1.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, 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, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to 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 often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, 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, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
2.
Box
–
Box describes a variety of containers and receptacles for permanent use as storage, or for temporary use, often for transporting contents. Boxes may be made of materials such as wood or metal, or of corrugated fiberboard, paperboard. The size may vary from small to the size of a large appliance. A corrugated box is a common shipping container. A decorative or storage box may be opened by raising, pulling, sliding or removing the lid, several types of boxes are used in packaginlolg and storage. A corrugated box is a container made of corrugated fiberboard. These are most commonly used to transport and warehouse products during distribution, a folding carton is fabricated from paperboard. The paperboard is printed, die-cut and scored to form a blank and these are transported and stored flat, and erected at the point of filling. These are used to package a wide range of goods, intended either for use or as a storage box for the remaining goods. A set up box is made of paperboard, permanently glued together with paper skins that can be printed or colored. Unlike folding cartons, these are assembled at the point of manufacture, set-up boxes are more expensive than folding boxes and are typically used for protecting high value items such as cosmetics, watches or smaller consumer electronics. A crate is a heavy duty shipping container made of wood. Crates are distinct from wooden boxes, also used as heavy duty shipping containers, for a wooden container to be a crate, all six of its sides must be put in place to result in the rated strength of the container. The strength of a box, on the other hand, is rated based on the weight it can carry before the top or opening is installed. A bulk box is a large box often used in industrial environments and it is sized to fit well on a pallet. Depending on locale and specific usage, the carton and box are sometimes used interchangeably. The invention of large steel intermodal shipping containers has helped advance the globalization of commerce, boxes for storing various items in can often be very decorative, as they are intended for permanent use and sometimes are put on display in certain locations. A jewelry or jewellery box, is a box for trinkets or jewels and it can take a very modest form with paper covering and lining, covered in leather and lined with satin, or be larger and more highly decorated
3.
Sugar
–
Sugar is the generic name for sweet, soluble carbohydrates, many of which are used in food. There are various types of derived from different sources. Simple sugars are called monosaccharides and include glucose, fructose, the table sugar or granulated sugar most customarily used as food is sucrose, a disaccharide of glucose and fructose. Sugar is used in prepared foods and it is added to some foods, in the body, sucrose is hydrolysed into the simple sugars fructose and glucose. Other disaccharides include maltose from malted grain, and lactose from milk, longer chains of sugars are called oligosaccharides or polysaccharides. Some other chemical substances, such as glycerol may also have a sweet taste, low-calorie food substitutes for sugar, described as artificial sweeteners, include aspartame and sucralose, a chlorinated derivative of sucrose. Sugars are found in the tissues of most plants and are present in sufficient concentrations for efficient commercial extraction in sugarcane, the world production of sugar in 2011 was about 168 million tonnes. The average person consumes about 24 kilograms of sugar each year, equivalent to over 260 food calories per person, since the latter part of the twentieth century, it has been questioned whether a diet high in sugars, especially refined sugars, is good for human health. Sugar has been linked to obesity, and suspected of, or fully implicated as a cause in the occurrence of diabetes, cardiovascular disease, dementia, macular degeneration, the etymology reflects the spread of the commodity. The English word sugar ultimately originates from the Sanskrit शर्करा, via Arabic سكر as granular or candied sugar, the contemporary Italian word is zucchero, whereas the Spanish and Portuguese words, azúcar and açúcar, respectively, have kept a trace of the Arabic definite article. The Old French word is zuchre and the contemporary French, sucre, the earliest Greek word attested is σάκχαρις. The English word jaggery, a brown sugar made from date palm sap or sugarcane juice, has a similar etymological origin – Portuguese jagara from the Sanskrit शर्करा. Sugar has been produced in the Indian subcontinent since ancient times and it was not plentiful or cheap in early times and honey was more often used for sweetening in most parts of the world. Originally, people chewed raw sugarcane to extract its sweetness, sugarcane was a native of tropical South Asia and Southeast Asia. Different species seem to have originated from different locations with Saccharum barberi originating in India and S. edule, one of the earliest historical references to sugarcane is in Chinese manuscripts dating back to 8th century BC that state that the use of sugarcane originated in India. Sugar was found in Europe by the 1st century AD, but only as an imported medicine and it is a kind of honey found in cane, white as gum, and it crunches between the teeth. It comes in lumps the size of a hazelnut, sugar is used only for medical purposes. Sugar remained relatively unimportant until the Indians discovered methods of turning sugarcane juice into granulated crystals that were easier to store, crystallized sugar was discovered by the time of the Imperial Guptas, around the 5th century AD
4.
Regular dodecahedron
–
In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
5.
Regular icosahedron
–
In geometry, a regular icosahedron is a convex polyhedron with 20 faces,30 edges and 12 vertices. It is one of the five Platonic solids, and also the one with the most sides and it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35 and it is the dual of the dodecahedron, which is represented by, having three pentagonal faces around each vertex. A regular icosahedron is a pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedrons or icosahedra. The surface area A and the volume V of a regular icosahedron of edge length a are, note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ −1, the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, if two vertices are taken to be at the north and south poles, then the other ten vertices are at latitude ±arctan ≈ ±26. 57°. These ten vertices are at evenly spaced longitudes, alternating between north and south latitudes and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, an icosahedron has 43,380 distinct nets. To color the icosahedron, such that no two adjacent faces have the color, requires at least 3 colors. A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere, the problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers, as it turns out, the icosahedron occupies less of the spheres volume than the dodecahedron. The following construction of the icosahedron avoids tedious computations in the number field ℚ necessary in more elementary approaches, the existence of the icosahedron amounts to the existence of six equiangular lines in ℝ3. Indeed, intersecting such a system of lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of an icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. In order to such an equiangular system, we start with this 6 ×6 square matrix
6.
Platonic solid
–
In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements