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Sphinx Tiling
In geometry, the sphinx tiling is a tessellation of the plane using the "sphinx", a pentagonal hexiamond formed by gluing six equilateral triangles together. The resultant shape is named for its reminiscence to the Great Sphinx at Giza. A sphinx can be dissected into any square number of copies of itself, some of them mirror images, and repeating this process leads to a non-periodic tiling of the plane. The sphinx is therefore a rep-tile (a self-replicating tessellation). It is one of few known pentagonal rep-tiles and is the only known pentagonal rep-tile whose sub-copies are equal in size. See also * Mosaic A mosaic is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and were particularly pop ... References External links * Mathematics Centre Sphinx Album ..* {{Tessellation Pentagonal tilings Aperiodic ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-replication Of Sphynx Hexidiamonds
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 replication, DNA is replicated and can be transmitted to offspring during reproduction. virus (biology), Biological viruses can Viral replication, replicate, but only by commandeering the reproductive machinery of cells through a process of infection. Harmful prion proteins can replicate by converting normal proteins into rogue forms. Computer viruses reproduce using the hardware and software already present on computers. Self-replication in robotics has been an area of research and a subject of interest in science fiction. Any self-replicating mechanism which does not make a perfect copy (mutation) will experience genetic variation and will create variants of itself. These variants will be subject to natural selection, since some will be better ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mirror Image
A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures. In geometry and geometrical optics In two dimensions In geometry, the mirror image of an object or two-dimensional figure is the virtual image formed by reflection in a plane mirror; it is of the same size as the original object, yet different, unless the object or figure has reflection symmetry (also known as a P-symmetry). Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside-out. If we first look at an object that is effectively two-dimensional (such as the writing on a card) and then turn the card to face a mirror, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Tilings
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') 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 called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mosaic
A mosaic is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and were particularly popular in the Ancient Roman world. Mosaic today includes not just murals and pavements, but also artwork, hobby crafts, and industrial and construction forms. Mosaics have a long history, starting in Mesopotamia in the 3rd millennium BC. Pebble mosaics were made in Tiryns in Mycenean Greece; mosaics with patterns and pictures became widespread in classical times, both in Ancient Greece and Ancient Rome. Early Christian basilicas from the 4th century onwards were decorated with wall and ceiling mosaics. Mosaic art flourished in the Byzantine Empire from the 6th to the 15th centuries; that tradition was adopted by the Norman Kingdom of Sicily in the 12th century, by the eastern-influenced Republic of Venice, and among the Rus. Mosaic fell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Martin Gardner in his " Mathematical Games" column in the May 1963 issue of ''Scientific American''. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in '' Mathematics Magazine''. Terminology 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'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-replication
Self-replication is any behavior of a dynamical system that yields construction of an identical or similar copy of itself. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and can be transmitted to offspring during reproduction. Biological viruses can replicate, but only by commandeering the reproductive machinery of cells through a process of infection. Harmful prion proteins can replicate by converting normal proteins into rogue forms. Computer viruses reproduce using the hardware and software already present on computers. Self-replication in robotics has been an area of research and a subject of interest in science fiction. Any self-replicating mechanism which does not make a perfect copy (mutation) will experience genetic variation and will create variants of itself. These variants will be subject to natural selection, since some will be better at surviving in their current environment than others and will o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Martin Gardner in his " Mathematical Games" column in the May 1963 issue of ''Scientific American''. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in '' Mathematics Magazine''. Terminology 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'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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- periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Definition and illustration Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dissection (geometry)
In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection (of one polytope into another). It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid set-theoretic issues related to the Banach–Tarski paradox and Tarski's circle-squaring problem, the pieces are typically required to be well-behaved. For instance, they may be restricted to being the closures of disjoint open sets. The Bolyai–Gerwien theorem states that any polygon may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any polyhedron has a dissection into any other polyhedron of the same volume using polyhedral pieces (see Dehn invariant). This process ''is'' possible, however, for any two honeycomb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giza
Giza (; sometimes spelled ''Gizah'' arz, الجيزة ' ) is the second-largest city in Egypt after Cairo and fourth-largest city in Africa after Kinshasa, Lagos and Cairo. It is the capital of Giza Governorate with a total population of 9.2 million as of 2021. It is located on the west bank of the Nile, southwest of central Cairo, and is a part of the Greater Cairo metropolis. Giza lies less than north of Memphis (''Men-nefer''), which was the capital city of the first unified Egyptian state from the days of the first pharaoh, Narmer. Giza is most famous as the location of the Giza Plateau, the site of some of the most impressive ancient monuments in the world, including a complex of ancient Egyptian royal mortuary and sacred structures, including the Great Sphinx, the Great Pyramid of Giza, and a number of other large pyramids and temples. Giza has always been a focal point in Egypt's history due to its location close to Memphis, the ancient pharaonic capital of the O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Sphinx Of Giza
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 directly from west to east, it stands on the Giza Plateau on the west bank of the Nile in Giza, Egypt. The face of the Sphinx appears to represent the pharaoh Khafre. The original shape of the Sphinx was cut from the bedrock, and has since been restored with layers of limestone blocks. It measures long from paw to tail, high from the base to the top of the head and wide at its rear haunches. Its nose was broken off for unknown reasons between the 3rd and 10th centuries AD. The Sphinx is the oldest known monumental sculpture in Egypt and one of the most recognisable statues in the world. The archaeological evidence suggests that it was created by ancient Egyptians of the Old Kingdom during the reign of Khafre (). Names The original name the Old Kingdom creators gave the Sphinx is unknown, as the Sphinx temple, enclosure and p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
