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Sphinx Tiling
In geometry, the sphinx tiling is a tessellation of the plane using the "sphinx", a pentagonal polyiamond, hexiamond formed by gluing six equilateral triangles together. The resultant shape is named for its reminiscence to the Great Sphinx of Giza, Great Sphinx at Giza. A sphinx can be dissection (geometry), dissected into any square number of copies of itself, some of them mirror images, and repeating this process leads to a aperiodic tiling, non-periodic tiling of the plane. The sphinx is therefore a rep-tile (a self-replication, self-replicating tessellation). It is one of few known Rep-tile#Pentagonal rep-tiles, pentagonal rep-tiles 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 f ... [...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. 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] |
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 results from specular reflection off from surfaces of lustrous materials, especially a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3D structures. In geometry and geometrical optics In two dimensions In geometry, the mirror image of an object or 2D geometric model, two-dimensional figure is the virtual image formed by reflection (mathematics), 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 obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Tilings
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 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 &= \sqrt \cdot H\approx 1.051~H, \\ R &= \sqrt t\approx 0.8507~t, \\ D &= R\ = 2R\cos 18^\circ ... [...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 (masonry), mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and were particularly popular in the Ancient Rome, 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 civilisation, 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 dynasty, Norman Kingdom of Sicily in the 12th century, by th ... [...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 Tessellation, 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 set of prototiles, aperiodic if copies of these tiles can form only non-periodic tiling, periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. In March 2023, four researchers, David Smith (amateur mathematician), David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an Einstein problem, aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subs ... [...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. Polygon dissection problem 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, how ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Giza Governorate with a total population of 4,872,448 in the 2017 census. It is located on the west bank of the Nile opposite central Cairo, and is a part of the Greater Cairo metropolis. Giza lies less than north of Memphis (''Men-nefer,'' today the village of Mit Rahina), which was the capital city of the unified Egyptian state during the reign of pharaoh Narmer, roughly 3100 BC. 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, among which are 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 E ... [...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 east, it stands on the Giza Plateau on the west bank of the Nile in Giza, Egypt. The original shape of the Sphinx was cut from bedrock of the Eocene-aged Mokattam Formation, 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. The Sphinx is the oldest known monumental sculpture in Egypt and one of the most recognizable statues in the world. The face of the Sphinx remains a matter of scholarly dispute; it appears to represent the pharaoh Khufu or one of his sons, pharaohs Djedefre and Khafre. Archaeological evidence suggests that it was created by ancient Egyptians of the Old Kingdom during the reign of Khufu () or Khafre (). The circumstances surrounding the Sphinx's nose being broken off are uncertain, but close ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
