Not fractals – but still, pretty math pictures… These were created with the following Python program (MIT license; download from GitHub):

#!/usr/bin/python
'''    Spectral Harmonographs   Copyright 2014 Alan Richmond (Tuxar.uk)

    Trace of 4 decaying sine waves, 2 per axis (x & y)(i.e. 2-pendula), with rainbow colour.
    I did this in Java some decades ago (Encyclogram; I no longer have the source), this
    version is in Python, with PyGame.
    It randomly generates a sequence of harmonographs. It's fast, and can be set to go
    much faster (or slower) if you want.
    Tip: set the display window to fullscreen. On KDE Ubuntu right-click on the title bar,
    select More Actions -> Fullscreen
'''
print("Quit: q key, Screenshot: spacebar")

import pygame, sys, random as r
from pygame.locals import *
from math import pi, sin, cos, exp
#                        EDIT THESE:
width,height=1280,720       # YouTube HD
width,height=1920,1080      # my left monitor
width,height=1280,1024      # my right monitor
width,height=1680,1050      # Lucy's monitor
width,height=1200,800
#width,height=2560,1440      # YT channel art
dd=0.99995                  # decay factor
dt=0.02                     # time increment
speed=200                   # yes, speed
hui=57*2                    # Hue increment
hue,sat,val,aaa=0,100,100,0
sd=0.005                    # frequency spread (from integer)
mx=4                        # max range for amplitudes & frequencies
print("Hit SPACE to save")

def check_event():
    global save
    for event in pygame.event.get():
        if event.type == QUIT:
            sys.exit()
        elif event.type == KEYDOWN and event.key == K_q:
            sys.exit()
        elif event.type == KEYDOWN and event.key == K_SPACE:
            save=True
            print("Saving when finished...")

def scale(length):
    while True:
        a1,a2=r.randint(-mx,mx),r.randint(-mx,mx)
        max=abs(a1)+abs(a2)
        if max>0: break
    return a1,a2,length/(2*max)
steps=0
pygame.init()
pygame.event.set_allowed([QUIT, KEYDOWN])
screen = pygame.display.set_mode((width,height),DOUBLEBUF)
screen.set_alpha(None)
#fg=pygame.Color(0,0,0,0)
#fg=(0,0,0)
fg=(255,255,255)
save=False
while True:
#   Amplitudes & scales
    ax1,ax2,xscale=scale(width)
    ay1,ay2,yscale=scale(height)
    fx1, fx2 =  r.randint(1,mx) + r.gauss(0,sd), r.randint(1,mx) + r.gauss(0,sd)
    fy1, fy2 =  r.randint(1,mx) + r.gauss(0,sd), r.randint(1,mx) + r.gauss(0,sd)
    px1, px2 =  r.uniform(0,2*pi), r.uniform(0,2*pi)
    py1, py2 =  r.uniform(0,2*pi), r.uniform(0,2*pi)
    print(ax1,ax2,ay1,ay2)
    print(fx1,fx2,fy1,fy2)
    print(px1,px2,py1,py2)

    dec=1.0
    t=0.0                       # angle for sin
    first=True
    while dec>0.015:
                                # calculate next x,y point along line
        x = xscale * dec * (ax1*sin(t * fx1 + px1) + ax2*sin(t * fx2 + px2)) + width/2
        y = yscale * dec * (ay1*sin(t * fy1 + py1) + ay2*sin(t * fy2 + py2)) + height/2
        dec*=dd                 # decay
        if not first:           # ignore any complaint about prev_x,y being undefined
#            fg.hsva=(hue,sat,val,aaa)
#            hue = (hue + dt*hui) % 360      # cycle hue
            pygame.draw.aaline(screen, fg, (x, y), (prev_x, prev_y), 1)
        else:
            first=False
        
        prev_x = x              # save x,y for next line segment start
        prev_y = y
        if steps%speed==0: pygame.display.update()
        steps+=1
        t+=dt                   # increment angle for sin
        check_event()
        
    if save:                    # parameters are encoded into filename
        pars='shg-{0}_{1}-{2}_{3}-{4}_{5}'.format(ax1,ax2,fx1,fx2,px1,px2)
        pygame.image.save(screen, pars+'.jpg')
        print("Saved as "+pars+'.jpg')
        save=False

    screen.fill((0,0,0))
#    screen.fill((255,255,255))

You may need to install python and/or pygame, e.g. on Ubuntu/Debian style linuxes:

sudo apt-get install pygame
sudo apt-get install python
Go to the directory you saved it in and make it executable, then run it:

chmod +x harmonograph.py
./harmonograph.py
Set the display window to fullscreen (right click on title bar, maybe it’s under More Actions?).

Encyclogram was #1 in JavaBoutique’s Hall of Fame

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