# Aperiodic monotile

# Single-tile code by Otto Perdeck
# https://rpubs.com/OttoP/1019978
#
# Multi-tile code by Bryan Clair
# http://turtlegraphics.wordpress.com/

library(ggplot2)
library(dplyr)

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")

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


ggplot(monotile_df, aes(V1, V2)) + 
  geom_polygon(mapping = aes(x=x,y=y,group=kite),
               data = monotile_composite, fill='white', color="grey") +
  geom_path(data=as.data.frame(rbind(monotile, monotile[1,])), color="black", size=1) + 
  theme_void() +
  theme(aspect.ratio = 1) +
  xlim(c(-size,size)) + ylim(c(-size,size))


# Bryan Clair's code starts here
#
# Build a data frame "one_tile" with vertices, a kite variable for polygons,
# and an order variable to draw the outside edge.  This simplifys the structure
# of the data and makes it easier to handle multiple tiles
#
monotile_edge <- as.data.frame(rbind(monotile, monotile[1,])) %>% select(x=V1, y=V2)
monotile_edge %>% ggplot(aes(x=x,y=y)) + geom_path()

one_tile <- monotile_edge %>% mutate(order = row_number()) %>%
  left_join(monotile_composite, ., by=c('x','y')) %>%
  mutate(order = ifelse(duplicated(order),NA,order))

# Plot the kite polygons
one_tile %>% ggplot(aes(x=x,y=y,group=kite)) +
  geom_polygon( fill='white', color="grey")

# Plot the edge
one_tile %>% filter(!is.na(order)) %>% arrange(order) %>%
  ggplot(aes(x=x,y=y)) + geom_path()

# Plot the kite polygons and edge
one_tile %>% ggplot(aes(x=x,y=y)) +
  geom_polygon(aes(group=kite), fill='white', color="grey") +
  geom_path(data = one_tile %>% arrange(order) %>% filter(!is.na(order)))


# Function takes integer (dx,dy) coordinates and translates the
# one_tile by the appropriate amount to lie on a periodic tiling
# (which of course has holes)
#
tile_translate <- function(delta) {
  dx <- delta[1]
  dy <- delta[2]
  one_tile %>% mutate(x = x + (2.25)*dx,
                      y = y + sqrt(3)*dy +.425*dx)
}

# Kludgy way to build a list of translation coordinates
tilespots <- list()
for (x in 1:5) {
  for (y in 1:8) {
    tilespots <- append(tilespots, list(c(x,y)))
  }
}

# Create a new data frame with all the translations
multi_tiles <- bind_rows(lapply(tilespots, tile_translate), .id='tile')

# Plot it
multi_tiles %>% ggplot(aes(x=x,y=y)) +
  geom_polygon( aes(group=interaction(tile,kite)), fill='white', color="grey") +
  geom_path(data = multi_tiles %>% arrange(order) %>% filter(!is.na(order)),
            aes(group=tile)) +
  coord_fixed() +
  theme_void()