In geometry Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures and with properties of space. Geometry is one of the oldest 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—, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms "convex polytope" and "convex polyhedron" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a. There are two competing incompatible definitions of a cuboid in the mathematical literature. In the more general definition of a cuboid, the only additional requirement is that these six faces each be a quadrilateral In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon (6-sided) and so on. The word quadrilateral is made of the words quad and lateral. Quad means four and lateral means, and that the undirected graph In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set formed by the vertices and edges of the polyhedron should be isomorphic such that any two vertices u and v of G are adjacent in G if and only if ƒ and ƒ(v) are adjacent in H. This kind of bijection is commonly called "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection to the graph of a cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the.[1] Alternatively, the word “cuboid” is sometimes used to refer to a shape of this type in which each of the faces is a rectangle In Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle, and in which each pair of adjacent faces meets in a right angle In geometry and trigonometry, a right angle is an angle of 90 degrees, corresponding to a quarter turn . It can be defined as the angle such that twice that angle amounts to a half turn, or 180°; this more restrictive type of cuboid is also known as a right cuboid, rectangular box Box describes a variety of containers and receptacles for permanent use as storage, or for temporary use often for transporting contents, rectangular hexahedron A hexahedron is a polyhedron with six faces. A regular hexahedron, with all its faces square, is a cube, right rectangular prism, or rectangular parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. (The term rhomboid is also sometimes used with this meaning.) It is to a parallelogram as a cube is to a square: Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds. Three equivalent definitions of.[2]
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General cuboids
By Euler's formula In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by χ (Greek letter chi) the number of faces (F), vertices (V), and edges (E) of any convex polyhedron are related by the formula F + V = E + 2. In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has 6 faces In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix -hedron is derived from the Greek word hedra which means face, 8 vertices In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces (typically triangles) in 3D models, where each such point is given as a vector, and 12 edges.
Along with the rectangular cuboids, any parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. (The term rhomboid is also sometimes used with this meaning.) It is to a parallelogram as a cube is to a square: Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds. Three equivalent definitions of is a cuboid of this type, as is a square frustum In geometry, a frustum is the portion of a solid — normally a cone or pyramid — which lies between two parallel planes cutting it (the shape formed by truncation In mathematics, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones of the apex of a square pyramid).
Rectangular cuboid
In a rectangular cuboid, all angles are right angles In geometry and trigonometry, a right angle is an angle of 90 degrees, corresponding to a quarter turn . It can be defined as the angle such that twice that angle amounts to a half turn, or 180°, and opposite faces of a cuboid are equal In geometry, two sets of points are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions. It is also a right rectangular prism In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids. The term "rectangular or oblong prism" is ambiguous. Also the term rectangular parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. (The term rhomboid is also sometimes used with this meaning.) It is to a parallelogram as a cube is to a square: Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds. Three equivalent definitions of or orthogonal parallelepiped is used.
The square cuboid, square box ,or right square prism (also ambiguously called square prism) is a special case of the cuboid in which at least two faces are squares. The cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the is a special case of the square cuboid in which all six faces are squares.
If the dimensions of a cuboid are a, b and c, then its volume The volume of any solid, liquid, gas, plasma, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as cubic meters, cubic centimeters, liters, is abc and its surface area Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved then the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface is 2ab + 2bc + 2ac.
The length of the space diagonal is
Cuboid shapes are often used for boxes Box describes a variety of containers and receptacles for permanent use as storage, or for temporary use often for transporting contents, cupboards A cupboard or press (Hiberno-English) is a type of cabinet, often made of wood, used indoors to store household objects such as food and crockery, and protect them from dust and dirt, rooms A room, in architecture, is any distinguishable space within a structure. Most typically a room is separated by interior walls from other spaces or passageways; moreover, it is separated by an exterior wall from outdoor areas, sometimes with a door. Historically the use of rooms dates at least to early Minoan cultures about 2200 BC, where, buildings, etc. Cuboids are among those solids that can tessellate 3-dimensional space In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. The shape is fairly versatile in being able to contain multiple smaller cuboids, e.g. sugar Sugar is a class of edible crystalline substances, mainly sucrose, lactose, and fructose. Human taste buds interpret its flavor as sweet. Sugar as a basic food carbohydrate primarily comes from sugar cane and from sugar beet, but also appears in fruit, honey, sorghum, sugar maple , and in many other sources. It forms the main ingredient in candy cubes in a box, small boxes in a large box, a cupboard in a room, and rooms in a building.
A cuboid with integer edges as well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.
See also
- Hyperrectangle In geometry, an orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals
- Oblong
- Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the
- Trapezohedron The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent deltoids . The faces are symmetrically staggered
References
- ^ Robertson, Stewart Alexander (1984), Polytopes and Symmetry, Cambridge University Press, p. 75, ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 9780521277396 .
- ^ Dupuis, Nathan Fellowes (1893), Elements of Synthetic Solid Geometry, Macmillan, p. 53 .
External links
- Weisstein, Eric W. Eric W. Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc, "Cuboid" from MathWorld MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign.
- Rectangular prism and cuboid Paper models and pictures
- Computer Simulation of Cuboid Dice, an Oxford research project
- Cuboids, Rectangular Prisms and Cubes from Math Is Fun Categories: Educational websites | Online encyclopedias | Mathematics websites | Internet forums |
Categories: Elementary geometry Elementary geometry consists of topics from geometry frequently taught at the primary or secondary school level | Polyhedra Categories: Geometry | Geometric shapes | Euclidean solid geometry | Polytopes | Prismatoid polyhedra This category groups polyhedra which have all their vertices contained within two parallel planes. This set includes prisms, antiprisms, pyramids, cupolas, frustums, wedges, etc | Space-filling polyhedra | Zonohedra
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Cuboid has released an animations pack Forum Thread
Matt Banks
Mon, 16 Feb 2009 15:00:28 GM
The beauty of . Cuboid. is its simplicity. You take control of a . cuboid. , a rectangular block, that you must maneuver from a starting point of a puzzle to a glowing blue finish. The only controls you need are the d-pad buttons. ...
Q. Thank you
Asked by HAYLEY S - Sat Mar 1 12:33:01 2008 - - 3 Answers - 0 Comments
A. A cuboid is a box, and none of the edges are necessarily equal. Only a cube has equal edges. There are many possible values for the dimensions of this cuboid: 4 x 3 x 4 3 x 8 x 2 6 x 4 x 2 are just a few. And just for the sake of accuracy, 4 x 4 x 4 = 64.
Answered by Twiggy - Sat Mar 1 14:53:15 2008
