1.
Vertex configuration
–
In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
2.
Tessellation
–
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
3.
Regular polygon
–
In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
4.
Johannes Kepler
–
Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
5.
Harmonices Mundi
–
Harmonices Mundi is a book by Johannes Kepler. In the work Kepler discusses harmony and congruence in geometrical forms, the final section of the work relates his discovery of the so-called third law of planetary motion. Thus Kepler could reason that his relationships gave evidence for God acting as a geometer, rather than a Pythagorean numerologist. The concept of musical harmonies intrinsically existing within the spacing of the planets existed in medieval philosophy prior to Kepler. Musica Universalis was a philosophical metaphor that was taught in the quadrivium. He notes musical harmony as being a product of man, derived from angles, in turn, this allowed Kepler to claim the Earth has a soul because it is subjected to astrological harmony. Chapters 1 and 2 of The Harmony of the World contain most of Kepler’s contributions concerning polyhedra and he is primarily interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather and he returns to this concept later in Harmonices Mundi with relation to astronomical explanations. He describes polyhedra in terms of their faces, which is similar to the used in Plato’s Timaeus to describe the formation of Platonic solids in terms of basic triangles. While medieval philosophers spoke metaphorically of the music of the spheres and he found that the difference between the maximum and minimum angular speeds of a planet in its orbit approximates a harmonic proportion. For instance, the angular speed of the Earth as measured from the Sun varies by a semitone, from mi to fa. Venus only varies by a tiny 25,24 interval, Kepler explains the reason for the Earths small harmonic range, The Earth sings Mi, Fa, Mi, you may infer even from the syllables that in this our home misery and famine hold sway. The celestial choir Kepler formed was made up of a tenor, at very rare intervals all of the planets would sing together in perfect concord, Kepler proposed that this may have happened only once in history, perhaps at the time of creation. Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring orbits approximate musical harmonies within a margin of error of less than a diesis. The orbits of Mars and Jupiter produce the one exception to this rule, in fact, the cause of Keplers dissonance might be explained by the fact that the asteroid belt separates those two planetary orbits, as discovered in 1801,150 years after Keplers death. Keplers previous book Astronomia nova related the discovery of the first two of the principles that we know today as Keplers laws, a copy of the 1619 edition was stolen from the National Library of Sweden in the 1990s. A small number of recent compositions either make reference to or are based on the concepts of Harmonices Mundi or Harmony of the Spheres, the most notable of these are, Laurie Spiegel, Keplers Harmony of the Worlds. An excerpt of the piece was selected by Carl Sagan for inclusion on the Voyager Golden Record, mike Oldfield, Music of the Spheres
6.
Symmetry group
–
In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
7.
Group action
–
In mathematics, an action of a group is a way of interpreting the elements of the group as acting on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, actions of groups on vector spaces are called representations of the group. Some groups can be interpreted as acting on spaces in a canonical way, more generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. A common way of specifying non-canonical actions is to describe a homomorphism φ from a group G to the group of symmetries of a set X. The action of an element g ∈ G on a point x ∈ X is assumed to be identical to the action of its image φ ∈ Sym on the point x. The homomorphism φ is also called the action of G. Thus, if G is a group and X is a set, if X has additional structure, then φ is only called an action if for each g ∈ G, the permutation φ preserves the structure of X. The abstraction provided by group actions is a one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them, in particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, the group G is said to act on X. The set X is called a G-set. In complete analogy, one can define a group action of G on X as an operation X × G → X mapping to x. g. =. h for all g, h in G and all x in X, for a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula −1 = h−1g−1, one can construct an action from a right action by composing with the inverse operation of the group. Also, an action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality. The trivial action of any group G on any set X is defined by g. x = x for all g in G and all x in X, that is, every group element induces the identity permutation on X. In every group G, left multiplication is an action of G on G, g. x = gx for all g, x in G
8.
Vertex (geometry)
–
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
9.
Congruence (geometry)
–
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
10.
Equilateral triangle
–
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
11.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
12.
Hexagon
–
In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
13.
