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Rep-tile

January 01, 1970

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American.[1] In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.[2]

The "sphinx" polyiamond rep-tile. Four copies of the sphinx can be put together as shown to make a larger sphinx.
A selection of rep-tiles, including the sphinx, two fish and the 5-triangle

Terminology edit

 
The chair substitution (left) and a portion of a chair tiling (right)

A rep-tile is labelled rep-n if the dissection uses n copies. Such a shape necessarily forms the prototile for a tiling of the plane, in many cases an aperiodic tiling. A rep-tile dissection using different sizes of the original shape is called an irregular rep-tile or irreptile. If the dissection uses n copies, the shape is said to be irrep-n. If all these sub-tiles are of different sizes then the tiling is additionally described as perfect. A shape that is rep-n or irrep-n is trivially also irrep-(kn − k + n) for any k > 1, by replacing the smallest tile in the rep-n dissection by n even smaller tiles. The order of a shape, whether using rep-tiles or irrep-tiles is the smallest possible number of tiles which will suffice.[3]

Examples edit

 
Defining an aperiodic tiling (the pinwheel tiling) by repeatedly dissecting and inflating a rep-tile.

Every square, rectangle, parallelogram, rhombus, or triangle is rep-4. The sphinx hexiamond (illustrated above) is rep-4 and rep-9, and is one of few known self-replicating pentagons. The Gosper island is rep-7. The Koch snowflake is irrep-7: six small snowflakes of the same size, together with another snowflake with three times the area of the smaller ones, can combine to form a single larger snowflake.

A right triangle with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic pinwheel tiling. By Pythagoras' theorem, the hypotenuse, or sloping side of the rep-5 triangle, has a length of √5.

The international standard ISO 216 defines sizes of paper sheets using the √2, in which the long side of a rectangular sheet of paper is the square root of two times the short side of the paper. Rectangles in this shape are rep-2. A rectangle (or parallelogram) is rep-n if its aspect ratio is √n:1. An isosceles right triangle is also rep-2.

Rep-tiles and symmetry edit

Some rep-tiles, like the square and equilateral triangle, are symmetrical and remain identical when reflected in a mirror. Others, like the sphinx, are asymmetrical and exist in two distinct forms related by mirror-reflection. Dissection of the sphinx and some other asymmetric rep-tiles requires use of both the original shape and its mirror-image.

Rep-tiles and polyforms edit

Some rep-tiles are based on polyforms like polyiamonds and polyominoes, or shapes created by laying equilateral triangles and squares edge-to-edge.

Squares edit

If a polyomino is rectifiable, that is, able to tile a rectangle, then it will also be a rep-tile, because the rectangle will have an integer side length ratio and will thus tile a square. This can be seen in the octominoes, which are created from eight squares. Two copies of some octominoes will tile a square; therefore these octominoes are also rep-16 rep-tiles.

 
Rep-tiles based on rectifiable octominoes

Four copies of some nonominoes and nonakings will tile a square, therefore these polyforms are also rep-36 rep-tiles.

 
Rep-tiles created from rectifiable nonominoes and 9-polykings (nonakings)

Equilateral triangles edit

Similarly, if a polyiamond tiles an equilateral triangle, it will also be a rep-tile.

 
Rep-tiles created from equilateral triangles


 
A fish-like rep-tile based on three equilateral triangles
 
A rocket-like rep-tile created from a dodeciamond, or twelve equilateral triangles laid edge-to-edge (and corner-to-corner)

Right triangles edit

A right triangle is a triangle containing one right angle of 90°. Two particular forms of right triangle have attracted the attention of rep-tile researchers, the 45°-90°-45° triangle and the 30°-60°-90° triangle.

45°-90°-45° triangles edit

Polyforms based on isosceles right triangles, with sides in the ratio 1 : 1 : √2, are known as polyabolos. An infinite number of them are rep-tiles. Indeed, the simplest of all rep-tiles is a single isosceles right triangle. It is rep-2 when divided by a single line bisecting the right angle to the hypotenuse. Rep-2 rep-tiles are also rep-2n and the rep-4,8,16+ triangles yield further rep-tiles. These are found by discarding half of the sub-copies and permutating the remainder until they are mirror-symmetrical within a right triangle. In other words, two copies will tile a right triangle. One of these new rep-tiles is reminiscent of the fish formed from three equilateral triangles.

