1.
Fish anatomy
–
Fish anatomy is the study of the form or morphology of fishes. It can be contrasted with fish physiology, which is the study of how the component parts of fish function together in the living fish, the anatomy of fish is often shaped by the physical characteristics of water, the medium in which fish live. Water is much denser than air, holds a small amount of dissolved oxygen. The body of a fish is divided into a head, trunk and tail, the skeleton, which forms the support structure inside the fish, is either made of cartilage, in cartilaginous fish, or bone in bony fish. The main skeletal element is the column, composed of articulating vertebrae which are lightweight yet strong. The ribs attach to the spine and there are no limbs or limb girdles. The main external features of the fish, the fins, are composed of bony or soft spines called rays which. They are supported by the muscles which compose the main part of the trunk, the heart has two chambers and pumps the blood through the respiratory surfaces of the gills and on round the body in a single circulatory loop. The eyes are adapted for seeing underwater and have only local vision, there is an inner ear but no external or middle ear. Sharks and rays are basal fish with numerous primitive anatomical features similar to those of ancient fish and their bodies tend to be dorso-ventrally flattened, they usually have five pairs of gill slits and a large mouth set on the underside of the head. The dermis is covered with separate dermal placoid scales and they have a cloaca into which the urinary and genital passages open, but not a swim bladder. Cartilaginous fish produce a number of large, yolky eggs. Some species are ovoviviparous and the young develop internally but others are oviparous, the bony fish lineage shows more derived anatomical traits, often with major evolutionary changes from the features of ancient fish. They have a skeleton, are generally laterally flattened, have five pairs of gills protected by an operculum. The dermis is covered with overlapping scales, bony fish have a swim bladder which helps them maintain a constant depth in the water column, but not a cloaca. They mostly spawn a number of small eggs with little yolk which they broadcast into the water column. Fish have a variety of different body plans, at the broadest level their body is divided into head, trunk, and tail, although the divisions are not always externally visible. The body is fusiform, a streamlined body plan often found in fast-moving fish

Fish anatomy
–
Internal anatomy of a bony fish

Fish anatomy
–

Anatomical directions and axes

Fish anatomy
–
Skeleton of a

bony fish
Fish anatomy
–
Skeletal structure of a

bass showing the vertebral column running from the head to the tail

2.
Herring
–
Herring are forage fish, mostly belonging to the family Clupeidae. Herring often move in schools around fishing banks and near the coast. Three species of Clupea are recognised, and provide about 90% of all captured in fisheries. Most abundant of all is the Atlantic herring, providing half of all herring capture. Fishes called herring are found in India, in the Arabian Sea, Indian Ocean. Herring played a role in the history of marine fisheries in Europe. These oily fish also have a history as an important food fish. A number of different species, most belonging to the family Clupeidae, are referred to as herrings. The origins of the herring is somewhat unclear, though it may derive from the Old High German heri meaning a host, multitude. The type genus of the herring family Clupeidae is Clupea, Clupea contains three species, the Atlantic herring found in the north Atlantic, the Pacific herring found in the north Pacific, and the Araucanian herring found off the coast of Chile. Subspecific divisions have been suggested for both the Atlantic and Pacific herrings, but their biological basis remain unclear, in addition, a number of related species, all in the family Clupeidae, are commonly referred to as herrings. The table immediately below includes those members of the Clupeidae family referred to by FishBase as herrings which have been assessed by the International Union for Conservation of Nature. There are also a number of other species called herrings, which may be related to clupeids or just share some characteristics of herrings. Just which of these species are called herrings can vary with locality, some examples, The species of Clupea belong to the larger family Clupeidae, which comprises some 200 species that share similar features. These silvery-coloured fish have a dorsal fin, which is soft. They have no lateral line and have a lower jaw. At least one stock of Atlantic herring spawns in every month of the year, each spawns at a different time and place. Greenland populations spawn in 0–5 metres of water while North Sea herrings spawn at up to 200 metres in autumn

Herring
–
The

Atlantic herring, Clupea harengus

Herring
–
Global commercial capture of herrings in million tonnes reported by the

FAO 1950–2010

Herring

Herring
–

Clupea pallasii, the Pacific herring

3.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa

Rectangle
–
Running bond

Rectangle
–
Rectangle

Rectangle
–
Basket weave

4.
Parallelogram
–
In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, by comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped, rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles Rectangle – A parallelogram with four angles of equal size. Rhombus – A parallelogram with four sides of equal length, square – A parallelogram with four sides of equal length and angles of equal size. A simple quadrilateral is a if and only if any one of the following statements is true. Two pairs of opposite angles are equal in measure, one pair of opposite sides are parallel and equal in length. Each diagonal divides the quadrilateral into two congruent triangles, the sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2, the sum of the distances from any interior point to the sides is independent of the location of the point. Thus all parallelograms have all the properties listed above, and conversely, if just one of statements is true in a simple quadrilateral. Opposite sides of a parallelogram are parallel and so will never intersect, the area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area, any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2, if it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong. If it has four lines of symmetry, it is a square. The perimeter of a parallelogram is 2 where a and b are the lengths of adjacent sides, unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area, the diagonals of a parallelogram divide it into four triangles of equal area. All of the formulas for general convex quadrilaterals apply to parallelograms

Parallelogram
–
This parallelogram is a

rhomboid as it has no right angles and unequal sides.

5.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left

Symmetry
–
Symmetric arcades of a portico in the

Great Mosque of Kairouan also called the Mosque of Uqba, in

Tunisia.

Symmetry

Symmetry
–
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.

Symmetry
–
The ceiling of

Lotfollah mosque,

Isfahan,

Iran has 8-fold symmetries.

6.
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

Wallpaper group
–
Example of an

Egyptian design with wallpaper group p4mm

Wallpaper group
–
Example C: Painted

porcelain,

China
Wallpaper group
–

Mediæval wall

diapering
7.
Wallpaper
–
It is usually sold in rolls and is put onto a wall using wallpaper paste. Wallpapers can come plain as lining paper, textured, with a repeating pattern design, or, much less commonly today. Wallpaper printing techniques include surface printing, gravure printing, silk screen-printing, rotary printing, Wallpaper is made in long rolls, which are hung vertically on a wall. The number of times the pattern repeats horizontally across a roll does not matter for this purpose, a single pattern can be issued in several different colorways. The worlds most expensive wallpaper, Les Guerres DIndependence, was priced at £24,896.50 for a set of 32 panels, the wallpaper was designed by Zuber in France and is very popular in the United States. The main historical techniques are, hand-painting, woodblock printing, stencilling, the first three all date back to before 1700. Wallpaper, using the technique of woodcut, gained popularity in Renaissance Europe amongst the emerging gentry. The social elite continued to hang large tapestries on the walls of their homes and these tapestries added color to the room as well as providing an insulating layer between the stone walls and the room, thus retaining heat in the room. However, tapestries were extremely expensive and so only the rich could afford them. Less well-off members of the elite, unable to buy tapestries due either to prices or wars preventing international trade, turned to wallpaper to brighten up their rooms. Prints were very often pasted to walls, instead of being framed and hung, and the largest sizes of prints, some important artists made such pieces - notably Albrecht Dürer, who worked on both large picture prints and also ornament prints - intended for wall-hanging. The largest picture print was The Triumphal Arch commissioned by the Holy Roman Emperor Maximilian I, very few samples of the earliest repeating pattern wallpapers survive, but there are a large number of old master prints, often in engraving of repeating or repeatable decorative patterns. These are called ornament prints and were intended as models for wallpaper makers, England and France were leaders in European wallpaper manufacturing. Among the earliest known samples is one found on a wall from England and is printed on the back of a London proclamation of 1509, without any tapestry manufacturers in England, English gentry and aristocracy alike turned to wallpaper. During the Protectorate under Oliver Cromwell, the manufacture of wallpaper, in 1712, during the reign of Queen Anne, a wallpaper tax was introduced which was not abolished until 1836. By the mid-eighteenth century, Britain was the wallpaper manufacturer in Europe. However this trade was disrupted in 1755 by the Seven Years War and later the Napoleonic Wars. In 1748 the British Ambassador to Paris decorated his salon with blue flock wallpaper, in the 1760s the French manufacturer Jean-Baptiste Réveillon hired designers working in silk and tapestry to produce some of the most subtle and luxurious wallpaper ever made

Wallpaper
–
Hanging a traditional repeat pattern wallpaper

Wallpaper
–
The traditional hand-blocking technique, here in France in 1877

Wallpaper
–
Hand-painted Chinese wallpaper showing a funeral procession, made for the European market, c. 1780

Wallpaper
–
'Sauvages de la Mer Pacifique', panels 1-10 of woodblock printed wallpaper designed by Jean-Gabriel Charvet and manufactured by

Joseph Dufour
8.
Mosaic
–
A mosaic is a piece of art or image made from the assemblage of small pieces of colored glass, stone, or other materials. It is often used in art or as interior decoration. Most mosaics are made of small, flat, roughly square, pieces of stone or glass of different colors, some, especially floor mosaics, are made of small rounded pieces of stone, and called pebble mosaics. Others are made of other materials, mosaics have a long history, starting in Mesopotamia in the 3rd millennium BC. Pebble mosaics were made in Tiryns in Mycenean Greece, mosaics with patterns and pictures became widespread in classical times, Early Christian basilicas from the 4th century onwards were decorated with wall and ceiling mosaics. Mosaic fell out of fashion in the Renaissance, though artists like Raphael continued to practise the old technique, Roman and Byzantine influence led Jews to decorate 5th and 6th century synagogues in the Middle East with floor mosaics. Mosaic was widely used on buildings and palaces in early Islamic art, including Islams first great religious building, the Dome of the Rock in Jerusalem. Mosaic went out of fashion in the Islamic world after the 8th century, modern mosaics are made by professional artists, street artists, and as a popular craft. Many materials other than stone and ceramic tesserae may be employed, including shells, glass. The earliest known examples of made of different materials were found at a temple building in Abra, Mesopotamia. They consist of pieces of colored stones, shells and ivory, excavations at Susa and Chogha Zanbil show evidence of the first glazed tiles, dating from around 1500 BC. However, mosaic patterns were not used until the times of Sassanid Empire, mythological subjects, or scenes of hunting or other pursuits of the wealthy, were popular as the centrepieces of a larger geometric design, with strongly emphasized borders. Pliny the Elder mentions the artist Sosus of Pergamon by name, describing his mosaics of the left on a floor after a feast. Both of these themes were widely copied, most recorded names of Roman mosaic workers are Greek, suggesting they dominated high quality work across the empire, no doubt most ordinary craftsmen were slaves. Splendid mosaic floors are found in Roman villas across North Africa, in such as Carthage. The tiny tesserae allowed very fine detail, and an approach to the illusionism of painting, often small panels called emblemata were inserted into walls or as the highlights of larger floor-mosaics in coarser work. The normal technique was opus tessellatum, using larger tesserae, which was laid on site, there was a distinct native Italian style using black on a white background, which was no doubt cheaper than fully coloured work. In Rome, Nero and his architects used mosaics to cover surfaces of walls and ceilings in the Domus Aurea, built 64 AD

Mosaic
–
Irano-Roman floor mosaic detail from the palace of

Shapur I at

Bishapur.

Mosaic
–
Cone mosaic courtyard from

Uruk in

Mesopotamia 3000 BC

Mosaic
–
Ancient Greek mosaic of a deer hunt, in the House of the Abduction of Helen at

Pella, late 4th century BC

Mosaic
–
Roman mosaic of

Ulysses, from Carthage. Now in the

Bardo Museum,

Tunisia
9.
Herringbone (cloth)
–
Herringbone, also called Broken Twill Weave describes a distinctive V-shaped weaving pattern usually found in twill fabric. It is distinguished from a plain chevron by the break at reversal, the pattern is called herringbone because it resembles the skeleton of a herring fish. Herringbone-patterned fabric is usually wool, and is one of the most popular cloths used for suits, tweed cloth is often woven with a herringbone pattern

Herringbone (cloth)

Herringbone (cloth)
–

Woven
10.
Security printing
–
The main goal of security printing is to prevent forgery, tampering, or counterfeiting. A number of methods are used in the security printing industry. In November 2011 Canada joined the list of countries using polymer currency as it began the introduction of a new banknote series. A watermark is an image or pattern in paper that appears lighter or darker than surrounding paper when viewed with a light from behind the paper. A watermark is made by impressing a water coated metal stamp or dandy roll onto the paper during manufacturing, watermarks were first introduced in Bologna, Italy in 1282, as well as their use in security printing, they have also been used by papermakers to identify their product. Watermarks can also be made on polymer currency, for example, printed with white ink, simulated watermarks have a different reflectance than the base paper and can be seen at an angle. Because the ink is white, it cannot be photocopied or scanned, intaglio is a printing technique in which the image is incised into a surface. Normally, copper or zinc plates are used, and the incisions are created by etching or engraving the image, but one may also use mezzotint. In printing, the surface is covered in ink, and then rubbed vigorously with tarlatan cloth or newspaper to remove the ink from the surface, leaving it in the incisions. A damp piece of paper is placed on top, and the plate and paper are run through a press that, through pressure. The very sharp printing obtained from the process is hard to imitate by other means. Intaglio also allows for the creation of latent images which are visible when the document is viewed at a very shallow angle. A guilloché is a pattern formed of two or more curved bands that interlace to repeat a circular design. They are made with a geometric lathe and this involves the use of extremely small text, and is most often used on currency and bank checks. The text is generally enough to be indiscernible to the naked eye. Cheques, for example, use microprint as the signature line, color changing inks are made from mica. Color changing inks colored magnetizable inks are prepared by including chromatic pigments of high color strength, the magnetic pigments’ strong inherent color generally reduces the spectrum of achievable shades. Generally, pigments should be used at high concentrations to ensure that sufficient magnetizable material is applied even in thin offset coats, some magnetic pigment are best suited for colored magnetizable inks due to their lower blackness

Security printing
–
A guilloché

11.
Herringbone gear
–
A herringbone gear, a specific type of double helical gear, is a special type of gear that is a side to side combination of two helical gears of opposite hands. From the top, each helical groove of this looks like the letter V. Unlike helical gears, herringbone gears do not produce an additional axial load, like helical gears, they have the advantage of transferring power smoothly because more than two teeth will be in mesh at any moment in time. Their advantage over the helical gears is that the side-thrust of one half is balanced by that of the other half and this means that herringbone gears can be used in torque gearboxes without requiring a substantial thrust bearing. Because of this herringbone gears were an important step in the introduction of the turbine to marine propulsion. Precision herringbone gears are more difficult to manufacture than equivalent spur or helical gears and they are used in heavy machinery. The latter alignment is the defining characteristic of a Wuest type herringbone gear. A disadvantage of the gear is that it cannot be cut by simple gear hobbing machines. With the older method of fabrication, herringbone gears had a channel separating the two oppositely-angled courses of teeth. This was necessary to permit the shaving tool to run out of the groove, the development of the Sykes gear shaper made it possible to have continuous teeth with no central gap. Sunderland, also in England, also produced a herringbone cutting machine, the Sykes uses cylindrical guides and round cutters, the Sunderland uses straight guides and rack-type cutters. The W. E. Sykes Co. dissolved in 1983–84, since then it has been common practice to obtain an older machine and rebuild it if necessary to create this unique type of gear. Recently, the Bourn and Koch company has developed a CNC-controlled derivation of the W. E. Sykes design called the HDS1600-300. This machine, like the Sykes gear shaper, has the ability to generate a true apex without the need for a clearance groove cut around the gear and this allows the gears to be used in positive displacement pumping applications, as well as power transmission. Helical gears with low weight, accuracy and strength may be 3D printed, the logo of the car maker Citroën is a graphic representation of a herringbone gear, reflecting André Citroëns earlier involvement in the manufacture of these gears. Early Mors and Citroën cars used a bevel gear final drive in the rear axle. Panhard Dyna X and successor cars used double helical gears in the transaxle and for the camshaft timing gears in the engine

Herringbone gear
–
Contents

Herringbone gear
–
This double-helical bevel gear was made by Citroen and installed around 1927 in small Miřejovice hydropower plant on

Vltava in the

Czech Republic, connecting a Francis turbine to the generator. It worked flawlessly until 2011.

Herringbone gear
–

Citroën Type A final drive herringbone pinion and crownwheel

12.
Jewellery
–
Jewellery or jewelry consists of small decorative items worn for personal adornment, such as brooches, rings, necklaces, earrings, and bracelets. Jewellery may be attached to the body or the clothes, for many centuries metal, often combined with gemstones, has been the normal material for jewellery, but other materials such as shells and other plant materials may be used. It is one of the oldest type of archaeological artefact – with 100, historically, the most widespread influence on jewellery in terms of design and style have come from Asia. Jewellery may be made from a range of materials. Gemstones and similar such as amber and coral, precious metals, beads, and shells have been widely used. In most cultures jewellery can be understood as a symbol, for its material properties, its patterns. Jewellery has been made to nearly every body part, from hairpins to toe rings. The word jewellery itself is derived from the jewel, which was anglicised from the Old French jouel. In British English, Indian English, New Zealand English, Hiberno-English, Australian English, both are used in Canadian English, though jewelry prevails by a two to one margin. Numerous cultures store wedding dowries in the form of jewellery or make jewellery as a means to store or display coins, alternatively, jewellery has been used as a currency or trade good, an example being the use of slave beads. Many items of jewellery, such as brooches and buckles, originated as functional items. Jewellery can also symbolise group membership or status, wearing of amulets and devotional medals to provide protection or ward off evil is common in some cultures. These may take the form of symbols, stones, plants, animals, body parts, in creating jewellery, gemstones, coins, or other precious items are often used, and they are typically set into precious metals. Alloys of nearly every metal known have been encountered in jewellery, bronze, for example, was common in Roman times. Modern fine jewellery usually includes gold, white gold, platinum, palladium, titanium, most contemporary gold jewellery is made of an alloy of gold, the purity of which is stated in karats, indicated by a number followed by the letter K. American gold jewellery must be of at least 10K purity, many whimsical fashions were introduced in the extravagant eighteenth century. Cameos that were used in connection with jewellery were the attractive trinkets along with many of the objects such as brooches, ear-rings. Some of the necklets were made of pieces joined with the gold chains were in and bracelets were also made sometimes to match the necklet

Jewellery
–

Amber pendants

Jewellery
–
Diamond temptation design

Jewellery
–
The Queen

Farida of Egypt red coral parure by Ascione manufacture, 1938, Naples,

Coral Jewellery Museum
Jewellery
–
The

Daria-i-Noor (meaning: Sea of Light) Diamond from the collection of the national jewels of Iran at

Central Bank of Islamic Republic of Iran
13.
Sculpture
–
Sculpture is the branch of the visual arts that operates in three dimensions. It is one of the plastic arts, a wide variety of materials may be worked by removal such as carving, assembled by welding or modelling, or molded, or cast. However, most ancient sculpture was painted, and this has been lost. Those cultures whose sculptures have survived in quantities include the cultures of the ancient Mediterranean, India and China, the Western tradition of sculpture began in ancient Greece, and Greece is widely seen as producing great masterpieces in the classical period. During the Middle Ages, Gothic sculpture represented the agonies and passions of the Christian faith, the revival of classical models in the Renaissance produced famous sculptures such as Michelangelos David. Relief is often classified by the degree of projection from the wall into low or bas-relief, high relief, sunk-relief is a technique restricted to ancient Egypt. Relief sculpture may also decorate steles, upright slabs, usually of stone, techniques such as casting, stamping and moulding use an intermediate matrix containing the design to produce the work, many of these allow the production of several copies. The term sculpture is used mainly to describe large works. The very large or colossal statue has had an enduring appeal since antiquity, another grand form of portrait sculpture is the equestrian statue of a rider on horse, which has become rare in recent decades. The smallest forms of life-size portrait sculpture are the head, showing just that, or the bust, small forms of sculpture include the figurine, normally a statue that is no more than 18 inches tall, and for reliefs the plaquette, medal or coin. Sculpture is an important form of public art, a collection of sculpture in a garden setting can be called a sculpture garden. One of the most common purposes of sculpture is in form of association with religion. Cult images are common in cultures, though they are often not the colossal statues of deities which characterized ancient Greek art. The actual cult images in the innermost sanctuaries of Egyptian temples, of which none have survived, were rather small. The same is true in Hinduism, where the very simple. Some undoubtedly advanced cultures, such as the Indus Valley civilization, appear to have had no monumental sculpture at all, though producing very sophisticated figurines, the Mississippian culture seems to have been progressing towards its use, with small stone figures, when it collapsed. Other cultures, such as ancient Egypt and the Easter Island culture, from the 20th century the relatively restricted range of subjects found in large sculpture expanded greatly, with abstract subjects and the use or representation of any type of subject now common. Today much sculpture is made for intermittent display in galleries and museums, small sculpted fittings for furniture and other objects go well back into antiquity, as in the Nimrud ivories, Begram ivories and finds from the tomb of Tutankhamun

Sculpture
–

The Dying Gaul, or The Capitoline Gaul a Roman marble copy of a

Hellenistic work of the late 3rd century BCE

Capitoline Museums, Rome

Sculpture

Sculpture
–

Michelangelo 's

Moses, (c. 1513–1515), housed in the church of

San Pietro in Vincoli in

Rome. The sculpture was commissioned in 1505 by

Pope Julius II for his

tomb.

Sculpture
–
Assyrian

lamassu gate guardian from

Khorsabad, c. 721–800 BCE

14.
Salzburg, Austria
–
Salzburg is the fourth-largest city in Austria and the capital of the federal state of Salzburg. Salzburgs Old Town is internationally renowned for its architecture and is one of the best-preserved city centers north of the Alps. It was listed as a UNESCO World Heritage Site in 1997, the city has three universities and a large population of students. Tourists also visit Salzburg to tour the historic center and the scenic Alpine surroundings, Salzburg was the birthplace of 18th-century composer Wolfgang Amadeus Mozart. In the mid‑20th century, the city was the setting for the musical play, traces of human settlements have been found in the area, dating to the Neolithic Age. The first settlements in Salzburg continuous with the present were apparently by the Celts around the 5th century BC, around 15 BC the Roman Empire merged the settlements into one city. At this time, the city was called Juvavum and was awarded the status of a Roman municipium in 45 AD, Juvavum developed into an important town of the Roman province of Noricum. After the Norican frontiers collapse, Juvavum declined so sharply that by the late 7th century it became a ruin. The Life of Saint Rupert credits the 8th-century saint with the citys rebirth, when Theodo of Bavaria asked Rupert to become bishop c. 700, Rupert reconnoitered the river for the site of his basilica, Rupert chose Juvavum, ordained priests, and annexed the manor Piding. He traveled to evangelise among pagans, the name Salzburg means Salt Castle. The name derives from the barges carrying salt on the Salzach River, the Festung Hohensalzburg, the citys fortress, was built in 1077 by Archbishop Gebhard, who made it his residence. It was greatly expanded during the following centuries, independence from Bavaria was secured in the late 14th century. Salzburg was the seat of the Archbishopric of Salzburg, a prince-bishopric of the Holy Roman Empire, as the reformation movement gained steam, riots broke out among peasants in the areas in and around Salzburg. The city was occupied during the German Peasants War, and the archbishop had to flee to the safety of the fortress It was besieged for three months in 1525. It was in the 17th century that Italian architects rebuilt the city center as it is today along with many palaces,21,475 citizens refused to recant their beliefs and were expelled from Salzburg. Most of them accepted an offer by King Friedrich Wilhelm I of Prussia, the rest settled in other Protestant states in Europe and the British colonies in America. In 1772–1803, under archbishop Hieronymus Graf von Colloredo, Salzburg was a centre of late Illuminism, in 1803, the archbishopric was secularised by Emperor Napoleon, he transferred the territory to Ferdinando III of Tuscany, former Grand Duke of Tuscany, as the Electorate of Salzburg

Salzburg, Austria
–
Salzburg viewed from the

Festung Hohensalzburg
Salzburg, Austria
–

UNESCO World Heritage Site
Salzburg, Austria
–

Mozart was born in Salzburg, capital of the Archbishopric of Salzburg, a former ecclesiastical principality in what is now Austria, then part of the Holy Roman Empire of the German Nation

Salzburg, Austria
–
Salzburg in 1914

15.
Budapest, Hungary
–
Budapest is the capital and most populous city of Hungary, one of the largest cities in the European Union and sometimes described as the primate city of Hungary. It has an area of 525 square kilometres and a population of about 1.8 million within the limits in 2016. Budapest became a single city occupying both banks of the Danube river with the unification of Buda and Óbuda on the west bank, the history of Budapest began with Aquincum, originally a Celtic settlement that became the Roman capital of Lower Pannonia. Hungarians arrived in the territory in the 9th century and their first settlement was pillaged by the Mongols in 1241–1242. The re-established town became one of the centres of Renaissance humanist culture by the 15th century, following the Battle of Mohács and nearly 150 years of Ottoman rule, the region entered a new age of prosperity, and Budapest became a global city after its unification in 1873. It also became the co-capital of the Austro-Hungarian Empire, a power that dissolved in 1918. Budapest was the point of the Hungarian Revolution of 1848, the Hungarian Republic of Councils in 1919, the Battle of Budapest in 1945. Budapest is an Alpha- global city, with strengths in arts, commerce, design, education, entertainment, fashion, finance, healthcare, media, services, research, and tourism. Its business district hosts the Budapest Stock Exchange and the headquarters of the largest national and international banks and it is the highest ranked Central and Eastern European city on Innovation Cities Top 100 index. Budapest attracts 4.4 million international tourists per year, making it the 25th most popular city in the world, further famous landmarks include Andrássy Avenue, St. It has around 80 geothermal springs, the worlds largest thermal water system, second largest synagogue. Budapest is home to the headquarters of the European Institute of Innovation and Technology, the European Police College, over 40 colleges and universities are located in Budapest, including the Eötvös Loránd University, Central European University and Budapest University of Technology and Economics. Budapest is the combination of the city names Buda and Pest, One of the first documented occurrences of the combined name Buda-Pest was in 1831 in the book Világ, written by Count István Széchenyi. The origins of the names Buda and Pest are obscure, according to chronicles from the Middle Ages, the name Buda comes from the name of its founder, Bleda, brother of the Hunnic ruler Attila. The theory that Buda was named after a person is also supported by modern scholars, an alternative explanation suggests that Buda derives from the Slavic word вода, voda, a translation of the Latin name Aquincum, which was the main Roman settlement in the region. There are also theories about the origin of the name Pest. One of the states that the word Pest comes from the Roman times. According to another theory, Pest originates from the Slavic word for cave, or oven, the first settlement on the territory of Budapest was built by Celts before 1 AD

Budapest, Hungary

Budapest, Hungary
–
From top, left to right:

Hungarian Parliament,

Fisherman's Bastion,

Sándor Palace guard,

Heroes' Square,

National Theatre,

St. Stephen's Basilica and

Széchenyi Chain Bridge by night

Budapest, Hungary

Budapest, Hungary

16.
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

Geometry
–
Visual checking of the

Pythagorean theorem for the (3, 4, 5)

triangle as in the

Chou Pei Suan Ching 500–200 BC.

Geometry
–
An illustration of

Desargues' theorem, an important result in

Euclidean and

projective geometry
Geometry
–
Geometry lessons in the 20th century

Geometry
–
A

European and an

Arab practicing geometry in the 15th century.

