File:Lavaurs-12.png
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Summary
DescriptionLavaurs-12.png |
English: Topological model of Mandelbrot set using Lavaurs algorithm up to period 12. " The quadratic minor lamination, or QML, is defined by Thurston to be the union of S 1 with all minor leaves. ... The QML is conjecturally equivalent to the Mandelbrot set; this conjecture being equivalent to that stating that the Mandelbrot set is locally connected." [1] Français : modèle topologique de l'ensemble de Mandelbrot, utilisant l'algorithme de Lavaurs jusqu'à la période 12 Polski: Topologiczny model zbioru Mandelbrota dla okresów 1-12. Korzysta z algorytmu Lavaurs'a. |
Date | |
Source | own code based on the Lisp code by R Berenguel |
Author | Adam majewski |
compare with
-
quadratic invariant lamination
associated with rabbit Julia set
-
quadratic invariant lamination
associated with basilica Julia set
-
Shrub model
Programs :
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Src code
; console program
; Lisp / Common Lisp / SBCL
; based on :
; http://www.mostlymaths.net/2009/08/lavaurs-algorithm.html
; lisp code by R Berenguel
; reducible angle is not angle of lower period
; for example 9/15 = 3/5 has period 4
;
; Rules by Lavaures :
; no arc crosses any other arc
; arc connects 2 angles with the same period : first and second
; first angle is the smallest angles not yet connected, and second angle is the next smallest angle not yet connected
;
; orthogonal circles (x1,y1,r1) and (x2,y2,r2)
; r1^2 + r2^2 = (x2-x1)^2 +(y2-y1)^2
; http://planetmath.org/encyclopedia/OrthogonalCircle.html
; http://classes.yale.edu/fractals/Labs/NonLinTessLab/BasicConstr3.html
;
; example of use :
;
; sbcl
; (load "ls.lisp")
; (draw-lavaurs "lavaurs-5.png" 2000 5)
; ;look for lavaurs-5.png file in your home directory
;
; Adam Majewski
; fraktal.republika.pl
; 2010.09.04- 11.17
;
;
;; This program is free software: you can redistribute it and/or
;; modify it under the terms of the GNU General Public License as
;; published by the Free Software Foundation, either version 3 of the
;; License, or (at your option) any later version.
;; This program is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
;; General Public License for more details.
;; You should have received a copy of the GNU General Public License
;; along with this program. If not, see
;; <http://www.gnu.org/licenses/>.
; first run
;(require :asdf)
;(require :asdf-install)
;(asdf-install:install :zpng)
;(asdf-install:install :Vecto)
; http://www.xach.com/lisp/vecto/
; you must press 2 and 0 when the program asks
; next times load packages from disk
(asdf:operate 'asdf:load-op 'vecto)
(defun doubling-map (ratio-angle)
" period doubling map = The dyadic transformation (also known as the dyadic map,
bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map "
(let* ((n (numerator ratio-angle))
(d (denominator ratio-angle)))
(setq n (mod (* n 2) d)) ; (2 x n) modulo d = doubling
(/ n d)))
(defun give-period (ratio-angle)
"gives period of angle in turns (ratio) under doubling map"
(let* ((n (numerator ratio-angle))
(d (denominator ratio-angle))
(temp n)) ; temporary numerator
(loop for p from 1 to 100 do
(setq temp (mod (* temp 2) d)) ; (2 x n) modulo d = doubling)
when ( or (= temp n) (= temp 0)) return p )))
(defun give-list (period)
"Returns a list of all angles of given period
without angles with lower periods, which divide period.
period is integer > 0 "
(let* ((angles-list '())
(den (- (expt 2 period) 1 )) ; denominator of angle = (2^period)-1
a ) ; angle
(when (> period 0)
(dotimes (num (+ den 1)) ; from 0 to den
(setq a (/ num den ))
(when (= period (give-period a)) ;
(setq angles-list (append angles-list (list a)))))) ;
angles-list))
(defun not-crosses (new-pair old-pair)
"checks if new arc ( = between new-angle-1 and new-angle-2)
crosses old arc ( = between (first old-pair) and (second old-pair)).
It is a part of Lavaurs algorithm.