 
Rep-tiles based on right triangles
 
A fish-like rep-tile based on four isosceles right triangles

30°-60°-90° triangles edit

Polyforms based on 30°-60°-90° right triangles, with sides in the ratio 1 : √3 : 2, are known as polydrafters. Some are identical to polyiamonds.[4]

 
A tridrafter, or shape created by three triangles of 30°-60°-90°
 
The same tridrafter as a reptile
 
A tetradrafter, or shape created from four 30°-60°-90° triangles
 
The same tetradrafter as a reptile
 
A hexadrafter, or shape created by six 30°-60°-90° triangles
 
The same hexadrafter as a reptile

Multiple and variant rep-tilings edit

Many of the common rep-tiles are rep-n2 for all positive integer values of n. In particular this is true for three trapezoids including the one formed from three equilateral triangles, for three axis-parallel hexagons (the L-tromino, L-tetromino, and P-pentomino), and the sphinx hexiamond.[5] In addition, many rep-tiles, particularly those with higher rep-n, can be self-tiled in different ways. For example, the rep-9 L-tetramino has at least fourteen different rep-tilings. The rep-9 sphinx hexiamond can also be tiled in different ways.

 
Variant rep-tilings of the rep-9 L-tetromino
 
Variant rep-tilings of the rep-9 sphinx hexiamond

Rep-tiles with infinite sides edit

 
Horned triangle or teragonic triangle

The most familiar rep-tiles are polygons with a finite number of sides, but some shapes with an infinite number of sides can also be rep-tiles. For example, the teragonic triangle, or horned triangle, is rep-4. It is also an example of a fractal rep-tile.

Pentagonal rep-tiles edit

Triangular and quadrilateral (four-sided) rep-tiles are common, but pentagonal rep-tiles are rare. For a long time, the sphinx was widely believed to be the only example known, but the German/New-Zealand mathematician Karl Scherer and the American mathematician George Sicherman have found more examples, including a double-pyramid and an elongated version of the sphinx. These pentagonal rep-tiles are illustrated on the pages overseen by the American mathematician Erich Friedman.[6] However, the sphinx and its extended versions are the only known pentagons that can be rep-tiled with equal copies. See Clarke's Reptile pages.

 
A pentagonal rep-tile discovered by Karl Scherer

Rep-tiles and fractals edit

Rep-tiles as fractals edit

Rep-tiles can be used to create fractals, or shapes that are self-similar at smaller and smaller scales. A rep-tile fractal is formed by subdividing the rep-tile, removing one or more copies of the subdivided shape, and then continuing recursively. For instance, the Sierpinski carpet is formed in this way from a rep-tiling of a square into 27 smaller squares, and the Sierpinski triangle is formed from a rep-tiling of an equilateral triangle into four smaller triangles. When one sub-copy is discarded, a rep-4 L-triomino can be used to create four fractals, two of which are identical except for orientation.

 
Geometrical dissection of an L-triomino (rep-4)
 
A fractal based on an L-triomino (rep-4)
 
Another fractal based on an L-triomino
 
Another fractal based on an L-triomino

Fractals as rep-tiles edit

Because fractals are often self-similar on smaller and smaller scales, many may be decomposed into copies of themselves like a rep-tile. However, if the fractal has an empty interior, this decomposition may not lead to a tiling of the entire plane. For example, the Sierpinski triangle is rep-3, tiled with three copies of itself, and the Sierpinski carpet is rep-8, tiled with eight copies of itself, but repetition of these decompositions does not form a tiling. On the other hand, the dragon curve is a space-filling curve with a non-empty interior; it is rep-4, and does form a tiling. Similarly, the Gosper island is rep-7, formed from the space-filling Gosper curve, and again forms a tiling.

By construction, any fractal defined by an iterated function system of n contracting maps of the same ratio is rep-n.

 
A Sierpinski triangle based on three smaller copies of a Sierpinski triangle
 
A Sierpinski carpet based on eight smaller copies of a Sierpinski carpet
 
A dragon curve based on 4 smaller copies of a dragon curve

Infinite tiling edit

Among regular polygons, only the triangle and square can be dissected into smaller equally sized copies of themselves. However, a regular hexagon can be dissected into six equilateral triangles, each of which can be dissected into a regular hexagon and three more equilateral triangles. This is the basis for an infinite tiling of the hexagon with hexagons. The hexagon is therefore an irrep-∞ or irrep-infinity irreptile.


  •  
    Regular hexagon tiled with infinite copies of itself
  •  
    Fractal elongated Koch snowflake (Siamese) tiled with infinite copies of itself[7]

See also edit

Notes edit

  1. ^ A Gardner's Dozen—Martin's Scientific American Cover Stories
  2. ^ Sallows (2012).
  3. ^ Gardner (2001).
  4. ^ Polydrafter Irreptiling
  5. ^ Niţică (2003).
  6. ^ Math Magic, Problem of the Month (October 2002)
  7. ^ Pietrocola, Giorgio (2005). "Tartapelago. Arte tassellazione". Maecla.

References edit

  • Gardner, M. (2001), "Rep-Tiles", The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems, New York: W. W. Norton, pp. 46–58
  • Gardner, M. (1991), "Chapter 19: Rep-Tiles, Replicating Figures on the Plane", The Unexpected Hanging and Other Mathematical Diversions, Chicago, IL: Chicago University Press, pp. 222–233
  • Langford, C. D. (1940), "Uses of a Geometric Puzzle", The Mathematical Gazette, 24 (260): 209–211, doi:10.2307/3605717
  • Niţică, Viorel (2003), "Rep-tiles revisited", MASS selecta, Providence, RI: American Mathematical Society, pp. 205–217, MR 2027179
  • Sallows, Lee (2012), "On self-tiling tile sets", Mathematics Magazine, 85 (5): 323–333, doi:10.4169/math.mag.85.5.323, MR 3007213
  • Scherer, Karl (1987), A Puzzling Journey to the Reptiles and Related Animals
  • Wells, D. (1991), The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, pp. 213–214

External links edit

Rep-tiles edit

 
Wikimedia Commons has media related to Rep-tiles.

