Kokotsakis polyhedron is a
polyhedral surface
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...
in three-dimensional space consisting of any number sided of a
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
as its base, and
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
s are its lateral faces with triangles between the consecutive quadrilateral; for
-sided polygonal base of a polyhedron, 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
Crease may refer to:
* A line (geometry) or mark made by folding or doubling any pliable substance
* Crease (band), American hard rock band that formed in Ft. Lauderdale, Florida in 1994
* Crease pattern, origami diagram type that consists of al ...
, which satisfies the
Kawasaki's theorem
Kawasaki's theorem or Kawasaki–Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single Vertex (geometry), vertex that may be folded to form a flat figure. It states that the pattern is fl ...
. 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
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...
about the flexibility of a Kokotsakis polyhedron, which was later disproved it by showing the Kokotsakis polyhedron is ''orthodiagonal anti-involutive'', meaning the
planar angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing a ...
s has conditions such as all quadrilaterals are spherically
orthodiagonal
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicula ...
(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
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
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
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s consisting of quadrilateral with congruent convex that is not trapezoidal and parallelogram. In the case of a quadrangle mesh, it is
planar symmetric (exchanging adjacent vertices in-between),
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
(two adjacent vertices are translated, mapping the faces),
isogonal (opposite angles at every vertex is equal or
complementary
Complement may refer to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class collections into complementary sets
* Complementary color, in the visu ...
),
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
(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).
The Kokotsakis mesh can be used to construct cylindrical polyhedra.
See also
*
Bricard octahedron
In geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A ...
, a member of flexible polyhedra.
Notes
Bibliography
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{{refend
Flexible polyhedra