Angles are external angles mesured in turns
angle-1 < angle-2
test :
(setq old-p '(1/3 2/3))
(setq new-p '(4/15 6/15))
(not-crosses new-p old-p)
(not-crosses (list 1/7 2/7) old-p) ; t
(not-crosses (list 1/7 3/7) old-p) ; nil
"
(let ((old-angle-1 (first old-pair))
(old-angle-2 (second old-pair))
(new-angle-1 (first new-pair))
(new-angle-2 (second new-pair)))
; check the order of input angles
(when (< new-angle-2 new-angle-1) (rotatef new-angle-1 new-angle-2))
(when (< old-angle-2 old-angle-1) (rotatef old-angle-1 old-angle-2))
(cond
((< new-angle-1 new-angle-2 old-angle-1 old-angle-2) t)
((< old-angle-1 old-angle-2 new-angle-1 new-angle-2) t)
((< new-angle-1 old-angle-1 old-angle-2 new-angle-2) t)
((< old-angle-1 new-angle-1 new-angle-2 old-angle-2) t)
(t nil))))
(defun not-crosses-all (new-pair old-list)
"checks if new pair of rays do not crosses any of pairs from old list
test :
(setq old-pairs '((1/3 2/3) (1/7 2/7)))
(not-crosses-all (list 4/15 6/15) old-pairs) ; nil
"
(let ((b-result T))
(loop for old-pair in old-list do (setq b-result (and b-result (not-crosses new-pair old-pair))))
b-result ))
(defun give-pairs-up-to (period)
"gives list of external angles (pairs landing on the same root point)
for periods up to input period
period >= 3
examples of use :
(give-pairs-old-old 3)
(give-pairs-old-old 7)
"
(let* ((pairs (list (list 1/3 2/3))) ; list for period 2
angles-list ; temporary list
i
new-first-angle
new-second-angle)
( loop for p from 3 to period do
(setq angles-list (give-list p))
(loop for j from 0 to (/ (- (length angles-list) 1) 2) do
(setq new-first-angle (pop angles-list)) ; pop removes angle from list
; find second ( conjugate) angle
(setq i 0)
(loop do ;for i from 0 to (- (length angles-list) 1) do ; first = nth 0
(setq new-second-angle (nth i angles-list)) ; nth do not removes angle from list
(setq i (+ i 1))
until (not-crosses-all (list new-first-angle new-second-angle) pairs))
(setq angles-list (remove new-second-angle angles-list))
(push (list new-first-angle new-second-angle) pairs))) ; save new pair to pairs list
(reverse pairs)))
; it should be the same as number of components
;(loop for p from 3 to 10 do (print (length (give-pairs p))))
;3
;6
;15
;27
;63
;120
;252
;495
;
(defun give-pairs (period-max)
"gives list of external angles (pairs landing on the same root point)
for period = pariod-max
period >= 2
examples of use :
(give-pairs 3)
(give-pairs 7) ;
(time (give-pairs 16))
(compile 'give-pairs)
"
(let* ((pairs (list (list 1/3 2/3))) ; list for period 2
angles-list ; temporary list
(previous-list pairs) ; list for "previous period" = (period -1)
i
new-first-angle
new-second-angle)
( loop for period from 3 to period-max do
(setq angles-list (give-list period)) ; find all angles for given period
(setq previous-list pairs) ; update previous list
; match pairs of angles
(loop for j from 0 to (/ (- (length angles-list) 1) 2) do
(setq new-first-angle (pop angles-list)) ; pop removes angle from list
; find second ( conjugate) angle
(setq i 0)
(loop do ;for i from 0 to (- (length angles-list) 1) do ; first = nth 0
(setq new-second-angle (nth i angles-list)) ; nth do not removes angle from list
(setq i (+ i 1))
until (not-crosses-all (list new-first-angle new-second-angle) pairs))
(setq angles-list (remove new-second-angle angles-list))
(push (list new-first-angle new-second-angle) pairs))); save new pair to pairs list
(setq pairs (set-difference pairs previous-list :test 'equal)) ; remove previous angles
(reverse pairs)))
; -------------------------------------- drawing code ------------------------------------------------
(defun ttr (turn)
" Turns to Radians"
(* turn (* 2 pi) ))
; circle-list angle-list
(defun give-arc-list (circle-list angle-list)
"
Copyright 2009 Rubén Berenguel
ruben /at/ maia /dot/ ub /dot/ es
Find the ortogonal circle to the main circle, given the angles in
it.
