{-# LANGUAGE NoMonomorphismRestriction #-}
{-# LANGUAGE TypeFamilies #-}
module CreatingSymmetry where
import Control.Monad (replicateM)
import Control.Monad.Random
import Data.Colour.Palette.ColorSet (Brightness (..),
d3Colors2)
import Data.Complex
import Data.List.Split (chunksOf)
import Diagrams.Backend.Rasterific.CmdLine
import Diagrams.Prelude
type Term = (Complex Double, Int)
type Params = [Term]
-- | Draw the parametric curve in the complex plane f(t) = \sum_{j=1}^M a_j e^{n_j i t} .
drawCyclicCurve :: (TrailLike t, N t ~ Double, V t ~ V2) => Params -> t
drawCyclicCurve = drawCyclicCurve' 360
-- | Draw the parametric curve in the complex plane f(t) =
-- \sum_{j=1}^M a_j e^{n_j i t}. The first argument controls the
-- number of sample points to generate.
drawCyclicCurve' :: (TrailLike t, N t ~ Double, V t ~ V2) => Int -> Params -> t
drawCyclicCurve' samples params
= cubicSpline True (map (eval params) [0, incr .. tau - incr])
where
incr = tau / fromIntegral samples
eval params t = c2p . sum . map (evalComponent t) $ params
c2p :: Complex a -> P2 a
c2p (x :+ y) = (x ^& y)
evalComponent :: Double -> Term -> Complex Double
evalComponent t (a,n) = a * cis (fromIntegral n * t)
drawCircles :: Params -> Double -> Diagram B
drawCircles params t = drawCircles' (zip [0..] params) t
where
drawCircles' [] _ = circle 0.05 # lw none # fc red
drawCircles' ((i,(a,n)):params) t
= drawCircles' params t # moveTo (c2p a # rotate theta)
<> (circle (magnitude a) <> (origin ~~ c2p a)) # lc (d3Colors2 Light i) # rotate theta
where
theta = (t * fromIntegral n) @@ rad
randomCurve :: (MonadRandom m, Applicative m) => m Params
randomCurve = do
numComponents <- getRandomR (2, 4)
replicateM numComponents randomComponent
randomComponent :: (MonadRandom m, Applicative m) => m (Complex Double, Int)
randomComponent = (,) <$> ((:+) <$> getRandomR (-1,1) <*> getRandomR (-1,1)) <*> getRandomR (-30,30)
mysteryCurve :: Diagram B
mysteryCurve = drawCyclicCurve mysteryParams
mysteryParams :: Params
mysteryParams = [(1,1), (1/2, 6), (0 :+ 1/3, -14)]
randomCurves :: Int -> IO (Diagram B)
randomCurves n = do
curves <- evalRandIO $ replicateM n randomCurve
return
$ vsep 1 . map (hsep 1) . chunksOf 5
. map ( beneath (square 4 # lw none)
. centerXY . sized (dims2D 4 4) . drawCyclicCurve)
$ curves
main = mainWith $ zip (map (bg' white . frame 0.5 . withEnvelope bb) frames)
(repeat (3 :: Int))
bg' :: Colour Double -> Diagram B -> Diagram B
bg' c d = (d <> r # lw none # fc c # scale 1.1) # withEnvelope r
where
r = boundingRect d
bb = mconcat . (map boundingBox) $ frames
frames =
[ mconcat [ drawCircles mysteryParams (fromIntegral t * tau / fromIntegral numFrames)
, mysteryCurve
]
| t <- [0 .. numFrames - 1]
]
numFrames :: Int
numFrames = 720
hub.darcs.net :: byorgey -> mathlesstraveled -> files
Source for blog posts “Random cyclic curves” at “The Math Less Traveled” mathlesstraveled.com (http://mathlesstraveled.com) by Brent Yorgey
root / 2015-06-creating-symmetry-curves / CreatingSymmetry.hs
“Creating Symmetry” written in Haskell
http://hub.darcs.net/byorgey/mathlesstraveled/browse/2015-06-creating-symmetry-curves/CreatingSymmetry.hs#15
Random cyclic curves
by Brent Yorgey
The Math Less Traveled
http://mathlesstraveled.com/2015/06/04/random-cyclic-curves-5/
Princeton Press just sent me a review copy of a new book by Frank Farris called Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. It looks amazing and I’m super excited to read it. Apparently John Cook has been reading it as well, and posted some Python code for generating this curve, which shows up towards the beginning of the book:
Mike Croucher also posted an interactive version using Jupyter notebook, where you can play with sliders to control the parameters of the curve and watch it evolve.
This is a plot of the parametric equation
in the complex plane. In general, Farris considers parametric equations of the form
where
are complex numbers, 
are integers, and as usual 
. All such equations correspond to cyclic plots in the complex plane; he analyzes what sorts of symmetry they will have based on the parameters 
.
He also spends some time talking about the aesthetics of picking values for
and 
that result in beautiful curves. Instead of making carefully considered choices in this kind of situations, I often like to employ randomness to just generate a bunch of different instantiations and see what comes up (although there is still a certain amount of art in choosing the distributions for the random parameters). So, here are 50 random curves. Each one has
I really like seeing these all together. They are not all great individually, but as a group, it’s fun seeing their differences, similarities, and idiosyncracies.
Don’t ask me what the parameters are for an individual curve because the program I used to generate them does not save the parameters anywhere! The code is here if you want to see it; of course it is written in Haskell and uses the diagrams framework.
Creating Symmetry:
The Artful Mathematics of Wallpaper Patterns
by Frank A. Farris
http://press.princeton.edu/titles/10435.html
Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you’ll learn how to create breathtaking art images of your own.
Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.