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UK dance music traces its roots back to the Black British Sound System Culture and the New Age Traveller movement of the 60s and 70s,[448] it also has influences from New Wave and Synth-pop such as from bands New Order and Depeche Mode[449] and also has influences from the Chicago House and Detroit Techno scenes. In the late 80's, dance music exploded with Rave culture mainly Acid House tracks which were made mainstream with novelty records (such as Smart E's Sesame's Treet and the Prodigy's Charly)[450] and the Balearic sound brought back from the Ibiza club scene. This led on to genres such as UK Garage, Speed Garage, Drum and bass, Jungle, Trance and Dubstep. Influential UK dance acts past and present include 808 State, Orbital, the Prodigy, Underworld, Roni Size, Leftfield, Massive Attack, Groove Armada, Fatboy Slim, Faithless, Basement Jaxx, Chemical Brothers, Sub Focus, Chase & Status, Disclosure, Calvin Harris and Fred Again.[451] Other influential UK DJs include Judge Jules, Pete Tong, Carl Cox, Paul Oakenfold, John Digweed and Sasha.[452]

Major British artists include: the Romantics William Blake, John Constable, Samuel Palmer and J. M. W. Turner; the portrait painters Sir Joshua Reynolds and Lucian Freud; the landscape artists Thomas Gainsborough and L. S. Lowry; the pioneer of the Arts and Crafts Movement William Morris; the figurative painter Francis Bacon; the Pop artists Peter Blake, Richard Hamilton and David Hockney; the pioneers of Conceptual art movement Art & Language;[453] the collaborative duo Gilbert and George; the abstract artist Howard Hodgkin; and the sculptors Antony Gormley, Anish Kapoor and Henry Moore. During the late 1980s and 1990s the Saatchi Gallery in London helped to bring to public attention a group of multi-genre artists who would become known as the "Young British Artists": Damien Hirst, Chris Ofili, Rachel Whiteread, Tracey Emin, Mark Wallinger, Steve McQueen, Sam Taylor-Wood and the Chapman Brothers are among the better-known members of this loosely affiliated movement.

The Royal Academy in London is a key organisation for the promotion of the visual arts in the United Kingdom. Major schools of art in the UK include: the six-school University of the Arts London, which includes the Central Saint Martins College of Art and Design and Chelsea College of Art and Design; Goldsmiths, University of London; the Slade School of Fine Art (part of University College London); the Glasgow School of Art; the Royal College of Art; and The Ruskin School of Drawing and Fine Art (part of the University of Oxford). The Courtauld Institute of Art is a leading centre for the teaching of the history of art. Important art galleries in the United Kingdom include the National Gallery, National Portrait Gallery, Tate Britain and Tate Modern (the most-visited modern art gallery in the world, with around 4.7 million visitors per year).[454]

The United Kingdom has had a considerable influence on the history of the cinema. The British directors Alfred Hitchcock, whose film Vertigo is considered by some critics as the best film of all time,[455] and David Lean who directed Lawrence of Arabia are among the most critically acclaimed directors of all time.[456] Recent popular directors include: Christopher Nolan, Sam Mendes, Steve McQueen, Danny Boyle, Tony Scott and Ridley Scott.[457]

[459] Many British actors have achieved international fame and critical success. Some of the most commercially successful films of all time have been produced in the United Kingdom, including two of the highest-grossing film franchises (Harry Potter and James Bond).[460]

2019 was a particularly good year for British films which grossed around £10.3 billion globally which was 28.7% of global box office revenue.[461] UK box-office takings totalled £1.25 billion in 2019, with around 176 million admissions.[462] In 2023 UK film and television studio stage space stands at 6.9 million sq ft, with 1 million sq ft added in the past year with more in development.[463] The annual BAFTA Film Awards are hosted by the British Academy of Film and Television Arts.[464]

British cuisine developed from various influences reflective of its land, settlements, arrivals of new settlers and immigrants, trade and colonialism. The food of England has historically been characterised by its simplicity of approach and a reliance on the high quality of natural produce.[465] The traditional Sunday roast is one example, featuring a roasted joint, usually of beef, lamb, chicken or pork, often free range (and generally grass-fed, in the case of beef). Roasts are served with either roasted or boiled vegetables, Yorkshire pudding, and gravy. Other traditional meals include meat pies and various stews.

A 2019 YouGov poll rated classic British food, the following had more than 80% of people like them who had tried them: Sunday roast, Yorkshire pudding, Fish and chips, Crumpets, and Full English breakfast.[466]

The British Empire facilitated a knowledge of Indian cuisine with its "strong, penetrating spices and herbs". British cuisine has absorbed the cultural influence of those who have settled in Britain, producing hybrid dishes, such as chicken tikka masala.[467] The British have embraced world cuisine and regularly eat recipes or fast food from Europe, the Caribbean and Asia.

