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Fractal Canopy
In geometry, a fractal canopy, a type of fractal tree, is one of the easiest-to-create types of fractals. Each canopy is created by splitting a line segment into two smaller segments at the end (symmetric binary tree), and then splitting the two smaller segments as well, and so on, infinitely.Bello, Ignacio; Kaul, Anton; and Britton, Jack R. (2013). ''Topics in Contemporary Mathematics'', p.511. Cengage Learning. . Canopies are distinguished by the angle between concurrent adjacent segments and ratio between lengths of successive segments. A fractal canopy must have the following three properties: #The angle between any two neighboring line segments is the same throughout the fractal. #The ratio of lengths of any two consecutive line segments is constant. #Points all the way at the end of the smallest line segments are interconnected, which is to say the entire figure is a Graph (discrete mathematics)#Connected graph, connected graph. The pulmonary system used by humans to breath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H Tree
In fractal geometry, the H tree is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering. Construction An H tree can be constructed by starting with a line segment of arbitrary length, drawing two shorter segments at right angles to the first through its endpoints, and continuing in the same vein, reducing (dividing) the length of the line segments drawn at each stage by \sqrt2. A variant of this construction could also be defined in which the length at each iteration is multiplied by a ratio less than 1/\sqrt2, but for this variant the resulting shape covers only part of its bounding rectangle, with a fractal boundary. An alternative process that generate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oracle Thinkquest
ThinkQuest was an educational website, created 1996 and intended for primary and secondary schools. Beginning 2002 it was owned by the Oracle Education Foundation and was known as Oracle ThinkQuest. History ThinkQuest was created in 1996 by Allan H. Weis as part of his nonprofit Advanced Network and Services (ANS). The website and competition were acquired by the Oracle Education Foundation in 2002. Prior to being acquired by the Oracle Education Foundation, April Wennerstrom (née Buther) led the international training and evaluation program for the competition and worked with a cadre of international educators and judges to identify scholarship recipients. During its time under ANS' leadership, the organization gave over $1.5 million dollars annually for scholarships and support for the student-created sites and program participants spanned over 75 countries and six continents. As of July 1, 2013, the ThinkQuest initiative was discontinued.http://www.thinkquest.org/en/ -- or, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lichtenberg Figure
A Lichtenberg figure (German: ''Lichtenberg-Figur''), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of Electrical insulation, insulating materials. Lichtenberg figures are often associated with the progressive deterioration of high-voltage components and equipment. The study of planar Lichtenberg figures along insulating surfaces and 3D Electrical treeing, electrical trees within insulating materials often provides engineers with valuable insights for improving the long-term reliability of high-voltage equipment. Lichtenberg figures are now known to occur on or within solids, liquids, and gases during electrical breakdown. Lichtenberg figures are natural phenomena that exhibit fractal properties. History Lichtenberg figures are named after the German physicist Georg Christoph Lichtenberg, who originally discovered and studied them. When they were first discovered, it was thought that their characteristi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dendrite (crystal)
A crystal dendrite is a crystal that develops with a typical multi-branching form, resembling a fractal. The name comes from the Ancient Greek word (), which means "tree", since the crystal's structure resembles that of a tree. These crystals can be synthesised by using a supercooled pure liquid, however they are also quite common in nature. The most common crystals in nature exhibit dendritic growth are snowflakes and frost on windows, but many minerals and metals can also be found in dendritic structures. History Maximum velocity principle The first dendritic patterns were discovered in palaeontology and are often mistaken for fossils because of their appearance. The first theory for the creation of these patterns was published by Nash and Glicksman in 1974, they used a very mathematical method and derived a non-linear integro-differential equation for a classical needle growth. However they only found an inaccurate numerical solution close to the tip of the needle a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Tree
In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a random real tree that can be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three articles published in 1991 and 1993. This tree has since then been generalized. This random tree has several equivalent definitions and constructions: using sub-trees generated by finitely many leaves, using a Brownian excursion, Poisson separating a straight line or as a limit of Galton-Watson trees. Intuitively, the Brownian tree is a binary tree whose nodes (or branching points) are dense in the tree; which is to say that for any distinct two points of the tree, there will always exist a node between them. It is a fractal object which can be approximated with computers or by physical processes with dendritic structures. Definitions The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or 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 "wikt:oblong, oblong" is used to refer to a non-square rectangle. A rectangle with Vertex (geometry), 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 rectangles, 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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the ''principal'' square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a Unit square, square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational number, irrational. The fraction (≈ 1.4142857) is sometimes used as a good Diophantine approximation, rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 60 decimal places: : History The Babylonian clay tablet YBC 7289 (–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. ''Perpendicular'' is also used as a noun: a perpendicular is a line which is perpendicular to a given line or plane. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
