Arnold's_Cat_Map_animation_(74px,_zoomed,_labelled).gif


Description
English: Animation of Arnold's cat map (zoomed 3 times to make the pixels clearer, and labelled with the iteration number), using an image of cherries as the starting image. The image is 74 pixels square, and repeats after 114 iterations. Notice how the image sometimes contains superimposed cherries, many tiny cherries or a single perfectly reproduced but upside-down image.
Date
Source

Own work

Used File:Cherry Stella444.jpg as the starting image
Author Inductiveload
Permission
( Reusing this file )
w:en:Creative Commons
attribution share alike
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Other versions

Python source code

#!/usr/bin/env python
#-*- coding:utf-8 -*-
 
import Image #needs PIL for image handling
import ImageChops
import ImageFont
import ImageDraw
 
# checks if two images are equal
def equal(im1, im2):
    return ImageChops.difference(im1, im2).getbbox() is None
 
# add a text caption to an image (adds space at the bottom)
def addCaption(im, text, pointSize):
    size = im.size
    draw = ImageDraw.Draw(im)
 
    textFontLocation = "/usr/share/fonts/truetype/ttf-dejavu/DejaVuSans.ttf"
    topBottomMargin = 5
 
    textFont = ImageFont.truetype(textFontLocation ,pointSize)
    textSize = draw.textsize(text, font=textFont) # the size of the text box!
 
    newImage = Image.new(im.mode,\
            (im.size[0], im.size[1]+textSize[1]+2*topBottomMargin), "white" )
    newImage.paste( im, (0,0) )
    draw = ImageDraw.Draw(newImage)
 
    textX = (newImage.size[0] / 2.0) - (textSize[0] / 2.0)
    textY = newImage.size[1] - textSize[1] - topBottomMargin
 
    draw.text((textX, textY), text, fill="black", font=textFont)
    return newImage
 
# add a solid border to an image
def addBorder(im, color, thickness):
    newImage = Image.new(im.mode,\
            (im.size[0]+thickness*2, im.size[1]+thickness*2), color)
    newImage.paste( im, (1,1) )
    return newImage
 
# scale an image up or down (doesn't resample, so the pixels are clear)
def scaleImage(im, factor):
    newImage = Image.new(im.mode,\
            (im.size[0]*factor, im.size[1]*factor), "white" )
    newImage.paste( im.transform((im.size[0]*factor, im.size[1]*factor),\
            Image.AFFINE,\
            (1/float(factor),0,0,0,1/float(factor),0) ),\
            (0,0) )
    return newImage
 
def saveImages(im, iteration):
    im.save("Catmap%04d.png"%iteration) #save the simple image
 
    #resize image, add border and caption
    imageLarge = scaleImage(im, 6)
    imageLarge = addBorder(imageLarge, "black", 1)
    imageLarge = addCaption(imageLarge, str(iteration), 40)
    imageLarge.save("CatmapLargeNumbered%04d.png"%iteration)
 
# CONTROL STARTS HERE
 
inFile = "aa-catmap-orig.resized.jpg" #input image
 
image0 = Image.open(inFile, 'r')
 
#crop to square if required
if image0.size[0] != image0.size[1]:
    n = min(image0.size[0], image0.size[1] )
    image0 = image0.crop((0,0,n,n))
else:
    n = image0.size[0]
 
imageOrig = image0.copy() # keep to original image to see when we return to it
 
image1 = Image.new(image0.mode, (n,n)) # temp image to copy pixels to
pixels1 = image1.load()
 
# BEGIN THE CAT MAPPING PROCESS
 
iteration = 0
saveImages(image0, iteration) # save the first image
while True:
    pixels0 = image0.load() #reload the iterated image
 
    for x in range(n):# perform the mapping
        for y in range(n):
            newX = (2*x + y) % n #find new location
            newY = (x + y) % n
            pixels1[newX, newY] = pixels0[x, y] # copy the pixel over
 
    image0 = image1.copy() #transfer back to image0 for the next iteration
    iteration += 1
    saveImages(image0, iteration) # save this iteration's image
 
    if equal(image0, imageOrig):
        break

Captions

The phase portrait of Arnold's cat map.

Items portrayed in this file

depicts

16 September 2010

image/gif

15d48a9deeea7020f0ed1b1994353ca581c394b1

2,042,441 byte

46.7999999999999 second

263 pixel

224 pixel