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In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.
The art of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability and the use of paper folds to solve mathematical equations.
Modular origami or unit origami is a paperfolding technique which uses two or more sheets of paper to create a larger and more complex structure than would be possible using single-piece origami techniques. Each individual sheet of paper is folded into a module, or unit, and then modules are assembled into an integrated flat shape or three-dimensional structure, usually by inserting flaps into pockets created by the folding process. These insertions create tension or friction that holds the model together.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In geometry a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.
A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.
In geometry, an antiparallelogram is a quadrilateral having, like a parallelogram, two opposite pairs of equal-length sides, but in which the sides of one pair cross each other as in a scissors mechanism. The longer of the two pairs will always be the one that crosses. Antiparallelograms are also called contraparallelograms or crossed parallelograms.
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape.
In geometry, a flexible polyhedron is a polyhedral surface that allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex.
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.
In geometry, a Bricard octahedron is a member of a family of flexible polyhedra constructed by Raoul Bricard in 1897. That is, it is possible for the overall shape of this polyhedron to change in a continuous motion, without any changes to the lengths of its edges nor to the shapes of its faces. These octahedra were the first flexible polyhedra to be discovered.
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.”
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid, having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.
A History of Folding in Mathematics: Mathematizing the Margins is a book in the history of mathematics on the mathematics of paper folding. It was written by Michael Friedman and published in 2018 by Birkhäuser as volume 59 of their Historial Studies series.
Origami Polyhedra Design is a book on origami designs for constructing polyhedra. It was written by origami artist and mathematician John Montroll, and published in 2009 by A K Peters.
Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published in Russian in 1950, under the title Выпуклые многогранники. It was translated into German by Wilhelm Süss as Konvexe Polyeder in 1958. An updated edition, translated into English by Nurlan S. Dairbekov, Semën Samsonovich Kutateladze and Alexei B. Sossinsky, with added material by Victor Zalgaller, L. A. Shor, and Yu. A. Volkov, was published as Convex Polyhedra by Springer-Verlag in 2005.
References
- 1 2 3 4 5 6 Carbno, Collin (May 2009), "Review of Geometric Folding Algorithms", MAA Reviews, Mathematical Association of America
- 1 2 3 4 5 6 7 8 9 Paquete, Luís (November 2009), "Review of Geometric Folding Algorithms", European Journal of Operational Research, 199 (1): 311–313, doi:10.1016/j.ejor.2008.06.009
- 1 2 mbec (2011), "Review of Geometric Folding Algorithms", EMS Reviews, European Mathematical Society
- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fasy, Brittany Terese; Millman, David L. (March 2011), "Review of Geometric Folding Algorithms", SIGACT News, Association for Computing Machinery, 42 (1): 43–46, doi:10.1145/1959045.1959056, S2CID 6514501
- ↑ Uehara, Ryuhei, 幾何的な折りアルゴリズム リンケージ・折り紙・多面体 , retrieved 2020-02-02
- ↑ Hull, Tom (2012), "Other sources", Project Origami: Activities for Exploring Mathematics (2nd ed.), CRC Press, p. xviii