Sweet foods are common within British cuisine, and there is a long list of British desserts. Afternoon tea is a light afternoon meal served with tea in tea rooms and hotels around the United Kingdom, with the tradition dating back to around 1840.[468]

Vegan and vegetarian diets have increased in Britain in recent years. In 2021, a survey found that 8% of British respondents eat a plant-based diet and 36% of respondents have a favourable view of plant-based diets.[469]

The BBC, founded in 1922, is the UK's publicly funded radio, television and Internet broadcasting corporation, and is the oldest and largest broadcaster in the world.[470][471][472] It operates numerous television and radio stations in the UK and abroad and its domestic services are funded by the television licence.[473] The BBC World Service is an international broadcaster owned and operated by the BBC. It is the world's largest of any kind.[474] It broadcasts radio news, speech and discussions in more than 40 languages.[475]

Other major players in the UK media include ITV, which operates 11 of the 15 regional television broadcasters that make up the ITV Network,[476] and Sky.[477] Newspapers produced in the United Kingdom include the Daily Mail, The Guardian, The Telegraph, The Times, and the Financial Times.[478] Magazines and journals published in the United Kingdom that have achieved worldwide circulation include The Spectator, The Economist, New Statesman, and Radio Times.

London dominates the media sector in the UK: national newspapers and television and radio are largely based there, although MediaCityUK in Manchester is also a significant national media centre. Edinburgh and Glasgow, and Cardiff, are important centres of newspaper and broadcasting production in Scotland and Wales, respectively.[479] The UK publishing sector, including books, directories and databases, journals, magazines and business media, newspapers and news agencies, has a combined turnover of around £20 billion and employs around 167,000 people.[480] In 2015, the UK published 2,710 book titles per million inhabitants, more than any other country, much of this being exported to other Anglophone countries.[481]

In 2010, 82.5 per cent of the UK population were Internet users, the highest proportion among the 20 countries with the largest total number of users in that year.[482] The British video game industry is the largest in Europe, and, since 2022, the UK has the largest video game market in Europe by sales, overtaking Germany.[483] It is the world's third-largest producer of video games after Japan and the United States.[484]

Association football, tennis, table tennis, badminton, rugby union, rugby league, rugby sevens, golf, boxing, netball, water polo, field hockey, billiards, darts, rowing, rounders and cricket originated or were substantially developed in the UK, with the rules and codes of many modern sports invented and codified in late 19th-century Victorian Britain.[u]

A 2003 poll found that football is the most popular sport in the UK.[487] England is recognised by FIFA as the birthplace of club football, and the Football Association is the oldest of its kind, with the rules of football first drafted in 1863 by Ebenezer Cobb Morley.[488] Each of the Home Nations (England, Scotland, Wales and Northern Ireland) has its own football association, national team and league system, and each is individually a governing member of the International Football Association Board alongside FIFA. The English top division, the Premier League, is the most watched football league in the world.[489] The first international football match was contested by England and Scotland on 30 November 1872.[490] England, Scotland, Wales and Northern Ireland usually compete as separate countries in international competitions.[491]

In 2003, rugby union was ranked the second most popular sport in the UK.[487] The sport was created in Rugby School, Warwickshire, and the first rugby international took place on 27 March 1871 between England and Scotland.[492][493] England, Scotland, Wales, Ireland, France and Italy compete in the Six Nations Championship, which is the premier international rugby union tournament in the northern hemisphere. Sports governing bodies in England, Scotland, Wales and Ireland organise and regulate the game separately.[494] Every four years, the Home Nations make a combined team known as the British and Irish Lions which tours Australia, New Zealand and South Africa.

The United Kingdom hosted the Summer Olympic Games in 1908, 1948 and 2012, with London acting as the host city on all three occasions. Birmingham hosted the 2022 Commonwealth Games, the seventh time a constitute country in the United Kingdom hosted the Commonwealth Games (England, Scotland and Wales have each hosted the Commonwealth Games at least once).[495]

The flag of the United Kingdom is the Union Flag (also referred to as the Union Jack).[496] It was created in 1606 by the superimposition of the flag of England, representing Saint George, on the flag of Scotland, representing Saint Andrew, and was updated in 1801 with the addition of Saint Patrick's Flag.[497] Wales is not represented in the Union Flag, as Wales had been conquered and annexed to England prior to the formation of the United Kingdom. The possibility of redesigning the Union Flag to include representation of Wales has not been completely ruled out.[498] The national anthem of the United Kingdom is "God Save the King", with "King" replaced with "Queen" in the lyrics whenever the monarch is a woman.

Britannia is a national personification of the United Kingdom, originating from Roman Britain.[499] Beside The Lion and the Unicorn and the dragon of heraldry, the bulldog is an iconic animal and commonly represented with the Union Flag.[500] A now rare personification is a character originating in the 18th century, John Bull.[501]

England, Wales, and Scotland each have a number of their own national symbols, including their national flags. Northern Ireland also has a number of symbols, many of which are shared with Republic of Ireland.

A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude.[1] It is the simplest, oldest and most widely used of the various spatial reference systems that are in use, and forms the basis for most others. Although latitude and longitude form a coordinate tuple like a cartesian coordinate system, the geographic coordinate system is not cartesian because the measurements are angles and are not on a planar surface.[2]

A full GCS specification, such as those listed in the EPSG and ISO 19111 standards, also includes a choice of geodetic datum (including an Earth ellipsoid), as different datums will yield different latitude and longitude values for the same location.[3]

The invention of a geographic coordinate system is generally credited to Eratosthenes of Cyrene, who composed his now-lost Geography at the Library of Alexandria in the 3rd century BC.[4] A century later, Hipparchus of Nicaea improved on this system by determining latitude from stellar measurements rather than solar altitude and determining longitude by timings of lunar eclipses, rather than dead reckoning. In the 1st or 2nd century, Marinus of Tyre compiled an extensive gazetteer and mathematically plotted world map using coordinates measured east from a prime meridian at the westernmost known land, designated the Fortunate Isles, off the coast of western Africa around the Canary or Cape Verde Islands, and measured north or south of the island of Rhodes off Asia Minor. Ptolemy credited him with the full adoption of longitude and latitude, rather than measuring latitude in terms of the length of the midsummer day.[5]

Ptolemy's 2nd-century Geography used the same prime meridian but measured latitude from the Equator instead. After their work was translated into Arabic in the 9th century, Al-Khwārizmī's Book of the Description of the Earth corrected Marinus' and Ptolemy's errors regarding the length of the Mediterranean Sea,[note 1] causing medieval Arabic cartography to use a prime meridian around 10° east of Ptolemy's line. Mathematical cartography resumed in Europe following Maximus Planudes' recovery of Ptolemy's text a little before 1300; the text was translated into Latin at Florence by Jacopo d'Angelo around 1407.

In 1884, the United States hosted the International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt the longitude of the Royal Observatory in Greenwich, England as the zero-reference line. The Dominican Republic voted against the motion, while France and Brazil abstained.[6] France adopted Greenwich Mean Time in place of local determinations by the Paris Observatory in 1911.

The "latitude" (abbreviation: Lat., ϕ, or phi) of a point on Earth's surface is the angle between the equatorial plane and the straight line that passes through that point and through (or close to) the center of the Earth.[note 2] Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they are parallel to the Equator and to each other. The North Pole is 90° N; the South Pole is 90° S. The 0° parallel of latitude is designated the Equator, the fundamental plane of all geographic coordinate systems. The Equator divides the globe into Northern and Southern Hemispheres.

The "longitude" (abbreviation: Long., λ, or lambda) of a point on Earth's surface is the angle east or west of a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses (often called great circles), which converge at the North and South Poles. The meridian of the British Royal Observatory in Greenwich, in southeast London, England, is the international prime meridian, although some organizations—such as the French Institut national de l'information géographique et forestière—continue to use other meridians for internal purposes. The prime meridian determines the proper Eastern and Western Hemispheres, although maps often divide these hemispheres further west in order to keep the Old World on a single side. The antipodal meridian of Greenwich is both 180°W and 180°E. This is not to be conflated with the International Date Line, which diverges from it in several places for political and convenience reasons, including between far eastern Russia and the far western Aleutian Islands.

The combination of these two components specifies the position of any location on the surface of Earth, without consideration of altitude or depth. The visual grid on a map formed by lines of latitude and longitude is known as a graticule.[7] The origin/zero point of this system is located in the Gulf of Guinea about 625 km (390 mi) south of Tema, Ghana, a location often facetiously called Null Island.

In order to be unambiguous about the direction of "vertical" and the "horizontal" surface above which they are measuring, map-makers choose a reference ellipsoid with a given origin and orientation that best fits their need for the area to be mapped. They then choose the most appropriate mapping of the spherical coordinate system onto that ellipsoid, called a terrestrial reference system or geodetic datum.

