Penrose tiling
A Penrose tiling can cover an entire plane without creating a pattern that periodically repeats.
There are many tile sets that can create non-periodic tilings, but those can typically also be used to create a periodic
tiling. What makes Penrose tiles special is that they can only be used to produce non-periodic tilings.
The two best-known Penrose tile sets are Kite and Dart (P2) and Thin Rhombus and Fat Rhombus (P3)
These so-called prototiles are usually depicted with smooth edges, but in reality Penrose tiles have interlocking tabs and cut-outs like the pieces of a jigsaw puzzle. For convenience these deformations are often replaced with matching rules, which ensure that the tiles are only connected in ways that guarantee a non-periodic tiling. (Otherwise, for instance, you could combine the kite and dart to form a rhombus, and easily create a periodic tiling from there.)
You can construct a Penrose tiling by setting up some prototiles, and adding tiles through trial and error, backtracking whenever you get stuck.
More commonly a method is used that takes advantage of the fact that Penrose tilings, like fractals, have a self-similarity on different levels. When zooming out it can be observed that groups of tiles are enclosed in areas that form exactly the same pattern as the tiles on the lower level. Departing from an inflated level, the prototiles can be subdivided into smaller tiles, always observing the matching rules. The subdivision may have to be repeated several times, before the desired level of detail is reached. This process is called deflation.
More information can be found through the links below.
The task: fill a rectangular area with a Penrose tiling.
- See also
- A good introduction (ams.org)
- Deflation explained for both sets (tartarus.org)
- Deflation explained for Kite and Dart, includes Python code (preshing.com)
Go
<lang go>package main
import (
"github.com/fogleman/gg" "math"
)
type tiletype int
const (
kite tiletype = iota dart
)
type tile struct {
tt tiletype x, y float64 angle, size float64
}
var gr = (1 + math.Sqrt(5)) / 2 // golden ratio
const theta = math.Pi / 5 // 36 degrees in radians
func setupPrototiles(w, h int) []tile {
var proto []tile
// sun
for a := math.Pi/2 + theta; a < 3*math.Pi; a += 2 * theta {
ww := float64(w / 2)
hh := float64(h / 2)
proto = append(proto, tile{kite, ww, hh, a, float64(w) / 2.5})
}
return proto
}
func distinctTiles(tls []tile) []tile {
tileset := make(map[tile]bool)
for _, tl := range tls {
tileset[tl] = true
}
distinct := make([]tile, len(tileset))
for tl, _ := range tileset {
distinct = append(distinct, tl)
}
return distinct
}
func deflateTiles(tls []tile, gen int) []tile {
if gen <= 0 {
return tls
}
var next []tile
for _, tl := range tls {
x, y, a, size := tl.x, tl.y, tl.angle, tl.size/gr
var nx, ny float64
if tl.tt == dart {
next = append(next, tile{kite, x, y, a + 5*theta, size})
for i, sign := 0, 1.0; i < 2; i, sign = i+1, -sign {
nx = x + math.Cos(a-4*theta*sign)*gr*tl.size
ny = y - math.Sin(a-4*theta*sign)*gr*tl.size
next = append(next, tile{dart, nx, ny, a - 4*theta*sign, size})
}
} else {
for i, sign := 0, 1.0; i < 2; i, sign = i+1, -sign {
next = append(next, tile{dart, x, y, a - 4*theta*sign, size})
nx = x + math.Cos(a-theta*sign)*gr*tl.size
ny = y - math.Sin(a-theta*sign)*gr*tl.size
next = append(next, tile{kite, nx, ny, a + 3*theta*sign, size})
}
}
}
// remove duplicates
tls = distinctTiles(next)
return deflateTiles(tls, gen-1)
}
func drawTiles(dc *gg.Context, tls []tile) {
dist := [2][3]float64{{gr, gr, gr}, {-gr, -1, -gr}}
for _, tl := range tls {
angle := tl.angle - theta
dc.MoveTo(tl.x, tl.y)
ord := tl.tt
for i := 0; i < 3; i++ {
x := tl.x + dist[ord][i]*tl.size*math.Cos(angle)
y := tl.y - dist[ord][i]*tl.size*math.Sin(angle)
dc.LineTo(x, y)
angle += theta
}
dc.ClosePath()
if ord == kite {
dc.SetHexColor("FFA500") // orange
} else {
dc.SetHexColor("FFFF00") // yellow
}
dc.FillPreserve()
dc.SetHexColor("A9A9A9") // dark gray
dc.SetLineWidth(1)
dc.Stroke()
}
}
func main() {
w, h := 700, 450
dc := gg.NewContext(w, h)
dc.SetRGB(1, 1, 1)
dc.Clear()
tiles := deflateTiles(setupPrototiles(w, h), 5)
drawTiles(dc, tiles)
dc.SavePNG("penrose_tiling.png")
}</lang>
- Output:
Image same as Java entry.
