An icosahedral number is a figurate number that represents an icosahedron. The nth icosahedral number is given by the formula
The first such numbers are 1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, … (sequence A006564 in the OEIS ).
The first study of icosahedral numbers appears to have been by René Descartes, around 1630, in his De solidorum elementis. Prior to Descartes, figurate numbers had been studied by the ancient Greeks and by Johann Faulhaber, but only for polygonal numbers, pyramidal numbers, and cubes. Descartes introduced the study of figurate numbers based on the Platonic solids and some semiregular polyhedra; his work included the icosahedral numbers. However, De solidorum elementis was lost, and not rediscovered until 1860. In the meantime, icosahedral numbers had been studied again by other mathematicians, including Friedrich Wilhelm Marpurg in 1774, Georg Simon Klügel in 1808, and Sir Frederick Pollock in 1850. [1]
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.
In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.
A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent, regular, polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:
René Descartes was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy.
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions. The term can mean
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The nth octahedral number can be obtained by the formula:
A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n(n + 1). The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.
In geometry, the (angular) defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess.
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing. As in the case of two-dimensional circumscribed circles, the radius of a sphere circumscribed around a polyhedron P is called the circumradius of P, and the center point of this sphere is called the circumcenter of P.
A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
Pasquale ("Pat") Joseph Federico was a lifelong mathematician and longtime high-ranking official of the United States Patent Office.
A centered octahedral number or Haüy octahedral number is a figurate number that counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. The Haüy octahedral numbers are named after René Just Haüy.
A centered icosahedral number is a centered figurate number that represents an icosahedron. The centered icosahedral number for a specific n is given by
A dodecahedral number is a figurate number that represents a dodecahedron. The nth dodecahedral number is given by the formula
Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the Royal Society. These conjectures are a partial extension of the Fermat polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers.
In three-dimensional hyperbolic geometry, an ideal polyhedron is, intuitively, a convex polyhedron all of whose vertices are ideal points. More formally, it can be defined as the convex hull of any finite set of ideal points that do not lie on a single plane of the three-dimensional hyperbolic space. That is, it is the intersection of all closed half-spaces that have the given ideal points as limit points. The resulting subset of hyperbolic space has ideal polygons as its faces, connected to each other along entire lines of the hyperbolic space.
Descartes on Polyhedra: A Study of the "De solidorum elementis" is a book in the history of mathematics, concerning the work of René Descartes on polyhedra. Central to the book is the disputed priority for Euler's polyhedral formula between Leonhard Euler, who published an explicit version of the formula, and Descartes, whose De solidorum elementis includes a result from which the formula is easily derived.
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