Wythoff symbol
In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.
A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.
The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...
The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con
Weaving
Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. Other methods are knitting, crocheting and braiding or plaiting; the longitudinal threads are called the warp and the lateral threads are the weft or filling. The method in which these threads are inter-woven affects the characteristics of the cloth. Cloth is woven on a loom, a device that holds the warp threads in place while filling threads are woven through them. A fabric band which meets this definition of cloth can be made using other methods, including tablet weaving, back strap loom, or other techniques without looms; the way the warp and filling threads interlace with each other is called the weave. The majority of woven products are created with one of three basic weaves: plain weave, satin weave, or twill. Woven cloth can be woven in decorative or artistic design. In general, weaving involves using a loom to interlace two sets of threads at right angles to each other: the warp which runs longitudinally and the weft that crosses it.
One warp thread is called. The warp threads are held taut and in parallel to each other in a loom. There are many types of looms. Weaving can be summarized as a repetition of these three actions called the primary motion of the loom. Shedding: where the warp threads are separated by raising or lowering heald frames to form a clear space where the pick can pass Picking: where the weft or pick is propelled across the loom by hand, an air-jet, a rapier or a shuttle. Beating-up or battening: where the weft is pushed up against the fell of the cloth by the reed; the warp is divided into two overlapping groups, or lines that run in two planes, one above another, so the shuttle can be passed between them in a straight motion. The upper group is lowered by the loom mechanism, the lower group is raised, allowing to pass the shuttle in the opposite direction in a straight motion. Repeating these actions form a fabric mesh but without beating-up, the final distance between the adjacent wefts would be irregular and far too large.
The secondary motion of the loom are the: Let off Motion: where the warp is let off the warp beam at a regulated speed to make the filling and of the required design Take up Motion: Takes up the woven fabric in a regulated manner so that the density of filling is maintainedThe tertiary motions of the loom are the stop motions: to stop the loom in the event of a thread break. The two main stop motions are the warp stop motion weft stop motionThe principal parts of a loom are the frame, the warp-beam or weavers beam, the cloth-roll, the heddles, their mounting, the reed; the warp-beam is a wooden or metal cylinder on the back of the loom. The threads of the warp extend in parallel order from the warp-beam to the front of the loom where they are attached to the cloth-roll; each thread or group of threads of the warp passes through an opening in a heddle. The warp threads are separated by the heddles into two or more groups, each controlled and automatically drawn up and down by the motion of the heddles.
In the case of small patterns the movement of the heddles is controlled by "cams" which move up the heddles by means of a frame called a harness. Where a complex design is required, the healds are raised by harness cords attached to a Jacquard machine; every time the harness moves up or down, an opening is made between the threads of warp, through which the pick is inserted. Traditionally the weft thread is inserted by a shuttle. On a conventional loom, the weft thread is carried on a pirn, in a shuttle that passes through the shed. A handloom weaver could propel the shuttle by throwing it from side to side with the aid of a picking stick; the "picking" on a power loom is done by hitting the shuttle from each side using an overpick or underpick mechanism controlled by cams 80–250 times a minute. When a pirn is depleted, it is ejected from the shuttle and replaced with the next pirn held in a battery attached to the loom. Multiple shuttle boxes allow more than one shuttle to be used; each can carry a different colour.
The rapier-type weaving machines do not have shuttles, they propel the weft by means of small grippers or rapiers that pick up the filling thread and carry it halfway across the loom where another rapier picks it up and pulls it the rest of the way. Some carry the filling yarns across the loom at rates in excess of 2,000 metres per minute. Manufacturers such as Picanol have reduced the mechanical adjustments to a minimum, control all the functions through a computer with a graphical user interface. Other types use compressed air to insert the pick, they are all fast and quiet. The warp is sized in a starch mixture for smoother running; the loom warped by passing the sized warp threads through two or more heddles attached to harnesses. The power weavers. Most looms used for industrial purposes have a machine that ties new warps threads to the waste of used warps threads, while still on the loom an operator rolls the old and new threads back on the warp beam; the harnesses are controlled by dobbies or a Jacquard head.
