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.Template:Sfnp
Properties and history
The polyhedron was discovered when Template:Harvtxt 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.Template:Sfnmp More examples of this special case of a Kokotsakis polyhedron were discovered by other mathematicians.Template:Sfnp 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.Template:Sfnp The work was done by Template:Harvtxt in which classifying the folding angle for a Kokotsakis polyhedron in the case of a quadrangular base.Template:Sfnmp
Template:Harvtxt conjectured that there exists no polynomial system of irreducible resultant about the flexibility of a Kokotsakis polyhedron, which was later Template:Harvtxt disproved it by showing the Kokotsakis polyhedron is orthodiagonal anti-involutive, meaning the planar angles has conditions such as all quadrilaterals are spherically orthodiagonal (any intersecting two diagonals in a quadrilateral form a right angle) and elliptic (the sum and difference of the edges of a quadrilateral are not equal to ), and the involution at common vertices are opposite.Template:Sfnmp
Kokotsakis mesh
As mentioned above, the Kokotsakis mesh was studied by Template:Harvtxt, showing an infinitesimally flexible polyhedron in the case of a quadrilateral base.Template:Sfnmp In general, the Kokotsakis mesh is defined as the infinite tessellations consisting of quadrilateral with congruent convex that is not trapezoidal and parallelogram.Template:Sfnp In the case of a quadrangle mesh, it is planar symmetric (exchanging adjacent vertices in-between), translation (two adjacent vertices are translated, mapping the faces), isogonal (opposite angles at every vertex is equal or complementary), orthogonal (the faces are parallel to the planes), and line-symmetric (appearance is symmetrical by half-rotating between two adjacent vertices around an axis passing through).Template:Sfnp
The Kokotsakis mesh can be used to construct cylindrical polyhedra.Template:Sfnmp
See also
- Bricard octahedron, a member of flexible polyhedra.