a world full of choices Tiling by regular polygons Regular tilings[edit] Archimedean, uniform or semiregular tilings[edit] Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second. Grünbaum and Shephard distinguish the description of these tilings as Archimedean as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as uniform as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Combinations of regular polygons that can meet at a vertex[edit] For Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees. -gon has internal angle degrees. With 3 polygons at a vertex: Below are diagrams of such vertices: With 4 polygons at a vertex: With 5 polygons at a vertex: With 6 polygons at a vertex: 36 - regular, Triangular tiling Other edge-to-edge tilings[edit]

Coxeter–Dynkin diagram Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups Coxeter–Dynkin diagrams for the fundamental affine Coxeter groups Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. See Dynkin diagrams for details. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.[1] Description[edit] Branches of a Coxeter–Dynkin diagram are labeled with a rational number p, representing a dihedral angle of 180°/p. Diagrams can be labeled by their graph structure. Schläfli matrix[edit] Finite and affine groups are also called elliptical and parabolic respectively.

Still Life and Street Still Life and Street is a woodcut print by the Dutch artist M. C. Escher which was first printed in March, 1937. It was his first print of an impossible reality. In this artwork we have two quite distinctly recognizable realities bound together in a natural, and yet at the same time a completely impossible, way. Looked at from the window, the houses make book-rests between which tiny dolls are set up. A small street in Savona, Italy, was the inspiration for this work.[1] Escher said it was one of his favorite drawings but thought he could have drawn it better. See also[edit] Printmaking References[edit] Sources[edit]

Lattice (group) In mathematics, especially in geometry and group theory, a lattice in is a discrete subgroup of which spans the real vector space . Every lattice in A lattice is the symmetry group of discrete translational symmetry in n directions. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translate of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense. A simple example of a lattice in is the subgroup . , and the Leech lattice in . is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions. A typical lattice in thus has the form where {v1, ..., vn} is a basis for . See also: Integer points in polyhedra Five lattices in the Euclidean plane A lattice in .

Architect and Graphic Designer Andy Hau This afternoon we take a look at the impressive work of architect and designer Andy Hau. Having completed his architecture degree three years ago at the University of Westminster and The Bartlett (UCL), Andy has since learnt to apply his knowledge and creativity in other areas including graphic and product design. “I know the breadth of my job description sounds like an attempt to be vertiginously eclectic but in reality it stems from a very simple and practical synergy that exists between the two different disciplines. “I fully qualified as an architect three years ago, having studied at the University of Westminster and The Bartlett (UCL). “In 2009, I was commissioned to work with Imogen Heap on her “Ellipse” album promo packaging and logo. “Currently I am expanding into furniture design, creating a coffee table using parametric technology, whereby extremely complex geometry can be produced and optimised through a set of rules and algorithms.

Stephen Collins - Penrose Tiling Generator Bob - Penrose Tiling Generator and Explorer Bob is a Microsoft Windows program designed to produce and explore rhombic Penrose tiling comprising two types of rhombus which together form an infinite, aperiodic plane. In particular, Bob allows the user to discover and examine geodesic "walks" within the tiling, some of which display beautiful, complex, five-fold symmetrical patterns - "Flowers". These Flowers appear to increase indefinitely in size and complexity as the tiling grows in extent. Bob is so named after my father, Dr. Bob Collins, who discovered these walks within rhombic Penrose tiling while working as a member of the Physics Department of the University of York in England. Dr. Feel free to email me with any problems or comments with the software. Back to Top There is a wealth of information online about Penrose Tiling, which was discovered by the British mathematician and physicist Roger Penrose, which I do not propose to duplicate here. Now comes the interesting bit. Dr.

List of convex uniform tilings This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane , and their dual tilings. There are three regular, and eight semiregular, tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. Uniform tilings are listed by their vertex configuration , the sequence of faces that exist on each vertex. For example means one square and two octagons on a vertex. These 11 uniform tilings have 32 different . In addition to the 11 convex uniform tilings, there are also 14 nonconvex tilings , using star polygons , and reverse orientation vertex configurations . Dual tilings are listed by their face configuration , the number of faces at each vertex of a face. In the 1987 book, , Branko Grünbaum calls the vertex-uniform tilings in parallel to the Archimedean solids , and the dual tilings in honor of crystallographer Fritz Laves . [ edit ] Convex uniform tilings of the Euclidean plane Families:

M. C. Escher Maurits Cornelis Escher (/ˈɛʃər/, Dutch: [ˈmʌurɪts kɔrˈneːlɪs ˈɛʃər] ( );[1] 17 June 1898 – 27 March 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. Early life[edit] Maurits Cornelis,[2] was born in Leeuwarden, Friesland, in a house that forms part of the Princessehof Ceramics Museum today. He was a sickly child, and was placed in a special school at the age of seven and failed the second grade.[3] Although he excelled at drawing, his grades were generally poor. Later life[edit] In 1922, an important year of his life, Escher traveled through Italy (Florence, San Gimignano, Volterra, Siena, Ravello) and Spain (Madrid, Toledo, Granada). In Italy, Escher met Jetta Umiker, whom he married in 1924. In 1935, the political climate in Italy (under Mussolini) became unacceptable to Escher. Escher, who had been very fond of and inspired by the landscapes in Italy, was decidedly unhappy in Switzerland. Works[edit] In his early years, Escher sketched landscapes and nature.

Rotational symmetry Generally speaking, an object with rotational symmetry, also known in biological contexts as radial symmetry, is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted. The degree of rotational symmetry is how many degrees the shape has to be turned to look the same on a different side or vertex. It can not be the same side or vertex. Formal treatment[edit] Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole E(m). Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. n-fold rotational symmetry[edit] The notation for n-fold symmetry is Cn or simply "n". The fundamental domain is a sector of 360°/n. Examples without additional reflection symmetry: Examples[edit] See also[edit] References[edit]

Interior Design | Designcollector Akoura Residence Youssef Tohme Architects “Conceived by Youssef Tohme as a break with the city, this house perched on a cliff edge in Lebanon is a monolith dressed in local stone mounted upon concrete walls. Up in the… Loft Apartment In Seattle by SHED SHED Architecture & Design have completed the remodel of a loft in the Capitol Hill area of Seattle, Washington. San Francisco Home Interior Interior designer Benedetta Amadi, one of The Room on the Roof by i29 Architects Located in Amsterdam’s Dam Square, the De Bijenkorf department store was first built in 1909 as a gorgeously cresselated neo-gothic structure pierced by a distinctive central turret. The Chamber of Curiosity by Gestalten “The Chamber of Curiosity by Gestalten is more than a collection of some of the best interiors from around the world. Artist Residence London Pennethorne’s Café, London, UK Simple Interior Design Photography by Anya Garienchick Living Spaces: Inspiration Set 57 It was good time, it was BED time!

Penrose tiling A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right. A Penrose tiling has many remarkable properties, most notably: It is non-periodic, which means that it lacks any translational symmetry.It is self-similar, so the same patterns occur at larger and larger scales. Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules, cut and project schemes and coverings. Background and history[edit] Periodic and aperiodic tilings[edit] Figure 1. Earliest aperiodic tilings[edit] Robinson's six prototiles Figure 2. Notes[edit]