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Polygon Tiling Examples

Triangular tiling
Square tiling
Hexagonal tiling
Elongated triangular tiling
Snub square tiling
Truncated square tiling
Truncated hexagonal tiling
Trihexagonal tiling
Snub trihexagonal tiling
Truncated trihexagonal tiling
Rhombitrihexagonal tiling
Additional uniform tilings with star polygons

Non-uniform tilings

Herringbone tiling
Pythagorean tiling
Rhombille tiling
Tetrakis square tiling
Additional tilings using rhombi

Several uniform regular polygon tiling patterns can be achieved by use of grid.pattern_regular_polygon() plus occasionally grid.polygon() to set a background color. This vignette highlights several such tiling patterns plus a couple notable non-uniform tiling patterns.

library("grid")
library("gridpattern")
add_borders <- function() {
    pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
    grid.rect(gp = gpar(fill = NA, col = "white", lwd = 6))
    popViewport()
    pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
    grid.rect(gp = gpar(fill = NA, col = "white", lwd = 6))
    popViewport()
    pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
    grid.rect(gp = gpar(fill = NA, col = "white", lwd = 6))
    popViewport()
    pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
    grid.rect(gp = gpar(fill = NA, col = "white", lwd = 6))
    popViewport()
}
x_sq <- c(0, 0, 1, 1)
y_sq <- c(0, 1, 1, 0)
# colorblind accessible scheme https://jfly.uni-koeln.de/color/
blue <- grDevices::rgb(0.35, 0.70, 0.90)
yellow <- grDevices::rgb(0.95, 0.90, 0.25)
red <- grDevices::rgb(0.80, 0.40, 0.00)
green <- grDevices::rgb(0.00, 0.60, 0.50)
orange <- grDevices::rgb(0.90, 0.60, 0.00)

Uniform tilings

Triangular tiling

When we consider the background to be a triangle fill color we can straight-forwardly achieve several of the uniform colorings of a triangular tiling using the “convex3” shape with the “hex” grid.

grid.triangular_tiling <- function(...) {
    grid.pattern_regular_polygon(..., shape = "convex3", density = 1, grid = "hex",
                                 spacing = 0.2, angle = 0, rot = 180)
}
# 1 color uniform triangular tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = yellow, col = "black")
grid.triangular_tiling(gp = gp)
popViewport()

# 2 color uniform triangular tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = red, col = "black")
grid.triangular_tiling(gp = gp)
popViewport()

# 3 color uniform triangular tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = c(blue, red), col = "black")
grid.triangular_tiling(gp = gp)
popViewport()

# 4 color uniform triangular tiling
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = c(red, blue, green), col = "black")
grid.triangular_tiling(gp = gp)
popViewport()
add_borders()

$Four different uniform colorings of a triangular tiling$

Square tiling

We can achieve the nine non-staggered uniform colorings of a square tiling using the “square” shape and the appropriate “type” (and perhaps “subtype”).

# 2 color uniform square tiling (1212)
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = c(yellow, red), col = "black")
grid.pattern_regular_polygon(shape = "square", density = 1, 
                             spacing = 0.2, angle = 0, gp = gp)
popViewport()
# 2 color uniform square tiling (1122)
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = c(yellow, red), col = "black")
grid.pattern_regular_polygon(shape = "square", density = 1, type = "horizontal",
                             spacing = 0.2, angle = 0, gp = gp)
popViewport()
# 3 color uniform square tiling (1213)
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = c(yellow, red, blue), col = "black")
grid.pattern_regular_polygon(shape = "square", density = 1,
                             spacing = 0.2, angle = 0, gp = gp)
popViewport()
# 4 color uniform square tiling (1234)
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = c(yellow, red, blue, green), col = "black")
grid.pattern_regular_polygon(shape = "square", density = 1,
                             spacing = 0.2, angle = 0, gp = gp)
popViewport()
add_borders()

Four different uniform colorings of a square tiling

Hexagonal tiling

We can achieve all the uniform colorings of a hexagonal tiling using the “convex6” shape with the “hex” grid and the appropriate “type”. Replacing the “convex6” shape with an appropriately scaled “star6” shape results in a uniform tiling using star polygons.

grid.hexagonal_tiling <- function(..., shape = "convex6", spacing = 0.2) {
    scale <- star_scale(6, 30)
    grid.pattern_regular_polygon(..., shape = shape, density = 1, grid = "hex",
                                 spacing = spacing, scale = scale, angle = 0)
}
# 1 color uniform hexagonal tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = "black")
grid.hexagonal_tiling(gp = gp)
popViewport()

# One of several 2 color uniform hexagonal tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = c(red, yellow), col = "black")
grid.hexagonal_tiling(gp = gp)
popViewport()

# 3 color uniform hexagonal tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = c(yellow, red, blue), col = "black")
grid.hexagonal_tiling(gp = gp)
popViewport()

# (4.6_{π/6})^3
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = red, col = "black")
grid.hexagonal_tiling(shape = "star6", spacing = 0.3, gp = gp)
popViewport()
add_borders()

Four different uniform colorings of a hexagonal tiling

Elongated triangular tiling

We can achieve the uniform colorings of a elongated triangular tiling using the “elongated_triangle” grid and the “convex4” and “convex3” shapes.

grid.elongated_triangular_tiling <- function(...) {
    grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
    grid.pattern_regular_polygon(..., shape = rep(c("convex4", "convex3"), each = 2), 
                                 density = rep(c(1.41, 1.15), each = 2), 
                                 grid = "elongated_triangle",
                                 type = "square_tiling", subtype = "3412",
                                 spacing = 0.2, angle = 0, 
                                 rot = rep(c(45, 0), each = 2))
}
# 1 color elongated triangular tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = "black")
grid.elongated_triangular_tiling(gp = gp)
popViewport()

# 2 color elongated triangular tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = c(red, red, yellow, yellow), col = "black")
grid.elongated_triangular_tiling(gp = gp)
popViewport()

# 3 color elongated triangular tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = c(red, blue, yellow, yellow), col = "black")
grid.elongated_triangular_tiling(gp = gp)
popViewport()
add_borders()

Three different uniform colorings of an elongated triangular tiling

Snub square tiling

We can achieve the two uniform colorings of a snub square tiling using a “star4” shape and a “convex4” shape.

grid.snub_square <- function(..., gp_sq, gp_sq2 = gp_sq, gp_tri = gp_sq) {
    scale_star <- star_scale(4, 90 + 60, external = TRUE)
    grid.polygon(x_sq, y_sq, gp = gp_sq)
    grid.pattern_regular_polygon(shape = "star4", scale = scale_star,
                                 angle = 0, rot = 15, spacing = 0.2,
                                 density = 1.4, gp = gp_tri)
    grid.pattern_regular_polygon(shape = "convex4", scale = scale_star,
                                 angle = 0, rot = 60, spacing = 0.2,
                                 density = scale_star * 1.4, gp = gp_sq2)
}

# 2 color uniform snub square tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp_y <- gpar(fill = yellow, col = "black")
gp_r <- gpar(fill = red, col = "black")
grid.snub_square(gp_sq = gp_y, gp_tri = gp_r)
popViewport()

# 3 color uniform snub square tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp_b <- gpar(fill = blue, col = "black")
grid.snub_square(gp_sq = gp_r, gp_sq2 = gp_b, gp_tri = gp_y)
popViewport()

# 1 color uniform hexagonal tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.snub_square(gp_sq = gp_y)
popViewport()

pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
popViewport()
add_borders()

Three different uniform colorings of a snub square tiling

Truncated square tiling

We can achieve the two uniform colorings of a truncated square tiling using the “convex8” shape rotated 22.5 degrees. Adding in an extra (possibly alternating) “star8” shape results in a uniform tiling using star polygons.

# 2 color uniform truncated square tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = red, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = yellow, col = "black")
grid.pattern_regular_polygon(shape = "convex8", density = 1.082, rot = 22.5, 
                             angle = 0, spacing = 0.3, gp = gp)
popViewport()

# 3 color uniform truncated square tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = blue, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = c(red, yellow), col = "black")
grid.pattern_regular_polygon(shape = "convex8", density = 1.082, rot = 22.5, 
                             angle = 0, spacing = 0.3, gp = gp)
popViewport()

# 3 color uniform truncated square tiling with star
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = red, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = yellow, col = "black")
grid.pattern_regular_polygon(shape = "convex8", density = 1.082, rot = 22.5, 
                             angle = 0, spacing = 0.3, gp = gp)
gp <- gpar(fill = orange, col = "black")
scale <- star_scale(8, 60, external = TRUE)
grid.pattern_regular_polygon(shape = "star8", density = 1.082, rot = 22.5, 
                             angle = 0, scale = scale, spacing = 0.3, gp = gp)
popViewport()

# 4 color uniform truncated square tiling with an alternating star
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = blue, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = c(red, yellow), col = "black")
grid.pattern_regular_polygon(shape = "convex8", density = 1.082, rot = 22.5, 
                             angle = 0, spacing = 0.3, gp = gp)
gp <- gpar(fill = orange, col = c("black"))
grid.pattern_regular_polygon(shape = c("null", "star8"), 
                             density = 1.082, rot = 22.5, 
                             angle = 0, scale = scale, spacing = 0.3, gp = gp)
popViewport()
add_borders()

Four different uniform colorings of a truncated square tiling

Truncated hexagonal tiling

We can achieve a truncated hexagonal tiling using the “convex12” shape with a 15 degree rotation in a “hex_circle” grid. The related “star12” variations are fun tilings as well.

# 2 color uniform truncated hexagonal tiling
grid.truncated_hexagonal_tiling <- function(..., shape = "convex12") {
    scale <- star_scale(12, 60, external = TRUE)
    grid.pattern_regular_polygon(..., shape = shape, density = 1.034, spacing = 0.3,
                                 angle = 0, rot = 15, scale = scale, 
                                 grid = "hex_circle")
}
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = red, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = yellow, col = "black")
grid.truncated_hexagonal_tiling(gp = gp)
popViewport()

# Add in a "convex12" star within
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = red, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = yellow, col = "black")
grid.truncated_hexagonal_tiling(gp = gp)
gp <- gpar(fill = blue, col = "black")
grid.truncated_hexagonal_tiling(gp = gp, shape = "star12")
popViewport()
add_borders()

A truncated hexagonal tiling and a related 'star12' variation

Trihexagonal tiling

We can achieve one of the uniform colorings of a trihexagonal tiling using the “convex6” shape with a “hex_circle” grid and rotating the shape 30 degrees.

There are topologically equivalent tilings using the “star6”, “star3”, and “convex3” shapes that also provides an interesting tiling effect.

grid.trihexagonal_tiling <- function(..., shape = "convex6", density = 1) {
    grid.pattern_regular_polygon(..., shape = shape, density = density, grid = "hex_circle",
                                 spacing = 0.2, rot = 30, angle = 0)
}
# 2 color uniform trihexagonal tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = yellow, col = "black")
grid.trihexagonal_tiling(gp = gp)
popViewport()

# 'star6' trihexagonal tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
scale <- star_scale(6, 120, external = TRUE)
grid.trihexagonal_tiling(shape = "star6", scale = scale, gp = gp)
popViewport()

# 'star3' trihexagonal tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
scale <- star_scale(3, 30)
grid.trihexagonal_tiling(shape = "star3", scale = scale, density=1.57, gp = gp)
popViewport()

# 'convex3' trihexagonal tiling
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
grid.trihexagonal_tiling(shape = "convex3", density=1.33, gp = gp)
popViewport()
add_borders()

A trihexagonal tiling plus 'star6', 'star3', and 'convex3' variations

Snub trihexagonal tiling

We can achieve the one uniform colorings of a snub (tri)hexagonal tiling using a “star6” shape and a “convex6” shape.

