Uniform polyhedronIsogonal polyhedron with regular facesSome examples of uniform polyhedra: triangular prism, square antiprism, regular tetrahedron (a Platonic solid), cuboctahedron (an Archimedean solid), small stellated dodecahedron (a Kepler–Poinsot polyhedron), and snub dodecadodecahedron (a uniform star polyhedron).In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other.
Vertex (geometry)In geometry, a vertex (pl.: vertices or vertexes), also called a corner, is a point where two or more curves, lines, or line segmentsmeet or intersect. For example, the point where two lines meet to form an angle and the point where edges of polygons and polyhedra meet are vertices. The term vertex is also used for a different type of point in geometry, a vertex of a curve, meaning a point of extreme curvature.
TriangleShape with three sidesTriangle, a polygon with three corners (vertices) and three lines (sides)A triangle or trigon is a polygon consisting of three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensionalpoints while the sides connecting them, also called edges, are one-dimensional line segments.
Isogonal figureIn geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second.
TessellationCovering by shapes without overlaps or gapsZelligeterracotta tiles in Marrakesh, forming edge‑to‑edge, regular and other tessellationsA wall sculpture in Leeuwarden celebrating the artistic tessellations of M. C. EscherAn example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tilesA 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.
PolyhedronIn geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek(poly-)'many' and (-hedron)'base, seat') is a three-dimensional figure with flat polygonalfaces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts.
PentagonIn geometry, a pentagon (from Greekπέντε (pente)'five' and γωνία (gonia)'angle') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram. Regular pentagonsA regular pentagon has Schläfli symbol {5} and interior angles of 108°.
Chirality (mathematics)In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral. A chiral object and its mirror image are said to be enantiomorphs.
Mathematical Models (Cundy and Rollett)Mathematical Models is a book on the construction of physical models of mathematical objects for educational purposes. It was written by Martyn Cundy and A. P. Rollett, and published by the Clarendon Press in 1951, with a second edition in 1961. Tarquin Publications published a third edition in 1981.
GeometryBranch of mathematics GeometryProjecting a sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry ConceptsFeaturesDimension Straightedge and compass constructions Angle Polytope Centroid Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional Point One-dimensional Line Line segment Ray Curve Geodesic Length Two-dimensional Surface Plane Area Polygon Simple Convex Concave Star Regular Reuleaux Triangle Centers Altitude Hypotenuse Pythagorean theorem Circular Reuleaux Hyperbolic Ideal Spherical Quadrilateral Parallelogram Rectangle Square Rhombus Rhomboid Trapezoid Kite Circle Radius Diameter Circumference Disk Area Three-dimensional Surface area Volume Polyhedron Platonic Solid Tetrahedron Reuleaux cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Star Toroidal Ideal Solid of revolution Sphere Great circle Cylinder Cone Four - other-dimensional 4-polytope Simplex 5-cell Hypercube Tesseract n-sphere Geometers by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov vte Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
Archimedean solidThe Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons and are vertex-transitive, although they are not face-transitive and are generally not edge-transitive. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices.
Face (geometry)In solid geometry, a face is a flat surface (a planarregion) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense.*As an Amazon Associate I earn from qualifying purchases.