title: The Rundown

text: In December 2009, Berg acknowledged that discussions for a sequel had begun shortly after the film's release and were ongoing. The filmmaker citied scheduling conflicts as the reason why the project hadn't been green-lit yet. By November 2010, Dwayne Johnson corroborated this; stating that he and Berg are always discussing options for a second film. Johnson stated that the pair are also in negotiations with Universal Pictures, and that if a good enough story is written the film will be made.


title: The Rundown

text: In December 2013, Berg announced that he was officially developing a second "Rundown" film. The filmmaker confirmed that Dwayne Johnson also was invested in a sequel, while stating that Scott would not return in his co-starring role. A number of unnamed screenwriters were hired for the project, by the filmmaker. By September 2016, Berg stated that the script for the sequel had been written with Kevin Hart in mind as the co-star, while also stating that he would also consider making the film with Jonah Hill alternatively in the supporting role. Johnson publicly confirmed his involvement with the project, while proposing "HellDorado" - which was the working title of the first movie - as a possible title for the sequel. The pair asked fans to campaign for Hill to sign on as co-star. In August 2018, the Berg revealed that the current draft of the script included a story that took place in Alaska with a comedic scene involving walruses, similar to the scene with baboons from the first movie. Acknowledging Johnson's disinterest in the ice-covered state, the filmmaker stated that a new draft is being written with a less frigid location being considered for the plot.


title: The Rundown

text: In September 2020, Berg stated that development for a sequel is ongoing, while citing Johnson's busy production schedule as one of the reasons the film hasn't been made yet. The filmmaker also stated that they are currently "rethink[ing] the script".


title: Electronica

text: Electronica is both a broad group of electronic-based music styles intended for listening rather than strictly for dancing and a music scene that started in the early 1990s in the United Kingdom. In the United States, the term is mostly used to refer to electronic music generally.


title: Electronica

text: The original wide-spread use of the term "electronica" derives from the influential English experimental techno label New Electronica, which was one of the leading forces of the early 1990s introducing and supporting dance-based electronic music oriented towards home listening rather than dance-floor play, although the word "electronica" had already begun to be associated with synthesizer generated music as early as 1983, when a "UK Electronica Festival" was first held. At that time electronica became known as "electronic listening music", also becoming more or less synonymous to ambient techno and intelligent techno, and was considered distinct from other emerging genres such as jungle and trip hop.


title: Electronica

text: Electronica artists that would later become commercially successful began to record in the late 1980s, before the term had come into common usage, including for example The Prodigy, Fatboy Slim, Daft Punk, The Chemical Brothers, The Crystal Method, Moby, Underworld and Faithless.


title: Electronica

text: Around the mid-1990s, with the success of the big beat-sound exemplified by The Chemical Brothers and The Prodigy in the UK, and spurred by the attention from mainstream artists, including Madonna in her collaboration with William Orbit on her album "Ray of Light" and Australian singer Dannii Minogue with her 1997 album "Girl", music of this period began to be produced with a higher budget, increased technical quality, and with more layers than most other forms of dance music, since it was backed by major record labels and MTV as the "next big thing".


title: Electronica

text: According to a 1997 "Billboard" article, "the union of the club community and independent labels" provided the experimental and trend-setting environment in which electronica acts developed and eventually reached the mainstream. It cites American labels such as Astralwerks (The Chemical Brothers, Fatboy Slim, The Future Sound of London, Fluke), Moonshine (DJ Keoki), Sims, and City of Angels (The Crystal Method) for playing a significant role in discovering and marketing artists who became popularized in the electronica scene.


title: Electronica

text: Madonna and Björk are said to be responsible for electronica's thrust into mainstream culture, with their albums "Ray of Light" (Madonna), "Post" and "Homogenic" (Björk).


title: Electronica

text: In 1997, the North American mainstream music industry adopted and to some extent manufactured "electronica" as an umbrella term encompassing styles such as techno, big beat, drum and bass, trip hop, downtempo, and ambient, regardless of whether it was curated by indie labels catering to the "underground" nightclub and rave scenes, or licensed by major labels and marketed to mainstream audiences as a commercially viable alternative to alternative rock music.


title: Electronica

text: New York City became one center of experimentation and growth for the electronica sound, with DJs and music producers from areas as diverse as Southeast Asia and Brazil bringing their creative work to the nightclubs of that city.


title: Electronica

text: By the early 2010s, the industry abandoned "electronica" in favor of "electronic dance music" (EDM), a term with roots in academia and an increasing association with outdoor music festivals and relatively mainstream, post-rave electro house and dubstep music.


title: Electronica

text: Electronica benefited from advancements in music technology, especially electronic musical instruments, synthesizers, music sequencers, drum machines, and digital audio workstations. As the technology developed, it became possible for individuals or smaller groups to produce electronic songs and recordings in smaller studios, even in project studios. At the same time, computers facilitated the use of music "samples" and "loops" as construction kits for sonic compositions. This led to a period of creative experimentation and the development of new forms, some of which became known as "electronica". Wide ranges of influences, both sonic and compositional, are combined in electronica recordings.


title: Electronica

text: Electronica includes a wide variety of musical acts and styles, linked by a penchant for overtly electronic production; a range which includes more popular acts such as Björk, Madonna, Goldfrapp and IDM artists such as Autechre, and Aphex Twin.


title: Electronica

text: The North American mainstream music industry uses the term as an umbrella category to refer any dance-based electronic music styles with a potential for pop appeal. However, U.S.-based "AllMusic" still categorizes electronica as a top-level genre, stating that it includes danceable grooves, as well as music for headphones and chillout areas.


title: Electronica

text: In other parts of the world, especially in the UK, "electronica" is also a broad term, but is associated with non-dance-oriented music, including relatively experimental styles of listening electronic music. It partly overlaps what is known chiefly outside the UK as "intelligent dance music" (IDM).


title: Electronica

text: In the late 1990s and early 2000s, electronica was increasingly used as background scores for television advertisements, initially for automobiles. It was also used for various video games, including the "Wipeout" series, for which the soundtrack was composed of many popular electronica tracks that helped create more interest in this type of music—and later for other technological and business products such as computers and financial services.


title: Electronica

text: Then in 2011, Hyundai Veloster, in association with The Grammys, produced a project that became known as Re:Generation.


title: Kalahari Desert

text: The Kalahari Desert is a large semi-arid sandy savanna in Southern Africa extending for , covering much of Botswana, and parts of Namibia and South Africa.


title: Kalahari Desert

text: It is not to be confused with the Angolan, Namibian, and South African Namib coastal desert, whose name is of Khoekhoegowab origin and means "vast place".


title: Kalahari Desert

text: "Kalahari" is derived from the Tswana word "Kgala", meaning "the great thirst", or "Kgalagadi", meaning "a waterless place"; the Kalahari has vast areas covered by red sand without any permanent surface water.


title: Kalahari Desert

text: The Kalahari Desert was not always a dry desert. The fossil flora and fauna from Gcwihaba Cave in Botswana indicates that the region was much wetter and cooler at least from 30 to 11 thousand BP (before present) especially after 17,500 BP.


title: Kalahari Desert

text: Drainage of the desert is by dry black valleys, seasonally inundated pans and the large salt pans of the Makgadikgadi Pan in Botswana and Etosha Pan in Namibia. The only permanent river, the Okavango, flows into a delta in the northwest, forming marshes that are rich in wildlife. Ancient dry riverbeds—called omuramba—traverse the central northern reaches of the Kalahari and provide standing pools of water during the rainy season.


title: Kalahari Desert

text: A semi-desert, with huge tracts of excellent grazing after good rains, the Kalahari supports more animals and plants than a true desert, such as the Namib Desert to the west. There are small amounts of rainfall and the summer temperature is very high. The driest areas usually receive of rain per year, and the wettest just a little over . The surrounding Kalahari Basin covers over extending farther into Botswana, Namibia and South Africa, and encroaching into parts of Angola, Zambia and Zimbabwe.


title: Kalahari Desert

text: Numerous pans exist within the Kalahari, including the Groot-vloer Pan and Verneukpan where evidence of a wetter climate exists in the form of former contouring for capturing of water. This and other pans, as well as river bottoms, were written about extensively at Sciforums by an article by Walter Wagner regarding the extensive formerly wet areas of the Kalahari. The Kalahari is extensive and extends farther north where abandoned extensive roadways also exist.


title: Kalahari Desert

text: North and east, approximately where the dry forests, savannahs and salt lakes prevail, the climate is sub-humid rather than semi-arid. South and west, where the vegetation is predominantly xeric savanna or even a semi-desert, the climate is "Kalaharian" semi-arid. The Kalaharian climate is subtropical (average annual temperature greater than or equal to 18 °C, at peaks reaching 40 °C and above, with mean monthly temperature of the coldest month strictly below 18 °C), and is semi-arid with the dry season during the "cold" season, the coldest six months of the year. It is the southern tropical equivalent of the Sahelian climate with the wet season during summer. The altitude has been adduced as the explanation why the Kalaharian climate is not tropical; its altitude ranges from 600 to 1600 meters (and generally from 800 to 1200 meters), resulting in a cooler climate than that of the Sahel or Sahara. For example, winter frost is common from June to August, something rarely seen in the warmer Sahelian regions. For the same reason, summer temperatures certainly can be very hot, but not in comparison to regions of low altitude in the Sahel or Sahara, where some stations record average temperatures of the warmest month around 38 °C, whereas the average temperature of the warmest month in any region in the Kalahari never exceeds 29 °C, though daily temperatures occasionally reach up to close to (44.8 °C at Twee Rivieren Rest Camp in 2012).


title: Kalahari Desert

text: The dry season lasts eight months or more, and the wet season typically from less than one month to four months, depending on location. The southwestern Kalahari is the driest area, in particular a small region located towards the west-southwest of Tsaraxaibis (Southeast of Namibia). The average annual rainfall ranging from around 110 mm (close to aridity) to more than 500 mm in some areas of the north and east. During summer time in all regions rainfall may go with heavy thunderstorms. In the driest and sunniest parts of the Kalahari, over 4,000 hours of sunshine are recorded annually on average.


title: Kalahari Desert

text: In the Kalahari, there are two main mechanisms of atmospheric circulation, dominated by the Kalahari High anticyclone:


title: Kalahari Desert

text: There are huge subterranean water reserves beneath parts of the Kalahari; the Dragon's Breath Cave, for example, is the largest documented non-subglacial underground lake. Such reserves may be in part the residues of ancient lakes; the Kalahari Desert was once a much wetter place. The ancient Lake Makgadikgadi dominated the area, covering the Makgadikgadi Pan and surrounding areas, but it drained or dried out some 10,000 years ago. It may have once covered as much as . In ancient times, there was sufficient moisture for farming, with dikes and dams collecting the water. These are now filled with sediment, breached, or no longer in use, though they can be readily seen via Google Earth.


title: Kalahari Desert

text: The Kalahari has had a complex climatic history over the past million or so years, in line with major global changes. Changes in the last 250,000 years have been reconstructed from various data sources, and provide evidence of both former extensive lakes and periods drier than now. During the latter the area of the Kalahari has expanded to include parts of western Zimbabwe, Zambia and Angola.


title: Kalahari Desert

text: Due to its low aridity, the Kalahari supports a variety of flora. The native flora includes acacia trees and many other herbs and grasses. The kiwano fruit, also known as the horned melon, melano, African horned cucumber, jelly melon, or hedged gourd, is endemic to a region in the Kalahari Desert (specific region unknown).


title: Kalahari Desert

text: Even where the Kalahari "desert" is dry enough to qualify as a desert in the sense of having low precipitation, it is not strictly speaking a desert because it has too dense a ground cover. The main region that lacks ground cover is in the southwest Kalahari (southeast of Namibia, northwest of South Africa and southwest of Botswana) in the south of the Kgalagadi Transfrontier Park. For instance in the ZF Mgcawu District Municipality of South Africa, total vegetation cover may be as low as 30.72% on non-protected (from cattle grazing) farmlands south of Twee Rivieren Rest Camp and 37.74% in the protected (from cattle grazing) South African side of the Kgalagadi Transfrontier Park: these southernmost Kalahari xeric savanna areas are truly semi-deserts. However, in all the remaining Kalahari, except on salt pans during the dry season, the vegetation cover can be clearly denser, up to almost 100% in some limited areas.


title: Kalahari Desert

text: In an area of about 600,000 km in the south and west of the Kalahari, the vegetation is mainly xeric savanna. This area is the ecoregion identified by World Wide Fund for Nature as Kalahari xeric savanna AT1309. Typical savanna grasses include "Schmidtia", "Stipagrostis", "Aristida", and "Eragrostis"; these are interspersed with trees such as camelthorn ("Acacia erioloba"), grey camelthorn ("Acacia haematoxylon"), shepherd's tree ("Boscia albitrunca"), blackthorn ("Acacia mellifera"), and silver cluster-leaf ("Terminalia sericea").


title: Kalahari Desert

text: In certain areas where the climate is drier, it becomes a true semi-desert with ground not entirely covered by vegetation: "open" as opposed to "closed" vegetation. Examples include the north of the ZF Mgcawu District Municipality, itself in the north of South Africa, and the Keetmanshoop Rural in the southeast of Namibia. In the north and east, there are dry forests covering an area of over 300,000 km in which Rhodesian teak and several species of acacia are prominent. These regions are termed Kalahari Acacia-Baikiaea woodlands AT0709.


title: Kalahari Desert

text: Outside the Kalahari "desert", but in the Kalahari basin, a halophytic vegetation to the north is adapted to pans, lakes that are completely dry during the dry season, and maybe for years during droughts, such as in Etosha (Etosha Pan halophytics AT0902) and Makgadikgadi (Zambezian halophytics AT0908).


title: Kalahari Desert

text: A totally different vegetation is adapted to the perennial fresh water of the Okavango Delta, an ecoregion termed Zambezian flooded grasslands AT0907.


title: Kalahari Desert

text: The Kalahari is home to many migratory birds and animals. Previously havens for wild animals from elephants to giraffes, and for predators such as lions and cheetahs, the riverbeds are now mostly grazing spots, though leopards and cheetahs can still be found. The area is now heavily grazed and cattle fences restrict the movement of wildlife. Among deserts of the Southern Hemisphere, the Kalahari most closely resembles some Australian deserts in its latitude and its mode of formation.


title: Kalahari Desert

text: Although there are few endemic species, a wide variety of species are found in the region, including large predators such as the lion ("Panthera leo"), cheetah ("Acinonyx jubatus"), leopard ("Panthera pardus"), spotted hyena ("Crocuta crocuta"), brown hyena ("Parahyaena brunnea"), and African wild dog ("Lycaon pictus pictus"). Birds of prey include the secretary bird ("Sagittarius serpentarius"), martial eagle ("Polemaetus bellicosus") and other eagles, the giant eagle owl ("Bubo lacteus") and other owls, falcons, goshawks, kestrels, and kites. Other animals include wildebeest, springbok, gemsbok and other antelopes, porcupines ("Hystrix africaeaustralis") and ostriches ("Struthio camelus").


title: Kalahari Desert

text: Some of the areas within the Kalahari are seasonal wetlands, such as the Makgadikgadi Pans of Botswana. This area, for example, supports numerous halophilic species, and in the rainy season, tens of thousands of flamingos visit these pans.


title: Kalahari Desert

text: The biggest threat to wildlife are the fences erected to manage herds of grazing cattle, a practice which also removes the plant cover of the savanna itself. Cattle ranchers will also poison or hunt down predators from the rangeland, particularly targeting jackals and wild dogs.


title: Kalahari Desert

text: The San people have lived in the Kalahari for 20,000 years as hunter-gatherers. They hunt wild game with bows and poison arrows and gather edible plants, such as berries, melons and nuts, as well as insects. The San get most of their water requirements from plant roots and desert melons found on or under the desert floor. They often store water in the blown-out shells of ostrich eggs. The San live in huts built from local materials—the frame is made of branches, and the roof is thatched with long grass. Most of their hunting and gathering techniques replicate pre-historic tribes. Bantu-speaking Tswana, Kgalagadi, and Herero and a small number of European settlers also live in the Kalahari desert. The city of Windhoek is situated in the Kalahari Basin.


title: Kalahari Desert

text: In 1996, De Beers evaluated the potential of diamond mining at Gope. In 1997, the eviction of the San and Bakgalagadi tribes in the Central Kalahari Game Reserve from their land began.


title: Kalahari Desert

text: In 2006, a Botswana High Court ruled in favor of the San and Bakgalagadi tribes in the reserve, finding that their eviction was unlawful. The Government of Botswana granted a permit to De Beers' Gem Diamonds/Gope Exploration Company (Pty) Ltd. to conduct mining activities within the reserve.


title: Katie Piper

text: Kate Elizabeth Sutton (née Piper; born 12 October 1983) is an English writer, activist, television presenter and model from Andover, Hampshire.


title: Katie Piper

text: In March 2008, she was attacked with acid by her ex-boyfriend and an accomplice, causing major damage to her face and blindness in one eye. Piper underwent pioneering surgery to restore her face and vision. Both attackers were convicted and given life sentences. In 2018, one of the attackers was released after serving nine years in prison.