Triangular tiling
–
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the triangle is 60 degrees. The triangular tiling has Schläfli symbol of, Conway calls it a deltille, named from the triangular shape of the Greek letter delta. The triangular tiling can also be called a kishextille by a kis operation that adds a center point and it is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling, there are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors,111212 and 111112 from 121213 by combining 1 and 3, there is one class of Archimedean colorings,111112, which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows. The vertex arrangement of the tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb, the A*2 lattice can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice. + + = dual of = The vertices of the tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing, the packing density is π⁄√12 or 90. 69%. Since the union of 3 A2 lattices is also an A2 lattice, the voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling, with identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color, the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid and these can be expanded to Platonic solids, five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbols. It is also related as a part of sequence of Catalan solids with face configuration Vn.6.6. Like the uniform there are eight uniform tilings that can be based from the regular hexagonal tiling
14.
Hexagonal tiling
–
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t, english mathematician Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees and it is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling, the hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, however, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, tubular graphene sheets have been synthesised, these are known as carbon nanotubes. They have many applications, due to their high tensile strength. Chicken wire consists of a lattice of wires. The hexagonal tiling appears in many crystals, in three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest known sphere packings in three dimensions, and are believed to be optimal, structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a tiling, all generated from reflective symmetry of Wythoff constructions. The represent the periodic repeat of one colored tile, counting hexagonal distances as h first, the 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling, in the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling. The hexagons can be dissected into sets of 6 triangles and this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. This tiling is related to regular polyhedra with vertex figure n3. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6 and this tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry
15.
Square tiling
–
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of, meaning it has 4 squares around every vertex, the internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane, the other two are the triangular tiling and the hexagonal tiling. There are 9 distinct uniform colorings of a square tiling, naming the colors by indices on the 4 squares around a vertex,1111,1112,1112,1122,1123,1123,1212,1213,1234. Cases have simple reflection symmetry, and glide reflection symmetry, three can be seen in the same symmetry domain as reduced colorings, 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. This tiling is related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane. Like the uniform there are eight uniform tilings that can be based from the regular square tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. However treating faces identically, there are only three distinct forms, square tiling, truncated square tiling, snub square tiling. Other quadrilateral tilings can be made with topologically equivalent to the square tiling, isohedral tilings have identical faces and vertex-transitivity, there are 17 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two colinear edges. Symmetry given assumes all faces are the same color, the square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing, the packing density is π/4=78. 54% coverage. There are 4 uniform colorings of the circle packings, there are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices, Regular apeirogons pr are contrained by, 1/p + 2/q + 1/r =1. Edges have p vertices, and vertex figures are r-gonal, checkerboard List of regular polytopes List of uniform tilings Square lattice Tilings of regular polygons Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p.296, Table II, Regular honeycombs Klitzing, 2D Euclidean tilings o4o4x - squat - O1. The Geometrical Foundation of Natural Structure, A Source Book of Design, p36 Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Square Grid
16.
Isogonal figure
–
In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
17.
Symmetry operation
–
In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasize its usefulness, physical properties must be invariant with respect to symmetry operations. Symmetry operations can be collected together in groups which are isomorphous to permutation groups, wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property. These are denoted by Cnm and are rotations of 360°/n, performed m times, the superscript m is omitted if it is equal to one. C1, rotation by 360°, is called the Identity operation and is denoted by E or I, cnn, n rotations 360°/n is also an Identity operation. These are denoted by Snm and are rotations of 360°/n followed by reflection in a perpendicular to the rotation axis. S1 is usually denoted as σ, an operation about a mirror plane. S2 is usually denoted as i, an operation about an inversion centre. When n is an even number Snn = E, but when n is odd Sn2n = E. Rotation axes, mirror planes and inversion centres are symmetry elements, the rotation axis of highest order is known as the principal rotation axis. It is conventional to set the Cartesian z axis of the molecule to contain the principal rotation axis, there is a C2 rotation axis which passes through the carbon atom and the midpoints between the two hydrogen atoms and the two chlorine atoms. Define the z axis as co-linear with the C2 axis, the xz plane as containing CH2, a C2 rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the yz plane permutes the hydrogen atoms while reflection in the xz plane permutes the chlorine atoms, the four symmetry operations E, C2, σand σ form the point group C2v. Note that if any two operations are carried out in succession the result is the same as if an operation of the group had been performed. In addition to the rotations of order 2 and 3 there are three mutually perpendicular S4 axes which pass half-way between the C-H bonds and six mirror planes. In crystals screw rotations and/or glide reflections are additionally possible and these are rotations or reflections together with partial translation. The Bravais lattices may be considered as representing translational symmetry operations, combinations of operations of the crystallographic point groups with the addition symmetry operations produce the 230 crystallographic space groups. Cotton Chemical applications of theory, Wiley,1962,1971
18.
