geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the sphinx tiling is a

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 ...

of the plane using the "sphinx", a

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

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 ...

formed by gluing six

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s together. The resultant shape is named for its reminiscence to the

Great Sphinx The Great Sphinx of Giza is a limestone statue of a reclining sphinx, a mythical creature with the head of a human and the body of a lion. Facing east, it stands on the Giza Plateau on the west bank of the Nile in Giza, Egypt. The original sh ...

Giza Giza (; sometimes spelled ''Gizah, Gizeh, Geeza, Jiza''; , , ' ) is the third-largest city in Egypt by area after Cairo and Alexandria; and fourth-largest city in Africa by population after Kinshasa, Lagos, and Cairo. It is the capital of ...

. A sphinx can be dissected into any square number of copies of itself, some of them

mirror image A mirror image (in a plane mirror) is a reflection (physics), reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical phenomenon, optical effect, it r ...

s, and repeating this process leads to a non-periodic tiling of the plane. The sphinx is therefore a

rep-tile 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 ...

self-replicating Self-replication is any behavior of a dynamical system that yields construction of an identical or similar copy of itself. Cell (biology), Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA repli ...

tessellation). It is one of few known

pentagonal rep-tiles In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is cal ...

and is the only known pentagonal rep-tile whose sub-copies are equal in size.

General tilings

An outer boundary ("frame") in the shape of a sphinx can also be tiled in a non-recursive way for all orders. We define the order of a sphinx frame on a triangular lattice by the number of triangles at the "tail" end. An order-2 frame can be tiled by four sphinxes in exactly one way (as shown in the figure), an order-3 frame can be tiled by 9 sphinxes in 4 ways, etc. The number of tilings grows exponentially as

e^

with the order

n

of the frame, where

c \approx 0.425

References

External links

* Mathematics Centre Sphinx Album ..

* {{Tessellation Pentagonal tilings Aperiodic tilings Sphinxes

General tilings

See also

References

External links