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In one approach the preserved monotonicity trait is the line L. A three-dimensional polyhedron is called weakly monotonic in direction L if all cross-sections orthogonal to L are simple polygons. If the cross-sections are convex, then the polyhedron is called weakly monotonic in convex sense. [9] Both types may be recognized in polynomial time ...
A p-gonal regular polygon is represented by Schläfli symbol {p}. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.
Convex regular icosahedron. Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices.Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion.
X marks convex corners; O marks concave corners. Blue lines are knobs; red lines are anti-knobs; yellow lines are neither. A rectilinear polygon has corners of two types: corners in which the smaller angle (90°) is interior to the polygon are called convex and corners in which the larger angle (270°) is interior are called concave. [1]
A related theorem states that every simple polygon that is not a convex polygon has a mouth, a vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called anthropomorphic polygons. [16]
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)
The intersection of two convex polygons is a convex polygon. A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Helly's theorem: For every collection of at least three convex polygons: if all intersections of all but one polygon are nonempty ...
One might characterise the Greek definition as follows: A regular polygon is a planar figure with all edges equal and all corners equal. A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.