Uniform tiling
–
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane, Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group, a planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices. A fundamental domain triangle is, and a triangle, where p, q, r are whole numbers greater than 1. The triangle may exist as a triangle, a Euclidean plane triangle, or a hyperbolic plane triangle. There are a number of schemes for naming these figures, from a modified Schläfli symbol for right triangle domains. The Coxeter-Dynkin diagram is a graph with p, q, r labeled on the edges. If r =2, the graph is linear since order-2 domain nodes generate no reflections, the Wythoff symbol takes the 3 integers and separates them by a vertical bar. If the generator point is off the mirror opposite a domain node, finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex. All uniform tilings can be constructed from various operations applied to regular tilings and these operations as named by Norman Johnson are called truncation, rectification, and Cantellation. Omnitruncation is an operation that combines truncation and cantellation, snubbing is an operation of Alternate truncation of the omnitruncated form. Each is represented by a set of lines of reflection that divide the plane into fundamental triangles and these symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the tilings are repeated from different symmetry constructors. A prismatic symmetry group represented by represents by two sets of mirrors, which in general can have a rectangular fundamental domain. A further prismatic symmetry group represented by which has a fundamental domain. It constructs two uniform tilings, the prism and apeirogonal antiprism. The stacking of the faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane. It is called the triangular tiling, composed of alternating layers of squares and triangles
19.
Mirror image
–
A mirror image is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off of such as a mirror or water. It is also a concept in geometry and can be used as a process for 3-D structures. 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 that is effectively two-dimensional and then turn it towards a 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, then we compare the object with its reflection by turning ourselves 180 degrees, towards the mirror. Again we perceive a left-right reversal due to a change in orientation, so, in these examples the mirror does not actually cause the observed reversals. The concept of reflection can be extended to three-dimensional objects, including the inside parts, the term then relates to structural as well as visual aspects. A three-dimensional object is reversed in the 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 then the image of this point has coordinates, thus reflection is a reversal of the coordinate axis perpendicular to the mirrors surface. In everyday use, a mirror does not reverse right and left, however, there is often a perception of left-right reversal, probably because the left and right of an object are defined by its top and front. Reflection in a mirror does result in a change in chirality, as a consequence, if one looks in a mirror and lets two axes coincide with those in the mirror, then this gives a reversal of the third axis. Similarly, if you stand side-on to a mirror your left and its important to realise there are only two enantiomorphs, the object and its image. So, no matter how the object is oriented towards the mirror, all the images are fundamentally identical. In the photograph of the urn and mirror, the urn is fairly symmetrical front-back, so, its not surprising that no obvious reversal of the urn can be seen in the mirror image. A mirror image appears more obviously three-dimensional if the observer moves and this is because the relative position of objects changes as the observers perspective changes, or is different viewed with each eye
20.
Chirality (mathematics)
–
In geometry, a figure is chiral 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, in 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane, a chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ, the hand, the most familiar chiral object, a non-chiral figure is called achiral or amphichiral. Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. A right shoe is different from a left shoe only for being mirror images of each other, in contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped tetrominoes of the video game Tetris also exhibit chirality. Individually they contain no mirror symmetry in the plane, a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In three dimensions, every figure that possesses a plane of symmetry S1, an inversion center of symmetry S2. Note, however, that there are achiral figures lacking both plane and center of symmetry, an example is the figure F0 = which is invariant under the orientation reversing isometry ↦ and thus achiral, but it has neither plane nor center of symmetry. The figure F1 = also is achiral as the origin is a center of symmetry, note also that achiral figures can have a center axis. In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry and its symmetry group is a frieze group generated by a single glide reflection. A knot is called if it can be continuously deformed into its mirror image. For example the unknot and the knot are achiral, whereas the trefoil knot is chiral. Chirality in Metric Spaces, Symmetry, Culture and Science 21, pp. 27–36 Chiral Polyhedra by Eric W. Weisstein, when Topology Meets Chemistry by Erica Flapan. Chiral manifold at the Manifold Atlas
21.
Truncated hexagonal tiling
–
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex and it is given an extended Schläfli symbol of t. Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling. There are 3 regular and 8 semiregular tilings in the plane, there is only one uniform coloring of a truncated hexagonal tiling. The dodecagonal faces can be distorted into different geometries, like, Like the uniform there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. This tiling is related as a part of sequence of uniform truncated polyhedra with vertex configurations. Two 2-uniform tilings are related by dissected the dodecagons into a hexagonal and 6 surrounding triangles and squares. The truncated hexagonal tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing and this is the lowest density packing that can be created from a uniform tiling. The dodecagonal gaps can be filled perfectly with 7 circles, creating a denser 3-uniform packing, the triakis triangular tiling is a tiling of the Euclidean plane. It is a triangular tiling with each triangle divided into three obtuse triangles from the center point. It is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices, one with 3 triangles, and two with 12 triangles, Conway calls it a kisdeltille, constructed as a kis operation applied to a triangular tiling. In Japan the pattern is called asanoha for hemp leaf, although the name applies to other triakis shapes like the triakis icosahedron. It is the tessellation of the truncated hexagonal tiling which has one triangle. It is one of 7 dual uniform tilings in hexagonal symmetry, tilings of regular polygons List of uniform tilings John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. 2D Euclidean tilings o3x6x - toxat - O7
22.
Rhombitrihexagonal tiling
–
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex and it has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille and it can be considered a cantellated by Norman Johnsons terminology or an expanded hexagonal tiling by Alicia Boole Stotts operational language. There are 3 regular and 8 semiregular tilings in the plane, there is only one uniform coloring in a rhombitrihexagonal tiling. With edge-colorings there is a half symmetry form orbifold notation, the hexagons can be considered as truncated triangles, t with two types of edges. It has Coxeter diagram, Schläfli symbol s2, the bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, there is one related 2-uniform tilings, having hexagons dissected into 6 triangles. Every circle is in contact with 4 other circles in the packing, the translational lattice domain contains 6 distinct circles. The gap inside each hexagon allows for one circle, related to a 2-uniform tiling with the hexagons divided into 6 triangles, there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. This tiling is related as a part of sequence of cantellated polyhedra with vertex figure. These vertex-transitive figures have reflectional symmetry, the deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. The edges of this tiling can be formed by the overlay of the regular triangular tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90° and it is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling. The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling and its faces are deltoids or kites. It is one of 7 dual uniform tilings in hexagonal symmetry and this tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrillaterals. Ignoring the face colors below, the symmetry is p6m, and the lower symmetry is p31m with 3 mirrors meeting at a point. This tiling is related to the tiling by dividing the triangles and hexagons into central triangles
23.
Truncated trihexagonal tiling
–
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex and it has Schläfli symbol of tr. There is only one uniform coloring of a trihexagonal tiling. A 2-uniform coloring has two colors of hexagons, 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares. The truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling, the first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a hexagon and surrounding triangles and square, in two different orientations. The Truncated trihexagonal tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing, circles can be alternatedly colored in this packing with an even number of sides of all the regular polygons of this tiling. The gap inside each hexagon allows for one circle, and each dodecagon allows for 7 circles, the kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree right triangles with 4,6, conway calls it a kisrhombille for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings and it can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. It is labeled V4.6.12 because each right triangle face has three types of vertices, one with 4 triangles, one with 6 triangles, and one with 12 triangles, the kisrhombille tiling triangles represent the fundamental domains of p6m, symmetry. There are a number of small index subgroups constructed from by mirror removal, creates *333 symmetry, shown as red mirror lines. The commutator subgroup is, which is 333 symmetry, a larger index 6 subgroup constructed as, also becomes, shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12. There are eight uniform tilings that can be based from the hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. This tiling can be considered a member of a sequence of patterns with vertex figure. For p <6, the members of the sequence are omnitruncated polyhedra, for p >6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling
24.
Trihexagonal tiling
–
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles. The name derives from the fact that it combines a hexagonal tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an arrangement of lines. Its dual is the rhombille tiling and this pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi. The pattern has long used in Japanese basketry, where it is called kagome. The Japanese term for this pattern has taken up in physics. It occurs also in the structures of certain minerals. Conway calls it a hexadeltille, combining elements from a hexagonal tiling. Kagome is a traditional Japanese woven bamboo pattern, its name is composed from the words kago, meaning basket, and me, meaning eye, referring to the pattern of holes in a woven basket. It is an arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The weaved process gives the Kagome a chiral wallpaper group symmetry, the term kagome lattice was coined by Japanese physicist Kôdi Husimi, and first appeared in a 1951 paper by his assistant Ichirō Shōji. The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling, despite the name, these crossing points do not form a mathematical lattice. It is represented by the vertices and edges of the cubic honeycomb, filling space by regular tetrahedra. It contains four sets of planes of points and lines. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an orthorhombic-kagome lattice, the trihexagonal prismatic honeycomb represents its edges and vertices. Some minerals, namely jarosites and herbertsmithite, contain two layers or three dimensional kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism, for instance, the spin arrangement of the magnetic ions in Co3V2O8 rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures
25.
Truncated square tiling
–
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular polygons which contains an octagon. It has Schläfli symbol of t. Conway calls it a truncated quadrille, other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges. There are 3 regular and 8 semiregular tilings in the plane, there are two distinct uniform colorings of a truncated square tiling. The truncated square tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing, since there is an even number of sides of all the polygons, the circles can be alternately colored as shown below. One variations on this pattern, often called a Mediterranean pattern, is shown in stone tiles with smaller squares, other variations stretch the squares or octagons. The Pythagorean tiling alternates large and small squares, and may be seen as identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices, in the plane it can be represented by a compound tiling, or combined can be seen as a chamfered square tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. However treating faces identically, there are three unique topologically forms, square tiling, truncated square tiling, snub square tiling. The tetrakis square tiling is the tiling of the Euclidean plane dual to the square tiling. It can be constructed square tiling with each divided into four isosceles right triangles from the center point. Conway calls it a kisquadrille, represented by a kis operation that adds a center point and it is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices. CS1 maint, Multiple names, authors list Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. Dale Seymour and Jill Britton, Introduction to Tessellations,1989, ISBN 978-0866514613, pp. 50–56 http, //www. decrete. com/stencils/octagontile Weisstein, 2D Euclidean tilings o4x4x - tosquat - O6
26.
Snub square tiling
–
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex and its Schläfli symbol is s. Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling. There are 3 regular and 8 semiregular tilings in the plane, there are two distinct uniform colorings of a snub square tiling. The snub square tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing, the snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling. An alternate truncation deletes every other vertex, creating a new faces at the removed vertices. If the original tiling is made of regular faces the new triangles will be isosceles, starting with octagons which alternate long and short edge lengths will produce a snub tiling with perfect equilateral triangle faces. Example, This tiling is related to the triangular tiling which also has 3 triangles. The snub square tiling is third in a series of snub polyhedra, 2D Euclidean tilings s4s4s - snasquat - O10. Grünbaum, Branko, and Shephard, G. C, cS1 maint, Multiple names, authors list Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, p38 Dale Seymour and Jill Britton, Introduction to Tessellations,1989, ISBN 978-0866514613, pp. 50–56, dual p.115 Weisstein, Eric W. Semiregular tessellation
27.
Elongated triangular tiling
–
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex and it is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol, e. Conway calls it a isosnub quadrille. There are 3 regular and 8 semiregular tilings in the plane and this tiling is similar to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order. It is also the uniform tiling that cant be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal prisms and apeirogonal antiprisms, there is one uniform colorings of an elongated triangular tiling. The 2-uniform tilings are also called Archimedean colorings, there are infinite variations of these Archimedean colorings by arbitrary shifts in the square row colorings. The elongated triangular tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing and it is first in a series of symmetry mutations with hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4. n.4.3.3.3, and Coxeter diagram. Their duals have hexagonal faces in the plane, with face configuration V4. n.4.3.3.3. There are four related 2-uniform tilings, mixing 2 or 3 rows of triangles or squares, the prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 15 known isohedral pentagon tilings and it can be seen as a stretched hexagonal tiling with a set of parallel bisecting lines through the hexagons. Each of its faces has three 120° and two 90° angles. It is related to the Cairo pentagonal tiling with face configuration V3.3.4.3.4, tilings of regular polygons Elongated triangular prismatic honeycomb Gyroelongated triangular prismatic honeycomb Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. 2D Euclidean tilings elong - etrat - O4
28.
