Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.
Computer-Aided Design (CAD) is the use of computers to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. Designs made through CAD software help protect products and inventions when used in patent applications. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The terms computer-aided drafting (CAD) and computer-aided design and drafting (CADD) are also used.
A castle is a type of fortified structure built during the Middle Ages predominantly by the nobility or royalty and by military orders. Scholars usually consider a castle to be the private fortified residence of a lord or noble. This is distinct from a mansion, palace and villa, whose main purpose was exclusively for pleasance and are not primarily fortresses but may be fortified. Use of the term has varied over time and, sometimes, has also been applied to structures such as hill forts and 19th- and 20th-century homes built to resemble castles. Over the Middle Ages, when genuine castles were built, they took on a great many forms with many different features, although some, such as curtain walls, arrowslits, and portcullises, were commonplace.
An arch is a vertical curved structure that spans an elevated space and may or may not support the weight above it, or in case of a horizontal arch like an arch dam, the hydrostatic pressure against it.
A fractal landscape or fractal surface is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the surface resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.
A dome is an architectural element similar to the hollow upper half of a sphere. There is significant overlap with the term cupola, which may also refer to a dome or a structure on top of a dome. The precise definition of a dome has been a matter of controversy and there are a wide variety of forms and specialized terms to describe them.
Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
A motte-and-bailey castle is a European fortification with a wooden or stone keep situated on a raised area of ground called a motte, accompanied by a walled courtyard, or bailey, surrounded by a protective ditch and palisade. Relatively easy to build with unskilled labour, but still militarily formidable, these castles were built across northern Europe from the 10th century onwards, spreading from Normandy and Anjou in France, into the Holy Roman Empire, as well as the Low Countries it controlled, in the 11th century, when these castles were popularized in the area that became the Netherlands. The Normans introduced the design into England and Wales. Motte-and-bailey castles were adopted in Scotland, Ireland, and Denmark in the 12th and 13th centuries. By the end of the 13th century, the design was largely superseded by alternative forms of fortification, but the earthworks remain a prominent feature in many countries.
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or other objects such as tabletops. Alternatively, tile can sometimes refer to similar units made from lightweight materials such as perlite, wood, and mineral wool, typically used for wall and ceiling applications. In another sense, a tile is a construction tile or similar object, such as rectangular counters used in playing games. The word is derived from the French word tuile, which is, in turn, from the Latin word tegula, meaning a roof tile composed of fired clay.
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word graph is sometimes used as a synonym for diagram.
Garden design is the art and process of designing and creating plans for layout and planting of gardens and landscapes. Garden design may be done by the garden owner themselves, or by professionals of varying levels of experience and expertise. Most professional garden designers have some training in horticulture and the principles of design. Some are also landscape architects, a more formal level of training that usually requires an advanced degree and often a state license. Amateur gardeners may also attain a high level of experience from extensive hours working in their own gardens, through casual study, serious study in Master gardener programs, or by joining gardening clubs.
A keep is a type of fortified tower built within castles during the Middle Ages by European nobility. Scholars have debated the scope of the word keep, but usually consider it to refer to large towers in castles that were fortified residences, used as a refuge of last resort should the rest of the castle fall to an adversary. The first keeps were made of timber and formed a key part of the motte-and-bailey castles that emerged in Normandy and Anjou during the 10th century; the design spread to England, and south to Italy and Sicily. As a result of the Norman invasion of 1066, use spread into Wales during the second half of the 11th century and into Ireland in the 1170s. The Anglo-Normans and French rulers began to build stone keeps during the 10th and 11th centuries; these included Norman keeps, with a square or rectangular design, and circular shell keeps. Stone keeps carried considerable political as well as military importance and could take up to a decade or more to build.
Orford Castle is a castle in Orford in the English county of Suffolk, 12 miles (19 km) northeast of Ipswich, with views over Orford Ness. It was built between 1165 and 1173 by Henry II of England to consolidate royal power in the region. The well-preserved keep, described by historian R. Allen Brown as "one of the most remarkable keeps in England", is of a unique design and probably based on Byzantine architecture. The keep stands within the earth-bank remains of the castle's outer fortifications.
Pavement, in construction, is an outdoor floor or superficial surface covering. Paving materials include asphalt, concrete, stones such as flagstone, cobblestone, and setts, artificial stone, bricks, tiles, and sometimes wood. In landscape architecture, pavements are part of the hardscape and are used on sidewalks, road surfaces, patios, courtyards, etc.
Yantra (यन्त्र) is a geometrical diagram, mainly from the Tantric traditions of the Indian religions. Yantras are used for the worship of deities in temples or at home; as an aid in meditation; used for the benefits given by their supposed occult powers based on Hindu astrology and tantric texts. They are also used for adornment of temple floors, due mainly to their aesthetic and symmetric qualities. Specific yantras are traditionally associated with specific deities and/or certain types of energies used for accomplishment of certain tasks, vows, that may be materialistic or spiritual in nature. It becomes a prime tool in certain sadhanas performed by the sadhaka the spiritual seeker. Yantras hold great importance in Hinduism, Jainism, and Buddhism.
Geometric art is a phase of Greek art, characterized largely by geometric motifs in vase painting, that flourished towards the end of the Greek Dark Ages, c. 900–700 BC. Its center was in Athens, and from there the style spread among the trading cities of the Aegean. The Greek Dark Ages lasted from c. 1100 to 750 BC and include two periods, the Protogeometric period and the Geometric period, in reference to the characteristic pottery style. The vases had various uses or purposes within Greek society, including, but not limited to, funerary vases and symposium vases.
Zellij is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various patterns on the basis of tessellations, most notably elaborate Islamic geometric motifs such as radiating star patterns. This form of Islamic art is one of the main characteristics of architecture in the western Islamic world. It is found in the architecture of Morocco, the architecture of Algeria, early Islamic sites in Tunisia, and in the historic monuments of al-Andalus. From the 14th century onwards, zellij became a standard decorative element along lower walls, in fountains and pools, on minarets, and for the paving of floors.
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
A formal garden is a garden with a clear structure, geometric shapes and in most cases a symmetrical layout. Its origin goes back to the gardens which are located in the desert areas of Western Asia and are protected by walls. The style of a formal garden is reflected in the Persian gardens of Iran, and the monastic gardens from the Late Middle Ages. It has found its continuation in the Italian Renaissance gardens and has culminated in the French formal gardens from the Baroque period. Through its design, the garden conveys a sense of established order and transparency to the observer.
