Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has
In mathematics, the Klein bottle is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane. While a Möbius strip is a surface with a boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
Celtic knots are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular art. These knots are most known for their adaptation for use in the ornamentation of Christian monuments and manuscripts, such as the 8th-century St. Teilo Gospels, the Book of Kells and the Lindisfarne Gospels. Most are endless knots, and many are varieties of basket weave knots.
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.
A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. Since there are no true regular continuous hexagrams, the term is instead used to refer to a compound figure of two equilateral triangles. The intersection is a regular hexagon.
The plus–minus sign, ±, is a symbol with multiple meanings:
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called a parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation, or parametric system, or parameterization of the object.
A Lissajous curve, also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.
The valknut is a symbol consisting of three interlocked triangles. It appears on a variety of objects from the archaeological record of the ancient Germanic peoples. The term valknut is a modern development; it is not known what term or terms were used to refer to the symbol historically.
The endless knot or eternal knot is a symbolic knot and one of the Eight Auspicious Symbols. It is an important symbol in Hinduism, Jainism and Buddhism. It is an important cultural marker in places significantly influenced by Tibetan Buddhism such as Tibet, Mongolia, Tuva, Kalmykia, and Buryatia. It is also found in Celtic, Kazakh and Chinese symbolism.
The triquetra is a triangular figure composed of three interlaced arcs, or (equivalently) three overlapping vesicae piscis lens shapes. It is used as an ornamental design in architecture, and in medieval manuscript illumination. Its depiction as interlaced is common in Insular ornaments from about the 7th century. In this interpretation, the triquetra represents the topologically simplest possible knot.
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
The unicursal hexagram is a hexagram or six-pointed star that can be traced or drawn unicursally, in one continuous line rather than by two overlaid triangles. The hexagram can also be depicted inside a circle with the points touching it. It is often depicted in an interlaced form with the lines of the hexagram passing over and under one another to form a knot. It is a specific instance of the far more general shape discussed in Blaise Pascal's 1639 Hexagrammum Mysticum.
In knot theory, a Lissajous knot is a knot defined by parametric equations of the form
Solomon's knot is a traditional decorative motif used since ancient times, and found in many cultures. Despite the name, it is classified as a link, and is not a true knot according to the definitions of mathematical knot theory.
In knot theory, a Lissajous-toric knot is a knot defined by parametric equations of the form