Irrep-tiles edit

  • Math Magic - Problem of the Month 10/2002
  • Tanya Khovanova - L-Irreptiles

January 01, 1970
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In the geometry of tessellations a rep tile or reptile is a shape that can be dissected into smaller copies of the same shape The term was coined as a pun on animal reptiles by recreational mathematician Solomon W Golomb and popularized by Martin Gardner in his Mathematical Games column in the May 1963 issue of Scientific American 1 In 2012 a generalization of rep tiles called self tiling tile sets was introduced by Lee Sallows in Mathematics Magazine 2 The sphinx polyiamond rep tile Four copies of the sphinx can be put together as shown to make a larger sphinx A selection of rep tiles including the sphinx two fish and the 5 triangle Contents 1 Terminology 2 Examples 3 Rep tiles and symmetry 4 Rep tiles and polyforms 4 1 Squares 4 2 Equilateral triangles 4 3 Right triangles 4 3 1 45 90 45 triangles 4 3 2 30 60 90 triangles 5 Multiple and variant rep tilings 6 Rep tiles with infinite sides 7 Pentagonal rep tiles 8 Rep tiles and fractals 8 1 Rep tiles as fractals 8 2 Fractals as rep tiles 9 Infinite tiling 10 See also 11 Notes 12 References 13 External links 13 1 Rep tiles 13 2 Irrep tilesTerminology edit nbsp The chair substitution left and a portion of a chair tiling right A rep tile is labelled rep n if the dissection uses n copies Such a shape necessarily forms the prototile for a tiling of the plane in many cases an aperiodic tiling A rep tile dissection using different sizes of the original shape is called an irregular rep tile or irreptile If the dissection uses n copies the shape is said to be irrep n If all these sub tiles are of different sizes then the tiling is additionally described as perfect A shape that is rep n or irrep n is trivially also irrep kn k n for any k gt 1 by replacing the smallest tile in the rep n dissection by n even smaller tiles The order of a shape whether using rep tiles or irrep tiles is the smallest possible number of tiles which will suffice 3 Examples edit nbsp Defining an aperiodic tiling the pinwheel tiling by repeatedly dissecting and inflating a rep tile Every square rectangle parallelogram rhombus or triangle is rep 4 The sphinx hexiamond illustrated above is rep 4 and rep 9 and is one of few known self replicating pentagons The Gosper island is rep 7 The Koch snowflake is irrep 7 six small snowflakes of the same size together with another snowflake with three times the area of the smaller ones can combine to form a single larger snowflake A right triangle with side lengths in the ratio 1 2 is rep 5 and its rep 5 dissection forms the basis of the aperiodic pinwheel tiling By Pythagoras theorem the hypotenuse or sloping side of the rep 5 triangle has a length of 5 The international standard ISO 216 defines sizes of paper sheets using the 2 in which the long side of a rectangular sheet of paper is the square root of two times the short side of the paper Rectangles in this shape are rep 2 A rectangle or parallelogram is rep n if its aspect ratio is n 1 An isosceles right triangle is also rep 2 Rep tiles and symmetry editSome rep tiles like the square and equilateral triangle are symmetrical and remain identical when reflected in a mirror Others like the sphinx are asymmetrical and exist in two distinct forms related by mirror reflection Dissection of the sphinx and some other asymmetric rep tiles requires use of both the original shape and its mirror image Rep tiles and polyforms editSome rep tiles are based on polyforms like polyiamonds and polyominoes or shapes created by laying equilateral triangles and squares edge to edge Squares edit If a polyomino is rectifiable that is able to tile a rectangle then it will also be a rep tile because the rectangle will have an integer side length ratio and will thus tile a square This can be seen in the octominoes which are created from eight squares Two copies of some octominoes will tile a square therefore these octominoes are also rep 16 rep tiles nbsp Rep tiles based on rectifiable octominoes Four copies of some nonominoes and nonakings will tile a square therefore these polyforms are also rep 36 rep tiles nbsp Rep tiles created from rectifiable nonominoes and 9 polykings nonakings Equilateral triangles edit Similarly if a polyiamond tiles an equilateral triangle it will also be a rep tile nbsp Rep tiles created from equilateral triangles nbsp A fish like rep tile based on three equilateral triangles nbsp A rocket like rep tile created from a dodeciamond or twelve equilateral triangles laid edge to edge and corner to corner Right triangles edit A right triangle is a triangle containing one right angle of 90 Two particular forms of right triangle have attracted the attention of rep tile researchers the 45 90 45 triangle and the 30 60 90 triangle 45 90 45 triangles edit Polyforms based on isosceles right triangles with sides in the ratio 1 1 2 are known as polyabolos An infinite number of them are rep tiles Indeed the simplest of all rep tiles is a single isosceles right triangle It is rep 2 when divided by a single line bisecting the right angle to the hypotenuse Rep 2 rep tiles are also rep 2n and the rep 4 8 16 triangles yield further rep tiles These are found by discarding half of the sub copies and permutating the remainder until they are mirror symmetrical within a right triangle In other words two copies will tile a right triangle One of these new rep tiles is reminiscent of the fish formed from three equilateral triangles nbsp Rep tiles based on right triangles nbsp A fish like rep tile based on four isosceles right triangles 30 60 90 triangles edit Polyforms based on 30 60 90 right triangles with sides in the ratio 1 3 2 are known as polydrafters Some are identical to polyiamonds 4 nbsp A tridrafter or shape created by three triangles of 30 60 90 nbsp The same tridrafter as a reptile nbsp A tetradrafter or shape created from four 30 60 90 triangles nbsp The same tetradrafter as a reptile