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In 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 c ...
, a fractal canopy, a type of fractal tree, is one of the easiest-to-create types of fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s. 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 connected graph
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgrap ...
.
The pulmonary system
The respiratory system (also respiratory apparatus, ventilatory system) is a biological system consisting of specific organs and structures used for gas exchange in animals and plants. The anatomy and physiology that make this happen varies grea ...
used by humans to breathe resembles a fractal canopy, as do tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
s, blood vessel
The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide away ...
s, viscous fingering
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
, electrical breakdown
Electrical breakdown or dielectric breakdown is a process that occurs when an electrical insulating material, subjected to a high enough voltage, suddenly becomes an electrical conductor and electric current flows through it. All insulating mate ...
, and crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s with appropriately adjusted growth velocity from seed.Lines, M.E. (1994). ''On the Shoulders of Giants'', p.245. CRC Press. .
See also
*Brownian tree
In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a special case from random real trees which may be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three a ...
* Dendrite (crystal)
A crystal dendrite is a crystal that develops with a typical multi-branching form. The name comes from the Greek word dendron (δενδρον) which means "tree", since the crystal's structure resembles that of a tree. These crystals can be synth ...
* Lichtenberg figure
A Lichtenberg figure (German ''Lichtenberg-Figuren''), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of insulating materials. Lichtenberg figures are often associated wit ...
* 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 res ...
References
External links
* from a student-generatedOracle 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 All ...
website
Fractals
L-systems
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