make complex-plane Wegert plots (plot only the argument, knowing that the length=modulus works the same as in a "regular" Cartesian plot)

wegert.R

#run convenience functions first....
plat <- function(space,FUN,cex=2,chroma=45,a=66,b=4,c=3){		#a,b,c are tuning parameters
		func=FUN(space)						#eval for speedup
		coleur=arg(func)%%360
		light=a+b*mod(func)+c*l(coleur)
	       	plot(space, pch=46,cex=cex, col=hcl( h=coleur, c=chroma, l=light )  )
		}

#what this function does, is plot a boring series of points with nothing on them,
# moving the REAL ACTION to the colour parameter.

#So I'll draw for the space a square called Z,
# and then "pull back" the colours of what happens to the complex argument
# so all of the interesting computation happens in
# plot(...,col=hcl(HERE),...)
# instead of plot(HERE, ...).



#We could already see what happens to the modulus (size) with the usual Cartesian plots.

#The Wegert-plot shows us what's happening to the minus sign,
# with red being +, green being −, and other colours being "somewhere in between".


	#This is why we like the complex plane: instead of +/− being an "on/off" switch (ℤ₂),
	#it becomes either (in my %%360 way) a "circle" 𝕊₁ or
	#(in Wegert's mod(x)/log(mod(x)) way), a spiraling parking-lot ramp.

#It's notable that, around a root (where the function equals/totals/evals to zero),
#you'll see a colour spiral. Try it with
	plat(Z, function(x) (x-1)*(x-i))
#for example.



#convenience
l <- function(x) x/100 - floor(x/100)
arg <- function(x) Arg(x)*360/2/pi
mod <- function(x) l(Mod(x))
i <- complex(imaginary=1)					#sqrt(-1) doesn't work … kinda sad …



#make a complex-plane template square
Z <- matrix(1:500,500,500)		#you can do 501 if you want zero .. it's usually "bad" so I leave it out
Z + i*t(Z) -> Z
Z - complex(length.out=250, real=250,imaginary=250) -> Z
Z/100 -> Z


#fun things to plot in this way
plat( Z, zeta )
plat(Z, function(x) zeta(x,n=15) )

#airy
#bessel
plat( Z, log )
plat( Z, exp )
#(partial) geometric series
plat( Z, sinh )
#polynomials
poly.1 <- function(x) x**9 - 3*x**5 + x**2
poly.2 <- function(x) 5*x**4 + 2*i*x**3 - x**2/i
plat( Z, poly.1 )
plat( Z, poly.2 )



#ratios of polynomials
plat( Z, function(x) poly.1(x) / poly.2(x) )

#random polynomials
2*pi -> tau
rcomplex <- function(n=1L, length=5) complex( argument=runif(n=n, min=0, max=tau), modulus=rpois(n=n, lambda=length) )
rpoly <- function(x) rcomplex(1) * x**5 + rcomplex(1) * x**4 + rcomplex(1) * x**3 + rcomplex(1) * x**2 + rcomplex(1) * x

plat(Z, function(x) rpoly(x) - rpoly(x)		)			#re-rolls the random's every time
plat(Z, function(x) rpoly(x) - rpoly(x) + rpoly(x) - rpoly(x)	)
plat(Z, function(x) rpoly(x)**2 - i*rpoly(x) + rpoly(x) - rpoly(x)	)
plat(Z, function(x) rpoly(x)**2 - i*rpoly(x) + rpoly(x)**3 - rpoly(x)**9	)


#baseline reference
id <- function(x) x
plat( Z, id )







#codes for special functions
zeta <- function(x,n=5) eval( parse(text= paste0( 1:n, '**-x', collapse=' + ') ))	#tried it with sum(), but that discards imaginary parts *only* during plot( sum( ... ))
zeta <- function(x,M=1e9) (1-x)*( M**(1-x) - 1)
Lorentz <- function(x) 1/sqrt(1-x*x)
Airy <- function(x) exp(-2/3*x^1.5) / 2 / sqrt(pi) / x^.25
Airy.plus <- function(x) Airy(x) + x
Airy.sinh <- function(x) Airy(x) + sinh(x)
Dilogarithm <- function(M) integrate( function(t) -log(1-t)/t, 0, M)
polynomial.1 <- function(x) x^3 + 5*x^2 - 2*x+i
polynomial.2 <- function(x) x^4/14 + x^3/13
polynomial.1.2 <- function(x) polynomial.1(x) / polynomial.2(x)
polynomial.2.1 <- function(x) polynomial.2(x) / polynomial.1(x)
polynomial.222.11 <- function(x) polynomial.2(x)^3 / polynomial.1(x)^2
Blaschke <- function(x,k=seed,const=rot) prod( (x-k) / (1-Conj(k)*x ) ) * rot
Bessel.1 <- function(x,a=1) integrate( function(t) cos(a*t - x*sin(t)), 0, pi) / pi
#complex digamma

You can't copy-paste and run the code because some of the stuff is defined at the bottom. So either paste twice or make the Z and define the convenience functions first before running the whole code.

A polar plane:

U <- t( matrix( complex( modulus=rep(1:500,500), argument=1:500/500) ) )

Cool polynomials and a recursive way to generate them from graphs: en.wikipedia.org/wiki/Tutte_polynomial