roots of polynomials: watch the commutator, permutations, braids, as you twiddle the roots

quintic.dance.R

#OUTPUT: https://twitter.com/isomorphisms/status/566965475188670464
#AFTER: https://www.youtube.com/watch?v=RhpVSV6iCko
#ORIGIN: V. I. Arnol'd

#main parts are:
# — polyroot
# — concatenation of the list called "dance"



#convenience
rcomplex <- function(...) complex(real=rnorm(...), imaginary=rnorm(...))
i <- complex(imaginary=1)
arg <- function(z) Arg(z)/2/pi*360
require(magrittr)




#quintic
coefficients <- rcomplex(6)	#5+1 to move the origin around (x°=constant term)


#one set of answers:
options(digits=1)
coefficients %>% polyroot %>% arg %>% print


#(came from what again?)
coefficients %>% arg %>% print


#all right, now stop goofing off and get back on task...
coefficients %>% polyroot %>% polyroot %>% arg %>% print
coefficients %>% polyroot %>% polyroot %>% polyroot %>% arg %>% print
coefficients %>% polyroot %>% polyroot %>% polyroot %>% polyroot %>% arg %>% print







#LOGIC SECTION
#list concatenation is a bit strange in R. Test out with c( list(1,2,3), 1,2,3 ) in REPL.

	#I'm using a pattern like c( list(1:3), list(6:8) )
	#and then LongList <- c( LongList, list( NewStuff ) )
	#as well as NewList looking like: start + modifications**j


#so I'll load up a list-of-6-at-a-time, then loop through each 
dance <- list( coefficients )

for (j in 1:157) { 
	#list concatenation
	dance <- c( dance, coefficients + list(		#coefficients is the name of the INITIAL positions/values
	#TWEAK THIS TO CHANGE THE DANCE
	0,0,0,coefficients[4] + i^(.1 * j),0,0		#six components bc six coefficients
	) )
	}	#end of dance










#GRAPHICS SECTION
#polyroot does almost all the work of solving here

devtools::install_github('karthik/wesanderson'); require(wesanderson)


par(mfrow=c(2,1))	#side-by-side

for(j in 1:length(dance)) {
	#coeff plot
	plot( coefficients[[j]],
		pch=19, col=rgb(0,0,0,1-1:6/10),	#descending transparency
		xlab='ℝ', ylab='√−1', axes=F, mar=0,	#clear
		ylim=c(-3,3), xlim=c(-3,3)		#hold steady
		);
	
	#roots plot
	plot( coefficients[[j]] %>% polyroot,
		pch=19, cex=2, col=wes_palette('Zissou',5),
		xlab='ℝ', ylab='√−1', axes=F, mar=0,	#clear
		ylim=c(-3,3), xlim=c(-3,3)		#hold steady
		);
		
	Sys.sleep(.03)
	}	#end of animation
	
	
	
	
	
	
	
	
	
	
#you can then use require(animation); ?saveGIF
#when you want to save the plot. The command
#strings for interactive loop and saveGIF()
#differ a little, for example you'll need to
#move par(mfrow=c(2,1)) inside the loop, and
#take out Sys.sleep() from the loop.