In geometry, a cube is a three-dimensional solid object bounded by six square faces. It has twelve edges and eight vertices. It can be represented as a rectangular cuboid with six square faces, or a parallelepiped with equal edges. It is an example of many type of solids: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.
In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:
In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin modulus, a measure.
The adjective canonical is applied in many contexts to mean 'according to the canon' – the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, canonical example is often used to mean 'archetype'.
Solid modeling is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes (solids). Solid modeling is distinguished within the broader related areas of geometric modeling and computer graphics, such as 3D modeling, by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of 3D-computer-aided design, and in general, support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.
A Hessian is an inhabitant of the German state of Hesse.
A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.
Father Magnus J. Wenninger OSB was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction.
In mathematics Nef polygons and Nef polyhedra are the sets of polygons and polyhedra which can be obtained from a finite set of halfplanes (halfspaces) by Boolean operations of set intersection and set complement. The objects are named after the Swiss mathematician Walter Nef (1919–2013), who introduced them in his 1978 book on polyhedra.
Mathematical Models is a book on the construction of physical models of mathematical objects for educational purposes. It was written by Martyn Cundy and A. P. Rollett, and published by the Clarendon Press in 1951, with a second edition in 1961. Tarquin Publications published a third edition in 1981.