17.
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

Tessellation
–

Zellige terracotta tiles in

Marrakech, forming edge-to-edge, regular and other tessellations

Tessellation
–
A wall sculpture at

Leeuwarden celebrating the artistic tessellations of

M. C. Escher
Tessellation
–
A temple mosaic from the ancient Sumerian city of

Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles

Tessellation
–

Roman geometric mosaic

18.
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

Hexagonal tiling
–

Chicken wire fencing

Hexagonal tiling
–

Graphene
Hexagonal tiling
–

Periodic
Hexagonal tiling

19.
Parquetry
–
Parquet is a geometric mosaic of wood pieces used for decorative effect. The two main uses of parquetry are as wood veneer patterns on furniture and block patterns for flooring, parquet patterns are entirely geometrical and angular—squares, triangles, lozenges. The most popular parquet flooring pattern is herringbone, the word derives from the Old French parchet, literally meaning a small enclosed space. Such parquets en losange were noted by the Swedish architect Daniel Cronström at Versailles, while not technically a wood, bamboo is also a popular material for modern floors. Parquet floors were formerly usually adhered with hot bitumen, today modern cold adhesives are usually used. Wood floors may be brushed clean, and mopped when necessary, upright vacuum cleaners can scratch and wear the surface, as grit particles become embedded in the spinning brushes. Parquet floors are long lasting and require little or no maintenance. Bitumen-glued blocks require use of hot bitumen or a cold bitumen emulsion. Parquet floors are found in bedrooms and hallways. They are considered better than regular floor tiles since they feel warmer underfoot, however they do little to absorb sounds such as walking, vacuum cleaning and television, which can cause problems in multi-occupancy dwellings. One of the most famous parquet floors is the one used by the Boston Celtics of the NBA, the floor remained intact and in use until it was cut up and sold as souvenirs in 1999, after the 1998 demolition of Boston Garden. The Celtics today play on a parquet floor inside TD Garden that combines old, similar square-paneled parquet floors were made for the Orlando Magic, Minnesota Timberwolves, Denver Nuggets, and New Jersey Nets. Of the four, only the Magic continue to use a square-paneled parquet floor, the Nets debuted their parquet at the Meadowlands Arena in 1988, and continued to use the floor until 1997. The said floor remained in use with the Seton Hall basketball team until 2007, the Nuggets used a parquet floor from 1990 to 1993 at the McNichols Sports Arena, while the Timberwolves played on the parquet from 1996 to 2008 at the Target Center. In 1995, the Toronto Raptors debuted with a herringbone parquet, the Nets revived the use of the herringbone upon moving to the Barclays Center in 2012. While the Charlotte Hornets unveiled a parquet-like floor at the Time Warner Cable Arena for the 2014–15 season, it is not considered a true parquet floor. Instead, it simulated the pattern of the parquet by alternately painting light and dark trapezoid sections through the use of varnish, forming a beehive pattern that is synonymous with the franchise

Parquetry
–
Intricate parquet flooring in entry hall.

Parquetry
–
Parquet flooring, 18th century.

Parquetry
–
Parquet Versailles

Parquetry
–
The iconic parquet floor used by the

Boston Celtics at

TD Garden
20.
Masonry
–
Masonry is the building of structures from individual units, which are often laid in and bound together by mortar, the term masonry can also refer to the units themselves. The common materials of construction are brick, building stone such as marble, granite, travertine, and limestone, cast stone, concrete block, glass block. Masonry is generally a durable form of construction. However, the used, the quality of the mortar and workmanship. A person who constructs masonry is called a mason or bricklayer, Masonry is commonly used for walls and buildings. Brick and concrete block are the most common types of masonry in use in industrialized nations, Concrete blocks, especially those with hollow cores, offer various possibilities in masonry construction. They generally provide great strength, and are best suited to structures with light transverse loading when the cores remain unfilled. Filling some or all of the cores with concrete or concrete with steel reinforcement offers much greater tensile, the use of material such as bricks and stones can increase the thermal mass of a building and can protect the building from fire. Masonry walls are resistant to projectiles, such as debris from hurricanes or tornadoes. Extreme weather, under circumstances, can cause degradation of masonry due to expansion. Masonry tends to be heavy and must be built upon a foundation, such as reinforced concrete. Other than concrete, masonry construction does not lend well to mechanization. Masonry consists of components and has a low tolerance to oscillation as compared to other materials such as reinforced concrete, plastics, wood. Masonry has high compressive strength under vertical loads but has low tensile strength unless reinforced, the tensile strength of masonry walls can be increased by thickening the wall, or by building masonry piers at intervals. Where practical, steel reinforcements such as windposts can be added, a masonry veneer wall consists of masonry units, usually clay-based bricks, installed on one or both sides of a structurally independent wall usually constructed of wood or masonry. In this context the brick masonry is primarily decorative, not structural, the brick veneer is generally connected to the structural wall by brick ties. There is typically an air gap between the veneer and the structural wall. Concrete blocks, real and cultured stones, and veneer adobe are sometimes used in a very similar veneer fashion

Masonry
–
A mason laying mortar on top of a finished course of blocks, prior to placing the next course.

Masonry
–
Dry set masonry supports a rustic

log bridge, where it provides a well-drained support for the log (which will increase its service life).

Masonry
–
Masonry repair work done to a brick wall.

Masonry
–
Concrete masonry units (CMUs) or blocks in a basement wall before burial.

21.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

International Standard Book Number
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an

EAN-13 bar code
22.
Isohedral figure
–
In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007

Isohedral figure
–

Periodic
Isohedral figure
–
The

hexagonal bipyramid, V4.4.6 is a

nonregular example of an isohedral polyhedron.

Isohedral figure