ArchimedesArchimedes of Syracuse (ˌɑːrkᵻˈmiːdiːz, ; 287-212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.
Archimedean solidIn geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed of only one type of polygon), excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron. They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
Picard groupIn mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group For integral schemes the Picard group is isomorphic to the class group of Cartier divisors.
AntiprismIn geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An. Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron. Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are 2n triangles, rather than n quadrilaterals.
Volume formIn mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold of dimension , a volume form is an -form. It is an element of the space of sections of the line bundle , denoted as . A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form.
Ternary plotA ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equilateral triangle. It is used in physical chemistry, petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. In population genetics, a triangle plot of genotype frequencies is called a de Finetti diagram.
Kodaira dimensionIn algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X. Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation κ. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The canonical bundle of a smooth algebraic variety X of dimension n over a field is the line bundle of n-forms, which is the nth exterior power of the cotangent bundle of X.
Grigorchuk groupIn the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed by Grigorchuk in a 1980 paper and he then proved in a 1984 paper that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968.
CentroidIn mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in n-dimensional Euclidean space. In geometry, one often assumes uniform mass density, in which case the barycenter or center of mass coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.
Regular 4-polytopeIn mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.
