# Penrose tiling

Penrose tiling is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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.

The task: fill a rectangular area with a Penrose tiling.

## 11l

Translation of: Python
```F penrose(depth)
print(‘<svg viewBox="-100 -100 200 200" xmlns="http://www.w3.org/2000/svg">
<defs>
<path id="A0" d="M 80.90169943749474 58.778525229247315 L 0 0 100 0" stroke="black" fill="#8bc" />
<path id="B0" d="M 0 0 80.90169943749474 58.778525229247315 161.80339887498948 0" stroke="black" fill="#97e" />’)

L(d) 0 .< depth
print(‘  <g id="A’(d + 1)‘" transform="translate(100, 0) scale(0.6180339887498949)">
<use href="#A’d‘" transform="rotate(108)" />
<use href="#B’d‘" transform="scale(-1, 1)" />
</g>
<g id="B’(d + 1)‘">
<use href="#A’(d + 1)‘" />
<use href="#B’d‘" transform="translate(100, 0) scale(0.6180339887498949) rotate(144) translate(-80.90169943749474,-58.778525229247315)"/>
</g>’)

print(‘  <g id="G">
<use href="#A’depth‘"/>
<use href="#A’depth‘" transform="scale(1, -1)" />
</g>
</defs>
<g transform="scale(2, 2)">
<use href="#G" transform="rotate(-144)" />
<use href="#G" transform="rotate(-72)" />
<use href="#G" transform="rotate(0)" />
<use href="#G" transform="rotate(72)" />
<use href="#G" transform="rotate(144)" />
</g>
</svg>’)

penrose(6)```

## C++

Translation of: Nim
```#include <cmath>
#include <cstdlib>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>

int main() {
std::ofstream out("penrose_tiling.svg");
if (!out) {
std::cerr << "Cannot open output file.\n";
return EXIT_FAILURE;
}
std::string penrose("[N]++[N]++[N]++[N]++[N]");
for (int i = 1; i <= 4; ++i) {
std::string next;
for (char ch : penrose) {
switch (ch) {
case 'A':
break;
case 'M':
next += "OA++PA----NA[-OA----MA]++";
break;
case 'N':
next += "+OA--PA[---MA--NA]+";
break;
case 'O':
next += "-MA++NA[+++OA++PA]-";
break;
case 'P':
next += "--OA++++MA[+PA++++NA]--NA";
break;
default:
next += ch;
break;
}
}
penrose = std::move(next);
}
const double r = 30;
const double pi5 = 0.628318530717959;
double x = r * 8, y = r * 8, theta = pi5;
std::set<std::string> svg;
std::stack<std::tuple<double, double, double>> stack;
for (char ch : penrose) {
switch (ch) {
case 'A': {
double nx = x + r * std::cos(theta);
double ny = y + r * std::sin(theta);
std::ostringstream line;
line << std::fixed << std::setprecision(3) << "<line x1='" << x
<< "' y1='" << y << "' x2='" << nx << "' y2='" << ny << "'/>";
svg.insert(line.str());
x = nx;
y = ny;
} break;
case '+':
theta += pi5;
break;
case '-':
theta -= pi5;
break;
case '[':
stack.push({x, y, theta});
break;
case ']':
std::tie(x, y, theta) = stack.top();
stack.pop();
break;
}
}
out << "<svg xmlns='http://www.w3.org/2000/svg' height='" << r * 16
<< "' width='" << r * 16 << "'>\n"
<< "<rect height='100%' width='100%' fill='black'/>\n"
<< "<g stroke='rgb(255,165,0)'>\n";
for (const auto& line : svg)
out << line << '\n';
out << "</g>\n</svg>\n";
return EXIT_SUCCESS;
}
```
Output:

## EasyLang

```proc lsysexp level . axiom\$ rules\$[] .
for l to level
an\$ = ""
for c\$ in strchars axiom\$
for i = 1 step 2 to len rules\$[]
if rules\$[i] = c\$
c\$ = rules\$[i + 1]
break 1
.
.
an\$ &= c\$
.
swap axiom\$ an\$
.
.
stack[] = [ ]
proc lsysdraw axiom\$ x y ang lng . .
linewidth 0.3
move x y
for c\$ in strchars axiom\$
if c\$ = "E"
x += cos dir * lng
y += sin dir * lng
line x y
elif c\$ = "-"
dir -= ang
elif c\$ = "+"
dir += ang
elif c\$ = "["
stack[] &= x
stack[] &= y
stack[] &= dir
elif c\$ = "]"
l = len stack[]
x = stack[l - 2]
y = stack[l - 1]
dir = stack[l]
len stack[] -3
move x y
.
.
.
axiom\$ = "[b]++[b]++[b]++[b]++[b]"
rules\$[] = [ "a" "cE++dE----bE[-cE----aE]++" "b" "+cE--dE[---aE--bE]+" "c" "-aE++bE[+++cE++dE]-" "d" "--cE++++aE[+dE++++bE]--bE" "E" "" ]
lsysexp 6 axiom\$ rules\$[]
lsysdraw axiom\$ 50 50 36 4
```

## FreeBASIC

Translation of: Phython

Not much of a FreeBASIC program: majority of this code is string literals that dumps an SVG on output.

```Sub penrose(depth As Integer)
Dim As String svg
svg = "<svg viewBox=""-100 -100 200 200"" xmlns=""http://www.w3.org/2000/svg"">" + !"\n"
svg &= "  <defs>" + !"\n"
svg &= "	<path id=""A0"" d=""M 80.90169943749474 58.778525229247315 L 0 0 100 0"" stroke=""black"" fill=""#8bc"" />" + !"\n"
svg &= "	<path id=""B0"" d=""M 0 0 80.90169943749474 58.778525229247315 161.80339887498948 0"" stroke=""black"" fill=""#97e"" />" + !"\n"

For d As Integer = 0 To 5
svg &=  "	<g id=""A" : svg &= (d+1) : svg &= """ transform=""translate(100, 0) scale(0.6180339887498949)"">" + !"\n"
svg +=  "		<use href=" & """" & "#A" & d & """ transform=""rotate(108)"" />" + !"\n"
svg &=  "		<use href=" & """" & "#B" & d & """ transform=""scale(-1, 1)"" />" + !"\n"
svg &=  "	</g>" + !"\n"
svg &=  "	<g id=""B" & d+1 & """>" + !"\n"
svg &=  "		<use href=" & """" & "#A" & d+1 & """ />" + !"\n"
svg &=  "		<use href=" & """" & "#B" & d & """ transform=""translate(100, 0) scale(0.6180339887498949) rotate(144) translate(-80.90169943749474,-58.778525229247315)""/>" + !"\n"
svg &=  "	</g>" + !"\n"
Next

svg &=  "	<g id=""G"">" + !"\n"
svg &=  "		<use href=""#A6""/>" + !"\n"
svg &=  "		<use href=""#A6"" transform=""scale(1, -1)"" />" + !"\n"
svg &=  "	</g>" + !"\n"
svg &=  "  </defs>" + !"\n"

svg &=  "  <g transform=""scale(2, 2)"">" + !"\n"
svg &=  "	<use href=""#G"" transform=""rotate(-144)"" />" + !"\n"
svg &=  "	<use href=""#G"" transform=""rotate(-72)"" />" + !"\n"
svg &=  "	<use href=""#G"" transform=""rotate(0)"" />" + !"\n"
svg &=  "	<use href=""#G"" transform=""rotate(72)"" />" + !"\n"
svg &=  "	<use href=""#G"" transform=""rotate(144)"" />" + !"\n"
svg &=  "  </g>" + !"\n"
svg &=  "</svg>"
Print #1, svg
End Sub

Open "Penrose_tiling.svg" For Output As #1
penrose(6)
Close #1
```

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website.

In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.

Solution

It can be done using an L-system. There are generic functions written in Fōrmulæ to compute an L-system in the page L-system.

The program that creates a Penrose tiling is:

## Go

Library: Go Graphics
Translation of: Java
```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")
}
```
Output:
```Image same as Java entry.
```

## J

Translation of: perl
```require'format/printf'
penrosesvg=: {{
penrose=. rplc&(".{{)n
'A';'';
'M';'OA++PA----NA(-OA----MA)++';
'N';'+OA--PA(---MA--NA)+';
'O';'-MA++NA(+++OA++PA)-';
'P';'--OA++++MA(+PA++++NA)--NA'
}}-.LF)^:y '(N)++(N)++(N)++(N)++(N)'
LINE=. 2 2\$0
A=. a=. o.%5
R=. 20
LINES=. STACK=. EMPTY
for_ch. penrose do.
select. ch
case. 'A' do. LINES=. LINES,,LINE=. (R*0,:2 1 o. A)+"1 {:LINE
case. '+' do. A=. A+a
case. '-' do. A=. A-a
case. '(' do. STACK=. STACK, A;LINE
case. ')' do. STACK=. }: STACK [ 'A LINE'=. {: STACK
end.
end.
OFF=. 25+>.>./,LINES=. ~.LINES
assert 1<(F=.'penrose_tiling_%d.svg' sprintf y) fwrite~ {{)n
<svg xmlns="http://www.w3.org/2000/svg" height="%d" width="%d">
<rect height="100%%" width="100%%" style="fill:black" />
%s
</svg>
}} sprintf (2#<2*OFF),<}:,{{)n
<line x1="%.1f" y1="%.1f" x2="%.1f" y2="%.1f" style="stroke:rgb(255,165,0)"/>
}} sprintf"1 OFF+LINES
(jpathsep 1!:43''),'/',F
}}
```

Example images (linked): `penrosesvg 1`, `penrosesvg 2`, `penrosesvg 3`, `penrosesvg 4`, `penrosesvg 5`

## Java

Works with: Java version 8
```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.pack();
f.setLocationRelativeTo(null);
f.setVisible(true);
});
}
}
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

The SVG viewBox parameters are computed dynamically.

```def pi: 4 * (1|atan);

def rules:
{A : "",
M : "OA++PA----NA[-OA----MA]++",
N : "+OA--PA[---MA--NA]+",
O : "-MA++NA[+++OA++PA]-",
P : "--OA++++MA[+PA++++NA]--NA",
"": "[N]++[N]++[N]++[N]++[N]" } ;

# Apply the rules
def penrose(\$count):
rules as \$rules
| def repeat(\$count):
if \$count <= 0 then .
else gsub("M";"m") | gsub("N";"n") | gsub("O";"o")  | gsub("P";"p")
| gsub("A"; \$rules["A"])
| gsub("m"; \$rules["M"])
| gsub("n"; \$rules["N"])
| gsub("o"; \$rules["O"])
| gsub("p"; \$rules["P"])
| repeat(\$count-1)
end;
\$rules[""] | repeat(\$count) ;

# Update {svg, x, y, theta, stack, minx, maxx, miny, maxy}
def interpret(\$z):
def rnd: 1000*.|round/1000;
def minmax:
.minx = ([.minx, .x]|min)
| .miny = ([.miny, .y]|min)
| .maxx = ([.maxx, .x]|max)
| .maxy = ([.maxy, .y]|max) ;

if   \$z == "+" then .theta += pi/5
elif \$z == "-" then .theta -= pi/5
elif \$z == "[" then .stack += [ {x, y, theta} ]
elif \$z == "]" then .stack[-1] as {\$x, \$y, \$theta}
| .x = \$x | .y = \$y | .theta = \$theta
| .stack |= .[:-1]
elif \$z == "A"
then minmax
| .r as \$r
|.svg += "<line x1='\(.x|rnd)' y1='\(.y|rnd)' "
| .x += \$r * (.theta|cos)
| .y += \$r * (.theta|sin)
| .svg += "x2='\(.x|rnd)' y2='\(.y|rnd)' "
| .svg += "style='stroke:rgb(255,165,0)'/>\n"
| minmax
else .
end ;

def penrose_tiling(\$n):
penrose(\$n)
| split("")
| reduce .[] as \$action (
{x:160, y:160, theta: (pi/5), r: 20,
minx: infinite, miny: infinite,
maxx: -infinite, maxy: -infinite,
svg: "", stack: []};
interpret(\$action) ) ;

# viewBox = <min-x> <min-y> <width> <height>
# Input: {svg, minx, miny, maxx, maxy}
def svg:
([.minx, .miny] | min - 2 | floor) as \$min
| ([.maxx - .minx,  .maxy - .miny] | max + 2 | ceil) as \$size
| "<svg viewBox=\"\(\$min) \(\$min) \(\$size) \(\$size)\" xmlns=\"http://www.w3.org/2000/svg\">",
.svg,
"</svg>";

penrose_tiling(5)
| svg```
Output:

## Julia

Translation of: Perl
```using Printf

function drawpenrose()
lindenmayer_rules = Dict("A" => "",
"M" => "OA++PA----NA[-OA----MA]++", "N" => "+OA--PA[---MA--NA]+",
"O" => "-MA++NA[+++OA++PA]-", "P" => "--OA++++MA[+PA++++NA]--NA")

rul(x) = lindenmayer_rules[x]

penrose = replace(replace(replace(replace("[N]++[N]++[N]++[N]++[N]",
r"[AMNOP]" => rul), r"[AMNOP]" => rul), r"[AMNOP]" => rul), r"[AMNOP]" => rul)

