Penrose tiling: Difference between revisions
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=={{header|Perl 6}}== |
=={{header|Perl 6}}== |
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{{works with|Rakudo|2018.05}} |
{{works with|Rakudo|2018.05}} |
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Generated with a P3 |
Generated with a P3 tile set using a Lindenmayer system. |
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<lang perl6>use SVG; |
<lang perl6>use SVG; |
Revision as of 22:04, 22 June 2018
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)
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