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A convex polytope, like any compact convex subset of R n, is homeomorphic to a closed ball. [11] Let m denote the dimension of the polytope. If the polytope is full-dimensional, then m = n. The convex polytope therefore is an m-dimensional manifold with boundary, its Euler characteristic is 1, and its fundamental group is trivial.
A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra.
The outer protein shell of HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus. [26] Several species of radiolarians discovered by Ernst Haeckel , described its shells as the like-shaped various regular polyhedra; one of which is Circogonia icosahedra , whose skeleton is shaped like a regular icosahedron.
Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published in Russian in 1950, under the title Выпуклые многогранники. [1] [2] It was translated into German by Wilhelm Süss as Konvexe Polyeder in 1958. [3]
Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum , with contributions from Victor Klee , Micha Perles , and G. C. Shephard , and published in 1967 by John Wiley & Sons.
The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in a trirectangular tetrahedron. The snub cube is an Archimedean solid that can be constructed by separating away the cube square's face, and filling their gaps with twisted angle equilateral triangles;a process known as snub .
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset.
Download QR code; Print/export Download as PDF; Printable version; In other projects Wikidata item; Appearance. ... Schaum's Outline of Geometry (4th ed.). New York ...