**: Category: Kepler – Poinsot polyhedra for the four non-convex regular polyhedra.

Consider the non-convex polygon P shown in the figure, which is formed from a regular hexagon by adding projections on two of its sides and matching indentations on three sides.

* Star polygon, a non-convex shape

In geometry, a polygon can be either convex or concave ( non-convex or reentrant ).

A simple polygon that is not convex is called concave, non-convex or reentrant.

The branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called global optimization.

Nonlinear penalty functions have been used, particularly to reduce the effect of outliers on the classifier, but unless care is taken the problem becomes non-convex, and thus it is considerably more difficult to find a global solution.

The envelope of the infinitude of area bisectors is a deltoid ( broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set ).

A non-convex object might have a centroid that is outside the figure itself.

If non-convex bounding volumes are required, an approach is to represent them as a union of a number of convex bounding volumes.

If the sum of the angles exceeds a full circle, as occurs in some vertices of most ( not all ) non-convex polyhedra, then the defect is negative.

CSS is applicable to all optimization fields ; especially it is suitable for non-smooth or non-convex domains.

It is not known whether two non-convex analytic domains can have the same eigenvalues.

But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear.

This search was completed around 1620 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

A non-convex set, with a line-segment outside the set.

The Kepler-Poinsot polyhedra are examples of non-convex sets.

A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal solutions, and will treat the former as actual solutions to the original problem.

In contrast, non-linear least squares problems generally must be solved by an iterative procedure, and the problems can be non-convex with multiple optima for the objective function.

The sixteen dark-red points ( on the right ) form the Minkowski sum of the four non-convex sets ( on the left ), each of which consists of a pair of red points.

More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set.

However, if the room were non-convex ( for example, an L-shaped or partitioned room ), then there would be many isovists whose volume ( area ) would be less than that of the whole room, and perhaps some that were ; and many would have different, perhaps unique shapes: large and small, narrow and wide, centric and eccentric, whole and shredded.

Milton's description was quoted as the epigraph to the chapter " Markets with non-convex preferences and production " presenting in.

They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

For example, the perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides ( and decrease in the length of one side ).

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.

For example, every polygon is topologically self-dual ( it has the same number of vertices as edges, and these are switched by duality ), but will not in general be geometrically self-dual ( up to rigid motion, for instance ) – regular polygons are geometrically self-dual ( all angles are congruent, as are all edges, so under duality these congruences swap ), but irregular polygons may not be geometrically self-dual.

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i. e., not a Platonic solid, Archimedean solid, prism or antiprism.

Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

* Star polygon: a polygon which self-intersects in a regular way.

* Regular: A polygon is regular if it is both cyclic and equilateral.

Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides.

The area of a regular polygon is also given in terms of the radius r of its inscribed circle and its perimeter p by

While it is possible to construct analogies to the Penrose triangle with other regular polygons to create a Penrose polygon, the visual effect is not as striking, and as the sides increase, the object seems merely to be warped or twisted.

Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes.

Some regular polygons, like the heptagon, become constructible ; and John H. Conway gives constructions for several of them ; but the 11-sided polygon, the hendecagon, is still impossible, and infinitely many others.

* a regular polygon that can be constructed with compass and straightedge ; see constructible polygon.

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections.

In this article, D < sub > n </ sub > ( and sometimes Dih < sub > n </ sub >) refers to the symmetries of a regular polygon with n sides.