nbsp A hexadrafter or shape created by six 30 60 90 triangles nbsp The same hexadrafter as a reptileMultiple and variant rep tilings editMany of the common rep tiles are rep n2 for all positive integer values of n In particular this is true for three trapezoids including the one formed from three equilateral triangles for three axis parallel hexagons the L tromino L tetromino and P pentomino and the sphinx hexiamond 5 In addition many rep tiles particularly those with higher rep n can be self tiled in different ways For example the rep 9 L tetramino has at least fourteen different rep tilings The rep 9 sphinx hexiamond can also be tiled in different ways nbsp Variant rep tilings of the rep 9 L tetromino nbsp Variant rep tilings of the rep 9 sphinx hexiamondRep tiles with infinite sides edit nbsp Horned triangle or teragonic triangle The most familiar rep tiles are polygons with a finite number of sides but some shapes with an infinite number of sides can also be rep tiles For example the teragonic triangle or horned triangle is rep 4 It is also an example of a fractal rep tile Pentagonal rep tiles editTriangular and quadrilateral four sided rep tiles are common but pentagonal rep tiles are rare For a long time the sphinx was widely believed to be the only example known but the German New Zealand mathematician Karl Scherer and the American mathematician George Sicherman have found more examples including a double pyramid and an elongated version of the sphinx These pentagonal rep tiles are illustrated on the Math Magic pages overseen by the American mathematician Erich Friedman 6 However the sphinx and its extended versions are the only known pentagons that can be rep tiled with equal copies See Clarke s Reptile pages nbsp A pentagonal rep tile discovered by Karl SchererRep tiles and fractals editRep tiles as fractals edit Rep tiles can be used to create fractals or shapes that are self similar at smaller and smaller scales A rep tile fractal is formed by subdividing the rep tile removing one or more copies of the subdivided shape and then continuing recursively For instance the Sierpinski carpet is formed in this way from a rep tiling of a square into 27 smaller squares and the Sierpinski triangle is formed from a rep tiling of an equilateral triangle into four smaller triangles When one sub copy is discarded a rep 4 L triomino can be used to create four fractals two of which are identical except for orientation nbsp Geometrical dissection of an L triomino rep 4 nbsp A fractal based on an L triomino rep 4 nbsp Another fractal based on an L triomino nbsp Another fractal based on an L triomino Fractals as rep tiles edit Because fractals are often self similar on smaller and smaller scales many may be decomposed into copies of themselves like a rep tile However if the fractal has an empty interior this decomposition may not lead to a tiling of the entire plane For example the Sierpinski triangle is rep 3 tiled with three copies of itself and the Sierpinski carpet is rep 8 tiled with eight copies of itself but repetition of these decompositions does not form a tiling On the other hand the dragon curve is a space filling curve with a non empty interior it is rep 4 and does form a tiling Similarly the Gosper island is rep 7 formed from the space filling Gosper curve and again forms a tiling By construction any fractal defined by an iterated function system of n contracting maps of the same ratio is rep n nbsp A Sierpinski triangle based on three smaller copies of a Sierpinski triangle nbsp A Sierpinski carpet based on eight smaller copies of a Sierpinski carpet nbsp A dragon curve based on 4 smaller copies of a dragon curveInfinite tiling editAmong regular polygons only the triangle and square can be dissected into smaller equally sized copies of themselves However a regular hexagon can be dissected into six equilateral triangles each of which can be dissected into a regular hexagon and three more equilateral triangles This is the basis for an infinite tiling of the hexagon with hexagons The hexagon is therefore an irrep or irrep infinity irreptile nbsp Regular hexagon tiled with infinite copies of itself nbsp Fractal elongated Koch snowflake Siamese tiled with infinite copies of itself 7 See also editMosaic Self replication Self tiling tile set Reptiles M C Escher Notes edit A Gardner s Dozen Martin s Scientific American Cover Stories Sallows 2012 Gardner 2001 Polydrafter Irreptiling Niţică 2003 Math Magic Problem of the Month October 2002 Pietrocola Giorgio 2005 Tartapelago Arte tassellazione Maecla References editGardner M 2001 Rep Tiles The Colossal Book of Mathematics Classic Puzzles Paradoxes and Problems New York W W Norton pp 46 58 Gardner M 1991 Chapter 19 Rep Tiles Replicating Figures on the Plane The Unexpected Hanging and Other Mathematical Diversions Chicago IL Chicago University Press pp 222 233 Langford C D 1940 Uses of a Geometric Puzzle The Mathematical Gazette 24 260 209 211 doi 10 2307 3605717 Niţică Viorel 2003 Rep tiles revisited MASS selecta Providence RI American Mathematical Society pp 205 217 MR 2027179 Sallows Lee 2012 On self tiling tile sets Mathematics Magazine 85 5 323 333 doi 10 4169 math mag 85 5 323 MR 3007213 Scherer Karl 1987 A Puzzling Journey to the Reptiles and Related Animals Wells D 1991 The Penguin Dictionary of Curious and Interesting Geometry London Penguin pp 213 214External links editRep tiles edit nbsp Wikimedia Commons has media related to Rep tiles Mathematics Centre Sphinx Album http mathematicscentre com taskcentre sphinx htm Clarke A L Reptiles http www recmath com PolyPages PolyPages Reptiles htm Weisstein Eric W Rep Tile MathWorld http www uwgb edu dutchs symmetry reptile1 htm 1999 IFStile program for finding rep tiles https ifstile com Irrep tiles edit Math Magic Problem of the Month 10 2002 Tanya Khovanova L Irreptiles Retrieved from https en wikipedia org w index php title Rep tile amp oldid 1214541059, wikipedia, wiki, book, books, library,

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