title: Katie Piper

text: In 2009, Piper gave up her right to anonymity in order to increase awareness about burn victims. Her experience was documented in the 2009 Channel 4 documentary "Katie: My Beautiful Face" part of the "Cutting Edge" series. It has subsequently been sold internationally.


title: Katie Piper

text: More recently, Piper has appeared in follow-up series for Channel 4, released a best-selling autobiography, and had a regular column in weekly magazines. She also presents the Channel 4 television series "Bodyshockers" about members of the public about to undergo body-altering procedures and those who regret procedures they have had. Her main work has been for the Katie Piper Foundation, which she founded to help victims of burns and other disfigurement injuries. On 13 August 2018, she was announced as the first contestant for that year's "Strictly Come Dancing". On 11 September 2020 BBC media announced that Katie Piper would be joining their BBC1 "Songs of Praise" team as a feature presenter, scheduling her debut broadcast on 20 September. Piper had previously appeared as a guest, with her mother on the 2020 Mothers Day edition, when she explained the moment she found her faith. In July 2021, Piper became a panellist on "Loose Women".


title: Katie Piper

text: Piper was born in Andover, Hampshire, to David and Diane Piper and attended Harrow Way School and Portway Junior School as a child. She has an elder brother, Paul and a younger sister, Suzy.


title: Katie Piper

text: After leaving school, Piper trained as a beautician, aiming to build on her fondness for fashion and beauty, and to follow in the footsteps of her father, who worked as a barber.


title: Katie Piper

text: Piper subsequently began a career in modelling, taking part in various fashion, glamour and promotional photoshoots, including modelling for national newspapers. She also began work as a promotional model: appearing and carrying out publicity duties at live events, such as working as a ring-card girl at martial arts fixtures.


title: Katie Piper

text: Piper also conducted a career as a digital television presenter, working principally on web-TV shows and features, and on small digital television channels, primarily in the shopping and live-chat fields. As her career began to develop, Piper moved away from her family home in Hampshire, and began to live with friends in a flat in Golders Green, North London.


title: Katie Piper

text: Daniel Lynch, a martial arts enthusiast who had been tracking Piper's media and modelling career, met her through Facebook. The two first met in person in Reading, Berkshire, where Piper had been working, and initially Piper was pleased with the relationship.


title: Katie Piper

text: Two weeks into their relationship, the couple booked into a hotel in Bayswater, following a meal out. In the hotel room, Lynch raped and beat Piper, threatened to cut her with a razor and hang her, and stabbed her several times in the arms. After eight hours at the hotel, they drove back to Piper's Golders Green flat. Piper was treated for her wounds at Royal Free Hospital, but withheld the nature of the incident from the doctors and police, because she was afraid of Lynch.


title: Katie Piper

text: Piper received numerous phone calls and apologies from Lynch. On 31 March 2008, two days after the initial attack, Lynch persuaded Piper to go to an internet cafe to read an email he had sent to her Facebook account. Lynch gave her details to Stefan Sylvestre, who identified her on Golders Green Road. Wearing a hoodie to obscure his identity, Sylvestre approached Piper, who thought he was going to ask for money, and then threw sulphuric acid at her face.


title: Katie Piper

text: Lynch received two life sentences, and will serve a minimum of 16 years. Sylvestre received a life sentence, and was told he would serve a minimum of 6 years. Sylvestre's parole application for release was approved in 2018. In 2022 it was reported Sylvestre had breached the terms of his parole on numerous occasions and is now being hunted by police after going on the run and allegedly leaving the UK in August 2022, where he faces being recalled to prison when found.


title: Katie Piper

text: Immediately after the attack, Piper ran into a local café, where an ambulance was called. Piper was treated in Chelsea and Westminster Hospital, where her treatment was led by plastic surgeon Mohammad Ali Jawad. The acid, some of which Piper had swallowed, blinded her in her left eye, and caused partial thickness and full thickness burns. The surgeons completely removed the skin of Piper's face, and replaced it with a skin substitute, Matriderm, to build the foundations for a skin graft. This procedure was the first of its kind to be done in a single operation.


title: Katie Piper

text: Piper was put into an induced coma for 12 days. She has been through numerous surgical operations to treat her injuries, and wore a plastic face mask for 23 hours a day, which flattened the scars and helped retain moisture. As part of her care from the National Health Service, Piper was treated at a clinic in Southern France. The treatment she received there was designed to break down scar tissue, and prevent skin contraction.


title: Katie Piper

text: Subsequent to the attack, Piper moved out of her London flat and returned to Hampshire to live with her parents and younger sister Suzy.


title: Katie Piper

text: Near the end of 2009, Katie Piper established a charity, the Katie Piper Foundation, aimed at raising awareness of the plight of victims of burns and other disfigurement injuries: the charity also campaigns for the specialist treatment Piper received, such as the after-care scheme undertaken in France, to be more widely available to patients in Britain.


title: Katie Piper

text: Following her recovery and the successful establishment of the Katie Piper Foundation, Piper again moved out of the family home and returned to living alone in London, a process documented in episodes of "Katie: My Beautiful Face"; she later moved to live with her partner Richard Sutton, a carpenter.


title: Katie Piper

text: On 14 March 2014, she gave birth to her first child, Belle Elizabeth. In December 2014, she announced her engagement to Sutton. They married on 6 November 2015. On 13 December 2017, she gave birth to her second child, Penelope Diane.


title: Katie Piper

text: Although Piper had the right to remain anonymous because of the sexual assault, she chose to waive her anonymity, in an attempt to increase public awareness of the situation for burn victims, and also the treatment they go through. Piper also took part in a documentary about her experience, "Katie: My Beautiful Face", which was aired by Channel 4 on 29 October 2009, and, according to figures from Attentional, gained over 3.3 million viewers. The programme has been rebroadcast several times and in a number of territories.


title: Katie Piper

text: On 25 December 2009, Piper read the Alternative Christmas Message on Channel 4. The message was about Piper's own experiences, family and not judging people by their appearance.


title: Katie Piper

text: Piper also featured in one of the two 2011 Alternative Christmas Messages, alongside participants from "Seven Dwarves", "Beauty and the Beast: Ugly Face of Prejudice" and "My Transsexual Summer" - for a message with the theme "Just Be Yourself" (the other message featured senior staff from the school featured in "Educating Essex".)


title: Katie Piper

text: On 8 January 2010, the ABC (US)'s news-magazine television series "20/20" featured Piper as its primary subject. The programme consisted of an interview conducted by Elizabeth Vargas, and footage of Piper at home, including material which had appeared in "Katie: My Beautiful Face".


title: Katie Piper

text: Piper worked again with Mentorn Media, producer of "Katie: My Beautiful Face", in a series titled "Katie: My Beautiful Friends". In this, she met people who were disfigured, disabled or physically altered as a result of illness, injury, assault, accident or surgery. The films also chronicled the development and growth of the Katie Piper Foundation and began its four-part run on 22 March 2011. Items of supporting information and relevant associated content relating to the programme were placed on the Channel 4 website in tandem with the programme's broadcast. The series was broadcast in the US on the OWN channel from 16 August 2011.


title: Katie Piper

text: On 7 February 2012, Channel 4 broadcast a new one-off film featuring Piper as she prepared to undergo stem-cell surgery in a bid to restore sight in her damaged left eye. "Katie: The Science of Seeing Again" saw Piper look into the biology of the eye, visit the US to look into the religious and moral debate around the use of embryos in stem-cell research, and monitored her as she underwent the treatment by surgeon Sheraz Daya at Centre for Sight in East Grinstead, West Sussex. As with Piper's five previous Channel 4 films, this was produced by Mentorn Media. Within the programme Piper revealed she had undergone 109 medical operations in the four years since the acid attack, with the eye operation being her 110th.


title: Katie Piper

text: In spring 2013, at around the same time as Piper's participation in "Gok Live" was confirmed, her next solo Channel 4 project was announced, which was to be a new documentary series, provisionally titled "Undo Me", which would look at cosmetic and surgical procedures, and give advice to those looking to restore a more natural look having previously undergone treatments they now regret. It was initially slated for broadcast in late 2013 but eventually began to air weekly from 30 January 2014, by which time the proposed six-episode series had shrunk to four parts and acquired the name "Bodyshockers", with each episode having a sub-title linked to the episode's main theme.


title: Katie Piper

text: In summer 2014 it was confirmed that a second series of "Bodyshockers" had been commissioned, thus becoming the first of Piper's programmes to be reordered for a second run. This second series, which has seen the show acquire the sub-title "Nips, Tucks and Tattoos", began transmitting from 5 January 2015. In February 2015, it was announced that a third series had been ordered commissioned.


title: Katie Piper

text: Katie Piper has appeared on a number of television and radio programmes to talk about her experience; the Australian series "60 Minutes" featured Piper in November 2009. In the same month, she appeared on Channel 4's Krishnan Guru-Murthy-hosted television-led debate "The TV Show" to discuss the reaction to the original documentary.


title: Katie Piper

text: Piper has also appeared as a guest on a number of British magazine and news programmes including "Live From Studio Five", "Woman's Hour", "BBC Breakfast" and "This Morning".


title: Katie Piper

text: In October 2012, Piper participated in a new week-long series for Channel 4. "Hotel GB" saw a group of Channel 4 personalities - headed by Mary Portas, Gordon Ramsay, Gok Wan, Kirstie Allsopp, Phil Spencer, Christian Jessen, Kim Woodburn and Piper - running a London hotel with the aim of training a group of young unemployed people for potential careers in the hospitality industry, with viewers able to visit the hotel as guests and use the facilities. Piper drew on her past as a beautician to run the hotel's salon/spa area, assisted by one of the apprentices, Manisha.


title: Katie Piper

text: In 2013, Piper took part in "The Secret Millions", a "Secret Millionaire" spin-off in which each of five C4 hosts—Piper, Gok Wan, George Clarke, David Fishwick and Jimmy Doherty—work with a charitable or public-service organisation, running a scheme or pilot with the aid of disadvantaged or unemployed individuals, with the aim of convincing the Big Lottery Fund to give up to £2 million of funding (per scheme) for such projects to be rolled out more fully. Piper's episode aired on 14 April 2013 as the fourth of five transmitted; it featured a project in which prisoners were prepared for re-entry into the workplace by undertaking working placements in furniture manufacture. In the week leading up to the "Secret Millions" broadcast, The Community Channel repeated several of Piper's Channel 4 documentaries.


title: Katie Piper

text: In summer 2013, Piper worked alongside Gok Wan as a regular participant in a new live fashion series on Channel 4. The three-part "Gok Live: Stripping for Summer" was broadcast over three weeks from the plaza of MediaCityUK in Salford Quays, with Piper in attendance at each week's makeover, as well as filing filmed reports on summer beauty tips for each episode. The commission of "Undo Me" was confirmed around the same time as that of "Gok Live", but "Gok Live" went to air first.


title: Katie Piper

text: In 2018, Piper became a brand ambassador for Swarovski, alongside celebrity chef Nadiya Hussain and CoppaFeel founder, Kris Hallenga.


title: Katie Piper

text: On 13 August 2018, Piper was announced on BBC Breakfast as the first contestant in that year's "Strictly Come Dancing". She was partnered with professional dancer Gorka Marquez and was the third contestant to be eliminated.


title: Katie Piper

text: Piper's autobiography, called "Beautiful", was published in paperback by Ebury Press on 17 February 2011.


title: Katie Piper

text: Following the success of "Beautiful", Piper signed a three-book deal with publisher Quercus. The first of these was a self-help book titled "Things Get Better: If you believe then you will survive", followed by a page-a-day compilation of positive affirmations, quotes and mantras, "Start Your Day with Katie", both of which were published in 2012. A second volume of memoirs, titled "Beautiful Ever After", was released in 2014.


title: Katie Piper

text: In 2018, Piper and her mother published the book "From Mother to Daughter: The Things I'd Tell My Child" through Quercus.


title: Katie Piper

text: Piper wrote a regular column in "Reveal" magazine between April 2011 and July 2012. In October 2012 she began contributing a weekly column for "Now" magazine, which ran until January 2014. She has also written occasional one-off 'guest columns' for newspapers including the "Sunday Mirror" and its stablemate, the "Sunday People".


title: Katie Piper

text: "Katie: My Beautiful Face" was nominated for Best Single Documentary at the BAFTA Television Awards in June 2010, but did not win; the trophy was awarded to BBC One's "Wounded".


title: Katie Piper

text: Piper won the "Women to Watch: Inspiration" award at "Red" magazine's "Red's Hot Women Awards" in 2010.


title: Katie Piper

text: In February 2011, "Katie: My Beautiful Face" won the "Best Documentary Programme" award at the "Broadcast" Awards; Piper attended to collect the prize alongside the film-makers.


title: Katie Piper

text: Piper was appointed Officer of the Order of the British Empire (OBE) in the 2022 New Year Honours for services to charity and victims of burns and other disfigurement injuries.


title: Kelly criterion

text: In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. J. L. Kelly Jr, a researcher at Bell Labs, described the criterion in 1956. Because the Kelly Criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity), it is a scientific gambling method.


title: Kelly criterion

text: The practical use of the formula has been demonstrated for gambling and the same idea was used to explain diversification in investment management. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known successful investors including Warren Buffett and Bill Gross use Kelly methods. William Poundstone wrote an extensive popular account of the history of Kelly betting. Also see Intertemporal portfolio choice.


title: Kelly criterion

text: In a study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. But the behavior of the test subjects was far from optimal:


title: Kelly criterion

text: Using the Kelly criterion and based on the odds in the experiment (ignoring the cap of $250 and the finite duration of the test), the right approach would be to bet 20% of one's bankroll on each toss of the coin, which works out to a 2.034% average gain each round. This is a geometric mean, not the arithmetic rate of 4% (formula_1). The theoretical expected wealth after 300 rounds works out to $10,505 (formula_2) if it were not capped.


title: Kelly criterion

text: In this particular game, because of the cap, a strategy of betting only 12% of the pot on each toss would have even better results (a 95% probability of reaching the cap and an average payout of $242.03).


title: Kelly criterion

text: As an example, if a gamble has a 60% chance of winning (formula_10, formula_11), and the gambler receives 1-to-1 odds on a winning bet (formula_12), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (formula_13).


title: Kelly criterion

text: If the edge is negative (formula_15) the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in American roulette, the bettor is offered an even money payoff (formula_16) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (formula_17). The Kelly bet is formula_18, meaning the gambler should bet one-nineteenth of their bankroll that red will "not" come up. There is no explicit "anti-red" bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing.


title: Kelly criterion

text: A more general form of the Kelly formula allows for partial losses, which is relevant for investments:


title: Kelly criterion

text: Note that the Kelly Criterion is valid only for "known" outcome probabilities, which is not the case with investments. Risk averse investors should not invest the full Kelly fraction.


title: Kelly criterion

text: This formula can result in Kelly fractions higher than 1. In this case, it is theoretically advantageous to use leverage to purchase additional securities on margin.


title: Kelly criterion

text: Heuristic proofs of the Kelly criterion are straightforward. The Kelly criterion maximizes the expected value of the logarithm of wealth (the expectation value of a function is given by the sum, over all possible outcomes, of the probability of each particular outcome multiplied by the value of the function in the event of that outcome). We start with 1 unit of wealth and bet a fraction formula_28 of that wealth on an outcome that occurs with probability formula_29 and offers odds of formula_8. The probability of winning is formula_5, and in that case the resulting wealth is equal to formula_32. The probability of losing is formula_33, and in that case the resulting wealth is equal to formula_34. Therefore, the expected geometric growth rate formula_35 is:


title: Kelly criterion

text: We want to find the maximum "r" of this curve (as a function of "f"), which involves finding the derivative of the equation. This is more easily accomplished by taking the logarithm of each side first. The resulting equation is:


title: Kelly criterion

text: with formula_38 denoting logarithmic wealth growth. To find the value of formula_28 for which the growth rate is maximized, denoted as formula_4, we differentiate the above expression and set this equal to zero. This gives:


title: Kelly criterion

text: Notice that this expression reduces to the simple gambling formula when formula_44, when a loss results in full loss of the wager.


title: Kelly criterion

text: In a 1738 article, Daniel Bernoulli suggested that, when one has a choice of bets or investments, one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is entirely different (Bernoulli wanted to resolve the St. Petersburg paradox).


title: Kelly criterion

text: An English-language translation of the Bernoulli article was not published until 1954, but the work was well known among mathematicians and economists.


title: Kelly criterion

text: In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is "growth optimal."


title: Kelly criterion

text: Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty. If portfolio weights are largely a function of estimation errors, then "Ex-post" performance of a growth-optimal portfolio may differ fantastically from the "ex-ante" prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less than the Kelly criterion (e.g., half).


title: Kelly criterion

text: Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.


title: Kelly criterion

text: For a rigorous and general proof, see Kelly's original paper or some of the other references listed below. Some corrections have been published.


title: Kelly criterion

text: We give the following non-rigorous argument for the case with formula_16 (a 50:50 "even money" bet) to show the general idea and provide some insights.


title: Kelly criterion

text: When formula_16, a Kelly bettor bets formula_47 times their initial wealth formula_48, as shown above. If they win, they have formula_49 after one bet. If they lose, they have formula_50. Suppose they make formula_51 bets like this, and win formula_52 times out of this series of formula_51 bets. The resulting wealth will be:


title: Kelly criterion

text: Note that the ordering of the wins and losses does not affect the resulting wealth. Suppose another bettor bets a different amount, formula_55 for some value of formula_56 (where formula_56 may be positive or negative). They will have formula_58 after a win and formula_59 after a loss. After the same series of wins and losses as the Kelly bettor, they will have:


title: Kelly criterion

text: So in the long run, final wealth is maximized by setting formula_56 to zero, which means following the Kelly strategy.


title: Kelly criterion

text: This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the K win knowing that the rest of the bets will lose), one will end up with the most money if one bets:


title: Kelly criterion

text: each time. This is true whether formula_51 is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as formula_51 gets large, formula_52 will approach formula_71. Someone who bets more than Kelly can do better if formula_72 for a stretch; someone who bets less than Kelly can do better if formula_73 for a stretch, but in the long run, Kelly always wins.


title: Kelly criterion

text: In a single trial, if you invest the fraction formula_28 of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor formula_75, and, likewise, if the strategy fails, you end up having your capital decreased by the factor formula_34. Thus at the end of formula_51 trials (with formula_71 successes and formula_79 failures), the starting capital of $1 yields


title: Kelly criterion

text: Maximizing formula_81, and consequently formula_82, with respect to formula_28 leads to the desired result


title: Kelly criterion

text: Edward O. Thorp provided a more detailed discussion of this formula for the general case. There, it can be seen that the substitution of formula_5 for the ratio of the number of "successes" to the number of trials implies that the number of trials must be very large, since formula_5 is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting formula_4 each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and formula_5 and formula_8 are the same for each trial. In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. In the heuristic proof above, formula_71 successes and formula_79 failures are highly likely only for very large formula_51.


title: Kelly criterion

text: Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that the formula_93-th horse wins the race is formula_94, the total amount of bets placed on formula_93-th horse is formula_96, and


title: Kelly criterion

text: where formula_98 are the pay-off odds. formula_99, is the dividend rate where formula_100 is the track take or tax, formula_101 is the revenue rate after deduction of the track take when formula_93-th horse wins. The fraction of the bettor's funds to bet on formula_93-th horse is formula_104. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set formula_105 of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions formula_106 of bettor's wealth to be bet on the outcomes included in the optimal set formula_105.


title: Kelly criterion

text: If the optimal set formula_105 is empty then do not bet at all. If the set formula_105 of optimal outcomes is not empty, then the optimal fraction formula_106 to bet on formula_93-th outcome may be calculated from this formula:


title: Kelly criterion

text: where the right hand-side is the reserve rate. Therefore, the requirement formula_115 may be interpreted as follows: formula_93-th outcome is included in the set formula_105 of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction formula_106 may be interpreted as the excess of the expected revenue rate of formula_93-th horse over the reserve rate divided by the revenue after deduction of the track take when formula_93-th horse wins or as the excess of the probability of formula_93-th horse winning over the reserve rate divided by revenue after deduction of the track take when formula_93-th horse wins. The binary growth exponent is


title: Kelly criterion

text: This method of selection of optimal bets may be applied also when probabilities formula_94 are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that


title: Kelly criterion

text: The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be easily estimated from existing historical data – expected value and variance. This approximation leads to results that are robust and offer similar results as the original criterion.


title: Kelly criterion

text: For single assets(stock, index fund, etc.), and a risk-free rate, it is easy to obtain the optimal fraction to invest through geometric Brownian motion.


title: Kelly criterion

text: from the solution of the geometric Brownian motion where formula_145 is a Wiener process, and formula_146 (percentage drift) and formula_147 (the percentage volatility) are constants. Taking expectations of the logarithm:


title: Kelly criterion

text: For a portfolio made of an asset formula_141 and a bond paying risk-free rate formula_35, with fraction formula_28 invested in formula_141 and formula_155 in the bond, the expected one-period return is given by


title: Kelly criterion

text: however people seem to deal with the expected log return formula_157 for one-period instead in the context of Kelly:


title: Kelly criterion

text: formula_161 is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction.


title: Kelly criterion

text: Remember that formula_162 is different from the asset log return formula_149. Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion.


title: Kelly criterion

text: For multiple assets, consider a market with formula_164 correlated stocks formula_165 with stochastic returns formula_166, formula_167 and a riskless bond with return formula_35. An investor puts a fraction formula_169 of their capital in formula_165 and the rest is invested in the bond. Without loss of generality, assume that investor's starting capital is equal to 1.


title: Kelly criterion

text: where formula_175 and formula_176 are the vector of means and the matrix of second mixed noncentral moments of the excess returns.


title: Kelly criterion

text: There is also a numerical algorithm for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints.


title: Platonic solid

text: In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:


title: Platonic solid

text: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the "Timaeus", that the classical elements were made of these regular solids.


title: Platonic solid

text: It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetric.


title: Platonic solid

text: The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.


title: Platonic solid

text: The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue "Timaeus" 360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth.


title: Platonic solid

text: Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithēr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.


title: Platonic solid

text: Euclid completely mathematically described the Platonic solids in the "Elements", the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.


title: Platonic solid

text: Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the "Elements". Much of the information in Book XIII is probably derived from the work of Theaetetus.


title: Platonic solid

text: In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In "Mysterium Cosmographicum", published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the Kepler solids, which are two "nonconvex" regular polyhedra.


title: Platonic solid

text: For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below. The Greek letter "φ" is used to represent the golden ratio ≈ 1.6180.


title: Platonic solid

text: The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign (central symmetry), or, in the other cases, by exchanging two coordinates (reflection with respect to any of the three diagonal planes).


title: Platonic solid

text: These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or , one of two sets of 4 vertices in dual positions, as h{4,3} or . Both tetrahedral positions make the compound stellated octahedron.


title: Platonic solid

text: The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t{3,4} or , also called a "snub octahedron", as s{3,4} or , and seen in the compound of two icosahedra.


title: Platonic solid

text: Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the compound of five cubes.


title: Platonic solid

text: The symbol {"p", "q"}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.


title: Platonic solid

text: All other combinatorial information about these solids, such as total number of vertices ("V"), edges ("E"), and faces ("F"), can be determined from "p" and "q". Since any edge joins two vertices and has two adjacent faces we must have:


title: Platonic solid

text: This can be proved in many ways. Together these three relationships completely determine "V", "E", and "F":


title: Platonic solid

text: Swapping "p" and "q" interchanges "F" and "V" while leaving "E" unchanged. For a geometric interpretation of this property, see .


title: Platonic solid

text: The elements of a polyhedron can be expressed in a configuration matrix. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other.


title: Platonic solid

text: The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question—one that requires an explicit construction.


title: Platonic solid

text: A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that "V" − "E" + "F" = 2, and the fact that "pF" = 2"E" = "qV", where "p" stands for the number of edges of each face and "q" for the number of edges meeting at each vertex. Combining these equations one obtains the equation


title: Platonic solid

text: Using the fact that "p" and "q" must both be at least 3, one can easily see that there are only five possibilities for {"p", "q"}:


title: Platonic solid

text: There are a number of angles associated with each Platonic solid. The dihedral angle is the interior angle between any two face planes. The dihedral angle, "θ", of the solid {"p","q"} is given by the formula


title: Platonic solid

text: The quantity "h" (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.


title: Platonic solid

text: The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2. The defect, "δ", at any vertex of the Platonic solids {"p","q"} is


title: Platonic solid

text: By a theorem of Descartes, this is equal to 4 divided by the number of vertices (i.e. the total defect at all vertices is 4).


title: Platonic solid

text: The 3-dimensional analog of a plane angle is a solid angle. The solid angle, "Ω", at the vertex of a Platonic solid is given in terms of the dihedral angle by


title: Platonic solid

text: This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {"p","q"} is a regular "q"-gon.


title: Platonic solid

text: The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4 steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.


title: Platonic solid

text: The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in steradians. The constant "φ" = is the golden ratio.


title: Platonic solid

text: The radii of these spheres are called the "circumradius", the "midradius", and the "inradius". These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius "R" and the inradius "r" of the solid {"p", "q"} with edge length "a" are given by


title: Platonic solid

text: where "h" is the quantity used above in the definition of the dihedral angle ("h" = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in "p" and "q":


title: Platonic solid

text: The surface area, "A", of a Platonic solid {"p", "q"} is easily computed as area of a regular "p"-gon times the number of faces "F". This is:


title: Platonic solid

text: The volume is computed as "F" times the volume of the pyramid whose base is a regular "p"-gon and whose height is the inradius "r". That is,


title: Platonic solid

text: The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, "a", to be equal to 2.


title: Platonic solid

text: Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.


title: Platonic solid

text: For an arbitrary point in the space of a Platonic solid with circumradius "R", whose distances to the centroid of the Platonic solid and its "n" vertices are "L" and "d" respectively, and


title: Platonic solid

text: If "d" are the distances from the "n" vertices of the Platonic solid to any point on its circumscribed sphere, then


title: Platonic solid

text: A polyhedron "P" is said to have the "Rupert" property if a polyhedron of the same or larger size and the same shape as "P" can pass through a hole in "P".


title: Platonic solid

text: Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.


title: Platonic solid

text: If a polyhedron has Schläfli symbol {"p", "q"}, then its dual has the symbol {"q", "p"}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.


title: Platonic solid

text: One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.


title: Platonic solid

text: More generally, one can dualize a Platonic solid with respect to a sphere of radius "d" concentric with the solid. The radii ("R", "ρ", "r") of a solid and those of its dual ("R"*, "ρ"*, "r"*) are related by


title: Platonic solid

text: Dualizing with respect to the midsphere ("d" = "ρ") is often convenient because the midsphere has the same relationship to both polyhedra. Taking "d" = "Rr" yields a dual solid with the same circumradius and inradius (i.e. "R"* = "R" and "r"* = "r").


title: Platonic solid

text: In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant. The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the "full symmetry group", which includes reflections, and the "proper symmetry group", which includes only rotations.


title: Platonic solid

text: The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is "regular" if and only if it is vertex-uniform, edge-uniform, and face-uniform.


title: Platonic solid

text: There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are:


title: Platonic solid

text: The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are "centrally symmetric," meaning they are preserved under reflection through the origin.


title: Platonic solid

text: The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.


title: Platonic solid

text: The tetrahedron, cube, and octahedron all occur naturally in crystal structures. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the pyritohedron (named for the group of minerals of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular. Allotropes of boron and many boron compounds, such as boron carbide, include discrete B icosahedra within their crystal structures. Carborane acids also have molecular structures approximating regular icosahedra.


title: Platonic solid

text: In the early 20th century, Ernst Haeckel described (Haeckel, 1904) a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples include "Circoporus octahedrus", "Circogonia icosahedra", "Lithocubus geometricus" and "Circorrhegma dodecahedra". The shapes of these creatures should be obvious from their names.


title: Platonic solid

text: Many viruses, such as the herpes virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.


title: Platonic solid

text: In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which employ geodesic grids that are based on an icosahedron (refined by triangulation) instead of the more commonly used longitude/latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty.


title: Platonic solid

text: Geometry of space frames is often based on platonic solids. In the MERO system, Platonic solids are used for naming convention of various space frame configurations. For example, O+T refers to a configuration made of one half of octahedron and a tetrahedron.


title: Platonic solid

text: Several Platonic hydrocarbons have been synthesised, including cubane and dodecahedrane and not tetrahedrane.


title: Platonic solid

text: Platonic solids are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as d"n" where "n" is the number of faces (d8, d20, etc.); see dice notation for more details.


title: Platonic solid

text: These shapes frequently show up in other games or puzzles. Puzzles similar to a Rubik's Cube come in all five shapes – see magic polyhedra.


title: Platonic solid

text: For the intermediate material phase called liquid crystals, the existence of such symmetries was first proposed in 1981 by H. Kleinert and K. Maki.


title: Platonic solid

text: In aluminum the icosahedral structure was discovered three years after this by Dan Shechtman, which earned him the Nobel Prize in Chemistry in 2011.


title: Platonic solid

text: There exist four regular polyhedra that are not convex, called Kepler–Poinsot polyhedra. These all have icosahedral symmetry and may be obtained as stellations of the dodecahedron and the icosahedron.


title: Platonic solid

text: The next most regular convex polyhedra after the Platonic solids are the cuboctahedron, which is a rectification of the cube and the octahedron, and the icosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both "quasi-regular", meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen Archimedean solids, which are the convex uniform polyhedra with polyhedral symmetry. Their duals, the rhombic dodecahedron and rhombic triacontahedron, are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen Catalan solids.


title: Platonic solid

text: The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of regular or star polygons for faces. These include all the polyhedra mentioned above together with an infinite set of prisms, an infinite set of antiprisms, and 53 other non-convex forms.


title: Platonic solid

text: The Johnson solids are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex deltahedra, which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)


title: Platonic solid

text: The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the sphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the hosohedra, {2,"n"} with 2 vertices at the poles, and lune faces, and the dual dihedra, {"n",2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.


title: Platonic solid

text: One can show that every regular tessellation of the sphere is characterized by a pair of integers {"p", "q"} with  +  > . Likewise, a regular tessellation of the plane is characterized by the condition  +  = . There are three possibilities:


title: Platonic solid

text: In a similar manner, one can consider regular tessellations of the hyperbolic plane. These are characterized by the condition  +  < . There is an infinite family of such tessellations.


title: Platonic solid

text: In more than three dimensions, polyhedra generalize to polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids.


title: Platonic solid

text: In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids : 5-cell as {3,3,3}, 16-cell as {3,3,4}, 600-cell as {3,3,5}, tesseract as {4,3,3}, and 120-cell as {5,3,3}, and a sixth one, the self-dual 24-cell, {3,4,3}.


title: Platonic solid

text: In all dimensions higher than four, there are only three convex regular polytopes: the simplex as {3,3...,3}, the hypercube as {4,3...,3}, and the cross-polytope as {3,3...,4}. In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.


title: Monica Keena

text: Monica Keena (born May 28, 1979) is an American actress. She made her film debut in "While You Were Sleeping" (1995), and a lead role as Lilliana "Lilli" Hoffman in "" (1997).


title: Monica Keena

text: She later starred as Lori Campbell in "Freddy vs. Jason" (2003), and has had roles in the films "Brooklyn Rules" (2007) and the 2009 remake of "Night of the Demons". Keena has also worked in television, starring as Abby Morgan on "Dawson's Creek", Rachel Lindquist on the short-lived comedy "Undeclared", and Kristen on HBO's "Entourage".


title: Monica Keena

text: Keena was born and raised in Brooklyn, New York. Her parents are William Junior (a financial sales manager with the firm Paine Webber) and Mary Catherine Keena (a nurse). She has a sister, Samantha, who is two years older. Monica attended Saint Ann's School, a progressive private school in Brooklyn Heights.


title: Monica Keena

text: She auditioned for acceptance into LaGuardia High School of Performing Arts at age 13. Although she chose the drama department as her preference, she was accepted in both the dramatic and the vocal arts departments. Soon after starting her classes at LaGuardia, Keena played in her first role in a short film, "Burning Love".


title: Monica Keena

text: Keena portrayed Bertha in a stage reading of Strindberg's "The Father" and had her first starring role playing Lilliana "Lilli" Hoffman in the folktale-inspired film "". She has appeared in numerous television and movie projects including the TBS Original Movie "First Daughter" and "Crime and Punishment in Suburbia" which appeared at the 2000 Sundance Film Festival.


title: Monica Keena

text: Monica appeared in the film "While You Were Sleeping", "The Simian Line" with Lynn Redgrave, and "Bad Girls From Valley High" with the film eventually released straight to DVD in 2005. In 2003, she played the heroine Lori Campbell in "Freddy vs. Jason". In 2006, she played Celia in the film "Left in Darkness".


title: Monica Keena

text: Keena had a recurring role as Abby Morgan on "Dawson's Creek" and a starring role on the short-lived "Undeclared" as the college student Rachel. Keena also had guest roles on series such as "Law & Order", "", "", "Feds", and "Entourage". On the hit ABC medical drama "Grey's Anatomy", she appeared in the second season episode "Into You Like A Train" in which she played Bonnie, a patient who was severely injured in a massive train accident. She later reappeared in the third season episode "Some Kind of Miracle" to reprise her role as Bonnie.


title: Monica Keena

text: Keena appears as Maddie Curtis in Adam Gierasch's straight-to-video 2009 remake "Night of the Demons". It was during filming that she became romantically involved with Edward Furlong. He has been arrested three times for attacking her, and on January 15, 2013, was remanded in jail as he was unable to pay bail. Despite years of trouble the two have remained together and also continue to care for Furlong's son and have been seen together on both of their Instagrams and continue to attend fan conventions.


title: Abundance of the chemical elements

text: The abundance of the chemical elements is a measure of the occurrence of the chemical elements relative to all other elements in a given environment. Abundance is measured in one of three ways: by the mass-fraction (the same as weight fraction); by the mole-fraction (fraction of atoms by numerical count, or sometimes fraction of molecules in gases); or by the volume-fraction. Volume-fraction is a common abundance measure in mixed gases such as planetary atmospheres, and is similar in value to molecular mole-fraction for gas mixtures at relatively low densities and pressures, and ideal gas mixtures. Most abundance values in this article are given as mass-fractions.


title: Abundance of the chemical elements

text: For example, the abundance of oxygen in pure water can be measured in two ways: the "mass fraction" is about 89%, because that is the fraction of water's mass which is oxygen. However, the "mole-fraction" is about 33% because only 1 atom of 3 in water, HO, is oxygen. As another example, looking at the "mass-fraction" abundance of hydrogen and helium in both the Universe as a whole and in the atmospheres of gas-giant planets such as Jupiter, it is 74% for hydrogen and 23–25% for helium; while the "(atomic) mole-fraction" for hydrogen is 92%, and for helium is 8%, in these environments. Changing the given environment to Jupiter's outer atmosphere, where hydrogen is diatomic while helium is not, changes the "molecular" mole-fraction (fraction of total gas molecules), as well as the fraction of atmosphere by volume, of hydrogen to about 86%, and of helium to 13%.


title: Abundance of the chemical elements

text: The abundance of chemical elements in the universe is dominated by the large amounts of hydrogen and helium which were produced in the Big Bang. Remaining elements, making up only about 2% of the universe, were largely produced by supernovae and certain red giant stars. Lithium, beryllium, and boron, despite their low atomic number, are rare because, although they are produced by nuclear fusion, they are destroyed by other reactions in the stars. The elements from carbon to iron are relatively more abundant in the universe because of the ease of making them in supernova nucleosynthesis. Elements of higher atomic number than iron (element 26) become progressively rarer in the universe, because they increasingly absorb stellar energy in their production. Also, elements with even atomic numbers are generally more common than their neighbors in the periodic table, due to favorable energetics of formation.


title: Abundance of the chemical elements

text: The abundance of elements in the Sun and outer planets is similar to that in the universe. Due to solar heating, the elements of Earth and the inner rocky planets of the Solar System have undergone an additional depletion of volatile hydrogen, helium, neon, nitrogen, and carbon (which volatilizes as methane). The crust, mantle, and core of the Earth show evidence of chemical segregation plus some sequestration by density. Lighter silicates of aluminium are found in the crust, with more magnesium silicate in the mantle, while metallic iron and nickel compose the core. The abundance of elements in specialized environments, such as atmospheres, or oceans, or the human body, are primarily a product of chemical interactions with the medium in which they reside.


title: Abundance of the chemical elements

text: The elements – that is, ordinary (baryonic) matter made of protons, neutrons, and electrons, are only a small part of the content of the Universe. Cosmological observations suggest that only 4.6% of the universe's energy (including the mass contributed by energy, "E" = "mc" ⇔ "m" = "E" / "c") comprises the visible baryonic matter that constitutes stars, planets, and living beings. The rest is thought to be made up of dark energy (68%) and dark matter (27%). These are forms of matter and energy believed to exist on the basis of scientific theory and inductive reasoning based on observations, but they have not been directly observed and their nature is not well understood.


title: Abundance of the chemical elements

text: Most standard (baryonic) matter is found in intergalactic gas, stars, and interstellar clouds, in the form of atoms or ions (plasma), although it can be found in degenerate forms in extreme astrophysical settings, such as the high densities inside white dwarfs and neutron stars.


title: Abundance of the chemical elements

text: Hydrogen is the most abundant element in the Universe; helium is second. However, after this, the rank of abundance does not continue to correspond to the atomic number; oxygen has abundance rank 3, but atomic number 8. All others are substantially less common.


title: Abundance of the chemical elements

text: The abundance of the lightest elements is well predicted by the standard cosmological model, since they were mostly produced shortly (i.e., within a few hundred seconds) after the Big Bang, in a process known as Big Bang nucleosynthesis. Heavier elements were mostly produced much later, inside of stars.


title: Abundance of the chemical elements

text: Hydrogen and helium are estimated to make up roughly 74% and 24% of all baryonic matter in the universe respectively. Despite comprising only a very small fraction of the universe, the remaining "heavy elements" can greatly influence astronomical phenomena. Only about 2% (by mass) of the Milky Way galaxy's disk is composed of heavy elements.


title: Abundance of the chemical elements

text: These other elements are generated by stellar processes. In astronomy, a "metal" is any element other than hydrogen or helium. This distinction is significant because hydrogen and helium are the only elements that were produced in significant quantities in the Big Bang. Thus, the metallicity of a galaxy or other object is an indication of stellar activity after the Big Bang.


title: Abundance of the chemical elements

text: In general, elements up to iron are made in large stars in the process of becoming supernovae. Iron-56 is particularly common, since it is the most stable nuclide (in that it has the highest nuclear binding energy per nucleon) and can easily be made from alpha particles (being a product of decay of radioactive nickel-56, ultimately made from 14 helium nuclei). Elements heavier than iron are made in energy-absorbing processes in large stars, and their abundance in the universe (and on Earth) generally decreases with increasing atomic number.


title: Abundance of the chemical elements

text: The table shows the ten most common elements in our galaxy (estimated spectroscopically), as measured in parts per million, by mass.


title: Abundance of the chemical elements

text: Nearby galaxies that have evolved along similar lines have a corresponding enrichment of elements heavier than hydrogen and helium. The more distant galaxies are being viewed as they appeared in the past, so their abundances of elements appear closer to the primordial mixture. Since physical laws and processes are uniform throughout the universe, however, it is expected that these galaxies will likewise have evolved similar abundances of elements.


title: Abundance of the chemical elements

text: The abundance of elements is in keeping with their origin from the Big Bang and nucleosynthesis in a number of progenitor supernova stars. Very abundant hydrogen and helium are products of the Big Bang, while the next three elements are rare since they had little time to form in the Big Bang and are not made in stars (they are, however, produced in small quantities by breakup of heavier elements in interstellar dust, as a result of impact by cosmic rays).


title: Abundance of the chemical elements

text: Beginning with carbon, elements have been produced in stars by buildup from alpha particles (helium nuclei), resulting in an alternatingly larger abundance of elements with even atomic numbers (these are also more stable). The effect of odd-numbered chemical elements generally being more rare in the universe was empirically noticed in 1914, and is known as the Oddo-Harkins rule.


title: Abundance of the chemical elements

text: Loose correlations have been observed between estimated elemental abundances in the universe and the nuclear binding energy curve. Roughly speaking, the relative stability of various atomic nuclides has exerted a strong influence on the relative abundance of elements formed in the Big Bang, and during the development of the universe thereafter.


title: Abundance of the chemical elements

text: See the article about nucleosynthesis for an explanation of how certain nuclear fusion processes in stars (such as carbon burning, etc.) create the elements heavier than hydrogen and helium.


title: Abundance of the chemical elements

text: A further observed peculiarity is the jagged alternation between relative abundance and scarcity of adjacent atomic numbers in the elemental abundance curve, and a similar pattern of energy levels in the nuclear binding energy curve. This alternation is caused by the higher relative binding energy (corresponding to relative stability) of even atomic numbers compared with odd atomic numbers and is explained by the Pauli Exclusion Principle.


title: Abundance of the chemical elements

text: The semi-empirical mass formula (SEMF), also called "Weizsäcker's formula" or the "Bethe-Weizsäcker mass formula", gives a theoretical explanation of the overall shape of the curve of nuclear binding energy.


title: Abundance of the chemical elements

text: The Earth formed from the same cloud of matter that formed the Sun, but the planets acquired different compositions during the formation and evolution of the solar system. In turn, the natural history of the Earth caused parts of this planet to have differing concentrations of the elements.


title: Abundance of the chemical elements

text: The mass of the Earth is approximately 5.97 kg. In bulk, by mass, it is composed mostly of iron (32.1%), oxygen (30.1%), silicon (15.1%), magnesium (13.9%), sulfur (2.9%), nickel (1.8%), calcium (1.5%), and aluminium (1.4%); with the remaining 1.2% consisting of trace amounts of other elements.


title: Abundance of the chemical elements

text: The bulk composition of the Earth by elemental-mass is roughly similar to the gross composition of the solar system, with the major differences being that Earth is missing a great deal of the volatile elements hydrogen, helium, neon, and nitrogen, as well as carbon which has been lost as volatile hydrocarbons. The remaining elemental composition is roughly typical of the "rocky" inner planets, which formed in the thermal zone where solar heat drove volatile compounds into space. The Earth retains oxygen as the second-largest component of its mass (and largest atomic-fraction), mainly from this element being retained in silicate minerals which have a very high melting point and low vapor pressure.


