Brick
A brick is building material used to make walls and other elements in masonry construction. Traditionally, the term brick referred to a unit composed of clay, but it is now used to denote any rectangular units laid in mortar. A brick can be composed of clay-bearing soil and lime, or concrete materials. Bricks are produced in numerous classes, types and sizes which vary with region and time period, are produced in bulk quantities. Two basic categories of bricks are non-fired bricks. Block is a similar term referring to a rectangular building unit composed of similar materials, but is larger than a brick. Lightweight bricks are made from expanded clay aggregate. Fired bricks are one of the longest-lasting and strongest building materials, sometimes referred to as artificial stone, have been used since circa 4000 BC. Air-dried bricks known as mudbricks, have a history older than fired bricks, have an additional ingredient of a mechanical binder such as straw. Bricks are laid in courses and numerous patterns known as bonds, collectively known as brickwork, may be laid in various kinds of mortar to hold the bricks together to make a durable structure.
The earliest bricks were dried brick, meaning that they were formed from clay-bearing earth or mud and dried until they were strong enough for use. The oldest discovered bricks made from shaped mud and dating before 7500 BC, were found at Tell Aswad, in the upper Tigris region and in southeast Anatolia close to Diyarbakir; the South Asian inhabitants of Mehrgarh constructed, lived in, airdried mudbrick houses between 7000–3300 BC. Other more recent findings, dated between 7,000 and 6,395 BC, come from Jericho, Catal Hüyük, the ancient Egyptian fortress of Buhen, the ancient Indus Valley cities of Mohenjo-daro and Mehrgarh. Ceramic, or fired brick was used as early as 3000 BC in early Indus Valley cities like Kalibangan; the earliest fired bricks appeared in Neolithic China around 4400 BC at Chengtoushan, a walled settlement of the Daxi culture. These bricks were made of red clay, fired on all sides to above 600 °C, used as flooring for houses. By the Qujialing period, fired bricks were being used to pave roads and as building foundations at Chengtoushan.
Bricks continued to be used during 2nd millennium BC at a site near Xi'an. Fired bricks were found in Western Zhou ruins; the carpenter's manual Yingzao Fashi, published in 1103 at the time of the Song dynasty described the brick making process and glazing techniques in use. Using the 17th-century encyclopaedic text Tiangong Kaiwu, historian Timothy Brook outlined the brick production process of Ming Dynasty China: "...the kilnmaster had to make sure that the temperature inside the kiln stayed at a level that caused the clay to shimmer with the colour of molten gold or silver. He had to know when to quench the kiln with water so as to produce the surface glaze. To anonymous labourers fell the less skilled stages of brick production: mixing clay and water, driving oxen over the mixture to trample it into a thick paste, scooping the paste into standardised wooden frames, smoothing the surfaces with a wire-strung bow, removing them from the frames, printing the fronts and backs with stamps that indicated where the bricks came from and who made them, loading the kilns with fuel, stacking the bricks in the kiln, removing them to cool while the kilns were still hot, bundling them into pallets for transportation.
It was hot, filthy work." Early civilisations around the Mediterranean adopted the use of fired bricks, including the Ancient Greeks and Romans. The Roman legions operated mobile kilns, built large brick structures throughout the Roman Empire, stamping the bricks with the seal of the legion. During the Early Middle Ages the use of bricks in construction became popular in Northern Europe, after being introduced there from Northern-Western Italy. An independent style of brick architecture, known as brick Gothic flourished in places that lacked indigenous sources of rocks. Examples of this architectural style can be found in modern-day Denmark, Germany and Russia; this style evolved into Brick Renaissance as the stylistic changes associated with the Italian Renaissance spread to northern Europe, leading to the adoption of Renaissance elements into brick building. A clear distinction between the two styles only developed at the transition to Baroque architecture. In Lübeck, for example, Brick Renaissance is recognisable in buildings equipped with terracotta reliefs by the artist Statius von Düren, active at Schwerin and Wismar.