grid.snub_trihexagonal <- function(..., gp_tri, gp_tri2 = gp_tri, gp_hex = gp_tri) {
    scale_star <- star_scale(6, 60 + 60, external = TRUE)
    grid.polygon(x_sq, y_sq, gp = gp_tri)
    grid.pattern_regular_polygon(shape = "star6", scale = scale_star,
                                 grid = "hex_circle",
                                 angle = 0, rot = 19, spacing = 0.2,
                                 density = 1.305, gp = gp_tri2)
    grid.pattern_regular_polygon(shape = "convex6", grid = "hex_circle",
                                 angle = 0, rot = 19 + 30, spacing = 0.2,
                                 density = scale_star * 1.305, gp = gp_hex)
}

# 3 color uniform snub square tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp_y <- gpar(fill = yellow, col = "black")
gp_r <- gpar(fill = red, col = "black")
gp_b <- gpar(fill = blue, col = "black")
grid.snub_trihexagonal(gp_tri = gp_b, gp_tri2 = gp_r, gp_hex = gp_y)
popViewport()

# 2 color snub trihexagonal tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.snub_trihexagonal(gp_tri = gp_y, gp_hex = gp_r)
popViewport()

# 1 color snub trihexagonal tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.snub_trihexagonal(gp_tri = gp_y)
popViewport()

pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
popViewport()
add_borders()

Three different colorings of a snub trihexagonal tiling

Truncated trihexagonal

By carefully using grid.pattern_stripe() it is possible to achieve the truncated trihexagonal tiling.

# truncated trihexagonal
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = red, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.42, spacing = 0.3,
                    angle = 0, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.42, spacing = 0.3, 
                    angle = -60, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.42, spacing = 0.3, 
                    angle = 60, gp = gp)
gp <- gpar(fill = blue, col = "black")
grid.pattern_regular_polygon(shape = "convex6", grid = "hex_circle", 
                             density = 0.75, spacing = 0.3, angle = 0, gp = gp)
popViewport()

# truncated trihexagonal with 5 colors
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = green, col = NA))
gp <- gpar(fill = orange, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.42, spacing = 0.3, 
                    angle = 0, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.42, spacing = 0.3, 
                    angle = -60, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.42, spacing = 0.3, 
                    angle = 60, gp = gp)
gp <- gpar(fill = c(yellow, red, blue), col = "black")
grid.pattern_regular_polygon(shape = "convex6", grid = "hex_circle", 
                             density = 0.75, spacing = 0.3, angle = 0, gp = gp)
popViewport()
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
popViewport()
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
popViewport()
add_borders()

Two different colorings of a truncated trihexagonal tiling

Rhombitrihexagonal tiling

By carefully using grid.pattern_stripe() it is also possible to achieve the rhombitrihexagonal tiling. There are also variant tilings where we add in “star12” polygons.

# rhombitrihexagonal
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = red, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = 0, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = -60, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = 60, gp = gp)
gp <- gpar(fill = blue, col = "black")
grid.pattern_regular_polygon(shape = "convex12", grid = "hex_circle", rot = 15,
                             density = 0.82, spacing = 0.3, angle = 0, gp = gp)
popViewport()

# rhombitrihexagonal with 5 colors
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = green, col = NA))
gp <- gpar(fill = red, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = 0, gp = gp)
gp <- gpar(fill = yellow, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = -60, gp = gp)
gp <- gpar(fill = blue, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = 60, gp = gp)
gp <- gpar(fill = orange, col = "black")
grid.pattern_regular_polygon(shape = "convex12", grid = "hex_circle", rot = 15,
                             density = 0.82, spacing = 0.3, angle = 0, gp = gp)
popViewport()

# 3.4.6.3.12* rhombitrihexagonal plus star
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = red, col = "black")
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = 0, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = -60, gp = gp)
grid.pattern_stripe(grid = "hex_circle", density = 0.25, spacing = 0.3, 
                    angle = 60, gp = gp)
gp <- gpar(fill = blue, col = "black")
grid.pattern_regular_polygon(shape = "convex12", grid = "hex_circle", rot = 15,
                             density = 0.82, spacing = 0.3, angle = 0, gp = gp)
gp <- gpar(fill = green, col = "black")
scale <- star_scale(12, 30)
grid.pattern_regular_polygon(shape = "star12", grid = "hex_circle", rot = 15,
                             density = 0.82, spacing = 0.3, angle = 0, gp = gp,
                             scale = scale)
popViewport()

pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
popViewport()
add_borders()

Two different colorings of a rhombitrihexagonal tiling plus a 'star12' variation

Additional uniform tilings with star polygons

Some additional uniform tilings with star polygons not shown above.

# 12.12.4*
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = c(yellow, blue), col = "black")
grid.pattern_regular_polygon(shape="convex12", rot=15, density=1.035, 
                             angle=0, gp = gp, spacing=0.3)
popViewport()

# 12.3*.12.3*
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = c(yellow), col = "black")
grid.pattern_regular_polygon(shape="convex12", density=1, grid = "hex_circle",
                             angle=0, gp = gp, spacing=0.4)
popViewport()

# 4.4*.4**
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = c(yellow, blue), col = "black")
gp <- gpar(fill=yellow)
scale <- star_scale(4, 30)
grid.pattern_regular_polygon(shape="star4", density=1.60, angle=0, scale=scale, 
                             rot=-9.5, spacing=0.3, gp = gp)
popViewport()

# 3.3.8*.4**.8*
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
sp <- 0.35
gp <- gpar(fill = red, col = "black")
grid.pattern_regular_polygon(shape = "square", density = 1, angle = 0, 
                             spacing = sp, gp = gp)
gp <- gpar(fill = yellow, col = "black")
scale <- star_scale(4, 60)
grid.pattern_regular_polygon(shape = "star4", density = 1.14, rot = 45, 
                             xoffset = sp / 2, yoffset = sp / 2,
                             angle = 0, scale = scale, spacing = sp, gp = gp)
gp <- gpar(fill = blue, col = "black")
scale <- star_scale(8, 15)
grid.pattern_regular_polygon(shape = "star8", density = 1.082, rot = 22.5, 
                             angle = 0, scale = 0.30, spacing = sp, gp = gp)
popViewport()
add_borders()

# 3.6*.6**
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = yellow, col = "black")
scale <- star_scale(6, 30)
grid.pattern_regular_polygon(shape="star6", density=1.30, angle=0, scale=scale, rot=-22.0, 
                             spacing=0.3, grid="hex", gp=gp)
popViewport()

# 4.6.4*.6
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = c(blue, red), col = "black")
scale <- star_scale(4, 120, external = TRUE)
grid.pattern_regular_polygon(shape=c("star4", "convex4"), density=c(1.2, 0.8), 
                             scale = scale, 
                             spacing = 0.2, gp = gp, angle=45)
popViewport()

# 9.3.9.3*
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = blue, col = NA)
grid.pattern_stripe(spacing = 0.4, density = 0.35, angle = 0,
                    grid = "hex_circle", gp = gp, yoffset = 0.096)
gp <- gpar(fill = yellow, col = "black")
grid.pattern_regular_polygon(shape = "convex9", density = 1.01, angle = 0, 
                             grid = "hex_circle", spacing = 0.4, gp = gp)
popViewport()

# 8.4*.8.4*
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = red, col = NA))
gp <- gpar(fill = yellow, col = "black")
grid.pattern_regular_polygon(shape = c("convex8", "null"), density = 1.41, 
                             angle = 0, spacing = 0.2, gp = gp)
popViewport()
add_borders()

# 18.18.3*
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = blue, col = NA))
gp <- gpar(fill = c(red, yellow), col = "black")
grid.pattern_regular_polygon(shape = "convex18", density = 1, 
                             grid = "hex_circle", angle = 0,
                             spacing = 0.2, gp = gp)
popViewport()

# Only uniform if you consider the larger squares to be four-pointed stars...
# 4.8*.4**.8*
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = blue, col = NA))
gp <- gpar(fill = c(yellow, red), col = "black")
scale <- star_scale(8, 90, external = TRUE)
grid.pattern_regular_polygon(shape = c("star8", "convex4"), scale = scale,
                             density = c(1.53, 1.2), rot = c(22.5, 0), 
                             angle = 0, spacing = 0.2, gp = gp)
popViewport()
add_borders()

Non-uniform tilings

Herringbone tiling

We can achieve a herringbone tiling by using grid.pattern_weave() and particular “twill” weaves of the subtype “n/n(1)”.

grid.herringbone_tiling <- function(..., subtype = "2/2(1)") {
    grid.pattern_weave(..., type = "twill", subtype = subtype, density = 1, 
                       spacing = 0.1, angle = 45)
}
# 1 color 2:1 herringbone 
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = "black")
grid.herringbone_tiling(gp = gp)
popViewport()

# 2 color 2:1 herringbone
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = "black")
grid.herringbone_tiling(gp = gp, fill2 = red)
popViewport()

# 1 color 3:1 herringbone
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = "black")
grid.herringbone_tiling(gp = gp, subtype = "3/3(1)")
popViewport()

# 2 color 3:1 herringbone
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = "black")
grid.herringbone_tiling(gp = gp, fill2 = red, subtype = "3/3(1)")
popViewport()
add_borders()

Four different herringbone tilings

Pythagorean tiling

We can get a Pythagorean tiling by setting a background color and rotating “convex4” shapes

grid.pythagorean_tiling <- function(...) {
    grid.pattern_regular_polygon(..., shape = "convex4", density = 1.222, 
                                 rot = 15, spacing = 0.2, angle = -15)
}
# 1 color Pythagorean tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = yellow, col = "black")
grid.pythagorean_tiling(gp = gp)
popViewport()

# 2 color Pythagorean tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = red, col = "black")
grid.pythagorean_tiling(gp = gp)
popViewport()

# 3 color Pythagorean tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = c(blue, red), col = "black")
grid.pythagorean_tiling(gp = gp)
popViewport()

# 4 color Pythagorean tiling
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
grid.polygon(x_sq, y_sq, gp = gpar(fill = yellow, col = NA))
gp <- gpar(fill = c(red, blue, green), col = "black")
grid.pythagorean_tiling(gp = gp)
popViewport()
add_borders()

Four different colorings of a Pythagorean tiling

Rhombille tiling

We can get a rhombille tiling by using the “rhombille_rhombus” shape:

grid.rhombille_tiling <- function(gp1, gp2 = gp1, gp3 = gp2) {
    grid.pattern_regular_polygon(shape = "rhombille_rhombus", density = 1,
                       angle = 0, rot = -120, spacing = 0.2,
                       grid = "hex", gp = gp3)
    grid.pattern_regular_polygon(shape = "rhombille_rhombus", density = 1,
                       angle = 0, rot =  120, spacing = 0.2,
                       grid = "hex", gp = gp2)
    grid.pattern_regular_polygon(shape = "rhombille_rhombus", density = 1,
                       angle = 0, rot = 0, spacing = 0.2,
                       grid = "hex", gp = gp1)
}
gp1 <- gpar(fill = yellow, col = "black")
gp2 <- gpar(fill = blue, col = "black")
gp3 <- gpar(fill = red, col = "black")
# 1 color rhombille tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.rhombille_tiling(gp1)
popViewport()
# 2 color rhombille tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.rhombille_tiling(gp1, gp2)
popViewport()
# 3 color rhombille tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.rhombille_tiling(gp1, gp2, gp3)
popViewport()
add_borders()

Three different colorings of a rhombille tiling

Tetrakis square tiling

We can get a tetrakis square tiling by using the “tetrakis_left” and “tetrakis_right” shapes:

grid.tetrakis_tiling <- function(gp1, gp2 = gp1, gp3 = gp2, gp4 = gp1) {
    grid.pattern_regular_polygon(shape = "tetrakis_left", density = 1, 
                                 angle = 0, rot=0, gp = gp1, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_left", density = 1, 
                                 angle = 0, rot=90, gp = gp4, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_left", density = 1,
                                 angle = 0, rot=180, gp = gp1, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_left", density = 1,
                                 angle = 0, rot=270, gp = gp4, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_right", density = 1,
                                 angle = 0, rot=0, gp = gp2, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_right", density = 1,
                                 angle = 0, rot=90, gp = gp3, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_right", density = 1,
                                 angle = 0, rot=180, gp = gp2, spacing = 0.25)
    grid.pattern_regular_polygon(shape = "tetrakis_right", density = 1,
                                 angle = 0, rot=270, gp = gp3, spacing = 0.25)

}
gp1 <- gpar(fill = yellow, col = "black")
gp2 <- gpar(fill = blue, col = "black")
gp3 <- gpar(fill = red, col = "black")
gp4 <- gpar(fill = green, col = "black")
# 1 color tetrakis square tiling
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
grid.tetrakis_tiling(gp1)
popViewport()
# 2 color tetrakis square tiling
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
grid.tetrakis_tiling(gp1, gp2)
popViewport()
# 3 color tetrakis square tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
grid.tetrakis_tiling(gp1, gp2, gp3)
popViewport()
# 4 color tetrakis square tiling
pushViewport(viewport(x=0.75, y=0.25, width=0.5, height=0.5))
grid.tetrakis_tiling(gp1, gp2, gp3, gp4)
popViewport()
add_borders()

Four different colorings of a tetrakis square tiling

Additional tilings using rhombi

A rhombi may be produced by the “star2” shape:

# 2*.2**.2*.2**
pushViewport(viewport(x=0.25, y=0.75, width=0.5, height=0.5))
sp <- 0.3
ang <- 63.5
scale <- 0.5
gpy <- gpar(fill = yellow, col = "black")
grid.pattern_regular_polygon(shape = "star2", scale = scale, angle = ang,
                             density = 1, spacing = sp, gp = gpy)
grid.pattern_regular_polygon(shape = "star2", scale = scale, angle = ang,
                             density = 1, spacing = sp, gp = gpy,
                             xoffset = 0.5 * sp)
gpr <- gpar(fill = red, col = "black")
grid.pattern_regular_polygon(shape = "star2", scale = scale, angle = ang,
                             density = 1, spacing = sp, gp = gpr,
                             xoffset = 0.25 * sp, yoffset = 0.5 * sp)
grid.pattern_regular_polygon(shape = "star2", scale = scale, angle = ang,
                             density = 1, spacing = sp, gp = gpr,
                             xoffset = 0.75 * sp, yoffset = 0.5 * sp)
popViewport()

# 4.2*.4.2**
pushViewport(viewport(x=0.75, y=0.75, width=0.5, height=0.5))
dens_sq <- 0.73
scale <- star_scale(2, 60)
dens_rh <- 0.88
gpy <- gpar(fill = yellow, col = "black")
gpb <- gpar(fill = blue, col = "black")
gpr <- gpar(fill = red, col = "black")
grid.pattern_regular_polygon(shape = "convex4", scale = scale, angle = 0,
                             rot = 60,
                             density = dens_sq, spacing = sp, gp = gpb)
grid.pattern_regular_polygon(shape = "star2", scale = scale, angle = 0,
                             density = dens_rh, spacing = sp, gp = gpy,
                             rot = 45, xoffset = 0.5 * sp)
grid.pattern_regular_polygon(shape = "convex4", scale = scale, angle = 0,
                             rot = -60, yoffset = 0.5 * sp, xoffset = 0.5 * sp,
                             density = dens_sq, spacing = sp, gp = gpr)
grid.pattern_regular_polygon(shape = "star2", scale = scale, angle = 0,
                             density = dens_rh, spacing = sp, gp = gpy,
                             rot = -45, yoffset = 0.5 * sp)
popViewport()

# Use a "star12" instead of a "convex12" in truncated hexagonal tiling
pushViewport(viewport(x=0.25, y=0.25, width=0.5, height=0.5))
gp <- gpar(fill = yellow, col = NA)
grid.polygon(x_sq, y_sq, gp = gp)
gp <- gpar(fill = blue, col = "black")
grid.truncated_hexagonal_tiling(gp = gp, shape = "star12")
popViewport()
add_borders()

Three different tilings using rhombi

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.

Polygon Tiling Examples

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