In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

In geometry, the truncated dodecahedron is an Archimedean solid.

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces.

Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.

In geometry, a pentakis dodecahedron is a Catalan solid.

In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron.

In geometry, the augmented dodecahedron is one of the Johnson solids ( J < sub > 58 </ sub >), consisting of a dodecahedron with a pentagonal pyramid attached to one of the faces.

In geometry, the parabiaugmented dodecahedron is one of the Johnson solids ( J < sub > 59 </ sub >).

In geometry, the metabiaugmented dodecahedron is one of the Johnson solids ( J < sub > 60 </ sub >).

In geometry, the triaugmented dodecahedron is one of the Johnson solids ( J < sub > 61 </ sub >).

In geometry, the augmented truncated dodecahedron is one of the Johnson solids ( J < sub > 68 </ sub >).

In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids ( J < sub > 69 </ sub >).

In geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids ( J < sub > 70 </ sub >).

In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids ( J < sub > 71 </ sub >); of them, it has the greatest volume in proportion to side length.

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.

In geometry, a cube < ref > English cube from Old French < Latin cubus < Greek κύβος ( kubos ) meaning " a cube, a die, vertebra ".

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

The term “ Euclidean ” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems.

For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained.

Alternatively, the word " gematria " is generally held to derive from Greek geōmetriā, " geometry ", which was used as a translation of gēmaṭriyā, though some scholars believe it to derive from Greek grammateia, rather ; it's possible that both words had an influence on the formation of the Hebrew word.

Greek contributions to science — including works of geometry and mathematical astronomy, early accounts of biological processes and catalogs of plants and animals, and theories of knowledge and learning — were produced by philosophers and physicians, as well as practitioners of various trades.

The ancient Greek origins of the words " true " and " truth " have some consistent definitions throughout great spans of history that were often associated with topics of logic, geometry, mathematics, deduction, induction, and natural philosophy.

** Theodosius of Bithynia, Greek astronomer and mathematician who will write the Sphaerics, a book on the geometry of the sphere ( d. c. 100 BC )

* 300 Euclid, Greek mathematician, publishes Elements, treating both geometry and number theory ( see also Euclidean algorithm ).

Jean Racine's tragedies — inspired by Greek myths, Euripides, Sophocles and Seneca — condensed their plot into a tight set of passionate and duty-bound conflicts between a small group of noble characters, and concentrated on these characters ' double-binds and the geometry of their unfulfilled desires and hatreds.

While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.

In geometry, a hexagon ( from Greek ἕξ hex, " six ") is a polygon with six edges and six vertices.

* Theodosius of Bithynia, Greek astronomer and mathematician who will write the Sphaerics, a book on the geometry of the sphere ( d. c. 100 BC ), later translated from Arabic back into Latin to help restore knowledge of Euclidean geometry to the West.

* Apollonius of Perga ( Pergaeus ), Greek astronomer and mathematician specialising in geometry and noted for his writings on conic sections ( d. c. 190 BC )

Ancient Greek mathematics contributed many important developments to the field of mathematics, including the basic rules of geometry, the idea of formal mathematical proof, and discoveries in number theory, mathematical analysis, applied mathematics, and approached close to establishing integral calculus.

At the school he studied ancient history, French, Greek, Latin, geometry, algebra and trigonometry, achieving good marks in all subjects, but was bullied because of his strange appearance and unathletic body.

While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry.

** History of Geometry, on the early history of Greek geometry ( several quotes survive )

Early Greek philosophers disputed as to which is more basic, arithmetic or geometry.

The reflection of radiation originating from the anode holder and reflected back to it by the surrounding metal surfaces should also be small because of the peculiar characteristic of the metal surfaces and of the specific geometry.

The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments.

Among his major accomplishments were the 1940 proof, of the Riemann hypothesis for zeta-functions of curves over finite fields, and his subsequent laying of proper foundations for algebraic geometry to support that result ( from 1942 to 1946, most intensively ).

This letter and successive parts were distributed from Bangor ( see External Links below ): in an informal manner, as a kind of diary, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks.

Ideas from it have proved influential, and have been developed by others, in particular dessins d ' enfants and a new field emerging as anabelian geometry.

In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

A value of 0 means that the pixel does not have any coverage information and is transparent ; i. e. there was no color contribution from any geometry because the geometry did not overlap this pixel.

As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes ' numerical definitions and representations.

As the loads are usually fixed, an acceptable design will result from combination of a material selection procedure and geometry modifications, if possible.

Outdoor utility knives typically feature sturdy blades from four to six inches in length, with edge geometry designed to resist chipping and breakage.

These have frames with relaxed geometry, protecting the rider from shocks of the road and easing steering at low speeds.

This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

His friend Farkas Wolfgang Bolyai with whom Gauss had sworn " brotherhood and the banner of truth " as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.

A circle is a simple shape of Euclidean geometry that is the set of points in the plane that are equidistant from a given point, the

This image understanding can be seen as the disentangling of symbolic information from image data using models constructed with the aid of geometry, physics, statistics, and learning theory.

Therefore, four loose families of more-efficient light transport modelling techniques have emerged: rasterization, including scanline rendering, geometrically projects objects in the scene to an image plane, without advanced optical effects ; ray casting considers the scene as observed from a specific point-of-view, calculating the observed image based only on geometry and very basic optical laws of reflection intensity, and perhaps using Monte Carlo techniques to reduce artifacts ; and ray tracing is similar to ray casting, but employs more advanced optical simulation, and usually uses Monte Carlo techniques to obtain more realistic results at a speed that is often orders of magnitude slower.

In ray casting the geometry which has been modeled is parsed pixel by pixel, line by line, from the point of view outward, as if casting rays out from the point of view.

The most observed geometries are listed below, but there are many cases that deviate from a regular geometry, e. g. due to the use of ligands of different types ( which results in irregular bond lengths ; the coordination atoms do not follow a points-on-a-sphere pattern ), due to the size of ligands, or due to electronic effects ( see, e. g., Jahn-Teller distortion ):

The mass is normally measured with an appropriate scale or balance ; the volume may be measured directly ( from the geometry of the object ) or by the displacement of a fluid.

In one view, differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global.

The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.