Long-distance bulk transport of bricks and other construction equipment remained prohibitively expensive until the development of modern transportation infrastructure, with the construction of canal and railways. Production of bricks increased massively with the onset of the Industrial Revolution and the rise in factory building in England. For reasons of speed and economy, bricks were preferred as building material to stone in areas where the stone was available, it was at this time in London that bright red brick was chosen for construction to make the buildings more visible in the heavy fog and to help prevent traffic accidents. The transition from the traditional method of production known as hand-moulding to a mechanised form of mass-production took place during the first half of the nineteenth century; the first successful brick-making machine was patented by Henry Clayton, employed at the
Close-packing of equal spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is π 3 2 ≃ 0.74048. The same packing density can be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction; the Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only in case of 2, 3, 8 and 24 dimensions. Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them; the cubic and hexagonal arrangements are close to one another in energy, it may be difficult to predict which form will be preferred from first principles.
There are two simple regular lattices. They are called face-centered hexagonal close-packed, based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; the fcc lattice is known to mathematicians as that generated by the A3 root system. The problem of close-packing of spheres was first mathematically analyzed by Thomas Harriot around 1587, after a question on piling cannonballs on ships was posed to him by Sir Walter Raleigh on their expedition to America. Cannonballs were piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base; the cannonball problem asks which flat square arrangements of cannonballs can be stacked into a square pyramid. Édouard Lucas formulated the problem as the Diophantine equation ∑ n = 1 N n 2 = M 2 or 1 6 N = M 2 and conjectured that the only solutions are N = 1, M = 1, N = 24, M = 70.
Here N is the number of layers in the pyramidal stacking arrangement and M is the number of cannonballs along an edge in the flat square arrangement. In both the fcc and hcp arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres and two smaller gaps surrounded by four spheres; the distances to the centers of these gaps from the centers of the surrounding spheres is √3⁄2 for the tetrahedral, √2 for the octahedral, when the sphere radius is 1. Relative to a reference layer with positioning A, two more positionings B and C are possible; every sequence of A, B, C without immediate repetition of the same one is possible and gives an dense packing for spheres of a given radius. The most regular ones are fcc = ABC ABC ABC... hcp = AB AB AB AB.... There is an uncountably infinite number of disordered arrangements of planes that are sometimes collectively referred to as "Barlow packings", after crystallographer William BarlowIn close-packing, the center-to-center spacing of spheres in the xy plane is a simple honeycomb-like tessellation with a pitch of one sphere diameter.
The distance between sphere centers, projected on the z axis, is: pitch Z = 6 ⋅ d 3 ≈ 0.816 496 58 d, where d is the diameter of a sphere. The coordination number of hcp and fcc is 12 and their atomic packing factors are equal to the number mentioned above, 0.74. When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact; the distance between the centers along the shortest path namely that straight line will therefore be r1 + r2 where r1 is the radius of the first sphere and r2 is the radius of the second. In close packing all of the spheres share a common radius, r; therefore two centers would have a distance 2r. To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to hcp; the box would be placed on the x-y-z coordinate space.
First form a row of spheres. The centers will all lie on a straight line, their x-coordinate will vary by 2r since the distance between each center of the spheres are touching is 2r. The y-coordinate and z-coordinate wil
Hexagon
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple hexagon is 720°. A regular hexagon has Schläfli symbol and can be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equilateral and equiangular, it is bicentric, meaning that it is both tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 2 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, making up the dihedral group D6; the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations.
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 regular triangular lattice is the honeycomb tessellation of hexagons, it is not 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 inscribed circle, d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: 1 2 d = r = cos R = 3 2 R = 3 2 t and d = 3 2 D; the area of a regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 2.598 R 2 ≈ 3.464 r 2 ≈ 0.6495 D 2 ≈ 0.866 d 2. For any regular polygon, the area can be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, p = 6 R = 4 r 3, so A = a p 2 = r ⋅ 4 r 3 2 = 2 r 2 3 ≈ 3.464 r 2. The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C PE + PF = PA + PB + PC + PD. The regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, Dih1, 4 cyclic subgroups: Z6, Z3, Z2, Z1; these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a group order. R12 is full symmetry, a1 is no symmetry. P6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles; these two forms have half the symmetry order of the regular hexagon. The
Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, 8 cubes around each vertex, its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol. John Horton Conway calls this honeycomb a cubille. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps, it is an example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycombs are constructed in ordinary Euclidean space, like the convex uniform honeycombs, they may be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space; the Cartesian coordinates of the vertices are: for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1. It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form, starting with the square tiling, in the plane.