Java
<lang java>import java.awt.*; import java.util.List; import java.awt.geom.Path2D; import java.util.*; import javax.swing.*; import static java.lang.Math.*; import static java.util.stream.Collectors.toList;
public class PenroseTiling extends JPanel {
// ignores missing hash code
class Tile {
double x, y, angle, size;
Type type;
Tile(Type t, double x, double y, double a, double s) {
type = t;
this.x = x;
this.y = y;
angle = a;
size = s;
}
@Override
public boolean equals(Object o) {
if (o instanceof Tile) {
Tile t = (Tile) o;
return type == t.type && x == t.x && y == t.y && angle == t.angle;
}
return false;
}
}
enum Type {
Kite, Dart
}
static final double G = (1 + sqrt(5)) / 2; // golden ratio static final double T = toRadians(36); // theta
List<Tile> tiles = new ArrayList<>();
public PenroseTiling() {
int w = 700, h = 450;
setPreferredSize(new Dimension(w, h));
setBackground(Color.white);
tiles = deflateTiles(setupPrototiles(w, h), 5); }
List<Tile> setupPrototiles(int w, int h) {
List<Tile> proto = new ArrayList<>();
// sun
for (double a = PI / 2 + T; a < 3 * PI; a += 2 * T)
proto.add(new Tile(Type.Kite, w / 2, h / 2, a, w / 2.5));
return proto; }
List<Tile> deflateTiles(List<Tile> tls, int generation) {
if (generation <= 0)
return tls;
List<Tile> next = new ArrayList<>();
for (Tile tile : tls) {
double x = tile.x, y = tile.y, a = tile.angle, nx, ny;
double size = tile.size / G;
if (tile.type == Type.Dart) {
next.add(new Tile(Type.Kite, x, y, a + 5 * T, size));
for (int i = 0, sign = 1; i < 2; i++, sign *= -1) {
nx = x + cos(a - 4 * T * sign) * G * tile.size;
ny = y - sin(a - 4 * T * sign) * G * tile.size;
next.add(new Tile(Type.Dart, nx, ny, a - 4 * T * sign, size));
}
} else {
for (int i = 0, sign = 1; i < 2; i++, sign *= -1) {
next.add(new Tile(Type.Dart, x, y, a - 4 * T * sign, size));
nx = x + cos(a - T * sign) * G * tile.size;
ny = y - sin(a - T * sign) * G * tile.size;
next.add(new Tile(Type.Kite, nx, ny, a + 3 * T * sign, size));
}
}
}
// remove duplicates
tls = next.stream().distinct().collect(toList());
return deflateTiles(tls, generation - 1); }
void drawTiles(Graphics2D g) {
double[][] dist = {{G, G, G}, {-G, -1, -G}};
for (Tile tile : tiles) {
double angle = tile.angle - T;
Path2D path = new Path2D.Double();
path.moveTo(tile.x, tile.y);
int ord = tile.type.ordinal();
for (int i = 0; i < 3; i++) {
double x = tile.x + dist[ord][i] * tile.size * cos(angle);
double y = tile.y - dist[ord][i] * tile.size * sin(angle);
path.lineTo(x, y);
angle += T;
}
path.closePath();
g.setColor(ord == 0 ? Color.orange : Color.yellow);
g.fill(path);
g.setColor(Color.darkGray);
g.draw(path);
}
}
@Override
public void paintComponent(Graphics og) {
super.paintComponent(og);
Graphics2D g = (Graphics2D) og;
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
RenderingHints.VALUE_ANTIALIAS_ON);
drawTiles(g);
}
public static void main(String[] args) {
SwingUtilities.invokeLater(() -> {
JFrame f = new JFrame();
f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
f.setTitle("Penrose Tiling");
f.setResizable(false);
f.add(new PenroseTiling(), BorderLayout.CENTER);
f.pack();
f.setLocationRelativeTo(null);
f.setVisible(true);
});
}
}</lang>
Kotlin
<lang scala>// version 1.1.2
import java.awt.* import java.awt.geom.Path2D import javax.swing.*
class PenroseTiling(w: Int, h: Int) : JPanel() {
private enum class Type {
KITE, DART
}
private class Tile(
val type: Type,
val x: Double,
val y: Double,
val angle: Double,
val size: Double
) {
override fun equals(other: Any?): Boolean {
if (other == null || other !is Tile) return false
return type == other.type && x == other.x && y == other.y &&
angle == other.angle && size == other.size
}
}
private companion object {
val G = (1.0 + Math.sqrt(5.0)) / 2.0 // golden ratio
val T = Math.toRadians(36.0) // theta
}
private val tiles: List<Tile>
init {
preferredSize = Dimension(w, h)
background = Color.white
tiles = deflateTiles(setupPrototiles(w, h), 5)
}
private fun setupPrototiles(w: Int, h: Int): List<Tile> {
val proto = mutableListOf<Tile>()
var a = Math.PI / 2.0 + T
while (a < 3.0 * Math.PI) {
proto.add(Tile(Type.KITE, w / 2.0, h / 2.0, a, w / 2.5))
a += 2.0 * T
}
return proto
}
private fun deflateTiles(tls: List<Tile>, generation: Int): List<Tile> {
if (generation <= 0) return tls
val next = mutableListOf<Tile>()
for (tile in tls) {
val x = tile.x
val y = tile.y
val a = tile.angle
var nx: Double
var ny: Double
val size = tile.size / G
if (tile.type == Type.DART) {
next.add(Tile(Type.KITE, x, y, a + 5.0 * T, size))
var sign = 1
for (i in 0..1) {
nx = x + Math.cos(a - 4.0 * T * sign) * G * tile.size
ny = y - Math.sin(a - 4.0 * T * sign) * G * tile.size
next.add(Tile(Type.DART, nx, ny, a - 4.0 * T * sign, size))
sign *= -1
}
}
else {
var sign = 1
for (i in 0..1) {
next.add(Tile(Type.DART, x, y, a - 4.0 * T * sign, size))
nx = x + Math.cos(a - T * sign) * G * tile.size
ny = y - Math.sin(a - T * sign) * G * tile.size
next.add(Tile(Type.KITE, nx, ny, a + 3.0 * T * sign, size))
sign *= -1
}
}
}
// remove duplicates and deflate
return deflateTiles(next.distinct(), generation - 1)
}
private fun drawTiles(g: Graphics2D) {
val dist = arrayOf(
doubleArrayOf(G, G, G),
doubleArrayOf(-G, -1.0, -G)
)
for (tile in tiles) {
var angle = tile.angle - T
val path = Path2D.Double()
path.moveTo(tile.x, tile.y)
val ord = tile.type.ordinal
for (i in 0..2) {
val x = tile.x + dist[ord][i] * tile.size * Math.cos(angle)
val y = tile.y - dist[ord][i] * tile.size * Math.sin(angle)
path.lineTo(x, y)
angle += T
}
path.closePath()
with(g) {
color = if (ord == 0) Color.pink else Color.red
fill(path)
color = Color.darkGray
draw(path)
}
}
}
override fun paintComponent(og: Graphics) {
super.paintComponent(og)
val g = og as Graphics2D
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
RenderingHints.VALUE_ANTIALIAS_ON)
drawTiles(g)
}
}
fun main(args: Array<String>) {
SwingUtilities.invokeLater {
val f = JFrame()
with (f) {
defaultCloseOperation = JFrame.EXIT_ON_CLOSE
title = "Penrose Tiling"
isResizable = false
add(PenroseTiling(700, 450), BorderLayout.CENTER)
pack()
setLocationRelativeTo(null)
isVisible = true
}
}
}</lang>
Perl 6
Generated with a P3 tile set using a Lindenmayer system.
<lang perl6>use SVG;
role Lindenmayer {
has %.rules;
method succ {
self.comb.map( { %!rules{$^c} // $c } ).join but Lindenmayer(%!rules)
}
}
my $penrose = '[N]++[N]++[N]++[N]++[N]' but Lindenmayer(
{
A => ,
M => 'OA++PA----NA[-OA----MA]++',
N => '+OA--PA[---MA--NA]+',
O => '-MA++NA[+++OA++PA]-',
P => '--OA++++MA[+PA++++NA]--NA'
}
);
$penrose++ xx 4;
my @lines; my @stack;
for $penrose.comb {
state ($x, $y) = 300, 200;
state $d = 55 + 0i;
when 'A' { @lines.push: 'line' => [:x1($x.round(.01)), :y1($y.round(.01)), :x2(($x += $d.re).round(.01)), :y2(($y += $d.im).round(.01))] }
when '[' { @stack.push: ($x.clone, $y.clone, $d.clone) }
when ']' { ($x, $y, $d) = @stack.pop }
when '+' { $d *= cis -π/5 }
when '-' { $d *= cis π/5 }
default { }
}
say SVG.serialize(
svg => [
:600width, :400height, :style<stroke:rgb(250,12,210)>,
:rect[:width<100%>, :height<100%>, :fill<black>],
|@lines,
],
);</lang> See: Penrose tiling image
Phix
Translation of the original Python code <lang Phix>-- demo\rosetta\Penrose_tiling.exw -- Resizeable. Press space to iterate/subdivide, C to toggle colour scheme bool yellow_orange = true -- false = magenta on black, outlines only
include pGUI.e
Ihandle dlg, canvas cdCanvas cddbuffer, cdcanvas
include builtins\complex.e
constant golden_ratio = (1 + sqrt(5)) / 2
function subdivide(sequence triangles)
sequence result = {}
integer colour
complex A, B, C, P, Q, R
for i=1 to length(triangles) do
{colour, A, B, C} = triangles[i]
if colour == 0 then
-- Subdivide orange triangle
P = complex_add(A,complex_div(complex_sub(B,A),golden_ratio))
result &= {{0, C, P, B}, {1, P, C, A}}
else
-- Subdivide yellow triangle
Q = complex_add(B,complex_div(complex_sub(A,B),golden_ratio))
R = complex_add(B,complex_div(complex_sub(C,B),golden_ratio))
result &= {{1, R, C, A}, {1, Q, R, B}, {0, R, Q, A}}
end if
end for
return result
end function
function initial_wheel() -- Create an initial wheel of yellow triangles around the origin
sequence triangles = {}
complex B, C
atom phi
for i=0 to 9 do
phi = (2*i-1)*PI/10
B = {cos(phi),sin(phi)}
phi = (2*i+1)*PI/10
C = {cos(phi),sin(phi)}
if mod(i,2)==0 then
{B, C} = {C, B} -- mirror every second triangle
end if
triangles &= {{0, {0,0}, B, C}}
end for
return subdivide(triangles) -- ... and iterate once
end function
sequence triangles = initial_wheel()
integer hw, hh, h
procedure draw_one(sequence triangle, integer colour, mode)
if yellow_orange then
cdCanvasSetForeground(cddbuffer, colour)
end if
cdCanvasBegin(cddbuffer, mode)
for i=2 to 4 do
atom {x,y} = triangle[i]
cdCanvasVertex(cddbuffer, x*h+hw, y*h+hh)
end for
cdCanvasEnd(cddbuffer)
end procedure
function redraw_cb(Ihandle /*ih*/, integer /*posx*/, integer /*posy*/)
{hw, hh} = sq_floor_div(IupGetIntInt(canvas, "DRAWSIZE"),2)
h = min(hw,hh)
if yellow_orange then
cdCanvasSetBackground(cddbuffer, CD_WHITE)
else
cdCanvasSetBackground(cddbuffer, CD_BLACK)
cdCanvasSetForeground(cddbuffer, CD_MAGENTA)
end if
cdCanvasActivate(cddbuffer)
cdCanvasClear(cddbuffer)
for i=1 to length(triangles) do
sequence triangle = triangles[i]
if yellow_orange then
integer colour = iff(triangle[1]?CD_ORANGE:CD_YELLOW)
draw_one(triangle,colour,CD_FILL)
end if
draw_one(triangle,CD_DARK_GREY,CD_CLOSED_LINES)
end for
cdCanvasFlush(cddbuffer)
return IUP_DEFAULT
end function
function map_cb(Ihandle ih)
cdcanvas = cdCreateCanvas(CD_IUP, ih) cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas) return IUP_DEFAULT
end function
function esc_close(Ihandle /*ih*/, atom c)
if c=K_ESC then return IUP_CLOSE end if
if c=' ' then
triangles = subdivide(triangles)
IupUpdate(canvas)
elsif upper(c)='C' then
yellow_orange = not yellow_orange
IupUpdate(canvas)
end if
return IUP_CONTINUE
end function
procedure main()
IupOpen()
canvas = IupCanvas(NULL)
IupSetAttribute(canvas, "RASTERSIZE", "600x600") -- initial size
IupSetCallback(canvas, "MAP_CB", Icallback("map_cb"))
dlg = IupDialog(canvas)
IupSetAttribute(dlg, "TITLE", "Penrose tiling")
IupSetCallback(dlg, "K_ANY", Icallback("esc_close"))
IupSetCallback(canvas, "ACTION", Icallback("redraw_cb"))
IupMap(dlg) IupSetAttribute(canvas, "RASTERSIZE", NULL) -- release the minimum limitation IupShowXY(dlg,IUP_CENTER,IUP_CENTER) IupMainLoop() IupClose()
end procedure main()</lang> Output can be toggled to look like the java or perl output