Snub trihexagonal tiling
–
In geometry, the snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex and it has Schläfli symbol of sr. The snub tetrahexagonal tiling is a hyperbolic tiling with Schläfli symbol sr. Conway calls it a snub hextille. There are 3 regular and 8 semiregular tilings in the plane and this is the only one which does not have a reflection as a symmetry. There is only one uniform coloring of a trihexagonal tiling. The snub trihexagonal tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing, the lattice domain repeats 6 distinct circles. The hexagonal gaps can be filled by one circle, leading to the densest packing from the triangular tiling#circle packing. This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure and these figures and their duals have rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons, in geometry, the floret pentagonal tiling or rosette pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 15 known isohedral pentagon tilings and it is given its name because its six pentagonal tiles radiate out from a central point, like petals on a flower. Conway calls it a 6-fold pentille, each of its pentagonal faces has four 120° and one 60° angle. It is the dual of the tiling, snub trihexagonal tiling. The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, in one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling. Tilings of regular polygons List of uniform tilings John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Grünbaum, Branko, cS1 maint, Multiple names, authors list Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, 2D Euclidean tilings s3s6s - snathat - O11
29.
Wallpaper group
–
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, There are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the space groups. Wallpaper groups categorize patterns by their symmetries, subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. Consider the following examples, Examples A and B have the same group, it is called p4m in the IUC notation. Example C has a different wallpaper group, called p4g or 4*2, a complete list of all seventeen possible wallpaper groups can be found below. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance, think of shifting a set of vertical stripes horizontally by one stripe. Strictly speaking, a true symmetry only exists in patterns that repeat exactly, a set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end disappears and a new stripe is added at the other end. In practice, however, classification is applied to finite patterns, sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry, the types of transformations that are relevant here are called Euclidean plane isometries. This type of symmetry is called a translation, Examples A and C are similar, except that the smallest possible shifts are in diagonal directions. If we turn example B clockwise by 90°, around the centre of one of the squares, Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can also flip example B across a horizontal axis that runs across the middle of the image, example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A. However, example C is different and it only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a line, we do not get the same pattern back. This is part of the reason that the group of A and B is different from the wallpaper group of C. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891, the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done
30.
Internal and external angles
–
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple polygon, regardless of whether it is convex or non-convex, a polygon has exactly one internal angle per vertex. If every internal angle of a polygon is less than 180°. In contrast, an angle is an angle formed by one side of a simple polygon. The sum of the angle and the external angle on the same vertex is 180°. The sum of all the angles of a simple polygon is 180° where n is the number of sides. The formula can be proved using induction and starting with a triangle for which the angle sum is 180°. The sum of the angles of any simple convex or non-convex polygon is 360°. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles, in other words, 360k° represents the sum of all the exterior angles. For example, for convex and concave polygons k =1, since the exterior angle sum is 360°
31.
Heptagon
–
In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also referred to as the septagon, using sept- together with the Greek suffix -agon meaning angle. A regular heptagon, in all sides and all angles are equal, has internal angles of 5π/7 radians. The area of a regular heptagon of side length a is given by, the apothem is half the cotangent of π /7, and the area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be rewritten without complex components, since the indicated cubic function is casus irreducibilis. As 7 is a Pierpont prime but not a Fermat prime and this type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector, the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π7 ≈1.247 is a zero of the irreducible cubic x3 + x2 − 2x −1. Consequently, this polynomial is the polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0. 2% is shown in the drawing and it is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle, then B D =12 B C gives an approximation for the edge of the heptagon. Example to illustrate the error, At a circumscribed circle radius r =1 m, since 7 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z7, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptagon, john Conway labels these by a letter and group order. Full symmetry of the form is r14 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g7 subgroup has no degrees of freedom but can seen as directed edges. However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon and angles π /7,2 π /7, thus its sides coincide with one side and two particular diagonals of the regular heptagon. Two kinds of star heptagons can be constructed from regular heptagons, labeled by Schläfli symbols, blue, and green star heptagons inside a red heptagon
32.
Octagon
–
In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can also be constructed as a truncated square, t. A truncated octagon, t is a hexadecagon, t, the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°, the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. A regular octagon is a figure with sides of the same length. It has eight lines of symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol, the internal angle at each vertex of a regular octagon is 135°. The area of an octagon of side length a is given by A =2 cot π8 a 2 =2 a 2 ≃4.828 a 2. In terms of the circumradius R, the area is A =4 sin π4 R2 =22 R2 ≃2.828 R2. In terms of the r, the area is A =8 tan π8 r 2 =8 r 2 ≃3.314 r 2. These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as A = S2 − a 2, where S is the span of the octagon, or the second-shortest diagonal, and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside and then takes the corner triangles and places them with right angles pointed inward, the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈2.414 a. The area is then as above, A =2 − a 2 =2 a 2 ≈4.828 a 2, expressed in terms of the span, the area is A =2 S2 ≈0.828 S2. Another simple formula for the area is A =2 a S, more often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S /2.414, the two end lengths e on each side, as well as being e = a /2, may be calculated as e = /2. The circumradius of the octagon in terms of the side length a is R = a
33.
