Pythagorean tiling

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, it is used as a pattern for floor tiles. When used for this, it is known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern; this tiling has four-way rotational symmetry around each of its squares. When the ratio of the side lengths of the two squares is an irrational number such as the golden ratio, its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions have been studied; the Pythagorean tiling is the unique tiling by squares of two different sizes, both unilateral and equitransitive. Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagons.

The smaller squares in the Pythagorean tiling are adjacent to four larger tiles, as are the squares in the truncated square tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate between large and small, just as the octagons in the truncated square tiling. However, the two tilings have different sets of symmetries, because the truncated square tiling is symmetric under mirror reflections whereas the Pythagorean tiling isn't. Mathematically, this can be explained by saying that the truncated square tiling has dihedral symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic set of symmetries around the corresponding points, giving it p4 symmetry, it is a chiral pattern, meaning that it is impossible to superpose it on top of its mirror image using only translations and rotations. A uniform tiling is a tiling in which each tile is a regular polygon and in which every vertex can be mapped to every other vertex by a symmetry of the tiling.

Uniform tilings additionally are required to have tiles that meet edge-to-edge, but if this requirement is relaxed there are eight additional uniform tilings. Four are formed from infinite strips of squares or equilateral triangles, three are formed from equilateral triangles and regular hexagons; the remaining one is the Pythagorean tiling. This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth-century Islamic mathematicians Al-Nayrizi and Thābit ibn Qurra, by the 19th-century British amateur mathematician Henry Perigal. If the sides of the two squares forming the tiling are the numbers a and b the closest distance between corresponding points on congruent squares is c, where c is the length of the hypotenuse of a right triangle having sides a and b. For instance, in the illustration to the left, the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple.

By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides a and b into a single square of side c, showing that the two smaller squares have the same area as the larger one. Overlaying two Pythagorean tilings may be used to generate a six-piece dissection of two unequal squares into a different two unequal squares. Although the Pythagorean tiling is itself periodic its cross sections can be used to generate one-dimensional aperiodic sequences. In the "Klotz construction" for aperiodic sequences, one forms a Pythagorean tiling with two squares whose sizes are chosen to make the ratio between the two side lengths be an irrational number x. One chooses a line parallel to the sides of the squares, forms a sequence of binary values from the sizes of the squares crossed by the line: a 0 corresponds to a crossing of a large square and a 1 corresponds to a crossing of a small square. In this sequence, the relative proportion of 0s and 1s will be in the ratio x:1.

This proportion cannot be achieved by a periodic sequence of 0s and 1s, because it is irrational, so the sequence is aperiodic. If x is chosen as the golden ratio, the sequence of 0s and 1s generated in this way has the same recursive structure as the Fibonacci word: it can be split into substrings of the form "01" and "0" and if these two substrings are replaced by the shorter strings "0" and "1" another string with the same structure results. According to Keller's conjecture, any tiling of the plane by congruent squares must include two squares that meet edge-to-edge. None of the squares in the Pythagorean tiling meet edge-to-edge, but this fact does not violate Keller's conjecture because the tiles have different sizes, so they are not all congruent to each other; the Pythagorean tiling may be generalized to a three-dimensional tiling of Euclidean space by cubes of two different sizes, unilateral and equitransitive. Attila Bölcskei calls this three-dimensional tiling the Rogers filling, he conjectures that, in any dimension greater than three, there is again a unique unilateral and equitransitive way of tiling space by hypercubes of two different sizes.

Burns and Rigby found several prototiles, including the Koch snowflake, that may be used to tile the plane only by using copies of the prototile in two or more different sizes. An earlier paper by Danz

Rectification (geometry)

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope. A rectification operator is sometimes denoted by the letter r with a Schläfli symbol. For example, r is the rectified cube called a cuboctahedron, represented as, and a rectified cuboctahedron rr is a rhombicuboctahedron, represented as r. Conway polyhedron notation uses a for ambo as this operator. In graph theory this operation creates a medial graph; the rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron becoming an octahedron. As a special case, a square tiling will turn into another square tiling under a rectification operation. Rectification is the final point of a truncation process.

For example, on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form: Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, so on; this sequence shows a birectified cube as the final sequence from a cube to the dual where the original faces are truncated down to a single point: The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon; each platonic solid and its dual have the same rectified polyhedron. The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual: The rectified tetrahedron, whose dual is the tetrahedron, is the tetratetrahedron, better known as the octahedron.

The rectified octahedron, whose dual is the cube, is the cuboctahedron. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron. A rectified square tiling is a square tiling. A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling. Examples If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, from that graph one may form the medial graph by placing a vertex at each edge midpoint of the original graph, connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face; the resulting medial graph remains polyhedral, so by Steinitz's theorem it can be represented as a polyhedron. The Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, is Conway's expand operation, e, the same as Johnson's cantellation operation, t0,2 generated from regular polyhedral and tilings.

Each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. A regular 4-polytope has cells, its rectification will have two cell types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex. A rectified is not the same as a rectified, however. A further truncation, called bitruncation, is symmetric between its dual. See Uniform 4-polytope#Geometric derivations. Examples A first rectification truncates edges down to points. If a polytope is regular, this form is represented by r. A second rectification, or birectification, truncates faces down to points. If regular it has notation 2r. For polyhedra, a birectification creates a dual polyhedron. Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates n-faces to points. If an n-polytope is -rectified, its facets are reduced to points and the polytope becomes its dual. There are different equivalent notations for each degree of rectification.

These tables show the names by the two type of facets for each. Facets are edges, represented as. Facets are regular polygons. Facets are rectified polyhedra. Facets are rectified 4-polytopes. Dual polytope Quasiregular polyhedron List of regular polytopes Truncation Conway polyhedron notation Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto, 1966 John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Olshevsky, George. "Rectification". Glossary for Hyperspace. Archived from the original on 4 February 2007

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp

Cantellation (geometry)

In geometry, a cantellation is an operation in any dimension that bevels a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation applies to regular tilings and honeycombs; this is rectifying its rectification. This operation is called expansion by Alicia Boole Stott, as imagined by taking the faces of the regular form moving them away from the center and filling in new faces in the gaps for each opened vertex and edge, it is represented by r or rr. For polyhedra, a cantellation operation offers a direct sequence from a regular polyhedron and its dual. Example cantellation sequence between a cube and octahedron For higher-dimensional polytopes, a cantellation offers a direct sequence from a regular polytope and its birectified form. A cuboctahedron would be a cantellated tetrahedron, as another example. Uniform polyhedron Uniform 4-polytope Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N.

W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto, 1966 Weisstein, Eric W. "Expansion". MathWorld

Branko Grünbaum

Branko Grünbaum was a Yugoslavian-born mathematician of Jewish descent and a professor emeritus at the University of Washington in Seattle. He received his Ph. D. in 1957 from Hebrew University of Jerusalem in Israel. He authored over 200 papers in discrete geometry, an area in which he is known for various classification theorems, he wrote on the theory of abstract polyhedra. His paper on line arrangements may have inspired a paper by N. G. de Bruijn on quasiperiodic tilings. This paper is cited by the authors of a monograph on hyperplane arrangements as having inspired their research. Grünbaum devised a multi-set generalisation of Venn diagrams, he was a frequent contributor to Geombinatorics. Grünbaum's classic monograph Convex polytopes, first published in 1967, became the main textbook on the subject, his monograph Tilings and Patterns, coauthored with G. C. Shephard, helped to rejuvenate interest in this classic field, has proved popular with nonmathematical audiences, as well as with mathematicians.

In 1976 Grünbaum won a Lester R. Ford Award for his expository article Venn diagrams and independent families of sets. In 2004, Gil Kalai and Victor Klee edited a special issue of Discrete and Computational Geometry in his honor, the "Grünbaum Festschrift". In 2005, Grünbaum was awarded the Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society, he was a Guggenheim Fellow, a Fellow of the AAAS and in 2012 he became a fellow of the American Mathematical Society. Grünbaum supervised 19 Ph. D.s and has at least 200 mathematical "descendants". Grünbaum, Kaibel, Volker. Convex Polytopes, Graduate Texts in Mathematics, 221, Springer-Verlag, ISBN 0-387-00424-6. Grünbaum, Branko. Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Pentagram map Orlik, Peter.

Dual polyhedron

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 geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.

R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.

The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.

When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form

Alternation (geometry)

In geometry, an alternation or partial truncation, is an operation on a polygon, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed h, standing for half; because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, being degenerate, is reduced to a single edge. More any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation of a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces. So for example, the cube 4.4.4 is alternated as 2.3.2.3.2.3, reduced to 3.3.3, being the tetrahedron, all the 6 edges of the tetrahedra can be seen as the degenerate faces of the original cube. A snub can be seen as an alternation of a truncated truncated quasiregular polyhedron.

In general a polyhedron can be snubbed. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra; the snub square antiprism is an example of a general snub, can be represented by ss, with the square antiprism, s. This alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform; the voids created by the deleted vertices will not in general create uniform facets, there are not enough degrees of freedom to allow an appropriate rescaling of the new edges. Exceptions do exist, such as the derivation of the snub 24-cell from the truncated 24-cell. Examples: Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb. An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell. 4-honeycombs: An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube.

Cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract... Coxeter used the operator a, which contains both halves, so retains the original symmetry. For even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron. Norman Johnson extended the use of the altered operator a, b for blended, c for converted, as, respectively; the compound polyhedron, stellated octahedron can be represented by and. The star-polyhedron, small ditrigonal icosidodecahedron, can be represented by and. Here all the pentagons have been alternated into pentagrams, triangles have been inserted to take up the resulting free edges. A similar operation can truncate alternate vertices, rather than just removing them. Below is a set of polyhedra; these have two types of vertices. Truncating the "higher order" vertices and both vertex types produce these forms: Conway polyhedral notation Wythoff construction Coxeter, H.

S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto, 1966 Weisstein, Eric W. "Snubification". MathWorld. Richard Klitzing, alternated facetings, Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, George. "Alternation". Glossary for Hyperspace. Archived from the original on 4 February 2007. Polyhedra Names, snub