In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,
In mathematics, a square root of a number a is a number y such that y2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is a. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 · 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy.
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanislaw Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
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. It can also be defined as 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 ABCD.
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, , which is , where is approximately 1.618.
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space.
In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x:
Brahmagupta was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text.
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.
In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
In number theory, a congruum is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x2, y2, and z2 are three square numbers that are equally spaced apart from each other, then the spacing between them, z2 − y2 = y2 − x2, is called a congruum.
In geometry, the spiral of Theodorus is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
In combinatorial mathematics, the Lobb numberLm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.
In number theory, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin
