Mirror image

A mirror image is a reflected duplication of an object that appears identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off of substances such as water, it is a concept in geometry and can be used as a conceptualization process for 3-D structures. In geometry, the mirror image of an object or two-dimensional figure is the virtual image formed by reflection in a plane mirror. Two-dimensional mirror images can be seen in the reflections of mirrors or other reflecting surfaces, or on a printed surface seen inside-out. If we look at an object, two-dimensional and turn it towards a mirror, the object turns through an angle of 180º and we see a left-right reversal in the mirror. In this example, it is the change in orientation rather than the mirror itself that causes the observed reversal. Another example is when we stand with our backs to the mirror and face an object that's in front of the mirror. We compare the object with its reflection by turning ourselves 180º, towards the mirror.

Again we perceive a left-right reversal due to a change in orientation. So, in these examples the mirror does not cause the observed reversals; the concept of reflection can be extended to three-dimensional objects, including the inside parts if they are not transparent. The term relates to structural as well as visual aspects. A three-dimensional object is reversed in the direction perpendicular to the mirror surface. In physics, mirror images are investigated in the subject called geometrical optics. In chemistry, two versions of a molecule, one a "mirror image" of the other, are called enantiomers if they are not "superposable" on each other; that is an example of chirality. In general, an object and its mirror image are called enantiomorphs. If a point of an object has coordinates the image of this point has coordinates, thus reflection is a reversal of the coordinate axis perpendicular to the mirror's surface. Although a plane mirror reverses an object only in the direction normal to the mirror surface, there is a perception of a left-right reversal.

Hence, the reversal is called "lateral inversion". The perception of a left-right reversal is because the left and right of an object are defined by its perceived top and front, but there is still some debate about the explanation amongst psychologists; the psychology of the perceived left-right reversal is discussed in "Much ado about mirrors" by Professor Michael Corballis. Reflection in a mirror does result in a change in chirality, more from a right-handed to a left-handed coordinate system; as a consequence, if one looks in a mirror and lets two axes coincide with those in the mirror this gives a reversal of the third axis. If a person stands side-on to a mirror and right will be reversed directly by the mirror, because the person's left-right axis is normal to the mirror plane. However, it's important to understand that there are always only two enantiomorphs, the object and its image. Therefore, no matter how the object is oriented towards the mirror, all the resulting images are fundamentally identical.

In the picture of the mountain reflected in the lake, the reversal normal to the reflecting surface is obvious. Notice that there is no obvious front-back or left-right of the mountain. In the example of the urn and mirror, the urn is symmetrical front-back. Thus, no obvious reversal of any sort can be seen in the mirror image of the urn. A mirror image appears more three-dimensional if the observer moves, or if the image is viewed using binocular vision; this is because the relative position of objects changes as the observer's perspective changes, or is differently viewed with each eye. Looking through a mirror from different positions is like looking at the 3D mirror image of space. A mirror does not just produce an image of. A mirror hanging on the wall makes the room brighter because additional light sources appear in the mirror image. However, the appearance of additional light does not violate the conservation of energy principle, because some light no longer reaches behind the mirror, as the mirror re-directs the light energy.

In terms of the light distribution, the virtual mirror image has the same appearance and the same effect as a real, symmetrically arranged half-space behind a window. Shadows may extend from the mirror into the halfspace before it, vice versa. In mirror writing a text is deliberately displayed in mirror image, in order to be read through a mirror. For example, emergency vehicles such as ambulances or fire engines use mirror images in order to be read from a driver's rear-view mirror; some movie theaters take advantage of mirror writing in a Rear Window Captioning System used to assist individuals with heari

Pentagon

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be self-intersecting. A self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°. A regular pentagon has five lines of reflectional symmetry, rotational symmetry of order 5; the diagonals of a convex regular pentagon are in the golden ratio to its sides. Its height and width are given by Height = 5 + 2 5 2 ⋅ Side ≈ 1.539 ⋅ Side, Width = Diagonal = 1 + 5 2 ⋅ Side ≈ 1.618 ⋅ Side, Diagonal = R 5 + 5 2 = 2 R cos 18 ∘ = 2 R cos π 10 ≈ 1.902 R, where R is the radius of the circumcircle. The area of a convex regular pentagon with side length t is given by A = t 2 25 + 10 5 4 = 5 t 2 tan 4 ≈ 1.720 t 2. A pentagram or pentangle is a regular star pentagon, its Schläfli symbol is. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.

When a regular pentagon is circumscribed by a circle with radius R, its edge length t is given by the expression t = R 5 − 5 2 = 2 R sin 36 ∘ = 2 R sin π 5 ≈ 1.176 R, its area is A = 5 R 2 4 5 + 5 2. The area of any regular polygon is: A = 1 2 P r where P is the perimeter of the polygon, r is the inradius. Substituting the regular pentagon's values for P and r gives the formula A = 1 2 ⋅ 5 t ⋅ t tan 2 = 5 t 2 tan 4 with side length “f” Like every regular convex polygon, the regular convex pentagon has an inscribed circle; the apothem, the radius r of the inscribed circle, of a regular pentagon is related to the side length t by r = t 2 tan = t 2 5 − 20 ≈ 0.6882 ⋅ t. Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C PA + PD = PB + PC + PE; the regular pentagon is constructible with compass and straightedge. A variety of methods are known for constructing a regular pentagon.

Some are discussed below. One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's "Polyhedra."The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius, its center is located at point

Rhombille tiling

In geometry, the rhombille tiling known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 120 ° angles. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles; the rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling, it can be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals of each rhomb are in the ratio 1:√3; this is the dual tiling of the trihexagonal kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, in the face configuration for monohedral tilings it is denoted, it is one of 56 possible isohedral tilings by quadrilaterals, one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.

It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, more such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube; the rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion. In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. In another of his works, Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements and an upstairs patio tiled with the rhombille tiling.

A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so. These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more and includes a depiction of the reversible cubes illusion on a flag within the scene; the rhombille tiling is used as a design for parquetry and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. It appears in ancient Greek floor mosaics from Delos and from Italian floor tilings from the 11th century, although the tiles with this pattern in Siena Cathedral are of a more recent vintage. In quilting, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation; as a quilting pattern it has many other names including cubework, heavenly stairs, Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape.

See Quilts of the Underground Railroad. In these decorative applications, the rhombi may appear in multiple colors, but are given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms; the rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field.

The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers. In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice, it is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals, it has been studied in percolation theory. The rhombille tiling has *632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a *333 symmetry; the rhombille tiling is the dual of the trihexagonal tiling, as such is part of a set of uniform dual tilings. It is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry, starting from the cube, which can be seen as a rhombic hexahedron where the rhombi are squares; the nth element in this sequence has a face configuration of V3.n.3.n. The rhombille tiling is one of many different ways of tiling the plane by congruent rhombi.

Others include a diagonally flattened variation of the square tiling, the tiling used by the Miura-ori folding pattern, the Penrose tiling which

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

Ammann–Beenker tiling

In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker; because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings described in Tilings and Patterns; the Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings, most notably: They are nonperiodic, which means that they lack any translational symmetry. Their non-periodicity is implied by their hierarchical structure: the tilings are substitution tilings arising from substitution rules for growing larger and larger patches; this substitution structure implies that: Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, they are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction.

This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation." All of this infinite global structure is forced through local matching rules on a pair of tiles, among the simplest aperiodic sets of tiles found, Ammann's A5 set. Various methods to describe the tilings have been proposed: matching rules, substitutions and project schemes and coverings. In 1987 Wang and Kuo announced the discovery of a quasicrystal with octagonal symmetry. Amman's A and B tiles in his pair A5 a 45-135-degree rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic and quasiperiodic structures of each of the infinite number of individual Ammann-Beenker tilings. An alternate set of tiles discovered by Ammann, labelled "Ammann 4" in Grünbaum and Shephard, consists of two nonconvex right-angle-edged pieces. One consists of two squares overlapping on a smaller square, while the other consists of a large square attached to a smaller square.

The diagrams below show a portion of the tilings. This is the substitution rule for the alternate tileset; the relationship between the two tilesets. In addition to the edge arrows in the usual tileset, the matching rules for both tilesets can be expressed by drawing pieces of large arrows at the vertices, requiring them to piece together into full arrows. Katz has studied the additional tilings allowed by dropping the vertex constraints and imposing only the requirement that the edge arrows match. Since this requirement is itself preserved by the substitution rules, any new tiling has an infinite sequence of "enlarged" copies obtained by successive applications of the substitution rule; each tiling in the sequence is indistinguishable from a true Ammann–Beenker tiling on a successively larger scale. Since some of these tilings are periodic, it follows that no decoration of the tiles which does force aperiodicity can be determined by looking at any finite patch of the tiling; the orientation of the vertex arrows which force aperiodicity can only be deduced from the entire infinite tiling.

The tiling has an extremal property: among the tilings whose rhombuses alternate, the proportion of squares is found to be minimal in the Ammann–Beenker tilings. The Ammann–Beenker tilings are related to the silver ratio and the Pell numbers; the substitution scheme R → R r R. The eigenvalues of the substitution matrix are 1 + 2 and 1 − 2. In the alternate tileset, the long edges have 1 + 2 times longer sides than the short edges. One set of Conway worms, formed by the short and long diagonals of the rhombs, forms the above strings, with r as the short diagonal and R as the long diagonal. Therefore, the Ammann bars form Pell ordered grids; the Ammann bars for the usual tileset. If the bold outer lines are taken to have length 2 2, the bars split the edges into segments of length 1 + 2 and 2 − 1; the Ammann bars for the alternate tileset. Note that the bars for the asymmetric tile extend outside it; the tesseractic honeycomb has an eightfold rotational symmetry, corresponding to an eightfold rotational symmetry of the tesseract.

A rotation matrix representing this symmetry is: A = [ 0 0 0 − 1 1 0 0 0 0 − 1

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is equiangular, it is a regular polygon, so it is referred to as a regular triangle. Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: The area is A = 3 4 a 2 The perimeter is p = 3 a The radius of the circumscribed circle is R = a 3 The radius of the inscribed circle is r = 3 6 a or r = R 2 The geometric center of the triangle is the center of the circumscribed and inscribed circles The altitude from any side is h = 3 2 a Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that: The area of the triangle is A = 3 3 4 R 2 Many of these quantities have simple relationships to the altitude of each vertex from the opposite side: The area is A = h 2 3 The height of the center from each side, or apothem, is h 3 The radius of the circle circumscribing the three vertices is R = 2 h 3 The radius of the inscribed circle is r = h 3 In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, the medians to each side coincide.

A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc, where R and r are the radii of the circumcircle and incircle is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, knowing that any one of them is true directly implies that we have an equilateral triangle. A = b = c 1 a + 1 b + 1 c = 25 R r − 2 r 2 4 R r s = 2 R + r s 2 = 3 r 2 + 12 R r s 2 = 3 3 T s = 3 3 r s = 3 3 2 R A = B = C = 60 ∘ cos A + cos B + cos C = 3 2 sin A 2 sin B 2 sin C 2 = 1 8 T = a 2 + b 2 + c 2 4 3 T = 3 4 2 3 R = 2 r 9 R 2 = a 2 + b 2 + c 2 r = r a +

Pinwheel tiling

In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations. Let T be the right triangle with side length 1, 2 and 5. Conway noticed that T can be divided in five isometric copies of its image by the dilation of factor 1 / 5. By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of T; the union of all these triangles yields a tiling of the whole plane by isometric copies of T. In this tiling, isometric copies of T appears in infinitely many orientations. Despite this, all the vertices have rational coordinates. Radin relied on the above construction of Conway to define pinwheel tilings. Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of T, in which a tile may intersect another tile only either on a whole side or on half the length 2 side, such that the following property holds.

Given any pinwheel tiling P, there is a pinwheel tiling P ′ which, once each tile is divided in five following the Conway construction and the result is dilated by a factor 5, is equal to P. In other words, the tiles of any pinwheel tilings can be grouped in sets of five into homothetic tiles, so that these homothetic tiles form a new pinwheel tiling; the tiling constructed by Conway is a pinwheel tiling, but there are uncountably many other different pinwheel tiling. They are all locally undistinguishable, they all share with the Conway tiling the property that tiles appear in infinitely many orientations. The main result proven by Radin is that there is a finite set of so-called prototiles, with each being obtained by coloring the sides of T, so that the pinwheel tilings are the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point. In terms of symbolic dynamics, this means. Radin and Conway proposed a three-dimensional analogue, dubbed the quaquaversal tiling.

There are other generalizations of the original idea. One gets a fractal by iteratively dividing T in five isometrics copies, following the Conway construction, discarding the middle triangle; this "pinwheel fractal" has Hausdorff dimension d = ln 4 ln 5 ≈ 1.7227. Federation Square, a building complex in Melbourne, features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub-framing for the facades, allowing for the facades to be fabricated off-site, in a factory and erected to form the facades; the pinwheel tiling system was based on the single triangular element, composed of zinc, perforated zinc, sandstone or glass, joined to 4 other similar tiles on an aluminum frame, to form a "panel". Five panels were affixed to a galvanized steel frame, forming a "mega-panel", which were hoisted onto support frames for the facade; the rotational positioning of the tiles gives the facades a more random, uncertain compositional quality though the process of its construction is based on pre-fabrication and repetition.

The same pinwheel tiling system is used in the development of the structural frame and glazing for the "Atrium" at Federation Square, although in this instance, the pin-wheel grid has been made "3-dimensional" to form a portal frame structure. Pinwheel at the Tilings Encyclopedia Dynamic Pinwheel made in GeoGebra