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In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted). [ 1 ] [ 2 ] Parallelogons have an even number of sides and opposite sides that are equal in length.
An example of a convex polygon: a regular pentagon. In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). [1]
A parallelohedron is defined to be a polyhedron whose translated copies meet face-to-face to fill space, forming a honeycomb. [1] The resulting honeycomb must be periodic, having a three-dimensional system of global symmetries, because each translation from a copy of the polyhedron to an adjoining copy must apply to all copies, forming a symmetry of the whole tiling. [2]
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [ 3 ] [ 4 ] [ 5 ] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph ∪ {\displaystyle \cup } .
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.
A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices.
convex, zonohedron In geometry , a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square .
Convex conjugate - a dual of a real functional in a vector space. Can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. Convex curve - a plane curve that lies entirely on one side of each of its supporting lines. The interior of a closed convex curve is a convex set.