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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'' is tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-tiling Tile Set
A self-tiling tile set, or ''setiset'', of order ''n'' is a set of ''n'' shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of ''n'' shapes. That is, the ''n'' shapes can be assembled in ''n'' different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Figure 1 shows an example for ''n'' = 4 using distinctly shaped decominoes. The concept can be extended to include pieces of higher dimension. The name setisets was coined by Lee Sallows in 2012, but the problem of finding such sets for ''n'' = 4 was asked decades previously by C. Dudley Langford, and examples for polyaboloes (discovered by Martin Gardner, Wade E. Philpott and others) and polyominoes (discovered by Maurice J. Povah) were previously published by Gardner.''Polyhexes and Polyaboloes'' in ''Mathematical Magic Show'', by Martin Gardner, Knopf, 1977, pp 146-159 Examples and definitions From the above definition it fol ... [...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] |
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] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional spaces, higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and ''semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An ''aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern. A ''tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Games (column)
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, bringing his total to 297, as other authors wrote most of the "Mathematical Games" columns. The table below lists Gardner's columns. Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "over in the table with a hyperlink to the cover. Other articles by Gardner Gardner wrote 5 other articles for ''Scientific American''. His flexagon article in December 1956 was in all but name the first article in the series of ''Mathematical Games'' columns and led directly to the series which began the following month. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Recreational mathematics Sallows is an expert on the theory of magic squares and has invented several variations on them, including alphamagic squares and geomagic squares. The latter invention caught the attention of mathematician Peter Cameron who has said that he believes that "an even deeper structure may lie hidden beyond geomagic squares" In "The lost theorem" published in 1997 he showed that every 3 × 3 magic square is associated with a unique parallelogram on the complex plane, a discovery that had escaped all previous researchers from ancient times down to the present day. In 2014 Sallows discovered a previously unnoticed result involving the medians of a triangle. A golygon is a polygon containing only right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper Curve
The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7. It is a fractal curve similar in its construction to the dragon curve and the Hilbert curve. The Gosper curve can also be used for efficient hierarchical hexagonal clustering and indexing. Algorithm Lindenmayer system The Gosper curve can be represented using an L-system with rules as follows: * Angle: 60° * Axiom: A * Replacement rules: ** A \mapsto A-B--B+A++AA+B- ** B \mapsto +A-BB--B-A++A+B In this case both A and B mean to move forward, + means to turn left 60 degrees and - means to turn right 60 degrees - using a "turtle"-style program such as Logo. Logo A Logo program to draw the Gosper curve using turtle graphics: to rg :st :ln make "st :st - 1 make "ln :ln / sqrt 7 if :st > 0 g :st :ln rt 60 gl :st :ln rt 120 gl :st :ln lt 60 rg :st :ln lt 120 rg :st :ln rg :st :ln lt 60 gl :st :l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was also a leading authority on Lewis Carroll. ''The Annotated Alice'', which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of the Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematicsand by extension, mathematics in generalthroughout the latter half of the 20th century, principally through his "Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch. The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles. The areas enclosed by the successive stages in the construction of the snowflake converge to \tfrac times the area of the original triangle, while the perimeters of the successive stages increase without bound. Consequently, the snowflake encloses a finite area, but has an infinite perimeter. Construction The Koch snowflake can be constructed by starting with an equilateral triangle, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 angle (that is, a 90-degree angle), i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry. The side opposite to the right angle is called the ''hypotenuse'' (side ''c'' in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: ''cathetus''). Side ''a'' may be identified as the side ''adjacent to angle B'' and ''opposed to'' (or ''opposite'') ''angle A'', while side ''b'' is the side ''adjacent to angle A'' and ''opposed to angle B''. If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a ''Pythagor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 orientations. Conway's tessellation 250px, Conway's triangle decomposition into smaller similar triangles. Let T be the right triangle with side length 1, 2 and \sqrt. Conway noticed that T can be divided in five isometric copies of its image by the dilation of factor 1/\sqrt. 250px, The increasing sequence of triangles which defines Conway's tiling of the plane. By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of T. The union of all these triangles yields a tiling of the whole plane by isometric copies of T. In this tiling, isometric copies of T appear in infinitely many orientations (this is due to the angles \arc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