Input :
R: radius of the main circle
angle1, angle2 : angles of main circles (in turns)
(a, b) , (ba, bb) : points of main circle and new ortogonal circle
Output is a list for vecto:arc procedure
thru draw-arc procedure
http://classes.yale.edu/fractals/Labs/NonLinTessLab/BasicConstr3.html
"
(let* ((x0 (first circle-list))
(y0 (second circle-list))
(r0 (third circle-list))
(alpha (ttr ( first angle-list))) ; convert units from turns to radians
(balpha (ttr (second angle-list)))
(gamma (+ alpha (/ (- balpha alpha) 2))) ; angle between alpha and balpha
(ca (cos alpha))
(cg (cos gamma))
(sa (sin alpha))
(sg (sin gamma))
(temp (/ r0 (+ (* ca cg) (* sa sg))))
; first common point
(a (+ x0 (* r0 ca))) ; a = x0 + r0 * cos(alpha)
(b (+ y0 (* r0 sa))) ; b = y0 + r0 * sin(alpha)
; center of ortogonal circle
(x (+ x0 (* temp cg)))
(y (+ y0 (* temp sg)))
; center of middle circle
(xma (- x a))
(ymb (- y b))
; radius of ortogonal circle
(r (sqrt (+ (* xma xma) (* ymb ymb))))
; angle of point (a,b) measured in new circle units
(phi (atan r0 r)))
; result
(list x y r
(+ pi gamma phi) ; new balpha
(- (+ pi gamma) phi) ; new alpha
a b))) ; point (a,b)
(defun draw-arc (circle-list angle-list)
" computes otogonal circle
using give-arc-list
and draws arc using vecto:arcn procedure
vecto:arcn x y radius angle1 angle2 "
(let* ((arc-list (give-arc-list circle-list angle-list)))
(vecto:move-to ( sixth arc-list) (seventh arc-list)) ; beginning of arc is point (a,b)
(vecto:arcn
( first arc-list) ; x
(second arc-list) ; y
(third arc-list) ; radius
(fourth arc-list) ; angle1
(fifth arc-list))) ; angle2
(vecto:stroke))
(defun draw-arcs (circle-list angles-list)
"draws arc from angles-list
using draw-arc procedure"
(loop for angles in angles-list do (draw-arc circle-list angles)))
; example of use : (draw-lavaurs "a.png" 800 2)
(defun draw-lavaurs (file side period)
"computes
and draws Lavaurs model of Mandelbrot set. "
(vecto:with-canvas (:width side :height side) ; vecto
(vecto:set-rgb-stroke 0 0 0) ; vecto
(vecto:set-line-width 1) ; vecto
(let* ( (x0 (/ side 2))
(y0 x0) ;
(r0 (- x0 50)) ; leave place for titles
(main-circle-list (list x0 y0 r0))
arc-list)
; main circle
(vecto:centered-circle-path x0 y0 r0 )
; arcs ( chords)
(setq arc-list (give-pairs-up-to period)) ; compute
(draw-arcs main-circle-list arc-list) ; draw
(vecto:stroke)) ; before save ( vecto procedure)
(print (vecto:save-png file)))) ; save image
;----------global var ----------------------
(defparameter *period* 4 " maximal period. It is an integer >= 2 ")
(defparameter *size* 800 " size of image in pixels. It is an integer >= 0 ")
(defparameter *file-name*
(make-pathname
:name (concatenate 'string "lavaurs-" (write-to-string *period*))
:type "png")
"name (or pathname) of png file ")
;======================= run =====================================================================
(draw-lavaurs *file-name* *size* *period*)
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current | 17:58, 3 August 2023 | ![]() | 2,000 × 2,000 (356 KB) | Obscure2020 | Optimized with OxiPNG and ZopfliPNG. |
12:15, 7 November 2010 | ![]() | 2,000 × 2,000 (916 KB) | Soul windsurfer | {{Information |Description={{en|1=Topological model of Mandelbrot set using Lavaurs algorith up to period 12}} |Source={{own}} |Author=Adam majewski |Date=2010-11-07 |Permission= |other_versions= }} |
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