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Cairo Pentagonal Tiling
In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net after Percy Alexander MacMahon, who depicted it in his 1921 publication ''New Mathematical Pastimes''. John Horton Conway called it a 4-fold pentille. Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges. In architecture, beyond Cairo, the Cairo tiling has been used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex ( or more) and sphere with three pentagons; the latter produces a tiling topologically equivalent to the dodecahedron. Monohedral convex pentagonal tilings Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile). The most recent one was discovered in 2015. This list has been shown to be complete by (result subject to peer-review). showed that there are only eight edge-to-edge convex types, a result obtained independently by . Michaël Rao of the École normale supérieure de Lyon claimed in May 2017 to have found the proof that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isohedral Tiling
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites. Characterizations A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the ''incenter'' (the center of the incircle). There exists a tangential polygon of ''n'' sequential sides ''a''1, ..., ''a''''n'' if and only if the system of equations :x_1+x_2=a_1,\quad x_2+x_3=a_2,\quad \ldots,\quad x_n+x_1=a_n has a solution (''x''1, ..., ''x''''n'') in positive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice vers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Grid
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. Related polyhedra and tilings This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordination Sequence
In crystallography and the theory of infinite vertex-transitive graphs, the coordination sequence of a vertex v is an integer sequence that counts how many vertices are at each possible distance from v. That is, it is a sequence n_0, n_1, n_2,\dots where each n_i is the number of vertices that are i steps away from v. If the graph is vertex-transitive, then the sequence is an Graph property, invariant of the graph that does not depend on the specific choice of v. Coordination sequences can also be defined for sphere packings, by using either the contact graph of the spheres or the Delaunay triangulation of their centers, but these two choices may give rise to different sequences. As an example, in a square grid, for each positive integer i, there are 4i grid points that are i steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence 1,4,8,12,16,20,\dots\ . in which, except for the initial value of one, each number is a multiple of four. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taxicab Geometry
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, ''L''1 distance, ''L''1 distance or \ell_1 norm (see ''Lp'' space), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets. The geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In \mathbb^2 , the taxicab distance between two points (x_1, y_1) and (x_2, y_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-integer
In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers such as 1 (being half the integer 2). A name such as "integer-plus-half" may be more accurate, but even though not literally true, "half integer" is the conventional term. Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient. Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the dyadic rationals (numbers produced by dividing an integer by a power of two). Notation and algebraic structure The set of all half-integers is often denoted :\mathbb Z + \tfrac \quad = \quad \left( \tfrac \mathbb Z \right) \smallsetminus \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be ''achiral''. A chiral object and its mirror image are said to be enantiomorphs. The word ''chirality'' is derived from the Greek (cheir), the hand, the most familiar chiral object; the word ''enantiomorph'' stems from the Greek (enantios) 'opposite' + (morphe) 'form'. Examples Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped ''tetrominoes'' of the popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Paving
Block paving, also known as brick paving, is a commonly used decorative method of creating a pavement or hardstanding. The main benefit of bricks over other materials is that individual bricks can later be lifted up and replaced. This allows for remedial work to be carried out under the surface of the paving without leaving a lasting mark once the paving bricks have been replaced. Typical areas of use would be for driveways, pavement, patios, town centres, pedestrian precincts and more commonly in road surfacing. Bricks are typically made of concrete or clay, though other composite materials are also used. Each has its own means of construction. The biggest difference is the way they set hard ready for use. A clay brick has to be fired in a kiln to bake the brick hard. A concrete brick has to be allowed to set. The concrete paving bricks are a porous form of brick formed by mixing small stone hardcore, dyes, cement and sand and other materials in various amounts. Many block pavi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilateral Symmetry
Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a plane of symmetry down its centre, or a pine cone with a clear symmetrical spiral pattern. Internal features can also show symmetry, for example the tubes in the human body (responsible for transporting gases, nutrients, and waste products) which are cylindrical and have several planes of symmetry. Biological symmetry can be thought of as a balanced distribution of duplicate body parts or shapes within the body of an organism. Importantly, unlike in mathematics, symmetry in biology is always approximate. For example, plant leaves – while considered symmetrical – rarely match up exactly when folded in half. Symmetry is one class of patterns in nature whereby there is near-repetition of the pattern element, either by reflection or rotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
