Geometric group theory often revolves around the Cayley graph , which is a geometric representation of a group. Other important topics include quasi-isometries , Gromov-hyperbolic groups , and right angled Artin groups. Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics.
Convex geometry dates back to antiquity. The isoperimetric problem , a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato , Euclid , and later Kepler and Coxeter all studied convex polytopes and their properties.
From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature , algorithms , tilings and lattices. Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.
Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure. The golden ratio is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.
Tilings , or tessellations, have been used in art throughout history. Islamic art makes frequent use of tessellations, as did the art of Escher. This is still used in art theory today, although the exact list of shapes varies from author to author.
Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design. The field of astronomy , especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.
Riemannian geometry and pseudo-Riemannian geometry are used in general relativity. Calculus was strongly influenced by geometry. This played a key role in the emergence of infinitesimal calculus in the 17th century.
Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum. Another important area of application is number theory.
However, the discovery of incommensurable lengths contradicted their philosophical views. In essence, their propositions concerning the properties of quadrangles which they considered, assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries.
Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle.
By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines — made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics Kitab al-Manazir — was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration.
Wallis's and G. Saccheri's studies of the theory of parallel lines. From Wikipedia, the free encyclopedia. For other uses, see Geometry disambiguation. Branch of mathematics. Projecting a sphere to a plane. Outline History. Concepts Features. Line segment ray Length. Volume Cube cuboid Cylinder Pyramid Sphere. Tesseract Hypersphere. Main article: History of geometry. See also: Euclidean geometry and Axiom. Main article: Point geometry. Main article: Line geometry.
Main article: Plane geometry. Main article: Angle. Main article: Curve geometry. Main article: Surface mathematics. Main article: Manifold. Main articles: Length , Area , and Volume. Main articles: Metric mathematics and Measure mathematics. Main articles: Congruence geometry and Similarity geometry. Main article: Compass and straightedge constructions.
Main article: Dimension. Main article: Symmetry. Main article: Euclidean geometry. Main article: Differential geometry. Main article: Non-Euclidean geometry. Main article: Topology. Main article: Algebraic geometry.
Main article: Complex geometry. Main article: Discrete geometry. Main article: Computational geometry. Main article: Geometric group theory. Main article: Convex geometry. Main article: Mathematics and art. Main articles: Mathematics and architecture and Architectural geometry. Main article: Mathematical physics.
Turner; Jonathan M. Blackledge; Patrick R. Andrews Fractal geometry in digital imaging. Academic Press. Archived from the original on 6 September Scientific American. Bronze man and centaur. Terracotta neck-amphora. Terracotta krater Hirschfeld Workshop. Bronze horse. Terracotta pyxis box with lid. Citation Department of Greek and Roman Art. Geometric Greece. New York: St. Martin's Press, Garland, Robert.
The Greek Way of Death. Ithaca, N. Info Print Cite. Submit Feedback. Thank you for your feedback. Home Visual Arts Painting. The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree See Article History. Read More on This Topic. This can be obtained by elementary geometric considerations see 2.
Now, the compact closed structure itself has a very elegant geometric representation in the category of oriented bordisms. Translations of geometric in Chinese Traditional. Need a translator? Translator tool. What is the pronunciation of geometric?
Browse geologically. Test your vocabulary with our fun image quizzes. Image credits. Word of the Day childish. Blog Sparkling, glinting and glistening Words related to light, Part 2 August 12, If you were asked to find the class arithmetic average of test scores, you would simply add up all the test scores of the students and then divide that sum by the number of students.
This would be calculated as:. If one student happens to perform poorly on the exam, the next student's chances of performing poorly or well on the exam is not affected. Consider investment returns , for example. That's because when it comes to annual investment returns, the numbers are not independent of each other.Using simple geometric forms such as circles and squares in design and decoration. 3. Of or relating to properties in algebraic geometry involving algebraically closed fields.