Aperiodic Tiling

Aperiodic Tiling

Reproducing the patterns from the March ’23 discovery of a monotile.

https://aperiodical.com/2023/03/an-aperiodic-monotile-exists

library(ggplot2)

kite_longlen <- cos(pi/6)
kite_shortlen <- sin(pi/6)
a <- sin(pi/6)*kite_longlen # = sqrt(3)/4
b <- cos(pi/6)*kite_longlen # = 0.75
kite <- matrix(c(0, 0, a, b, 0, 1, -a, b), byrow=T, ncol=2)

kitedf <- as.data.frame(kite)
names(kitedf) <- c("x", "y")
kitedf$p <- 1:nrow(kitedf)
ggplot(kitedf, aes(x, y)) + geom_polygon(fill="blue") + 
  geom_label(aes(label=p))+
  theme_minimal() +
  theme(aspect.ratio = 1) +
  ggtitle("Kite")

rotationMatrix <- function(theta)
{
  matrix(c(cos(theta), -sin(theta), sin(theta), cos(theta)), byrow=T, nrow=2)  
}

transpose <- function(m, x, y)  
{
  m + matrix(c(rep(x, dim(m)[1]), rep(y, dim(m)[1])), byrow=F, ncol=2)
}

kite1 <- as.data.frame(kite%*%rotationMatrix(-pi/6))
kite1$kite <- 1
kite2 <- as.data.frame(kite%*%rotationMatrix(pi/6))
kite2$kite <- 2
kite3 <- as.data.frame(transpose(kite%*%rotationMatrix(5*pi/6), 0, 2*kite_longlen))
kite3$kite <- 3
kite4 <- as.data.frame(transpose(kite%*%rotationMatrix(-3*pi/6), 1+kite_shortlen, kite_longlen))
kite4$kite <- 4
kite5 <- as.data.frame(transpose(kite%*%rotationMatrix(3*pi/6), 0, 2*kite_longlen))
kite5$kite <- 5
kite6 <- as.data.frame(transpose(kite%*%rotationMatrix(-pi/6), 1+kite_shortlen, kite_longlen))
kite6$kite <- 6
kite7 <- as.data.frame(transpose(kite%*%rotationMatrix(-5*pi/6), 1+kite_shortlen, kite_longlen))
kite7$kite <- 7
kite8 <- as.data.frame(transpose(kite%*%rotationMatrix(5*pi/6), 1+kite_shortlen, kite_longlen))
kite8$kite <- 8

monotile_composite <- do.call("rbind", list(kite1, kite2, kite3, kite4, kite5, kite6, kite7, kite8))
names(monotile_composite) <- c("x", "y", "kite")
monotile_composite$kite <- as.factor(monotile_composite$kite)
offset_x <- mean(monotile_composite$x)
offset_y <- mean(monotile_composite$y)
monotile_composite$x <- monotile_composite$x-offset_x
monotile_composite$y <- monotile_composite$y-offset_y
size <- max(c(monotile_composite$x, monotile_composite$y))

ggplot(monotile_composite, aes(x=x,y=y,group=kite,fill=kite)) + 
  geom_polygon() + 
  theme_minimal() + 
  theme(aspect.ratio = 1) + 
  xlim(c(-size,size)) + ylim(c(-size,size)) +
  ggtitle("Composite Monotile")

refPoint <- function(k,i) { unlist(k[i,1:2]) }

monotile <- matrix(c(refPoint(kite1, 1),
                     refPoint(kite1, 4), 
                     refPoint(kite1, 3), 
                     refPoint(kite1, 2),
                     refPoint(kite3, 1),
                     refPoint(kite5, 4),
                     refPoint(kite5, 3),
                     refPoint(kite6, 2),
                     refPoint(kite6, 1),
                     refPoint(kite8, 4),
                     refPoint(kite8, 3),
                     refPoint(kite8, 2),
                     refPoint(kite7, 3),
                     refPoint(kite7, 2)),
                   byrow = T,
                   ncol=2)
monotile[,1] <- monotile[,1]-offset_x # mean(monotile[,1])
monotile[,2] <- monotile[,2]-offset_y # mean(monotile[,2])

ggplot(as.data.frame(monotile), aes(V1, V2)) + 
  geom_polygon(mapping = aes(x=x,y=y,group=kite),
               data = monotile_composite, fill="purple", color="grey") +
  geom_path(data=as.data.frame(rbind(monotile, monotile[1,])), color="black", size=1) + 
  theme_minimal() +
  theme(aspect.ratio = 1) +
  xlim(c(-size,size)) + ylim(c(-size,size)) +
  ggtitle("Monotile") + xlab("x") + ylab("y")