Icositetragon
–
In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragons interior angles is 3960 degrees, the regular icositetragon is represented by Schläfli symbol and can also be constructed as a truncated dodecagon, t, or a twice-truncated hexagon, tt, or thrice-truncated triangle, ttt. One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°, the area of a regular icositetragon is, A =6 t 2 cot π24 =6 t 2. The icositetragon appeared in Archimedes polygon approximation of pi, along with the hexagon, dodecagon, tetracontaoctagon, as 24 =23 ×3, a regular icositetragon is constructible using a compass and straightedge. As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon, the regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries, and, and 8 cyclic group symmetries and these 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a letter and group order, the full symmetry of the regular form is r48 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g24 subgroup has no degrees of freedom but can seen as directed edges. A regular triangle, octagon, and icositetragon can completely fill a plane vertex, an icositetragram is a 24-sided star polygon. There are 3 regular forms given by Schläfli symbols, and, there are also 7 regular star figures using the same vertex arrangement,2,3,4,6,8,3, and 2. There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon and these also generate two quasitruncations, t=, and t=. A skew icositetragon is a polygon with 24 vertices and edges. The interior of such an icositetragon is not generally defined, a skew zig-zag icositetragon has vertices alternating between two parallel planes. A regular skew icositetragon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D12d, symmetry, order 48. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including, Weisstein, Eric W. Icositetragon. Naming Polygons and Polyhedra polygon icosatetragon
34.
Nonagon
–
In geometry, a nonagon /ˈnɒnəɡɒn/ is a nine-sided polygon or 9-gon. The name nonagon is a hybrid formation, from Latin, used equivalently, attested already in the 16th century in French nonogone. The name enneagon comes from Greek enneagonon, and is more correct. A regular nonagon is represented by Schläfli symbol and has angles of 140°. Although a regular nonagon is not constructible with compass and straightedge and it can be also constructed using neusis, or by allowing the use of an angle trisector. The following is a construction of a nonagon using a straightedge. The regular enneagon has Dih9 symmetry, order 18, there are 2 subgroup dihedral symmetries, Dih3 and Dih1, and 3 cyclic group symmetries, Z9, Z3, and Z1. These 6 symmetries can be seen in 6 distinct symmetries on the enneagon, john Conway labels these by a letter and group order. Full symmetry of the form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g9 subgroup has no degrees of freedom but can seen as directed edges. The regular enneagon can tessellate the euclidean tiling with gaps and these gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H with H representing *632 hexagonal symmetry in the plane, the K9 complete graph is often drawn as a regular enneagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex and they Might Be Giants have a song entitled Nonagon on their childrens album Here Come the 123s. It refers to both an attendee at a party at which everybody in the party is a many-sided polygon, slipknots logo is also a version of a nonagon, being a nine-pointed star made of three triangles. King Gizzard & the Lizard Wizard have an album titled Nonagon Infinity, temples of the Bahai Faith are required to be nonagonal. The U. S. Steel Tower is an irregular nonagon, enneagram Trisection of the angle 60°, Proximity construction Weisstein, Eric W. Nonagon
35.
Octadecagon
–
An octadecagon or 18-gon is an eighteen-sided polygon. A regular octadecagon has a Schläfli symbol and can be constructed as a truncated enneagon, t. As 18 =2 ×32, a regular octadecagon cannot be constructed using a compass, however, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is similar to that of the enneagon. It is also feasible with exclusive use of compass and straightedge, the regular octadecagon has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries, Dih9, and, and 6 cyclic group symmetries and these 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and group order, full symmetry of the regular form is r36 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g18 subgroup has no degrees of freedom but can seen as directed edges. A regular triangle, nonagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of polygons with this property. The regular octadecagon can tessellate the plane with concave hexagonal gaps, and another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a hexagonal tiling. An octadecagram is an 18-sided star polygon, represented by symbol, there are two regular star polygons, and, using the same points, but connecting every fifth or seventh points. Deeper truncations of the regular enneagon and enneagrams can produce isogonal intermediate octadecagram forms with equally spaced vertices, other truncations form double coverings, t==2, t==2, t==2. The regular octadecagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in these orthogonal projections from Coxeter planes, octadecagon Weisstein
36.