The raising and lowering
Kissing number problem
In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch a common unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. Other names for kissing number that have been used are Newton number, contact number. In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in -dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space. Finding the kissing number when centers of spheres are confined to a line or a plane is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century. Solutions in higher dimensions are more challenging, only a handful of cases have been solved exactly. For others investigations have determined lower bounds, but not exact solutions.
In one dimension, the kissing number is 2: In two dimensions, the kissing number is 6: Proof: Consider a circle with center C, touched by circles with centers C1, C2.... Consider the rays C Ci; these rays all emanate from the same center C, so the sum of angles between adjacent rays is 360°. Assume by contradiction that there are more than six touching circles. At least two adjacent rays, say C C1 and C C2, are separated by an angle of less than 60°; the segments C Ci have the same length – 2r – for all i. Therefore, the triangle C C1 C2 is isosceles, its third side – C1 C2 – has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction. In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two, it is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of space left over, it is not obvious that there is no way to pack in a 13th sphere. This was the subject of a famous disagreement between mathematicians Isaac David Gregory.
Newton thought that the limit was 12. Some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one by Reinhold Hoppe, but the first correct proof did not appear until 1953; the twelve neighbors of the central sphere correspond to the maximum bulk coordination number of an atom in a crystal lattice in which all atoms have the same size. A coordination number of 12 is found in a hexagonal close-packed structure. In four dimensions, it was known for some time that the answer was either 24 or 25, it is easy to produce a packing of 24 spheres around a central sphere. As in the three-dimensional case, there is a lot of space left over—even more, in fact, than for n = 3—so the situation was less clear. In 2003, Oleg Musin proved the kissing number for n = 4 using a subtle trick; the kissing number in n dimensions is unknown for n > 4, except for n = 8, n = 24. The results in these dimensions stem from the existence of symmetrical lattices: the E8 lattice and the Leech lattice.
If arrangements are restricted to lattice arrangements, in which the centres of the spheres all lie on points in a lattice this restricted kissing number is known for n = 1 to 9 and n = 24 dimensions. For 5, 6, 7 dimensions the arrangement with the highest known kissing number found so far is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded; the following table lists some known bounds on the kissing number in various dimensions. The dimensions in which the kissing number is known are listed in boldface; the kissing number problem can be generalized to the problem of finding the maximum number of non-overlapping congruent copies of any convex body that touch a given copy of the body. There are different versions of the problem depending on whether the copies are only required to be congruent to the original body, translates of the original body, or translated by a lattice. For the regular tetrahedron, for example, it is known that both the lattice kissing number and the translative kissing number are equal to 18, whereas the congruent kissing number is at least 56.
There are several approximation algorithms on intersection graphs where the approximation ratio depends on the kissing number. For example, there is a polynomial-time 10-approximation algorithm to find a maximum non-intersecting subset of a set of rotated unit squares; the kissing number problem can be stated as the existence of a solution to a set of inequalities. Let x n be a set of N D-dimensional position vectors of the centres of the spheres; the condition that this set of spheres can lie round the centre sphere without overlapping is: ∃ x { ∀ n ∧ ∀ m, n: m ≠ n { (
Circle packing
This article describes the packing of circles on surfaces. For the related article on circle packing with a prescribed intersection graph, please see the circle packing theorem. In geometry, circle packing is the study of the arrangement of circles on a given surface such that no overlapping occurs and so that all circles touch one another; the associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which deals only with identical spheres. While the circle has a low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. The "worst" shape to pack onto a plane is not known, but the smoothed octagon has a packing density of about 0.902414, the lowest maximum packing density known of any centrally-symmetric convex shape. Packing densities of concave shapes such as star polygons can be arbitrarily small; the branch of mathematics known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.
In two dimensional Euclidean space, Joseph Louis Lagrange proved in 1773 that the highest-density lattice arrangement of circles is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice, each circle is surrounded by 6 other circles. The density of this arrangement is η h = π 3 6 ≈ 0.9069. Hexagonal packing of equal circles was found to fill a fraction Pi/Sqrt ≃ 0.91 of area—which was proved maximal for periodic packings by Carl Friedrich Gauss in 1831. Axel Thue provided the first proof that this was optimal in 1890, showing that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular. However, his proof was considered by some to be incomplete; the first rigorous proof is attributed to László Fejes Tóth in 1940. At the other extreme, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist. There are 11 circle packings based on the 11 uniform tilings of the plane.