Datums may be global, meaning that they represent the whole Earth, or they may be local, meaning that they represent an ellipsoid best-fit to only a portion of the Earth. Points on the Earth's surface move relative to each other due to continental plate motion, subsidence, and diurnal Earth tidal movement caused by the Moon and the Sun. This daily movement can be as much as a meter. Continental movement can be up to 10 cm a year, or 10 m in a century. A weather system high-pressure area can cause a sinking of 5 mm. Scandinavia is rising by 1 cm a year as a result of the melting of the ice sheets of the last ice age, but neighboring Scotland is rising by only 0.2 cm. These changes are insignificant if a local datum is used, but are statistically significant if a global datum is used.[8]

Examples of global datums include World Geodetic System (WGS 84, also known as EPSG:4326[9]), the default datum used for the Global Positioning System,[note 3] and the International Terrestrial Reference System and Frame (ITRF), used for estimating continental drift and crustal deformation.[10] The distance to Earth's center can be used both for very deep positions and for positions in space.[8]

Local datums chosen by a national cartographical organization include the North American Datum, the European ED50, and the British OSGB36. Given a location, the datum provides the latitude

. In the United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differs from about 120 m to 180 m.[8]

The latitude and longitude on a map made against a local datum may not be the same as one obtained from a GPS receiver. Converting coordinates from one datum to another requires a datum transformation such as a Helmert transformation, although in certain situations a simple translation may be sufficient.[11]

In popular GIS software, data projected in latitude/longitude is often represented as a Geographic Coordinate System. For example, data in latitude/longitude if the datum is the North American Datum of 1983 is denoted by 'GCS North American 1983'.

On the GRS 80 or WGS 84 spheroid at sea level at the Equator, one latitudinal second measures 30.715 m, one latitudinal minute is 1843 m and one latitudinal degree is 110.6 km. The circles of longitude, meridians, meet at the geographical poles, with the west–east width of a second naturally decreasing as latitude increases. On the Equator at sea level, one longitudinal second measures 30.92 m, a longitudinal minute is 1855 m and a longitudinal degree is 111.3 km. At 30° a longitudinal second is 26.76 m, at Greenwich (51°28′38″N) 19.22 m, and at 60° it is 15.42 m.

On the WGS 84 spheroid, the length in meters of a degree of latitude at latitude ϕ (that is, the number of meters you would have to travel along a north–south line to move 1 degree in latitude, when at latitude ϕ), is about

The returned measure of meters per degree latitude varies continuously with latitude.

Similarly, the length in meters of a degree of longitude can be calculated as

An alternative method to estimate the length of a longitudinal degree at latitude is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

Like any series of multiple-digit numbers, latitude-longitude pairs can be challenging to communicate and remember. Therefore, alternative schemes have been developed for encoding GCS coordinates into alphanumeric strings or words:

These are not distinct coordinate systems, only alternative methods for expressing latitude and longitude measurements.

An alternative method to estimate the length of a longitudinal degree at latitude is to assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively):

where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude equals 6,378,137 m and ; for the GRS 80 and WGS 84 spheroids, is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

Local datums chosen by a national cartographical organization include the North American Datum, the European ED50, and the British OSGB36. Given a location, the datum provides the latitude and longitude . In the United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differs from about 120 m to 180 m.[8]

where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is where Earth's equatorial radius equals 6,378,137 m and ; for the GRS 80 and WGS 84 spheroids, is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

Local datums chosen by a national cartographical organization include the North American Datum, the European ED50, and the British OSGB36. Given a location, the datum provides the latitude . In the United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differs from about 120 m to 180 m.[8]

where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is where Earth's equatorial radius equals 6,378,137 m and ; for the GRS 80 and WGS 84 spheroids, is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude where Earth's equatorial radius equals 6,378,137 m and ; for the GRS 80 and WGS 84 spheroids, is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

Local datums chosen by a national cartographical organization include the North American Datum, the European ED50, and the British OSGB36. Given a location, the datum provides the latitude. In the United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50, used by NATO, differs from about 120 m to 180 m.[8]

where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is where Earth's equatorial radius is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

An alternative method to estimate the length of a longitudinal degree at latitude

is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart.

A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space. A particular SRS specification (for example, "Universal Transverse Mercator WGS 84 Zone 16N") comprises a choice of Earth ellipsoid, horizontal datum, map projection (except in the geographic coordinate system), origin point, and unit of measure. Thousands of coordinate systems have been specified for use around the world or in specific regions and for various purposes, necessitating transformations between different SRS.

