In the

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety o ...

s, a rep-tile or reptile is a shape that can be

dissected Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

into smaller copies of the same shape. The term was coined as a

pun A pun, also known as paronomasia, is a form of word play that exploits multiple meanings of a term, or of similar-sounding words, for an intended humorous or rhetorical effect. These ambiguities can arise from the intentional use of homophoni ...

on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...

in his "

Mathematical Games A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...

" column in the May 1963 issue of ''

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...

''. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by

Lee Sallows Lee Cecil Fletcher Sallows (born April 30, 1944) is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares. ...

in ''

Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...

''.

Terminology

A rep-tile is labelled rep-''n'' if the dissection uses ''n'' copies. Such a shape necessarily forms the

prototile In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint in ...

for a tiling of the plane, in many cases an

aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- peri ...

. 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.

Examples

Every

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, rectangle, parallelogram,

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

, or

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

is rep-4. The

sphinx A sphinx ( , grc, σφίγξ , Boeotian: , plural sphinxes or sphinges) is a mythical creature with the head of a human, the body of a lion, and the wings of a falcon. In Greek tradition, the sphinx has the head of a woman, the haunches of ...

hexiamond A polyiamond (also polyamond or simply iamond, or sometimes triangular polyomino) is a polyform whose base form is an equilateral triangle. The word ''polyiamond'' is a back-formation from ''diamond'', because this word is often used to describe ...

(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 The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

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 A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...

with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic

pinwheel tiling In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many or ...

. By

Pythagoras' theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

, the

hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...

, or sloping side of the rep-5 triangle, has a length of . The international standard ISO 216 defines sizes of paper sheets using the , in which the long side of a rectangular sheet of paper is the

square root of two The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...

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 :1. An

isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

right triangle is also rep-2.

Rep-tiles and symmetry

Some rep-tiles, like the

and

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

, are

symmetrical Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

and remain identical when reflected in a mirror. Others, like the

, are

asymmetrical Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in pre ...

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

Some rep-tiles are based on

polyform In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle ...

s like

polyiamond A polyiamond (also polyamond or simply iamond, or sometimes triangular polyomino) is a polyform whose base form is an equilateral triangle. The word ''polyiamond'' is a back-formation from ''diamond'', because this word is often used to describe ...

s and

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...

es, or shapes created by laying

s and

s edge-to-edge.

Squares

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

. This can be seen in the

octomino An octomino (or 8-omino) is a polyomino of order 8, that is, a polygon in the plane made of 8 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 369 different ''free'' ...

es, 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 constructed from rectifiable octominoes

Rep-tiles constructed from rectifiable octominoes

Four copies of some

nonomino A nonomino (or enneomino or 9-omino) is a polyomino of order 9, that is, a polygon in the plane made of 9 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflectio ...

es and nonakings will tile a square, therefore these

s are also rep-36 rep-tiles. Nonominoes

Equilateral triangles

Similarly, if a

tiles an equilateral triangle, it will also be a rep-tile. Equilateral triangle reptiles

Right triangles

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

Polyforms based on

s, with sides in the ratio 1 : 1 : , are known as

polyabolo In recreational mathematics, a polyabolo (also known as a polytan) is a shape formed by gluing isosceles right triangles edge-to-edge, making a polyform with the isosceles right triangle as the base form. Polyaboloes were introduced by Martin Gar ...

s. 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

. 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

30°-60°-90° triangles

Polyforms based on 30°-60°-90° right triangles, with sides in the ratio 1 : : 2, are known as polydrafters. Some are identical to polyminoes and

s, others are distinct.Polydrafter Irreptiling
/ref>

Multiple and variant rep-tilings

Many of the common rep-tiles are rep- for all positive integer values of . In particular this is true for three

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...

s 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. 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.

Rep-tiles with infinite sides

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

Triangular and quadrilateral (four-sided) rep-tiles are common, but pentagonal rep-tiles are rare. For a long time, the

was widely believed to be the only example known, but the

German German(s) may refer to: * Germany (of or related to) ** Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...

/ New-Zealand mathematicia
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 th
Math Magic
pages overseen by the American mathematician Erich Friedman.Math Magic, Problem of the Month (October 2002)
/ref> However, the sphinx and its extended versions are the only known pentagons that can be rep-tiled with equal copies. See Clarke'

Rep-tiles and fractals

Rep-tiles as fractals

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 Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

. 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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...

Fractals as rep-tiles

Because fractals are 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 A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repe ...

is a

space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...

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.

Infinite tiling

Among regular polygons, only the triangle and square can be dissected into smaller equally sized copies of themselves. However, a regular

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

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 Tiling may refer to: *The physical act of laying tiles * Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester T ...

of the hexagon with hexagons. The hexagon is therefore an irrep-∞ or irrep-infinity irreptile. File:Regular hexagon tiled with infinite copies of itself.gif, Regular hexagon tiled with infinite copies of itself File:Frattale infinito rep-tile.gif, Fractal elongated

(Siamese) tiled with infinite copies of itself

Notes

References

* * * * * * *

External links

Rep-tiles

*Mathematics Centre Sphinx Album: http://mathematicscentre.com/taskcentre/sphinx.htm * Clarke, A. L. "Reptiles." http://www.recmath.com/PolyPages/PolyPages/Reptiles.htm. * *http://www.uwgb.edu/dutchs/symmetry/reptile1.htm (1999) *IFStile - program for finding rep-tiles: https://ifstile.com

Irrep-tiles

Math Magic - Problem of the Month 10/2002

Tanya Khovanova - L-Irreptiles
{{Tessellation Tessellation Fractals