Kokotsakis polyhedron
A Kokotsakis polyhedron is a polyhedral surface in three-dimensional space consisting of any polygon as its base and quadrilaterals alternating with triangles as its lateral faces; for an -sided polygonal base, there are quadrilaterals and triangles.
Properties and history
The polyhedron was discovered when studied the meshes wherein the perimeter of a polygon is surrounded by other polygons, showing an infinitesimally flexible in the case of a quadrilateral base, which was later known as Kokotsakis mesh. More examples of this special case of a Kokotsakis polyhedron were discovered by other mathematicians. Here, a polyhedron is flexible if the shape can be continuously changed while preserving the faces unchanged. Each of its vertexes is said to be "developable", meaning the sum of its plane angle is, resulting in the polyhedral surface being an origami crease pattern, which satisfies Kawasaki's theorem. The work was done by in which classifying the folding angle for a Kokotsakis polyhedron in the case of a quadrangular base.conjectured that there exists no polynomial system of irreducible resultant about the flexibility of a Kokotsakis polyhedron, which was later disproved it by showing the Kokotsakis polyhedron is orthodiagonal anti-involutive, meaning the planar angles has conditions such as all quadrilaterals are spherically orthodiagonal and elliptic, and the involution at common vertices are opposite.
Kokotsakis mesh
As mentioned above, the Kokotsakis mesh was studied by, showing an infinitesimally flexible polyhedron in the case of a quadrilateral base. In general, the Kokotsakis mesh is defined as the infinite tessellations consisting of quadrilateral with congruent convex that is not trapezoidal and parallelogram. In the case of a quadrangle mesh, it is planar symmetric, translation, isogonal, orthogonal, and line-symmetric.The Kokotsakis mesh can be used to construct cylindrical polyhedra.