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| CONES | |
|---|---|
| Name | CONES |
| Caption | A generic right circular geometric cone |
| Classification | Geometric solid |
| Dimensions | height, radius, slant height |
| Related | René Descartes, Isaac Newton, Euclid, Pythagoras, Archimedes |
CONES
Cones are three-dimensional geometric solids characterized by a curved lateral surface that tapers smoothly from a circular or polygonal base to a single apex; they appear across mathematics, physics, engineering, and culture. Their study connects classic figures such as Euclid, Archimedes, René Descartes, Isaac Newton, and modern researchers at institutions like Massachusetts Institute of Technology, University of Cambridge, and California Institute of Technology. Cones serve as models in optics, mechanics, materials science, and architecture, and recur in artifacts and symbols from Ancient Egypt to contemporary design.
In geometry a cone is typically defined as the union of line segments joining all points of a base region to a single point, the apex; classical sources include propositions in Euclid and volume arguments used by Archimedes. Standard terms include base, apex (vertex), axis, height, slant height, lateral surface, and generatrix, discussed in texts influenced by Pythagoras and later formalized by analytic geometry from René Descartes. Cones are classified by the shape of the base (circular, elliptical, polygonal) and by alignment of the apex with the base center (right versus oblique), concepts present in curricula at institutions like Harvard University and Stanford University. In calculus and differential geometry the surface of a cone is treated as a ruled surface, with connections to the work of Bernhard Riemann and Carl Friedrich Gauss on curvature.
Common types include right circular cones, oblique cones, elliptical cones, conic sections produced by plane intersections, and frustums formed by truncation—definitions appearing in works cited by scholars at University of Oxford and Princeton University. Specialized variants are double cones (two nappes) studied in optics at Bell Labs and particle physics at CERN, circular cones used in manufacturing lines at Siemens, and polyhedral cones central to convex analysis developed by researchers at Centre National de la Recherche Scientifique and Institute for Advanced Study. In algebraic geometry cones over projective varieties are investigated in seminars at Institute Henri Poincaré and Max Planck Institute for Mathematics.
Volumes and surface areas of right circular cones follow classical formulas proven by Archimedes and formalized by integral calculus in the works of Isaac Newton and Gottfried Wilhelm Leibniz; the volume equals one-third the base area times height, a fact used in problems at Massachusetts Institute of Technology and Imperial College London. Slant height relates to radius and height by the Pythagorean relation attributed to Pythagoras. Cones are examples of ruled surfaces and developable surfaces, analyzed in differential geometry by Bernhard Riemann and Henri Poincaré. In convex geometry cones define convex cones and pointed cones central to optimization theory advanced at Bell Labs, AT&T, IBM Research, and universities such as Columbia University.
Higher-dimensional analogues include n-dimensional cones and tangent cones studied in algebraic geometry at Princeton University and variational analysis at Soviet Academy of Sciences schools. Topological cones, obtained by collapsing a product space, are examined in algebraic topology courses at University of Chicago and ETH Zurich; constructions like reduced suspension link cones to work by Henri Poincaré and Luitzen Egbertus Jan Brouwer. Singularities modeled by cones appear in research at Institute for Advanced Study and in the study of metrics with conical singularities on surfaces analyzed in papers connected to Fields Medal winners.
Conical geometries are foundational in optics (reflectors and lenses used in Bell Labs inventions), acoustics (horns employed by Bells and whistles manufacturers), and aerodynamics (nose cones of rockets at NASA and SpaceX). In civil and mechanical engineering conical frustums appear in silos and turbine components fabricated by firms like General Electric and Siemens. In geology and volcanology, stratovolcanoes and cinder cones are studied in fieldwork associated with United States Geological Survey and Smithsonian Institution. In materials science, conical defects and tip geometries influence scanning probe microscopes designed at IBM Research and Max Planck Institute for Solid State Research.
Cones are manufactured via spinning, casting, stamping, and additive manufacturing processes used in industries including Boeing, Airbus, Tesla, Inc., and General Electric. Techniques such as lathe turning for rotational symmetry are taught at technical schools affiliated with California Institute of Technology and MIT. Materials range from metals (aluminum alloys used by Boeing), polymers utilized by Dow Chemical Company, to ceramics developed in labs at Oak Ridge National Laboratory; surface finishing and tolerancing follow standards promulgated by organizations like International Organization for Standardization.
Conical forms hold symbolic and ceremonial roles from the pyramidal cone variants in Ancient Egypt to modern architecture by firms such as Foster + Partners and Zaha Hadid Architects. Conical hats and headgear appear in cultural items studied by museums like the British Museum and Metropolitan Museum of Art, while cone metaphors feature in linguistic corpora analyzed at Oxford University Press and Cambridge University Press. Conical imagery recurs in popular culture, design competitions at Cooper Hewitt and awards contexts including the Pritzker Prize for architecture.
Category:Geometric shapes