In these packings, every circle can be mapped to every other circle by rotations. The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings; the truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing. The snub hexagonal tiling has two mirror-image forms. A related problem is to determine the lowest-energy arrangement of identically interacting points that are constrained to lie within a given surface; the Thomson problem deals with the lowest energy distribution of identical electric charges on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere; this is analogous to distributing non-point charges on a sphere. Packing circles in simple bounded shapes is a common type of problem in recreational mathematics; the influence of the container walls is important, hexagonal packing is not optimal for small numbers of circles.
There are a range of problems which permit the sizes of the circles to be non-uniform. One such extension is to find the maximum possible density of a system with two specific sizes of circle. Only nine particular radius ratios permit compact packing, when every pair of circles in contact is in mutual contact with two other circles. For seven of these radius ratios a compact packing is known that achieves the maximum possible packing fraction for mixtures of discs with that radius ratio. All nine have ratio-specific packings denser than the uniform triangular packing, as do some radius ratios without compact packingsIt is known that if the radius ratio is above 0.742, a binary mixture cannot pack better than uniformly-sized discs. Upper bounds for the density that can be obtained in such binary packings at smaller ratios have been obtained. Quadrature amplitude modulation is based on packing circles into circles within a phase-amplitude space. A modem transmits data as a series of points in a 2-dimensional phase-amplitude plane.
The spacing between the points determines the noise tolerance of the transmission, while the circumscribing circle diameter determines the transmitter power required. Performance is maximized when the constellation of code points are at the centres of an efficient circle packing. In practice, suboptimal rectangular packings are used to simplify decoding. Circle packing has become an essential tool in origami design, as each appendage on an origami figure requires a circle of paper. Robert J. Lang has used the mathematics of circle packing to develop computer programs that aid in the design of complex origami figures. Apollonian gasket Circle packing in a square Circle packing in a circle Inversive distance Kepler conjecture Malfatti circles Packing problem Wells D; the Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. Pp. 30–31, 167. ISBN 0-14-011813-6. Stephenson, Kenneth. "Circle Packing: A Mathematical Tale". Notices of the American Mathematical Society. 50
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Brickwork
Brickwork is masonry produced by a bricklayer, using bricks and mortar. Rows of bricks—called courses— are laid on top of one another to build up a structure such as a brick wall. Bricks may be differentiated from blocks by size. For example, in the UK a brick is defined as a unit having dimensions less than 337.5x225x112.5mm and a block is defined as a unit having one or more dimensions greater than the largest possible brick. Brick is a popular medium for constructing buildings, examples of brickwork are found through history as far back as the Bronze Age; the fired-brick faces of the ziggurat of ancient Dur-Kurigalzu in Iraq date from around 1400 BC, the brick buildings of ancient Mohenjo-daro in Pakistan were built around 2600 BC. Much older examples of brickwork made with dried bricks may be found in such ancient locations as Jericho in Judea, Çatal Hüyük in Anatolia, Mehrgarh in Pakistan; these structures have survived from the Stone Age to the present day. Brick dimensions are expressed in construction or technical documents in two ways as co-ordinating dimensions and working dimensions.
Coordination dimensions are the actual physical dimensions of the brick with the mortar required on one header face, one stretcher face and one bed. Working dimensions is the size of a manufactured brick, it is called the nominal size of a brick. Brick size may be different due to shrinkage or distortion due to firing etc. An example of a co-ordinating metric used for bricks in the UK is as follows: Bricks of dimensions 215 mm × 102.5 mm × 65 mm. In this case the co-ordinating metric works because the length of a single brick is equal to the total of the width of a brick plus a perpend plus the width of a second brick. There are many other brick sizes worldwide, many of them use this same co-ordinating principle; as the most common bricks are cuboids, six surfaces are named as followed: Top and bottom surfaces are called Beds Ends or narrow surfaces are called Headers or header faces Sides or wider surfaces are called Stretchers or stretcher faces Mortar placed between bricks is given separate names with respect to their position.