Although they date to the Hellenic Period, spatial reference systems are now a crucial basis for the sciences and technologies of Geoinformatics, including cartography, geographic information systems, surveying, remote sensing, and civil engineering. This has led to their standardization in international specifications such as the EPSG codes[1] and ISO 19111:2019 Geographic information—Spatial referencing by coordinates, prepared by ISO/TC 211, also published by the Open Geospatial Consortium as Abstract Specification, Topic 2: Spatial referencing by coordinate.[2]

The thousands of spatial reference systems used today are based on a few general strategies, which have been defined in the EPSG, ISO, and OGC standards:[1][2]

These standards acknowledge that standard reference systems also exist for measuring elevation using vertical datums and time (e.g. ISO 8601), which may be combined with a spatial reference system to form a compound coordinate system for representing three-dimensional and/or spatio-temporal locations. There are also internal systems for measuring location within the context of an object, such as the rows and columns of pixels in a raster image, Linear referencing measurements along linear features (e.g., highway mileposts), and systems for specifying location within moving objects such as ships. The latter two are often classified as subcategories of engineering coordinate systems.

The goal of any spatial reference system is to create a common reference frame in which locations can be measured precisely and consistently as coordinates, which can then be shared unambiguously, so that any recipient can identify the same location that was originally intended by the originator.[3] To accomplish this, any coordinate reference system definition needs to be composed of several specifications:

Thus, a CRS definition will typically consist of a "stack" of dependent specifications, as exemplified in the following table:

A Spatial Reference System Identifier (SRID) is a unique value used to unambiguously identify projected, unprojected, and local spatial coordinate system definitions. These coordinate systems form the heart of all GIS applications.

Virtually all major spatial vendors have created their own SRID implementation or refer to those of an authority, such as the EPSG Geodetic Parameter Dataset.

SRIDs are the primary key for the Open Geospatial Consortium (OGC) spatial_ref_sys metadata table for the Simple Features for SQL Specification, Versions 1.1 and 1.2, which is defined as follows:

In spatially enabled databases (such as IBM Db2, IBM Informix, Ingres, Microsoft SQL Server, MonetDB, MySQL, Oracle RDBMS, Teradata, PostGIS, SQL Anywhere and Vertica), SRIDs are used to uniquely identify the coordinate systems used to define columns of spatial data or individual spatial objects in a spatial column (depending on the spatial implementation). SRIDs are typically associated with a well-known text (WKT) string definition of the coordinate system (SRTEXT, above).

Here are two common coordinate systems with their EPSG SRID value followed by their WKT:

SRID values associated with spatial data can be used to constrain spatial operations — for instance, spatial operations cannot be performed between spatial objects with differing SRIDs in some systems, or trigger coordinate system transformations between spatial objects in others.

In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

It has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices, and 120 edges.

Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral rhombus, with icosidodecahedral rhombus being his name for a rhombic triacontahedron.[1][2] There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the rectification of the dual icosidodecahedron (right), which is the core of the dual compound.

For a rhombicosidodecahedron with edge length a, its surface area and volume are:

If you expand an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, or do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either.

Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of squares as five cubes.

Two clusters of faces of the bilunabirotunda, the lunes (each lune featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron.

The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of six or twelve pentagrammic prisms.

The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.

Twelve of the 92 Johnson solids are derived from the rhombicosidodecahedron, four of them by rotation of one or more pentagonal cupolae: the gyrate, parabigyrate, metabigyrate, and trigyrate rhombicosidodecahedron. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae.

Cartesian coordinates for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all even permutations of:[3]

where φ = 1 + √5/2 is the golden ratio. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √φ6+2 = √8φ+7 for edge length 2. For unit edge length, R must be halved, giving

The rhombicosidodecahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, and three types of faces: triangles, squares, and pentagons. The last two correspond to the A2 and H2 Coxeter planes.

The rhombicosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

This polyhedron is topologically related as a part of a sequence of cantellated polyhedra with vertex figure (3.4.n.4), which continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

There are 12 related Johnson solids, 5 by diminishment, and 8 including gyrations:

The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).

It also shares its vertex arrangement with the uniform compounds of six or twelve pentagrammic prisms.

In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph.[5]

In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids (each with only one type of polygon face), and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive.[1] An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

In these polyhedra, the vertices are identical, in the sense that a global isometry of the entire solid takes any one vertex to any other. Branko Grünbaum (2009) observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means

merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.[2] During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra,[3] as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See Schreiber, Fischer & Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance.)

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.[2]

There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images of two enantiomorphs, the snub cube and snub dodecahedron, are counted separately).

Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of 4.6.8 means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".[4]

The number of vertices is 720° divided by the vertex angle defect.

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds.)