folk talk about when discussing chaos and the formula x <- rx(1-x). It is interesting to plot these orbits for cases c) and d). Those of c), for example (0.25,0.5), form a star with 4 spikes, closing in on a point a=(0,0.5), _______ 1 - \/ 1 -4c a = -------------- 2 You may wish to look at orbits for case d). They are quite entertaining. To generate orbit paths I use 65k to 1M iterations. Here (Table I) are some for which starting values are exactly denotable in decimal. The columns labeled Ur, Ui are on a circle raduis 1/2. I use them to generate pairs of C, C1 on the Cardioid, C2 on the period 2 bud. C1 = -U(1 + U) C2 = U/2 - 1 I like to plot orbits for C1 and C2 (for the same U) in the same color. They form 3 similar patterns, two meeting at (0,0). The columns labeled Cr, Ci are included for convience, C2 = (Cr,Ci). Table I Ur Ui Cr Ci 0.49856 0.03792 0.2514358528 0.0001092096 0.4892352 0.1031936 0.26053303816192 0.00222171693056 0.48 0.14 0.2692 0.0056 0.468 0.176 0.279952 0.011264 0.4216 0.2688 0.31610688 0.04214784 0.4 0.3 0.33 0.06 0.376096 0.329472 0.343199597568 0.081645797376 0.329472 0.376096 0.362368402432 0.128269797376 0.3 0.4 0.37 0.16 0.2688 0.4216 0.37429312 0.19494784 0.176 0.468 0.364048 0.303264 0.14 0.48 0.3508 0.3456 0.1031936 0.4892352 0.33189576183808 0.38826331693056 0.03792 0.49856 0.2850441472 0.4607492096* -0.03792 0.49856 0.2092041472 0.5363707904 -0.1031936 0.4892352 0.12550856183808 0.59020708306944 -0.14 0.48 0.0708 0.6144 -0.176 0.468 0.012048 0.632736 -0.2688 0.4216 -0.16330688 0.64825216 -0.3 0.4 -0.23 0.64 -0.329472 0.376096 -0.296575597568 0.623922202624 -0.376096 0.329472 -0.408992402432 0.577298202624 -0.4 0.3 -0.47 0.54 -0.4216 0.2688 -0.52709312 0.49545216 -0.468 0.176 -0.656048 0.340736 -0.48 0.14 -0.6908 0.2744 -0.4892352 0.1031936 -0.71793736183808 0.20416548306944 -0.49856 0.03792 -0.7456841472 0.0757307904 Warmly, Jay -- { hilljr (at) jupiter } begin writeln(3*ln(640320)/sqrt(163):17:15) end. void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0; for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<3;n++); printf("%17.15lf\n",1/a);}revisions: 20031230 oldest on record; 20120422 brief intro for Jay Hill article
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.
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