In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes from 1 and 2. Starting from 0 and 1, the sequence begins
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. Expressed algebraically, for quantities and with , is in a golden ratio to if
In physics and mathematics, the phase of a wave or other periodic function of some real variable is an angle-like quantity representing the fraction of the cycle covered up to . It is expressed in such a scale that it varies by one full turn as the variable goes through each period. It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or as the variable completes a full period.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit, defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as rad = m/m. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral and the logarithmic spiral. Fermat spirals are named after Pierre de Fermat.
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes.
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.
A Fibonacci word is a specific sequence of binary digits. The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
A circular sector, also known as circle sector or disk sector or simply a sector, is the portion of a disk enclosed by two radii and an arc, with the smaller area being known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle, the radius of the circle, and is the arc length of the minor sector.
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.
A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:
A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians ; and the other arc, the major arc, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.
The belt problem is a mathematics problem which requires finding the length of a crossed belt that connects two circular pulleys with radius r1 and r2 whose centers are separated by a distance P. The solution of the belt problem requires trigonometry and the concepts of the bitangent line, the vertical angle, and congruent angles.
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
The goat grazing problem is either of two related problems in recreational mathematics involving a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing an exterior of a circular area. For the exterior problem, the constraint that the rope can not enter the circular area dictates that the grazing area forms an involute. If the goat were instead tethered to a post on the edge of a circular path of pavement that did not obstruct the goat, the interior and exterior problem would be complements of a simple circular area.
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles. These patterns contain spiral arms formed by circles linked through opposite points of tangency, with their centers on logarithmic spirals of three different shapes.
The chessboard paradox or paradox of Loyd and Schlömilch is a falsidical paradox based on an optical illusion. A chessboard or a square with a side length of 8 units is cut into four pieces. Those four pieces are used to form a rectangle with side lengths of 13 and 5 units. Hence the combined area of all four pieces is 64 area units in the square but 65 area units in the rectangle, this seeming contradiction is due an optical illusion as the four pieces don't fit exactly in the rectangle, but leave a small barely visible gap around the rectangle's diagonal. The paradox is sometimes attributed to the American puzzle inventor Sam Loyd (1841–1911) and the German mathematician Oskar Schlömilch (1832–1901).