x, y, theta, r, svglines, stack = 160, 160, π / 5, 20.0, String[], Vector{Real}[]

for c in split(penrose, "")
if c == "A"
xx, yy = x + r * cos(theta), y + r * sin(theta)
line = @sprintf("<line x1='%.1f' y1='%.1f' x2='%.1f' y2='%.1f' style='stroke:rgb(255,165,0)'/>\n", x, y, xx, yy)
x, y = xx, yy
push!(svglines, line)
elseif c == "+"
theta += π / 5
elseif c == "-"
theta -= π / 5
elseif c == "["
push!(stack, [x, y, theta])
elseif c == "]"
x, y, theta = pop!(stack)
end
end

svg = join(unique(svglines), "\n")
fp = open("penrose_tiling.svg", "w")
write(fp, """<svg xmlns="http://www.w3.org/2000/svg" height="350" width="350"> <rect height="100%" """ *
"""width="100%" style="fill:black" />""" * "\n\$svg</svg>")
close(fp)
end

drawpenrose()
```

## Kotlin

Translation of: Java
```// 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
pack()
setLocationRelativeTo(null)
isVisible = true
}
}
}
```

## Nim

Translation of: Phix

This is a translation of the Lindenmayer Phix version translated itself from Perl.

```import math, strformat, tables

const Lindenmayer = {'A': "",
'M': "OA++PA----NA[-OA----MA]++",
'N': "+OA--PA[---MA--NA]+",
'O': "-MA++NA[+++OA++PA]-",
'P': "--OA++++MA[+PA++++NA]--NA"}.toTable

var penrose = "[N]++[N]++[N]++[N]++[N]"

for _ in 1..4:
var next = ""
for ch in penrose:
penrose = move(next)

var
x, y = 160.0
theta = PI / 5
r = 20.0

var svg = ""
var stack: seq[(float, float, float)]

for ch in penrose:
case ch
of 'A':
let (nx, ny) = (x + r * cos(theta), y + r * sin(theta))
svg.add &"<line x1='{x:.1f}' y1='{y:.1f}' x2='{nx:.1f}' y2='{ny:.1f}'"
(x, y) = (nx, ny)
of '+':
theta += PI / 5
of '-':
theta -= PI / 5
of '[':
of ']':
(x, y, theta) = stack.pop()
else:

let svgFile = "penrose_tiling.svg".open(fmWrite)
svgFile.write """
<svg xmlns="http://www.w3.org/2000/svg" height="350" width="350">
<rect height="100%%" width="100%%" style="fill:black" />
"""
svgFile.write svg, "</svg>"
svgFile.close()
```
Output:

Same output as Perl.

## Perl

```use constant pi => 2 * atan2(1, 0);

# Generated with a P3 tile set using a Lindenmayer system.
%rules = (
A => '',
M => 'OA++PA----NA[-OA----MA]++',
N => '+OA--PA[---MA--NA]+',
O => '-MA++NA[+++OA++PA]-',
P => '--OA++++MA[+PA++++NA]--NA'
);
\$penrose = '[N]++[N]++[N]++[N]++[N]';
\$penrose =~ s/([AMNOP])/\$rules{\$1}/eg for 1..4;

# Draw the curve in SVG
(\$x, \$y) = (160, 160);
\$theta   = pi/5;
\$r       = 20;

for (split //, \$penrose) {
if (/A/) {
\$line  = sprintf "<line x1='%.1f' y1='%.1f' ", \$x, \$y;
\$line .= sprintf "x2='%.1f' ", \$x += \$r * cos(\$theta);
\$line .= sprintf "y2='%.1f' ", \$y += \$r * sin(\$theta);
\$line .= "style='stroke:rgb(255,165,0)'/>\n";
\$SVG{\$line} = 1;
} elsif (/\+/) { \$theta += pi/5
} elsif (/\-/) { \$theta -= pi/5
} elsif (/\[/) { push @stack, [\$x, \$y, \$theta]
} elsif (/\]/) { (\$x, \$y, \$theta) = @{pop @stack} }
}
\$svg .= \$_ for keys %SVG;
open  \$fh, '>', 'penrose_tiling.svg';
print \$fh  qq{<svg xmlns="http://www.w3.org/2000/svg" height="350" width="350"> <rect height="100%" width="100%" style="fill:black" />\n\$svg</svg>};
close \$fh;
```

Penrose tiling (offsite image)

## Phix

Translation of the original Python code. Output can be toggled to look like the java or perl output.

Library: Phix/pGUI
Library: Phix/online

You can run this online here.

```--
-- 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

with javascript_semantics
constant title = "Penrose tiling"
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
result &= {{0, C, P, B}, {1, P, C, A}}
else
-- Subdivide yellow triangle
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*/, /*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 key_cb(Ihandle /*ih*/, atom c)
if c=K_ESC then return IUP_CLOSE end if
if c=' ' then
if length(triangles)<=6100 then -- sane limit
triangles = subdivide(triangles)
IupUpdate(canvas)
else
IupSetAttribute(dlg,"TITLE",title & " (sane limit reached)")
end if
elsif upper(c)='C' then
yellow_orange = not yellow_orange
IupUpdate(canvas)
end if
return IUP_CONTINUE
end function

procedure main()
IupOpen()
canvas = IupCanvas("RASTERSIZE=600x600")
IupSetCallbacks(canvas, {"MAP_CB", Icallback("map_cb"),
"ACTION", Icallback("redraw_cb")})
dlg = IupDialog(canvas, `TITLE="%s"`,{title})
IupSetCallback(dlg, "KEY_CB", Icallback("key_cb"))
IupShow(dlg)
IupSetAttribute(canvas, "RASTERSIZE", NULL) -- release the minimum limitation
if platform()!=JS then
IupMainLoop()
IupClose()
end if
end procedure

main()
```

### Lindenmayer/svg

Translation of: Perl

Same output, obviously the resulting file can be opened in a separate browser.

```without js
constant Lindenmayer = new_dict({{'A',""},
{'M',"OA++PA----NA[-OA----MA]++"},
{'N',"+OA--PA[---MA--NA]+"},
{'O',"-MA++NA[+++OA++PA]-"},
{'P',"--OA++++MA[+PA++++NA]--NA"}})
string penrose = "[N]++[N]++[N]++[N]++[N]"
for n=1 to 4 do
string next = ""
for i=1 to length(penrose) do
integer ch = penrose[i]
object l = getd(ch,Lindenmayer)
next &= iff(l=NULL?ch:l)
end for
penrose = next
end for

atom x=160, y=160, theta=PI/5, r = 20
string svg = ""
constant line = "<line x1='%.1f' y1='%.1f' x2='%.1f' y2='%.1f' style='stroke:rgb(255,165,0)'/>\n"
sequence stack = {}
for i=1 to length(penrose) do
integer ch = penrose[i]
switch ch do
case 'A':   atom nx = x+r*cos(theta),
ny = y+r*sin(theta)
svg &= sprintf(line,{x,y,nx,ny})
{x,y} = {nx,ny}
case '+':   theta += PI/5
case '-':   theta -= PI/5
case '[':   stack = append(stack,{x,y,theta})
case ']':   {x,y,theta} = stack[\$]
stack = stack[1..\$-1]
end switch
end for
constant svgfmt = """
<svg xmlns="http://www.w3.org/2000/svg" height="350" width="350">
<rect height="100%%" width="100%%" style="fill:black" />
%s
</svg>"""
integer fn = open("penrose_tiling.svg","w")
printf(fn,svgfmt,svg)
close(fn)
```

## Processing

### LSystem

```class LSystem
{
int steps = 0;

String axiom;
String rule;
String production;

float startLength;
float drawLength;
float theta;

int generations;

LSystem() {
axiom = "F";
rule = "F+F-F";
startLength = 190.0;
reset();
}

void reset() {
production = axiom;
drawLength = startLength;
generations = 0;
}

int getAge() {
return generations;
}

void render() {
translate(width/2, height/2);
steps += 5;
if (steps > production.length()) {
steps = production.length();
}
for (int i = 0; i < steps; i++) {
char step = production.charAt(i);
if (step == 'F') {
rect(0, 0, -drawLength, -drawLength);
noFill();
translate(0, -drawLength);
}
else if (step == '+') {
rotate(theta);
}
else if (step == '-') {
rotate(-theta);
}
else if (step == '[') {
pushMatrix();
}
else if (step == ']') {
popMatrix();
}
}
}

void simulate(int gen) {
while (getAge() < gen) {
production = iterate(production, rule);
}
}

String iterate(String prod_, String rule_) {
drawLength = drawLength * 0.6;
generations++;
String newProduction = prod_;
newProduction = newProduction.replaceAll("F", rule_);
return newProduction;
}
}
```

### PenroseLSystem

```class PenroseLSystem extends LSystem {

int steps = 0;
float somestep = 0.1;
String ruleW;
String ruleX;
String ruleY;
String ruleZ;

PenroseLSystem() {
axiom = "[X]++[X]++[X]++[X]++[X]";
ruleW = "YF++ZF4-XF[-YF4-WF]++";
ruleX = "+YF--ZF[3-WF--XF]+";
ruleY = "-WF++XF[+++YF++ZF]-";
ruleZ = "--YF++++WF[+ZF++++XF]--XF";
startLength = 460.0;
reset();
}

void useRule(String r_) {
rule = r_;
}

void useAxiom(String a_) {
axiom = a_;
}

void useLength(float l_) {
startLength = l_;
}

void useTheta(float t_) {
}

void reset() {
production = axiom;
drawLength = startLength;
generations = 0;
}

int getAge() {
return generations;
}

void render() {
translate(width/2, height/2);
int pushes = 0;
int repeats = 1;
steps += 12;
if (steps > production.length()) {
steps = production.length();
}

for (int i = 0; i < steps; i++) {
char step = production.charAt(i);
if (step == 'F') {
stroke(255, 60);
for (int j = 0; j < repeats; j++) {
line(0, 0, 0, -drawLength);
noFill();
translate(0, -drawLength);
}
repeats = 1;
}
else if (step == '+') {
for (int j = 0; j < repeats; j++) {
rotate(theta);
}
repeats = 1;
}
else if (step == '-') {
for (int j =0; j < repeats; j++) {
rotate(-theta);
}
repeats = 1;
}
else if (step == '[') {
pushes++;
pushMatrix();
}
else if (step == ']') {
popMatrix();
pushes--;
}
else if ( (step >= 48) && (step <= 57) ) {
repeats = (int)step - 48;
}
}

// Unpush if we need too
while (pushes > 0) {
popMatrix();
pushes--;
}
}

String iterate(String prod_, String rule_) {
String newProduction = "";
for (int i = 0; i < prod_.length(); i++) {
char step = production.charAt(i);
if (step == 'W') {
newProduction = newProduction + ruleW;
}
else if (step == 'X') {
newProduction = newProduction + ruleX;
}
else if (step == 'Y') {
newProduction = newProduction + ruleY;
}
else if (step == 'Z') {
newProduction = newProduction + ruleZ;
}
else {
if (step != 'F') {
newProduction = newProduction + step;
}
}
}

drawLength = drawLength * 0.5;
generations++;
return newProduction;
}

}
```

### PenroseTile

```PenroseLSystem ds;

void setup() {
size(1000, 1000);
ds = new PenroseLSystem();
ds.simulate(5);
}

void draw() {
background(0);
ds.render();
}
```

## Python

Not much of a python program: majority of this code is string literals that dumps an SVG on output.

```def penrose(depth):
print('''<svg viewBox="-100 -100 200 200" xmlns="http://www.w3.org/2000/svg">
<defs>
<path id="A0" d="M 80.90169943749474 58.778525229247315 L 0 0 100 0" stroke="black" fill="#8bc" />
<path id="B0" d="M 0 0 80.90169943749474 58.778525229247315 161.80339887498948 0" stroke="black" fill="#97e" />''')

for d in range(depth):
print(f'''	<g id="A{d+1}" transform="translate(100, 0) scale(0.6180339887498949)">
<use href="#A{d}" transform="rotate(108)" />
<use href="#B{d}" transform="scale(-1, 1)" />
</g>
<g id="B{d+1}">
<use href="#A{d+1}" />
<use href="#B{d}" transform="translate(100, 0) scale(0.6180339887498949) rotate(144) translate(-80.90169943749474,-58.778525229247315)"/>
</g>''')

print(f'''	<g id="G">
<use href="#A{d+1}"/>
<use href="#A{d+1}" transform="scale(1, -1)" />
</g>
</defs>
<g transform="scale(2, 2)">
<use href="#G" transform="rotate(-144)" />
<use href="#G" transform="rotate(-72)" />
<use href="#G" transform="rotate(0)" />
<use href="#G" transform="rotate(72)" />
<use href="#G" transform="rotate(144)" />
</g>
</svg>''')

penrose(6)
```

## Racket

Translation of: Perl
```#lang racket

(require racket/draw)

(define rules '([M . (O A + + P A - - - - N A < - O A - - - - M A > + +)]
[N . (+ O A - - P A < - - - M A - - N A > +)]
[O . (- M A + + N A < + + + O A + + P A > -)]
[P . (- - O A + + + + M A < + P A + + + + N A > - - N A)]
[S . (< N > + + < N > + + < N > + + < N > + + < N >)]))

