# Polygon Tiling Examples

### Table of Contents

Uniform tilings

- 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()

## 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()