Mortar placed horizontally below or top of a brick is called a bed, mortar Placed vertically between bricks is called a perpend. A brick made with just rectilinear dimensions is called a solid brick. Bricks might have a depression on a single bed; the depression is called a frog, the bricks are known as frogged bricks. Frogs should never exceed 20 % of the total volume of the brick. Cellular bricks have depressions exceeding 20% of the volume of the brick. Perforated bricks have holes through the brick from bed to bed. Most of the building standards and good construction practices recommend the volume of holes should not exceeding 20% of the total volume of the brick. Parts of brickwork include bricks and perpends; the bed is the mortar upon. A perpend is a vertical joint between any two bricks and is usually—but not always—filled with mortar. A brick is given a classification based on how it is laid, how the exposed face is oriented relative to the face of the finished wall. Stretcher or stretching brick A brick laid flat with its long narrow side exposed.
Header or heading brick A brick laid flat with its width exposed. Soldier A brick laid vertically with its long narrow side exposed. Sailor A brick laid vertically with the broad face of the brick exposed. Rowlock A brick laid on the long narrow side with the short end of the brick exposed. Shiner or rowlock stretcher A brick laid on the long narrow side with the broad face of the brick exposed; the practice of laying uncut full-sized bricks wherever possible gives brickwork its maximum possible strength. In the diagrams below, such uncut full-sized bricks are coloured as follows: Stretcher HeaderOccasionally though a brick must be cut to fit a given space, or to be the right shape for fulfilling some particular purpose such as generating an offset—called a lap—at the beginning of a course. In some cases these special shapes or sizes are manufactured. In the diagrams below, some of the cuts most used for generating a lap are coloured as follows: Three-quarter bat, stretching A brick cut to three-quarters of its length, laid flat with its long, narrow side exposed.
Three-quarter bat, heading A brick cut to three-quarters of its length, laid flat with its short side exposed. Half bat A brick cut in half across its length, laid flat. Queen closer A brick cut in half down its width, laid with its smallest face exposed and standing vertically. A queen closer is used for the purpose of creating a lap. Less used cuts are all coloured as follows: Quarter bat A brick cut to a quarter of its length. Three-quarter queen closer A queen closer cut to three-quarters of its length. King closer A brick with one corner cut away. A nearly universal rule in brickwork is. Walls, extending upwards, can be of varying depth or thickness; the bricks are laid running linearly and extending upwards, forming wythes or leafs. It is as important; the dominant method for consolidating the leaves together was to lay bricks across them, rather than running linearly. Brickwork observing either or both of these two conventions is described as being laid in one or another bond. A leaf is as thick as the width of one brick, but a wall is said to be one brick thick if it as wide as the length of a brick.
Accordingly, a single-leaf wall
Octagon
In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t; the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon the midpoints of the segments connecting the centers of opposite squares form a quadrilateral, both equidiagonal and orthodiagonal; the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square. A regular octagon is a closed figure with sides of the same length and internal angles of the same size, it has eight lines of reflective 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 central angle is 45°. The area of a regular octagon of side length a is given by A = 2 cot π 8 a 2 = 2 a 2 ≃ 4.828 a 2. In terms of the circumradius R, the area is A = 4 sin π 4 R 2 = 2 2 R 2 ≃ 2.828 R 2. In terms of the apothem r, the area is A = 8 tan π 8 r 2 = 8 r 2 ≃ 3.314 r 2. These last two coefficients bracket the value of the area of the unit circle; the area can be expressed as A = S 2 − a 2, where S is the span of the octagon, or the second-shortest diagonal. This is proven if one takes an octagon, draws a square around the outside and takes the corner triangles and places them with right angles pointed inward, forming a square; 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 as above: A = 2 − a 2 = 2 a 2 ≈ 4.828 a 2. Expressed in terms of the span, the area is A = 2 S 2 ≈ 0.828 S 2.
Another simple formula for the area is A = 2 a S. More the span S is known, 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 regular octagon in terms of the side length a is R = a, the inradius is r = a. The regular octagon, in ter