(define (get-cmds n cmd)
(cond
[(= 0 n) (list cmd)]
[else (append-map (curry get-cmds (sub1 n))
(dict-ref rules cmd (list cmd)))]))

(define (make-curve DIM N R OFFSET COLOR BACKGROUND-COLOR)
(define target (make-bitmap DIM DIM))
(define dc (new bitmap-dc% [bitmap target]))
(send dc set-background BACKGROUND-COLOR)
(send dc set-pen COLOR 1 'solid)
(send dc clear)
(for/fold ([x 160] [y 160] [θ (/ pi 5)] [S '()])
([cmd (in-list (get-cmds N 'S))])
(define (draw/values x* y* θ* S*)
(send/apply dc draw-line (map (curry + OFFSET) (list x y x* y*)))
(values x* y* θ* S*))
(match cmd
['A (draw/values (+ x (* R (cos θ))) (+ y (* R (sin θ))) θ S)]
['+ (values x y (+ θ (/ pi 5)) S)]
['- (values x y (- θ (/ pi 5)) S)]
['<  (values x y θ (cons (list x y θ) S))]
['> (match-define (cons (list x y θ) S*) S)
(values x y θ S*)]
[_ (values x y θ S)]))
target)

(make-curve 500 4 20 80 (make-color 255 255 0) (make-color 0 0 0))
```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2018.05

Generated with a P3 tile set using a Lindenmayer system.

```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,
],
);
```

## Scala

### Java Swing Interoperability

Works with: Scala version 2.13
```import java.awt.{BorderLayout, Color, Dimension, Graphics, Graphics2D, RenderingHints}
import java.awt.geom.Path2D

import javax.swing.{JFrame, JPanel}

import scala.math._

object PenroseTiling extends App {
private val (φ, ϑ) = ((1 + sqrt(5)) / 2, toRadians(36)) // golden ratio and 36 degrees
private val dist: Array[Array[Double]] = Array(Array(φ, φ, φ), Array(-φ, -1, -φ))

class PenroseTiling extends JPanel {
private val (w, h) = (700, 450)
private val tiles: Set[Tile] = deflateTiles(setupPrototiles(w, h), 5)

override def paintComponent(og: Graphics): Unit = {
def drawTiles(g: Graphics2D): Unit =
for (tile <- tiles) {
val path: Path2D = new Path2D.Double()
val distL = dist(tile.tileType.id)

path.moveTo(tile.x, tile.y)
for {i <- 0 until 3
ω = tile.α + (i - 1) * ϑ}
path.lineTo(
tile.x + distL(i) * tile.size * cos(ω),
tile.y - distL(i) * tile.size * sin(ω))

path.closePath()
g.setColor(if (tile.tileType == Type.Kite) Color.orange else Color.yellow)
g.fill(path)
g.setColor(Color.darkGray)
g.draw(path)
}

super.paintComponent(og)
val g: Graphics2D = og.asInstanceOf[Graphics2D]
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON)
drawTiles(g)
}

private def setupPrototiles(w: Int, h: Int): Set[Tile] = (0 to 5).map(n =>
Tile(Type.Kite, (w / 2).toDouble, (h / 2).toDouble, Pi / 2 + ϑ + n * 2 * ϑ, w / 2.5)).toSet

@scala.annotation.tailrec
private def deflateTiles(tls: Set[Tile], generation: Int): Set[Tile] =
if (generation > 0) {
val next = for {
tile <- tls
size = tile.size / φ
} yield {

def nx(factor: Int) = tile.x + cos(tile.α - factor * ϑ) * φ * tile.size
def ny(factor: Int) = tile.y - sin(tile.α - factor * ϑ) * φ * tile.size

tile.tileType match {
case Type.Dart =>
Seq(Tile(Type.Kite, tile.x, tile.y, tile.α + 5 * ϑ, size)) ++
(for (sign <- -1 to 1 by 2)
yield Tile(Type.Dart, nx(sign * 4), ny(sign * 4), tile.α - 4 * ϑ * sign, size))

case Type.Kite => (for (sign <- 1 to -1 by -2) yield {
Seq(Tile(Type.Dart, tile.x, tile.y, tile.α - 4 * ϑ * sign, size),
Tile(Type.Kite, nx(sign), ny(sign), tile.α + 3 * ϑ * sign, size))
}).flatten
}
}
deflateTiles(next.flatten, generation - 1)
} else tls

private case class Tile(tileType: Type.Type, x: Double, y: Double, α: Double, size: Double)

private object Type extends Enumeration {
type Type = Value
val Kite, Dart = Value
}

setPreferredSize(new Dimension(w, h))
setBackground(Color.white)
}

new JFrame("Penrose Tiling") {
pack()
setDefaultCloseOperation(javax.swing.WindowConstants.EXIT_ON_CLOSE)
setLocationRelativeTo(null)
setResizable(false)
setVisible(true)
}

}
```

## Sidef

Using the LSystem class defined at Hilbert curve.