title: Abundance of the chemical elements

text: The mass-abundance of the nine most abundant elements in the Earth's crust is approximately: oxygen 46%, silicon 28%, aluminium 8.3%, iron 5.6%, calcium 4.2%, sodium 2.5%, magnesium 2.4%, potassium 2.0%, and titanium 0.61%. Other elements occur at less than 0.15%. For a complete list, see abundance of elements in Earth's crust.


title: Abundance of the chemical elements

text: The graph at right illustrates the relative atomic-abundance of the chemical elements in Earth's upper continental crust—the part that is relatively accessible for measurements and estimation.


title: Abundance of the chemical elements

text: Note that there are two breaks where the unstable (radioactive) elements technetium (atomic number 43) and promethium (atomic number 61) would be. These elements are surrounded by stable elements, yet their most stable isotopes have relatively short half lives (~4 million years and ~18 years respectively). These are thus extremely rare, since any primordial initial fractions of these in pre-Solar System materials have long since decayed. These two elements are now only produced naturally through the spontaneous fission of very heavy radioactive elements (for example, uranium, thorium, or the trace amounts of plutonium that exist in uranium ores), or by the interaction of certain other elements with cosmic rays. Both technetium and promethium have been identified spectroscopically in the atmospheres of stars, where they are produced by ongoing nucleosynthetic processes.


title: Abundance of the chemical elements

text: There are also breaks in the abundance graph where the six noble gases would be, since they are not chemically bound in the Earth's crust, and they are only generated in the crust by decay chains from radioactive elements, and are therefore extremely rare there.


title: Abundance of the chemical elements

text: The eight naturally occurring very rare, highly radioactive elements (polonium, astatine, francium, radium, actinium, protactinium, neptunium, and plutonium) are not included, since any of these elements that were present at the formation of the Earth have decayed away eons ago, and their quantity today is negligible and is only produced from the radioactive decay of uranium and thorium.


title: Abundance of the chemical elements

text: Oxygen and silicon are notably the most common elements in the crust. On Earth and in rocky planets in general, silicon and oxygen are far more common than their cosmic abundance. The reason is that they combine with each other to form silicate minerals. Other cosmically-common elements such as hydrogen, carbon and nitrogen form volatile compounds such as ammonia and methane that easily boil away into space from the heat of planetary formation and/or the Sun's light.


title: Abundance of the chemical elements

text: "Rare" earth elements is a historical misnomer. The persistence of the term reflects unfamiliarity rather than true rarity. The more abundant rare earth elements are similarly concentrated in the crust compared to commonplace industrial metals such as chromium, nickel, copper, zinc, molybdenum, tin, tungsten, or lead. The two least abundant rare earth elements (thulium and lutetium) are nearly 200 times more common than gold. However, in contrast to the ordinary base and precious metals, rare earth elements have very little tendency to become concentrated in exploitable ore deposits. Consequently, most of the world's supply of rare earth elements comes from only a handful of sources. Furthermore, the rare earth metals are all quite chemically similar to each other, and they are thus quite difficult to separate into quantities of the pure elements.


title: Abundance of the chemical elements

text: Differences in abundances of individual rare earth elements in the upper continental crust of the Earth represent the superposition of two effects, one nuclear and one geochemical. First, the rare earth elements with even atomic numbers (Ce, Nd, ...) have greater cosmic and terrestrial abundances than the adjacent rare earth elements with odd atomic numbers (La, Pr, ...). Second, the lighter rare earth elements are more incompatible (because they have larger ionic radii) and therefore more strongly concentrated in the continental crust than the heavier rare earth elements. In most rare earth ore deposits, the first four rare earth elements – lanthanum, cerium, praseodymium, and neodymium – constitute 80% to 99% of the total amount of rare earth metal that can be found in the ore.


title: Abundance of the chemical elements

text: The mass-abundance of the seven most abundant elements in the Earth's mantle is approximately: oxygen 44.3%, magnesium 22.3%, silicon 21.3%, iron 6.32%, calcium 2.48%, aluminium 2.29%, nickel 0.19%.


title: Abundance of the chemical elements

text: Due to mass segregation, the core of the Earth is believed to be primarily composed of iron (88.8%), with smaller amounts of nickel (5.8%), sulfur (4.5%), and less than 1% trace elements.


title: Abundance of the chemical elements

text: The most abundant elements in the ocean by proportion of mass in percent are oxygen (85.84%), hydrogen (10.82%), chlorine (1.94%), sodium (1.08%), magnesium (0.13%), sulfur (0.09%), calcium (0.04%), potassium (0.04%), bromine (0.007%), carbon (0.003%), and boron (0.0004%).


title: Abundance of the chemical elements

text: The order of elements by volume-fraction (which is approximately molecular mole-fraction) in the atmosphere is nitrogen (78.1%), oxygen (20.9%), argon (0.96%), followed by (in uncertain order) carbon and hydrogen because water vapor and carbon dioxide, which represent most of these two elements in the air, are variable components. Sulfur, phosphorus, and all other elements are present in significantly lower proportions.


title: Abundance of the chemical elements

text: According to the abundance curve graph, argon, a significant if not major component of the atmosphere, does not appear in the crust at all. This is because the atmosphere has a far smaller mass than the crust, so argon remaining in the crust contributes little to mass-fraction there, while at the same time buildup of argon in the atmosphere has become large enough to be significant.


title: Abundance of the chemical elements

text: "For a complete list of the abundance of elements in urban soils, see Abundances of the elements (data page)#Urban soils."


title: Abundance of the chemical elements

text: By mass, human cells consist of 65–90% water (HO), and a significant portion of the remainder is composed of carbon-containing organic molecules. Oxygen therefore contributes a majority of a human body's mass, followed by carbon. Almost 99% of the mass of the human body is made up of six elements: hydrogen (H), carbon (C), nitrogen (N), oxygen (O), calcium (Ca), and phosphorus (P) . The next 0.75% is made up of the next five elements: potassium (K), sulfur (S), chlorine (Cl), sodium (Na), and magnesium (Mg). Only 17 elements are known for certain to be necessary to human life, with one additional element (fluorine) thought to be helpful for tooth enamel strength. A few more trace elements may play some role in the health of mammals. Boron and silicon are notably necessary for plants but have uncertain roles in animals. The elements aluminium and silicon, although very common in the earth's crust, are conspicuously rare in the human body.


title: Protectorate

text: A protectorate, in the context of international relations, is a state that is under protection by another state for defence against aggression and other violations of law. It is a dependent territory that enjoys autonomy over most of its internal affairs, while still recognizing the suzerainty of a more powerful sovereign state without being a possession. In exchange, the protectorate usually accepts specified obligations depending on the terms of their arrangement. Usually protectorates are established de jure by a treaty. Under certain conditions—as with Egypt under British rule (1882–1914)—a state can also be labelled as a de facto protectorate or a veiled protectorate.


title: Protectorate

text: A protectorate is different from a colony as it has local rulers, is not directly possessed, and rarely experiences colonization by the suzerain state. A state that is under the protection of another state while retaining its "international personality" is called a "protected state", not a protectorate.


title: Protectorate

text: Protectorates are one of the oldest features of international relations, dating back to the Roman Empire. "Civitates foederatae" were cities that were subordinate to Rome for their foreign relations. In the Middle Ages, Andorra was a protectorate of France and Spain. Modern protectorate concepts were devised in the nineteenth century.


title: Protectorate

text: In practice, a protectorate often has direct foreign relations only with the protector state, and transfers the management of all its more important international affairs to the latter. Similarly, the protectorate rarely takes military action on its own but relies on the protector for its defence. This is distinct from annexation, in that the protector has no formal power to control the internal affairs of the protectorate.


title: Protectorate

text: Protectorates differ from League of Nations mandates and their successors, United Nations Trust Territories, whose administration is supervised, in varying degrees, by the international community. A protectorate formally enters into the protection through a bilateral agreement with the protector, while international mandates are stewarded by the world community-representing body, with or without a administering power.


title: Protectorate

text: A protected state has a form of protection where it continues to retain an "international personality" and enjoys an agreed amount of independence in conducting its foreign policy.


title: Protectorate

text: For political and pragmatic reasons, the protection relationship is not usually advertised, but described with euphemisms such as "an independent state with special treaty relations" with the protecting state. A protected state appears on world maps just as any other independent state.


title: Protectorate

text: International administration of a state can also be regarded as an internationalized form of protection, where the protector is an international organisation rather than a state.