### Machine learning unravels mysteries of atomic geometry

New research has used machine learning to find the properties of atomic pieces of geometry, in pioneering work that could drive the development of new results in mathematics.

New research has used machine learning to find the properties of atomic pieces of geometry, in pioneering work that could drive the development of new results in mathematics.

Mathematics

21 hours ago

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8

In math class, you probably learned how to compute the area of lots of different shapes by memorizing algebraic formulas. Remember "base x height" for rectangles and "Â˝ base x height" for triangles? Or "đťś‹ x radiusÂ˛" ...

Mathematics

Aug 9, 2023

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16

Science and math skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimization or medical imaging. Pure mathematics often yields ...

Mathematics

Jan 3, 2023

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50

When the nuclei of atoms are about to collide in an experiment, their centers never perfectly align along the direction of relative motion. This leads to collisions with complex three-dimensional geometry. Emissions from ...

General Physics

Dec 19, 2022

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16

Recently, NASA's Double Asteroid Redirection Test (DART) spacecraft crashed into a 170 m asteroid Dimorphos at 6.6 km/s, as the first on-orbit demonstration of deflecting an asteroid by kinetic impact. The DART spacecraft ...

Space Exploration

Nov 17, 2022

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50

Recent efforts using computational modeling to understand how melting ice in Antarctica will impact the planet's oceans have focused on ice-sheet geometry, fracture, and surface meltingâ€”processes that could potentially ...

Earth Sciences

Sep 14, 2022

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A new study corrects an important error in the 3D mathematical space developed by the Nobel Prize-winning physicist Erwin SchrĂ¶dinger and others, and used by scientists and industry for more than 100 years to describe how ...

Mathematics

Aug 10, 2022

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Scientists at the University of Bristol have uncovered the deadly workings of a carnivorous plant.

Plants & Animals

Aug 2, 2022

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160

Processes in nature can often be described by equations. In many non-trivial cases, it is impossible to find the exact solutions to these equations. However, some equations are much simpler to deal with because of their extreme ...

Mathematics

Feb 7, 2022

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355

An international team of scientists from Austria and Germany has launched a new paradigm in magnetism and superconductivity, putting effects of curvature, topology, and 3D geometry into the spotlight of next-decade research. ...

Superconductivity

Nov 3, 2021

2

2045

**Geometry** (Ancient Greek: *γεωμετρία*; *geo-* "earth", *-metria* "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer.

The introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, 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. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).

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