```var rules = Hash(
a => 'cE++dE----bE[-cE----aE]++',
b => '+cE--dE[---aE--bE]+',
c => '-aE++bE[+++cE++dE]-',
d => '--cE++++aE[+dE++++bE]--bE',
E => '',
)

var lsys = LSystem(
width:  1000,
height: 1000,

scale: 1,
xoff: -500,
yoff: -500,

len:   40,
angle: 36,
color: 'dark blue',
)

lsys.execute('[b]++[b]++[b]++[b]++[b]', 5, "penrose_tiling.png", rules)
```

Output image: Penrose tiling

## Wren

Translation of: Kotlin
Library: DOME
Library: Wren-dynamic
Library: Wren-set
Library: Wren-polygon
```import "graphics" for Canvas, Color
import "dome" for Window
import "math" for Math
import "./dynamic" for Enum, Tuple
import "./set" for Set
import "./polygon" for Polygon

var Type = Enum.create("Type", ["KITE", "DART"])

var Tile = Tuple.create("Tile", ["type", "x", "y", "angle", "size"])

var DistinctTiles = Fn.new { |tiles|
var tileStr = tiles.map { |t| t.toString }.toList
var tileSet = Set.new(tileStr)
var tileDst = []
for (tile in tiles) {
var str = tile.toString
if (tileSet.contains(str)) {
tileSet.remove(str)
}
}
return tileDst
}

var Radians = Fn.new { |d| d * Num.pi / 180 }

var G = (1 + 5.sqrt) / 2   // golden ratio
var T = Radians.call(36)   // theta

class PenroseTiling {
construct new(width, height) {
Window.title = "Penrose Tiling"
Window.resize(width, height)
Canvas.resize(width, height)
_w = width
_h = height
}

init() {
var tiles = deflateTiles_(setupPrototiles_(_w, _h), 5)
drawTiles(tiles)
}

setupPrototiles_(w, h) {
var proto = []
var a = Num.pi / 2 + T
while (a < 3 * Num.pi) {
proto.add(Tile.new(Type.KITE, w / 2, h / 2, a, w / 2.5))
a = a + 2 * T
}
return proto
}

deflateTiles_(tiles, generation) {
if (generation <= 0) return tiles
var next = []
for (tile in tiles) {
var x = tile.x
var y = tile.y
var a = tile.angle
var nx
var ny
var size = tile.size / G
if (tile.type == Type.DART) {
next.add(Tile.new(Type.KITE, x, y, a + 5 * T, size))
var sign = 1
for (i in 0..1) {
nx = x + Math.cos(a - 4 * T * sign) * G * tile.size
ny = y - Math.sin(a - 4 * T * sign) * G * tile.size
next.add(Tile.new(Type.DART, nx, ny, a - 4 * T * sign, size))
sign = -sign
}
} else {
var sign = 1
for (i in 0..1) {
next.add(Tile.new(Type.DART, x, y, a - 4 * 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.new(Type.KITE, nx, ny, a + 3 * T * sign, size))
sign = -sign
}
}
}
// remove duplicates and deflate
return deflateTiles_(DistinctTiles.call(next), generation - 1)
}

drawTiles(tiles) {
var dist = [ [G, G, G], [-G, -1, -G] ]
for (tile in tiles) {
var angle = tile.angle - T
var x0 = tile.x
var y0 = tile.y
var ord = tile.type
var vertices = [[x0, y0]]
for (i in 0..2) {
var x1 = tile.x + dist[ord][i] * tile.size * Math.cos(angle)
var y1 = tile.y - dist[ord][i] * tile.size * Math.sin(angle)
angle = angle + T
x0 = x1
y0 = y1
}
var poly = Polygon.quick(vertices)
poly.drawfill((ord == 0) ? Color.orange : Color.yellow)
poly.draw(Color.darkgray)
}
}

update() {}

draw(alpha) {}
}

var Game = PenroseTiling.new(700, 450)
```