Decagon
–
In geometry, a decagon is a ten-sided polygon or 10-gon. A regular decagon has all sides of length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a pentagon, t. By simple trigonometry, d =2 a, and it can be written algebraically as d = a 5 +25. The side of a regular decagon inscribed in a circle is −1 +52 =1 ϕ. As 10 =2 ×5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon. An alternative method is as follows, Construct a pentagon in a circle by one of the shown in constructing a pentagon. Extend a line from each vertex of the pentagon through the center of the circle to the side of that same circle. Where each line cuts the circle is a vertex of the decagon, the five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon, both in the construction with given circumcircle as well as with given side length is the golden ratio dividing a line segment by exterior division the determining construction element. In the construction with given circumcircle the circular arc around G with radius GE3 produces the segment AH, a M ¯ M H ¯ = A H ¯ A M ¯ =1 +52 = Φ ≈1.618. In the construction with side length the circular arc around D with radius DA produces the segment E10F. E1 E10 ¯ E1 F ¯ = E10 F ¯ E1 E10 ¯ = R a =1 +52 = Φ ≈1.618, the regular decagon has Dih10 symmetry, order 20. There are 3 subgroup dihedral symmetries, Dih5, Dih2, and Dih1, and 4 cyclic group symmetries, Z10, Z5, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the decagon, john Conway labels these by a letter and group order. Full symmetry of the form is r20 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g10 subgroup has no degrees of freedom but can seen as directed edges
37.
Pentadecagon
–
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. A regular pentadecagon is represented by Schläfli symbol, a regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by A =154 a 2 cot π15 =15 a 28 ≃17.6424 a 2. A regular triangle, decagon, and pentadecagon can completely fill a plane vertex. The point H divides the radius A M ¯ in golden ratio, compared with the first animation are in the following two images the two circular arcs rotated 90° counterclockwise shown. They do not use the segment C G ¯, but rather they use segment M G ¯ as radius A H ¯ for the circular arc. A compass and straightedge construction for a side length.618. Dih15 has 3 dihedral subgroups, Dih5, Dih3, and Dih1, and four more cyclic symmetries, Z15, Z5, Z3, and Z1, with Zn representing π/n radian rotational symmetry. On the pentadecagon, there are 8 distinct symmetries, john Conway labels these symmetries with a letter and order of the symmetry follows the letter. He gives r30 for the full symmetry, Dih15. These lower symmetries allows degrees of freedoms in defining irregular pentadecagons, only the g15 subgroup has no degrees of freedom but can seen as directed edges. There are three regular polygons, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, fourth. There are also three regular star figures, the first being a compound of three pentagons, the second a compound of five triangles, and the third a compound of three pentagrams. Calculation of the circumradius Weisstein, Eric W. Pentadecagon
38.
Pentagon
–
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or 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, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan 4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan 2 =5 t 2 tan 4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2
39.
Icosagon
–
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagons interior angles is 3240 degrees, the regular icosagon has Schläfli symbol, and can also be constructed as a truncated decagon, t, or a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°, the area of a regular icosagon with edge length t is A =5 t 2 ≃31.5687 t 2. In terms of the radius R of its circumcircle, the area is A =5 R22, since the area of the circle is π R2, the Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section. The Globe, the outdoor theater used by William Shakespeares acting company, was discovered to have built on an icosagonal foundation when a partial excavation was done in 1989. As a golygonal path, the swastika is considered to be an irregular icosagon, a regular square, pentagon, and icosagon can completely fill a plane vertex. E20 E1 ¯ E1 F ¯ = E20 F ¯ E20 E1 ¯ =1 +52 = φ ≈1.618 The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries, and, and 6 cyclic group symmetries and these 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order, full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g20 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular icosagon. An icosagram is a 20-sided star polygon, represented by symbol, there are three regular forms given by Schläfli symbols, and. There are also five regular star figures using the vertex arrangement,2,4,5,2,4. Deeper truncations of the regular decagon and decagram can produce isogonal intermediate icosagram forms with equally spaced vertices, a regular icosagram, can be seen as a quasitruncated decagon, t=. Similarly a decagram, has a quasitruncation t=, and finally a simple truncation of a decagram gives t=
40.
Dodecagon
–
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=