It is one of 28 uniform honeycombs using convex uniform polyhedral cells. Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems: There is a large number of uniform colorings, derived from different symmetries; these include: The cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest symmetry form projects into a triangular tiling. A square symmetry projection forms a square tiling, it is related to the regular 4-polytope tesseract, Schläfli symbol, which exists in 4-space, only has 3 cubes around each edge. It's related to the order-5 cubic honeycomb, Schläfli symbol, of hyperbolic space with 5 cubes around each edge, it is in a sequence of honeycomb with octahedral vertex figures. It in a sequence of regular polytopes and honeycombs with cubic cells. The, Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb; the expanded cubic honeycomb is geometrically identical to the cubic honeycomb.
The, Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the A ~ 3 Coxeter group; the symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams: The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1. John Horton Conway calls this honeycomb a cuboctahedrille, its dual an oblate octahedrille; the rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, Wythoff construction name, the Coxeter diagram below; this honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra, creating two triangular cupolae.
This scaliform honeycomb is represented by Coxeter diagram, symbol s3, with coxeter notation symmetry.. The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, it is composed of truncated cubes and octahedra in a ratio of 1:1. John Horton Conway calls this honeycomb a truncated cubille, its dual pyramidille; the truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. There is a second uniform coloring by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells; the alternated bitruncated cubic honeycomb or bisnub cubic honeycomb can be creating regular icosahedron from the truncated octahedra with irregular tetrahedral cells created in the gaps. There are three constructions from three related Coxeter diagrams:, and; these have symmetry, + respectively. The first and last symmetry can be doubled as and +; this honeycomb is represented in the boron atoms of the α-rhombihedral crystal.
The centers of the icosahedra are located at the fcc positions of the lattice. The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, it is composed of rhombicuboctahedra and cubes in a ratio of 1:1:3. John Horton Conway calls this honeycomb a 2-RCO-trille, its dual quarter oblate octahedrille; the cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells; the dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram, containing faces from two of four hyperplanes of the cubic fundamental domain. It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, 2 vertices; the cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedra, truncated octah
Octahedron
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, six vertices. The term is most used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube, it is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations, it is a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric. If the edge length of a regular octahedron is a, the radius of a circumscribed sphere is r u = a 2 2 ≈ 0.707 ⋅ a and the radius of an inscribed sphere is r i = a 6 6 ≈ 0.408 ⋅ a while the midradius, which touches the middle of each edge, is r m = a 2 = 0.5 ⋅ a The octahedron has four special orthogonal projections, centered, on an edge, vertex and normal to a face. The second and third correspond to A2 Coxeter planes.
The octahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates and radius r is the set of all points such that | x − a | + | y − b | + | z − c | = r; the surface area A and the volume V of a regular octahedron of edge length a are: A = 2 3 a 2 ≈ 3.464 a 2 V = 1 3 2 a 3 ≈ 0.471 a 3 Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice. If an octahedron has been stretched so that it obeys the equation | x x m | + | y y m | + | z z m | = 1, the formulas for the surface area and volume expand to become A = 4 x m y m z m × 1 x m 2 + 1 y m 2 + 1 z m 2, V = 4 3 x m y m z m.
Additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 2 2. {\displaystyle x_=y_=z_=
Rhombic dodecahedral honeycomb
The rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space, it consists of copies of the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:√2. Three cells meet at each edge; the honeycomb is thus face-transitive and edge-transitive. The vertices with the obtuse rhombic face angles have 4 cells; the vertices with the acute rhombic face angles have 6 cells. The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing. Cells can be given 4 colors in square layers of 2-colors where neighboring faces have different colors, 6 colors in hexagonal layers of 3 colors where same-colored cells have no contact at all; the rhombic dodecahedral honeycomb can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons.
Each rhombic dodecahedra can be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb. The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space, it consists of copies of the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi, it is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb. The rhombic pyramidal honeycomb or half oblate octahedrille is a uniform space-filling tessellation in Euclidean 3-space; this honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 12 rhombic pyramids. It is dual to the cantic cubic honeycomb: catoptric tessellation Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 168. ISBN 0-486-23729-X. Weisstein, Eric W. "Space-filling polyhedron". MathWorld. Examples of Housing Construction using this geometry
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp