Engineering

Predicting the complex propagation of 3D fractures

Fracture propagation is ubiquitous across different temporal and spatial scales. Examples include the breaking of a vase, fatigue cracks in machine parts, and scars left by strong earthquakes. Understanding 3D fracture propagation ...

Computer Sciences

Computer vision technique to enhance 3D understanding of 2D images

Upon looking at photographs and drawing on their past experiences, humans can often perceive depth in pictures that are, themselves, perfectly flat. However, getting computers to do the same thing has proved quite challenging.

Engineering

New 3D body-mapping tech helps consumers, the environment

Online shopping for clothing offers consumers convenience but comes with some notable downsides for them and the environment. Size and fit issues often prompt consumers to return the items, which leads to increased carbon ...

Engineering

Snakeskin can inspire safer buildings

Despite human inventiveness and ingenuity, we still lag far behind the elegant and efficient solutions forged by nature over millions of years of evolution.

Engineering

Electric pulses precisely shape 3-D-printed metal parts

Professor Dirk Bähre and his research team at Saarland University have developed a non-contact method of transforming metal parts fabricated by a 3-D printer into high-precision technical components for specialist applications. ...

Energy & Green Tech

Research shows fractals could be pleasing in solar panels

Stress reduction and improved solar electricity could someday come together in an unexpected package, and a University of Oregon study suggests that a new design of eye-pleasing, fractal-patterned rooftop solar panels could ...

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Geometry

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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