Symmetry (geometry)In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable.
Reflection symmetryIn mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called . In conclusion, a line of symmetry splits the shape in half and those halves should be identical.
Inverse trigonometric functionsIn mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
Symmetry (physics)In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries.
Product (business)In marketing, a product is an object, or system, or service made available for consumer use as of the consumer demand; it is anything that can be offered to a market to satisfy the desire or need of a customer. In retailing, products are often referred to as merchandise, and in manufacturing, products are bought as raw materials and then sold as finished goods. A service is also regarded as a type of product. In project management, products are the formal definition of the project deliverables that make up or contribute to delivering the objectives of the project.
Matrix similarityIn linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that Similar matrices represent the same linear map under two (possibly) different bases, with P being the change of basis matrix. A transformation A ↦ P−1AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that P be chosen to lie in H.
Upper classUpper class in modern societies is the social class composed of people who hold the highest social status, usually are the wealthiest members of class society, and wield the greatest political power. According to this view, the upper class is generally distinguished by immense wealth which is passed on from generation to generation. Prior to the 20th century, the emphasis was on aristocracy, which emphasized generations of inherited noble status, not just recent wealth.
Dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
Icosahedral symmetryIn mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120.
Symmetry in biologySymmetry 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, the face of a human being has a plane of symmetry down its centre, or a pine cone displays 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.
FormulaIn science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be either formulas (from the most common English plural noun form) or, under the influence of scientific Latin, formulae (from the original Latin). In mathematics, a formula generally refers to an equation relating one mathematical expression to another, with the most important ones being mathematical theorems.
Arithmetical hierarchyIn mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946). The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.
