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Tricylinder
In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is called a bicylinder. Topologically, it is equivalent to a square hosohedron. The intersection of three cylinders is called a tricylinder. A bisected bicylinder is called a vault, and a cloister vault in architecture has this shape. Steinmetz solids are named after mathematician Charles Proteus Steinmetz, who solved the problem of determining the volume of the intersection. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. They appear prominently in the sculptures of Frank Smullin. Bicylinder A bicylinder generated by two cylinders with radius has the volume V = \frac r^3, and the surface area A = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex (geometry), apex. Each base edge (geometry), edge and apex form a triangle, called a lateral face. A pyramid is a cone, conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid. All pyramids are Self-dual polyhedron, self-dual. Etymology The word "pyramid" derives from the ancient Greek term "πυραμίς" (pyramis), which referred to a pyramid-shaped structure and a type of wheat cake. The term is rooted in the Greek "πυρ" (pyr, 'fire') and "άμις" (amis, 'vessel'), highlighting the shape's pointed, flame-like appearance. In Byzantine Greek, the term evolved to "πυραμίδα" (pyramída), continuing to denote pyramid structures. The Greek term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavalieri's Principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. * 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross section (geometry), cross-sections of equal area, then the two regions have equal volumes. Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem and layer cake representation, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangular Cuboid
A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these "cuboids", without qualifying them as being rectangular, but others use cuboid to refer to a more general class of polyhedra with six quadrilateral faces. Properties A rectangular cuboid is a convex polyhedron with six rectangle faces. The dihedral angles of a rectangular cuboid are all right angles, and its opposite faces are congruent. Because of the faces' orthogonality, the rectangular cuboid is classified as convex orthogonal polyhedron. By definition, this makes it a ''right rectangular prism''. Rectangular cuboids may be referred to colloquially as "boxes" (after the physical object). If two opposite faces become square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liu Hui
Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state of Cao Wei during the Three Kingdoms period (220–280 CE) of China. His major contributions as recorded in his commentary on ''The Nine Chapters on the Mathematical Art'' include a proof of the Pythagorean theorem, theorems in solid geometry, an improvement on Archimedes's approximation of , and a systematic method of solving linear equations in several unknowns. In his other work, '' Haidao Suanjing (The Sea Island Mathematical Manual)'', he wrote about geometrical problems and their application to surveying. He probably visited Luoyang, where he measured the sun's shadow. Mathematical work Liu Hui expressed mathematical results in the form of decimal fractions that utilized metrological units (i.e., related units of length wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... is Vadim Ponomarenko ( San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements, Euclid's ''Elements''. Euclid defined the term in Book XI as "a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms". However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
