title: Predation

text: A range of mathematical models have been developed by relaxing the assumptions made in the Lotka-Volterra model; these variously allow animals to have geographic distributions, or to migrate; to have differences between individuals, such as sexes and an age structure, so that only some individuals reproduce; to live in a varying environment, such as with changing seasons; and analysing the interactions of more than just two species at once. Such models predict widely differing and often chaotic predator-prey population dynamics. The presence of refuge areas, where prey are safe from predators, may enable prey to maintain larger populations but may also destabilize the dynamics.


title: Predation

text: Predation dates from before the rise of commonly recognized carnivores by hundreds of millions (perhaps billions) of years. Predation has evolved repeatedly in different groups of organisms. The rise of eukaryotic cells at around 2.7 Gya, the rise of multicellular organisms at about 2 Gya, and the rise of mobile predators (around 600 Mya - 2 Gya, probably around 1 Gya) have all been attributed to early predatory behavior, and many very early remains show evidence of boreholes or other markings attributed to small predator species. It likely triggered major evolutionary transitions including the arrival of cells, eukaryotes, sexual reproduction, multicellularity, increased size, mobility (including insect flight) and armoured shells and exoskeletons.


title: Predation

text: The earliest predators were microbial organisms, which engulfed or grazed on others. Because the fossil record is poor, these first predators could date back anywhere between 1 and over 2.7 Gya (billion years ago). Predation visibly became important shortly before the Cambrian period—around —as evidenced by the almost simultaneous development of calcification in animals and algae, and predation-avoiding burrowing. However, predators had been grazing on micro-organisms since at least , with evidence of selective (rather than random) predation from a similar time.


title: Predation

text: "Auroralumina attenboroughii" is an Ediacaran crown-group cnidarian (557–562 mya, some 20 million years before the Cambrian explosion) from Charnwood Forest, England. It is thought to be one of the earliest predatory animals, catching small prey with its nematocysts as modern cnidarians do.


title: Predation

text: The fossil record demonstrates a long history of interactions between predators and their prey from the Cambrian period onwards, showing for example that some predators drilled through the shells of bivalve and gastropod molluscs, while others ate these organisms by breaking their shells.


title: Predation

text: Among the Cambrian predators were invertebrates like the anomalocaridids with appendages suitable for grabbing prey, large compound eyes and jaws made of a hard material like that in the exoskeleton of an insect.


title: Predation

text: Some of the first fish to have jaws were the armoured and mainly predatory placoderms of the Silurian to Devonian periods, one of which, the "Dunkleosteus", is considered the world's first vertebrate "superpredator", preying upon other predators.


title: Predation

text: Insects developed the ability to fly in the Early Carboniferous or Late Devonian, enabling them among other things to escape from predators.


title: Predation

text: Among the largest predators that have ever lived were the theropod dinosaurs such as "Tyrannosaurus" from the Cretaceous period. They preyed upon herbivorous dinosaurs such as hadrosaurs, ceratopsians and ankylosaurs.


title: Predation

text: Humans, as omnivores, are to some extent predatory, using weapons and tools to fish, hunt and trap animals. They also use other predatory species such as dogs, cormorants, and falcons to catch prey for food or for sport.


title: Predation

text: Two mid-sized predators, dogs and cats, are the animals most often kept as pets in western societies.


title: Predation

text: Human hunters, including the San of southern Africa, use persistence hunting, a form of pursuit predation where the pursuer may be slower than prey such as a kudu antelope over short distances, but follows it in the midday heat until it is exhausted, a pursuit that can take up to five hours.


title: Predation

text: In biological pest control, predators (and parasitoids) from a pest's natural range are introduced to control populations, at the risk of causing unforeseen problems. Natural predators, provided they do no harm to non-pest species, are an environmentally friendly and sustainable way of reducing damage to crops and an alternative to the use of chemical agents such as pesticides.


title: Predation

text: In film, the idea of the predator as a dangerous if humanoid enemy is used in the 1987 science fiction horror action film "Predator" and its three sequels. A terrifying predator, a gigantic man-eating great white shark, is central, too, to Steven Spielberg's 1974 thriller "Jaws".


title: Predation

text: Among poetry on the theme of predation, a predator's consciousness might be explored, such as in Ted Hughes's "Pike". The phrase "Nature, red in tooth and claw" from Alfred, Lord Tennyson's 1849 poem "In Memoriam A.H.H." has been interpreted as referring to the struggle between predators and prey.


title: Predation

text: In mythology and folk fable, predators such as the fox and wolf have mixed reputations. The fox was a symbol of fertility in ancient Greece, but a weather demon in northern Europe, and a creature of the devil in early Christianity; the fox is presented as sly, greedy, and cunning in fables from Aesop onwards. The big bad wolf is known to children in tales such as "Little Red Riding Hood", but is a demonic figure in the Icelandic Edda sagas, where the wolf Fenrir appears in the apocalyptic ending of the world. In the Middle Ages, belief spread in werewolves, men transformed into wolves. In ancient Rome, and in ancient Egypt, the wolf was worshipped, the she-wolf appearing in the founding myth of Rome, suckling Romulus and Remus. More recently, in Rudyard Kipling's 1894 "The Jungle Book", Mowgli is raised by the wolf pack. Attitudes to large predators in North America, such as wolf, grizzly bear and cougar, have shifted from hostility or ambivalence, accompanied by active persecution, towards positive and protective in the second half of the 20th century.


title: Luis von Ahn

text: Luis von Ahn (; born 19 August 1978) is a German-Guatemalan entrepreneur and a consulting professor in the Computer Science Department at Carnegie Mellon University in Pittsburgh, Pennsylvania. He is known as one of the pioneers of crowdsourcing. He is the founder of the company reCAPTCHA, which was sold to Google in 2009, and the co-founder and CEO of Duolingo.


title: Luis von Ahn

text: Luis von Ahn was born in and grew up in Guatemala City. Von Ahn grew up in a wealthy household with both of his parents working as physicians. He is a Guatemalan of German-Jewish descent. His mother was one of the first women in Guatemala to complete medical school, and decided to get pregnant with Von Ahn despite being single at age 42. He attended the American School of Guatemala, a private English-language school in Guatemala City, an experience he cites as a great privilege. When von Ahn was eight years old, his mother bought him a Commodore 64 computer, beginning his fascination with technology and computer science. When he applied to colleges in the United States, Von Ahn had to spend more than $1,200 to fly to neighboring El Salvador to take the Toefl. This experience left him with a negative impression of "extractive" testing industry, ripe for disruption.


title: Luis von Ahn

text: At age 18, von Ahn began studying at Duke University, where he received a Bachelor of Science (BS) in Mathematics, "summa cum laude," in 2000. He later earned his PhD in Computer Science at Carnegie Mellon University in 2005.


title: Luis von Ahn

text: Von Ahn's early research was in the field of cryptography. With Nicholas J. Hopper and John Langford, he was the first to provide rigorous definitions of steganography and to prove that private-key steganography is possible.


title: Luis von Ahn

text: In 2000, he did early pioneering work with Manuel Blum on CAPTCHAs, computer-generated tests that humans are routinely able to pass but that computers have not yet mastered. These devices are used by web sites to prevent automated programs, or bots, from perpetrating large-scale abuse, such as automatically registering for large numbers of accounts or purchasing huge numbers of tickets for resale by scalpers. CAPTCHAs brought von Ahn his first widespread fame among the general public due to their coverage in the "New York Times" and "USA Today" and on the Discovery Channel, NOVA scienceNOW, and other mainstream outlets.


title: Luis von Ahn

text: Von Ahn's PhD thesis, completed in 2005, was the first publication to use the term "human computation" that he had coined, referring to methods that combine human brainpower with computers to solve problems that neither could solve alone. Von Ahn's PhD thesis is also the first work on Games With A Purpose, or GWAPs, which are games played by humans that produce useful computation as a side effect. The most famous example is the ESP Game, an online game in which two randomly paired people are simultaneously shown the same picture, with no way to communicate. Each then lists a number of words or phrases that describe the picture within a time limit, and are rewarded with points for a match. This match turns out to be an accurate description of the picture, and can be successfully used in a database for more accurate image search technology. The ESP Game was licensed by Google in the form of the Google Image Labeler, and is used to improve the accuracy of the Google Image Search. Von Ahn's games brought him further coverage in the mainstream media. His thesis won the Best Doctoral Dissertation Award from Carnegie Mellon University's School of Computer Science.


title: Luis von Ahn

text: In July 2006, von Ahn gave a tech talk at Google on "Human Computation" (i.e., crowdsourcing) which was watched by over one million viewers.


title: Luis von Ahn

text: In 2007, von Ahn invented reCAPTCHA, a new form of CAPTCHA that also helps digitize books. In reCAPTCHA, the images of words displayed to the user come directly from old books that are being digitized; they are words that optical character recognition could not identify and are sent to people throughout the web to be identified. ReCAPTCHA is currently in use by over 100,000 web sites and is transcribing over 40 million words per day.


title: Luis von Ahn

text: In 2009, von Ahn and his graduate student Severin Hacker began to develop Duolingo, a language education platform. They founded a company of the same name, with von Ahn as chief executive officer and Hacker as chief technology officer. In November 2011, a private beta test of Duolingo was launched and the app was released to the public in June 2012. As of May 2020, Duolingo was valued at $1.5 billion. In a talk with NPR, von Ahn shared that Duolingo saw a spike in users during the COVID-19 pandemic. von Ahn has a chapter giving advice in Tim Ferriss' book Tools of Titans.


title: Luis von Ahn

text: In May 2021 von Ahn joined the executive committee of Partnership for Central America, an entity bringing together a variety of businesses, academic organizations and nonprofit organizations "to advance economic opportunity, address urgent climate, education and health challenges, and promote long-term investments and workforce capability building to support a vision of hope for Central America". The Partnership for Central America was presented in the context of the United States' Vice President Kamala Harris's "call to action" to address irregular migration from Central America to the United States by "deepening investment in the Northern Triangle" (a term coined to refer to Guatemala, El Salvador and Honduras).


title: Luis von Ahn

text: His research on CAPTCHAs and human computation has earned him international recognition and numerous honors. He was awarded a MacArthur Fellowship in 2006, the David and Lucile Packard Foundation Fellowship in 2009, a Sloan Fellowship in 2009, and a Microsoft New Faculty Fellowship in 2007, and the Presidential Early Career Award for Scientists and Engineers in 2012. He has also been named one of the 50 Best Brains in Science by "Discover", and has made it to many recognition lists that include "Popular Science"<nowiki>'</nowiki>s Brilliant 10, Silicon.com's 50 Most Influential People in Technology, "MIT Technology Review"'s TR35: Young Innovators Under 35, and "Fast Company<nowiki>'</nowiki>"s 100 Most Innovative People in Business.


title: Luis von Ahn

text: "Siglo Veintiuno", one of the biggest newspapers in Guatemala, chose him as the person of the year in 2009. In 2011, Foreign Policy Magazine in Spanish named him the most influential intellectual of Latin America and Spain.


title: Luis von Ahn

text: In 2011, he was awarded the A. Nico Habermann development chair in computer science, which is awarded every three years to a junior faculty member of unusual promise in the School of Computer Science.


title: Luis von Ahn

text: In 2017, he was awarded the Distinguished Leadership Award for Innovation and Social Impact by the Inter-American Dialogue.


title: Luis von Ahn

text: In 2018, von Ahn was awarded the Lemelson-MIT prize for his "dedication to improving the world through technology."


title: Luis von Ahn

text: Von Ahn has used a number of unusual techniques in his teaching, which have won him multiple teaching awards at Carnegie Mellon University. In the fall of 2008, he began teaching a new course at Carnegie Mellon entitled "Science of the Web". A combination of graph theory and social science, the course covers topics from network and game theory to auction theory.


title: Luis von Ahn

text: In 2021, von Ahn established the Luis von Ahn Foundation. The focus of the foundation is to support Guatemalans, especially women and girls, through financial support to local community leaders and nonprofit organizations. According to the foundation website, in 2022 the Luis von Ahn Foundation will give US$3 million to various organizations that focus on "women's and girls equality, conservation of the environment, and democracy and youth participation."


title: Indra Kumar

text: Indra Kumar is an Indian film director and producer. He had five Filmfare nominations, has also appeared in number of Gujarati. Kumar's real name is Indra Irani. His latest film produced is Thank God (2022).


title: Indra Kumar

text: Kumar debuted into direction as well as production with "Dil" (1990) and then directed the different concept-based drama films "Beta" (1992), "Raja" (1995) and "Ishq" (1997); all four being among the highest-grossing Hindi films of respective years and critically as well as commercially successful. He received two Filmfare Award for Best Director nominations for "Beta" and "Raja". He followed this with a number of average grossers or commercial failures, causing a setback, except the comedy "Masti" (2004), which ended with "Dhamaal" (2007). Kumar went on to make commercially successful comedy film sequels including "Double Dhamaal" (2011), "Grand Masti" (2013) and "Total Dhamaal" (2019). The latter ranks among the highest-grossing Hindi films and remains his highest-grossing release till date.


title: Indra Kumar

text: In 1990, Kumar made his directorial debut with the romance "Dil" starring Aamir Khan and Madhuri Dixit. The film, which was co-produced by him and based on a tale of parental opposition of teenage love, emerged as a major commercial success, and finished up as India's highest-grossing Indian film of the year, with domestic gross earnings of . The film was nominated for eight awards at the 36th Filmfare Awards, including a nomination in the Best Film category for Kumar.


title: Indra Kumar

text: The drama "Beta" (1992), based on the concept of how greediness overtakes family values, was Kumar's next directorial venture. The film starred Madhuri Dixit, Anil Kapoor, and Kumar's elder sister Aruna Irani, and proved to be a major commercial success as well as the highest-grossing Indian film of the year just like "Dil". For directing the film, Kumar received his first Best Director nomination at Filmfare.


title: Indra Kumar

text: In 1995, Kumar once again collaborated with Dixit in the romantic action film "Raja", which also starred Sanjay Kapoor. The film, which narrated the story of a man and woman who were about to get married in their childhood but fall in love and get married in their youth, proved to be the 3rd highest-grossing Indian film of the year. Kumar received a second Best Director nomination at Filmfare.


title: Indra Kumar

text: In 1997, Kumar directed the action romantic comedy "Ishq" starring Ajay Devgn, Aamir Khan, Kajol, and Juhi Chawla. The film, based on the subject of true love, received negative reviews from critics but proved to be one of the year's top-grossing productions with grossing revenues of worldwide.


title: Indra Kumar

text: From 1999 to 2006, Kumar continued to direct a number of films but experienced a setback. These films were "Mann", "Aashiq", "Rishtey", "Masti" (the first film of the "Masti franchise"), and "Pyare Mohan". With the exception of "Masti" which was moderately successful, all these films were commercial failures, despite being critical successes.


title: Indra Kumar

text: Kumar's directorial career prospects briefly improved when he directed the ensemble comedy "Dhamaal" (2007), the first film of the "Dhamaal franchise", starring Sanjay Dutt, Riteish Deshmukh, Aashish Chaudhary, Arshad Warsi, and Javed Jaffrey. The film received unfavourable reactions from critics and was a commercial success, grossing over worldwide.


title: Indra Kumar

text: After a four-year absence from directing films, Kumar directed "Double Dhamaal" (2011). A sequel to "Dhamaal" (2007), it retained the five previous lead actors, and also added Kangana Ranaut and Mallika Sherawat in the roles of Sanjay Dutt's sister and wife respectively. Collecting totally worldwide, "Double Dhamaal" proved to be a box office hit, despite being critically panned unlike its predecessor


title: Indra Kumar

text: Kumar's next directorial venture was yet another comedy film sequel–"Grand Masti" (2013)–a sequel to "Masti". It saw Riteish Deshmukh, Aftab Shivdasani, and Vivek Oberoi, reprising their roles from "Masti", with Karishma Tanna, Manjari Fadnis, and Sonali Kulkarni entering the cast as the new female leads. The film received positive response from critics and became the most successful entry in the "Masti franchise", with a gross revenue of more than .


title: Indra Kumar

text: The following year, Kumar directed the drama "Super Nani" (2014), which starred Sharman Joshi, Rekha, Randhir Kapoor, Anupam Kher, and Kumar's daughter Shweta Kumar. It failed to do well at the box office. He next directed the third installment of the "Masti franchise", titled "Great Grand Masti" (2016), again starring the previous three male leads. Unlike the success of the previous two films, "Great Grand Masti" was a major financial failure grossing worldwide on a budget of .


title: Indra Kumar

text: In 2019, Kumar directed the third installment of the "Dhamaal franchise", titled "Total Dhamaal", which was not a sequel to "Double Dhamaal" and was a reboot to it, having a different story and a different genre of adventure comedy. Ajay Devgn, Madhuri Dixit, Anil Kapoor, Riteish Deshmukh, Javed Jaffrey, Arshad Warsi, Boman Irani, and Esha Gupta played the lead roles (Deshmukh, Jaffrey, and Warsi reprised their roles). "Total Dhamaal" proved to be Kumar's highest-grossing film release, collecting within 12 days of its release, thus attaining a super hit status.


title: Indra Kumar

text: Kumar will next direct "Hera Pheri 3", the third installment of the comedy franchise "Hera Pheri", starring Akshay Kumar, Paresh Rawal, and Suniel Shetty. Kumar announced to direct the film after the death of Neeraj Vora, who directed the second installment and was to direct "Hera Pheri 3" too. Later, reports suggested that Kumar will not direct "Hera Pheri 3", so Priyadarshan, who was the director of the first installment, will return for "Hera Pheri 3". But it was later finally confirmed by Kumar that he will direct the film. Furthermore, Kumar will also direct the fourth installment of the "Dhamaal franchise", titled "Full on Dhamaal", which will feature Ajay Devgn, Riteish Deshmukh, Javed Jaffrey, and Arshad Warsi reprising their roles, with Sanjay Dutt and Aashish Chaudhary returning to play the lead roles in this installment though they did not were seen in "Total Dhamaal".


title: Midrash

text: Midrash (; ; or מִדְרָשׁוֹת "midrashot") is expansive Jewish Biblical exegesis using a rabbinic mode of interpretation prominent in the Talmud. The word itself means "textual interpretation", "study", or "exegesis", derived from the root verb (), which means "resort to, seek, seek with care, enquire, require", forms of which appear frequently in the Hebrew Bible.


title: Midrash

text: Midrash and rabbinic readings "discern value in texts, words, and letters, as potential revelatory spaces", writes the Hebrew scholar Wilda Gafney. "They reimagine dominant narratival readings while crafting new ones to stand alongside—not replace—former readings. Midrash also asks questions of the text; sometimes it provides answers, sometimes it leaves the reader to answer the questions". Vanessa Lovelace defines midrash as "a Jewish mode of interpretation that not only engages the words of the text, behind the text, and beyond the text, but also focuses on each letter, and the words left unsaid by each line".


title: Midrash

text: The term is also used of a rabbinic work that interprets Scripture in that manner. Such works contain early interpretations and commentaries on the Written Torah and Oral Torah (spoken law and sermons), as well as non-legalistic rabbinic literature () and occasionally Jewish religious laws (), which usually form a running commentary on specific passages in the Hebrew Scripture (Tanakh).


title: Midrash

text: The word "Midrash", especially if capitalized, can refer to a specific compilation of these rabbinic writings composed between 400 and 1200 CE. According to Gary Porton and Jacob Neusner, "midrash" has three technical meanings:


title: Midrash

text: The Hebrew word "midrash" is derived from the root of the verb (), which means "resort to, seek, seek with care, enquire, require", forms of which appear frequently in the Bible.


title: Midrash

text: The word "midrash" occurs twice in the Hebrew Bible: 2 Chronicles 13:22 "in the "midrash" of the prophet Iddo", and 24:27 "in the "midrash" of the book of the kings". Both the King James Version (KJV) and English Standard Version (ESV) translate the word as "story" in both instances; the Septuagint translates it as (book) in the first, as (writing) in the second. The meaning of the Hebrew word in these contexts is uncertain: it has been interpreted as referring to "a body of authoritative narratives, or interpretations thereof, concerning historically important figures" and seems to refer to a "book", perhaps even a "book of interpretation", which might make its use a foreshadowing of the technical sense that the rabbis later gave to the word.


title: Midrash

text: Since the early Middle Ages the function of much of midrashic interpretation has been distinguished from that of , straight or direct interpretation aiming at the original literal meaning of a scriptural text.


title: Midrash

text: A definition of "midrash" repeatedly quoted by other scholars is that given by Gary G. Porton in 1981: "a type of literature, oral or written, which stands in direct relationship to a fixed, canonical text, considered to be the authoritative and revealed word of God by the midrashist and his audience, and in which this canonical text is explicitly cited or clearly alluded to".


title: Midrash

text: Lieve M. Teugels, who would limit midrash to rabbinic literature, offered a definition of midrash as "rabbinic interpretation of Scripture that bears the lemmatic form", a definition that, unlike Porton's, has not been adopted by others. While some scholars agree with the limitation of the term "midrash" to rabbinic writings, others apply it also to certain Qumran writings, to parts of the New Testament, and of the Hebrew Bible (in particular the superscriptions of the Psalms, Deuteronomy, and Chronicles), and even modern compositions are called midrashim.


title: Midrash

text: Midrash is now viewed more as method than genre, although the rabbinic midrashim do constitute a distinct literary genre. According to the "Encyclopaedia Britannica", "Midrash was initially a philological method of interpreting the literal meaning of biblical texts. In time it developed into a sophisticated interpretive system that reconciled apparent biblical contradictions, established the scriptural basis of new laws, and enriched biblical content with new meaning. Midrashic creativity reached its peak in the schools of Rabbi Ishmael and Akiba, where two different hermeneutic methods were applied. The first was primarily logically oriented, making inferences based upon similarity of content and analogy. The second rested largely upon textual scrutiny, assuming that words and letters that seem superfluous teach something not openly stated in the text."


title: Midrash

text: Many different exegetical methods are employed to derive deeper meaning from a text. This is not limited to the traditional thirteen textual tools attributed to the Tanna Rabbi Ishmael, which are used in the interpretation of (Jewish law). The presence of words or letters which are seen to be apparently superfluous, and the chronology of events, parallel narratives or what are seen as other textual "anomalies" are often used as a springboard for interpretation of segments of Biblical text. In many cases, a handful of lines in the Biblical narrative may become a long philosophical discussion


title: Midrash

text: Numerous Jewish midrashim previously preserved in manuscript form have been published in print, including those denominated as smaller or minor midrashim. Bernard H. Mehlman and Seth M. Limmer deprecate this usage claiming that the term "minor" seems judgmental and "small" is inappropriate for midrashim some of which are lengthy. They propose instead the term "medieval midrashim", since the period of their production extended from the twilight of the rabbinic age to the dawn of the Age of Enlightenment.


title: Midrash

text: Generally speaking, rabbinic midrashim either focus on religious law and practice () or interpret biblical narrative in relation to non-legal ethics or theology, creating homilies and parables based on the text. In the latter case they are described as .


title: Midrash

text: "Midrash halakha" is the name given to a group of tannaitic expositions on the first five books of the Hebrew Bible. These midrashim, written in Mishnahic Hebrew, clearly distinguish between the Biblical texts that they discuss, and the rabbinic interpretation of that text. They often go well beyond simple interpretation and derive or provide support for halakha. This work is based on pre-set assumptions about the sacred and divine nature of the text, and the belief in the legitimacy that accords with rabbinic interpretation.


title: Midrash

text: Although this material treats the biblical texts as the authoritative word of God, it is clear that not all of the Hebrew Bible was fixed in its wording at this time, as some verses that are cited differ from the Masoretic, and accord with the Septuagint, or Samaritan Torah instead.


title: Midrash

text: With the growing canonization of the contents of the Hebrew Bible, both in terms of the books that it contained, and the version of the text in them, and an acceptance that new texts could not be added, there came a need to produce material that would clearly differentiate between that text, and rabbinic interpretation of it. By collecting and compiling these thoughts they could be presented in a manner which helped to refute claims that they were only human interpretations -- the argument being that, by presenting the various collections of different schools of thought, each of which relied upon close study of the text, the growing difference between early biblical law and its later rabbinic interpretation could be reconciled.


title: Midrash

text: Midrashim that seek to explain the non-legal portions of the Hebrew Bible are sometimes referred to as or .


title: Midrash

text: Aggadic discussions of the non-legal parts of Scripture are characterized by a much greater freedom of exposition than the halakhic midrashim (midrashim on Jewish law). Aggadic expositors availed themselves of various techniques, including sayings of prominent rabbis. These aggadic explanations could be philosophical or mystical disquisitions concerning angels, demons, paradise, hell, the messiah, Satan, feasts and fasts, parables, legends, satirical assaults on those who practice idolatry, etc.


title: Midrash

text: Some of these midrashim entail mystical teachings. The presentation is such that the midrash is a simple lesson to the uninitiated, and a direct allusion, or analogy, to a mystical teaching for those educated in this area.


title: Midrash

text: A wealth of literature and artwork has been created in the 20th and 21st centuries by people aspiring to create "contemporary midrash". Forms include poetry, prose, Bibliodrama (the acting out of Bible stories), murals, masks, and music, among others. The Institute for Contemporary Midrash was formed to facilitate these reinterpretations of sacred texts. The institute hosted several week-long intensives between 1995 and 2004, and published eight issues of "Living Text: The Journal of Contemporary Midrash" from 1997 to 2000.


title: Midrash

text: According to Carol Bakhos, recent studies that use literary-critical tools to concentrate on the cultural and literary aspects of midrash have led to a rediscovery of the importance of these texts for finding insights into the rabbinic culture that created them. Midrash is increasingly seen as a literary and cultural construction, responsive to literary means of analysis.


title: Midrash

text: Frank Kermode has written that midrash is an imaginative way of "updating, enhancing, augmenting, explaining, and justifying the sacred text". Because the Tanakh came to be seen as unintelligible or even offensive, midrash could be used as a means of rewriting it in a way that both makes it more acceptable to later ethical standards and conforms more to later notions of plausibility.


title: Midrash

text: James L. Kugel, in "The Bible as It Was" (Cambridge, Massachusetts: Harvard University Press, 1997), examines a number of early Jewish and Christian texts that comment on, expand, or re-interpret passages from the first five books of the Tanakh between the third century BCE and the second century CE.


title: Midrash

text: Kugel traces how and why biblical interpreters produced new meanings by the use of exegesis on ambiguities, syntactical details, unusual or awkward vocabulary, repetitions, etc. in the text. As an example, Kugel examines the different ways in which the biblical story that God's instructions are not to be found in heaven (Deuteronomy 30:12) has been interpreted. Baruch 3:29-4:1 states that this means that divine wisdom is not available anywhere other than in the Torah. Targum Neophyti (Deuteronomy 30:12) and b. Baba Metzia 59b claim that this text means that Torah is no longer hidden away, but has been given to humans who are then responsible for following it.


title: Upstairs, Downstairs (1971 TV series)

text: Upstairs, Downstairs is a British television drama series produced by London Weekend Television (LWT) for ITV. It ran for 68 episodes divided into five series on ITV from 1971 to 1975.


title: Upstairs, Downstairs (1971 TV series)

text: Set in a large townhouse at 165, Eaton Place in Belgravia in central London, the series depicts the servants—"downstairs"—and their masters, the family—"upstairs"—between the years 1903 and 1930, and shows the slow decline of the British aristocracy. Great events feature prominently in each episode but minor or gradual changes are also noted. The show may be regarded as a document of the social and technological changes that occurred during those 27 years, including the Edwardian period, women's suffrage, the First World War, the Roaring Twenties, and the Wall Street Crash. It was a ratings success for ITV and received outstanding acclaim worldwide, winning multiple awards.


title: Upstairs, Downstairs (1971 TV series)

text: A BBC Wales and "Masterpiece"-produced continuation, "Upstairs Downstairs", was broadcast by BBC One in 2010.


title: Upstairs, Downstairs (1971 TV series)

text: "Upstairs, Downstairs" was originally an idea by two actress friends, Jean Marsh and Eileen Atkins, for a comedy called "Behind the Green Baize Door". It would focus on two housemaids, played by Marsh and Atkins, in a large English Country House in the Victorian era. They soon added a family upstairs, as Marsh recognised "Servants have to serve somebody". In summer 1969, they took this idea to Sagitta Productions, which was run by John Hawkesworth and John Whitney. They soon removed the comedy element, changed the setting to a large townhouse in Edwardian London and the title became "Below Stairs". It was first offered to Granada Television in Manchester, but they declined as they already had a period drama, called "A Family at War", about to start. However, Stella Richman, the Controller of Programmes at London Weekend Television, saw potential, and in April 1970, the first series was commissioned.


title: Upstairs, Downstairs (1971 TV series)

text: Characters were then developed, but when Alfred Shaughnessy, an old friend of John Hawkesworth, was called in as script editor, he changed much of the detail to make the characters more realistic. Honor Blackman was short-listed for the role of Lady Marjorie and George Cole for that of the butler, Hudson. Jean Marsh was already slated to take the part of Rose Buck, the head house parlourmaid. Eileen Atkins was scheduled to play the other maid, Sarah Moffat, opposite Jean Marsh's Rose, but was playing Queen Victoria in a stage show at the time, so Pauline Collins took the role. Gordon Jackson was offered the role of Hudson after it was decided that Londoner George Cole would not be suitable to play a Scotsman. The programme took many names, including "Two Little Maids in Town", "The Servants' Hall" and "That House in Eaton Square". It was called "165 Eaton Place" until just before the production of the first episode when it was changed to "Upstairs, Downstairs," following a suggestion from John Hawkesworth.


title: Upstairs, Downstairs (1971 TV series)

text: Despite having a champion in Stella Richman, the show suffered from internal politics at the station, most notably from the sales department who could not see the attraction of a period drama, and the programme's videotapes spent nearly a year in storage awaiting a transmission date. Eventually the network had a space in its schedule at the unfashionable time of Sunday nights at 10:15 and called upon LWT to fill it. They chose "Upstairs, Downstairs," and with no promotion of the show, there was little expectation of success. However, audiences steadily grew and the series became a hit. In the United States, "Upstairs, Downstairs" was aired as part of PBS' "Masterpiece Theatre".


title: Upstairs, Downstairs (1971 TV series)

text: The stories depict the lives of the wealthy Bellamy family ("upstairs"), who reside at 165 Eaton Place in London's fashionable Belgravia, and their servants ("downstairs").


title: Upstairs, Downstairs (1971 TV series)

text: The household is led by Lady Marjorie Bellamy (née Talbot-Carey), daughter of the Earl and Countess of Southwold, and her husband Richard Bellamy MP, the son of a country parson. They married despite the objections of her parents and set up house at 165 Eaton Place, one of several London properties owned by Lord Southwold. Richard is a politician, and several plots revolve around his political ambitions and conflicts arising from his desire to follow both his conscience and his allegiance to his father-in-law's politically conservative Tory party.


title: Upstairs, Downstairs (1971 TV series)

text: Richard and Lady Marjorie Bellamy have two children, James and Elizabeth, who are, respectively, in their early twenties and late teens when the series starts in 1903. In 1912, James's wife Hazel becomes the new mistress of the house following Lady Marjorie's death, and the following year, Richard's ward, Georgina, comes to live at 165 Eaton Place.


title: Upstairs, Downstairs (1971 TV series)

text: The original servants comprise the authoritarian butler Mr Angus Hudson, cook Mrs Kate Bridges, pragmatic head house parlourmaid Rose Buck, sweet Irish kitchen maid Emily, eccentric footman Alfred, mischievous under-house parlourmaid Sarah, coachman Pearce, and Lady Marjorie's lady's maid Maude Roberts. Over the years they are joined by Edward, a cheeky footman who later becomes a chauffeur; Daisy, the parlourmaid who eventually marries Edward; Thomas Watkins, the devious chauffeur who dallies with Sarah's affections; and Ruby, the slow-witted kitchen maid.


title: Upstairs, Downstairs (1971 TV series)

text: In the episode "Another Year" from series 4, Hazel Bellamy notes that there are two families living in the house, one upstairs and the other downstairs, which she likens to a related family: Mr Hudson and Mrs Bridges are the father and mother; Rose, the eldest daughter who lost her man at the front; their son Edward and his wife, their daughter-in-law Daisy; and Ruby, the youngest child.


title: Upstairs, Downstairs (1971 TV series)

text: The first and second series span the period 1903 to 1910, during the reign of Edward VII. In 1903, Sarah Moffat applies to be under-house-parlour maid for the Bellamy family, pretending to be of French parentage but soon revealed to be illiterate, English, and with no work history. Later in the year, Lady Marjorie poses for Bohemian artist Mr Scone (which he pronounces "skoon"), who over the same period paints an intimate portrait of Sarah and (an imagined) Rose, the Bellamy's head house maid; he exhibits both paintings juxtaposed at the Royal Academy. Fearing a scandal, Bellamy threatens to sack the two maids but is dissuaded by Scone. Later, the Bellamys holiday in Scotland. With Mr Hudson gone from London, the servants carouse drunkenly through the house only to be caught by the Bellamy son James, who promises not to disclose their misbehaviour. James and Sarah later have an affair that results in Sarah's pregnancy. James is banished to India, while Sarah is sent to the Southwold estate for the duration of her pregnancy.


title: Upstairs, Downstairs (1971 TV series)

text: Around 1905, daughter Elizabeth returns from Germany to be prepared to be presented to the King and the Queen Consort at a social event. Her rebellious, headstrong nature causes her to flee the event. She later talks with servant Rose, who lectures her about the importance of duty and how it applies to the entire household.


title: Upstairs, Downstairs (1971 TV series)

text: Elizabeth soon falls in love with German Baron Klaus von Rimmer, unaware that he is gay and that he is using her in a scheme to obtain naval secrets from her father. Rose catches him having sex with Alfred, the footman, and he flees Eaton Place before the police arrive and arrest them. To spare Elizabeth the real reason for his departure, she is told that he is a spy. Alfred is replaced by Edward Barnes, a young and naive footman whose fun-loving and immature nature initially annoys Mr Hudson.


title: Upstairs, Downstairs (1971 TV series)

text: In the summer of 1906, Lady Marjorie is enamoured with her son James's friend, Captain Charles Hammond. She has an affair with him, though guilt compels her to break it off. While Lady Bellamy and Rose are away in the country, a new under-house-parlour maid, Mary Stokes, arrives. She is pregnant, having been raped by Myles Radford, the son of a powerful politician and a Bellamy family friend. Richard Bellamy attempts to help Mary, but the Radfords refuse to take responsibility and the legal system proves ineffective. Mary quits, but departs with a small gift of money from some of the servants.


title: Upstairs, Downstairs (1971 TV series)

text: The following year, 1907, Mrs Van Groeben arrives from South Africa. Housemaid Emily falls madly in love with Mrs Van Groeben's footman, William. Mrs Van Groeben learns of the affair, and, considering Emily to be beneath William, forbids him to see her. William drops Emily when he learns of his mistress's disapproval, suggesting he never really cared about her. Emily commits suicide. Mrs. Bridges, distraught with remorse over Emily's death, steals a baby from its pram outside a shop and hides it in her room. The baby is returned to its parents by Richard and Lady Marjorie, and Mrs Bridges only escapes a jail sentence after Hudson agrees to marry her once they are no longer in service. The scullery maid position is first replaced by Doris, then by Nellie, and finally by Ruby, who is rather slow and is frequently scolded by Mrs Bridges.


title: Upstairs, Downstairs (1971 TV series)

text: Somewhat indifferent to his new responsibilities as a householder, Lawrence also avoids marital relations with Elizabeth, claiming he prefers her to be pure and muse-like, leaving their marriage unconsummated. Elizabeth has a brief affair with Lawrence's publisher, which Lawrence facilitates. It results in Elizabeth becoming pregnant, giving birth to a daughter, Lucy Elizabeth. Elizabeth joins militant suffragettes who attack the home of a prominent politician, resulting in her being arrested. She is bailed out shortly thereafter by a wealthy Armenian gentleman named Julius Karekin. They become romantically involved, and he buys a hat shop for her to manage. Karekin is more interested in buying his way into British society than he is in Elizabeth's affections. He buys the lease of 165 Eaton Place after the Bellamys are forced to sell in the aftermath of Lady Marjorie's father's death in 1909. He offers the lease to Elizabeth who then gives it to her parents. Richard and Lady Marjorie are now morally in Karekin's debt.


title: Upstairs, Downstairs (1971 TV series)

text: A pregnant Sarah returns to Eaton Place the night the Bellamys host the King for dinner; Sarah goes into labour, but the baby, a son, is stillborn. James eventually returns from India just before his mother's birthday on 6 May 1910 (which coincides with the death of King Edward VII). He brings his brash and gushing fiancée, Phyllis, the daughter of an army veterinarian. James eventually breaks off the engagement, however, realizing that Phyllis is not right for him.


title: Upstairs, Downstairs (1971 TV series)

text: Elizabeth eventually moves to America in 1910 after splitting from Karekin and divorcing Kirbridge. She is later said to have married a man named Dana. Sarah falls in love with chauffeur, Thomas Watkins, who previously worked as Lawrence Kirbridge's valet. Sarah becomes pregnant again and will be dismissed. Thomas offers to marry her, though no one knows it is actually his child. To avoid any scandal, Lord Bellamy arranges for them to leave service and start their own business. While in the "Upstairs, Downstairs" episode "A Family Gathering" it is stated that Thomas and Sarah were married, this fact is disregarded in the spin-off series "Thomas & Sarah".


title: Upstairs, Downstairs (1971 TV series)

text: In episode 9 of this series, "An Object Of Value", veteran British actress Cathleen Nesbitt made a guest appearance as Lady Southwold, Lady Marjorie's mother. At the end of the second season, actresses Rachel Gurney (as Lady Marjorie) and Nicola Pagett (as Elizabeth) permanently left the series, although Gurney would appear briefly in the premiere episode of the third season ("Miss Forrest"). Pagett's character of Elizabeth would never be seen again although she was constantly mentioned throughout the remainder of the series. Actresses Meg Wynn Owen as Hazel Forrest Bellamy, James's wife; and Lesley-Anne Down as Georgina Worsley, Richard's ward and James's cousin became the major female upstairs characters.


title: Upstairs, Downstairs (1971 TV series)

text: Lady Marjorie, her brother, Hugo Talbot-Carey (the Earl of Southwold), and his new wife (widow Marion Worsley) all perish in the RMS "Titanic" disaster. Miss Roberts, Lady Marjorie's maid, survives and returns to Eaton Place, refusing to let anyone touch Marjorie's jewellery box, believing she is keeping it for her. Richard's new secretary, Hazel Forrest, gently persuades Miss Roberts to open the box, with her accepting Lady Marjorie is gone; she breaks down into sobs saying she tried to save her.


title: Upstairs, Downstairs (1971 TV series)

text: Richard had hired Hazel Forrest to type the biography of his father-in-law, the old Earl of Southwold, which he is writing. She and James fall in love and eventually marry. Hazel, from a respectable middle-class family, becomes mistress of the household. She and James are happy for a time but grow apart due to their differing backgrounds. Their estrangement is worsened by Hazel's miscarriage in the spring of 1914.


title: Upstairs, Downstairs (1971 TV series)

text: Georgina Worsley, James's cousin, comes to live with the Bellamys at Christmas 1913, aged 18. Georgina is the orphaned stepdaughter of Lady Marjorie's brother Hugo. She befriends Daisy, the new parlour maid. Georgina decides she and Daisy should take food from the pantry to give to Daisy's struggling family. She is shocked to discover that Daisy's family live in miserable, abject poverty.


title: Upstairs, Downstairs (1971 TV series)

text: Rose, the head houseparlourmaid, is shocked when Alfred, the Bellamys' former footman, turns up at Eaton Place one night. He claims that he has been sacked by his former employer and is homeless. She agrees to hide him in one of the basement rooms, but is horrified when it later transpires Alfred is actually on the run from the police having murdered his previous employer. After taking Edward hostage in the coal cellar, Alfred is arrested and eventually hanged for murder. The following year, Rose briefly becomes engaged to an Australian sheep-farmer called Gregory Wilmot whom she meets on a tram. She agrees to move to Australia with him, but at embarkation she discovers that he has a wife in Australia and decides to remain in the life she knows at Eaton Place, a plot line that echoes the story "Eveline" in James Joyce's "Dubliners".


title: Upstairs, Downstairs (1971 TV series)

text: Richard, who has sold the house's lease to James after Lady Marjorie's death (all her money passed on to James and Elizabeth), makes money after a share tip-off from a member of his gentleman's club. Richard is later unjustly accused of insider dealing. It is only Hazel's and Hudson's intervention that saves his career and reputation. (This plot was inspired by the Marconi scandal of 1912).


title: Upstairs, Downstairs (1971 TV series)

text: During a visit to Somerby, the country house of James's school-friend Lord "Bunny" Newbury in the autumn of 1913, Edward unwittingly becomes a witness in an impending divorce when he spies rising Tory MP Lord Charles Gilmour leaving the bedroom of a fellow MP's wife. Edward is pressured to change his story, but he refuses, and the case is withdrawn due to Richard's influence.


title: Upstairs, Downstairs (1971 TV series)

text: The servants are offered a day's holiday in Herne Bay in Kent in August 1914. They enjoy a rare day out together. Hudson is persuaded to sing on the vaudeville stage but their enjoyment is curtailed by an announcement that Germany has invaded Belgium. Hudson sings "Rule Britannia" instead. Back in London they gather to mark the hour when Britain goes to war.


title: Upstairs, Downstairs (1971 TV series)

text: In the Christmas episode, "Goodwill to All Men", Cathleen Nesbitt returns for a second and final appearance as Lady Southwold.


title: Upstairs, Downstairs (1971 TV series)

text: James serves in World War I, is seriously injured in a trench battle and brought home, amid much difficulty, to recover. He is nursed by his step-cousin Georgina, who now volunteers as a VAD Nurse and serves in France during the war. Georgina and James develop a close understanding and affection during the war but do not go so far as to become lovers. Downstairs, Edward signs up and fights in the trenches, Hudson serves as a Special Constable, Rose works as a bus conductress, and Ruby leaves Eaton Place to work in a munitions factory in Silvertown, returning to service after the Silvertown explosion. Hazel, unknowingly echoing her late mother-in-law Lady Marjorie, has a brief affair with an RFC Lieutenant named Jack Dyson who, like herself, has risen from the ranks of the middle classes. He is killed while James is at home on leave. Rose meets up with Gregory Wilmot again. After overcoming their emotional and practical hurdles, she finally agrees to marry him on his return from the war and follow him to Australia. However, he is killed in action. Rose is heartbroken but Gregory has left her £1,275 in his will, enough to make her independent in her retirement. Edward returns, and while not physically wounded, suffers badly from shell shock and goes into hospital. He speaks to Richard, who comforts him, assuring him that mental wounds are as real as physical, and no mark on his character.


title: Upstairs, Downstairs (1971 TV series)

text: The fourth season ends in 1918 with the close of World War I. Both James and Georgina return from their wartime duties. Despite the celebration and relief that the war is over, Hazel, who is about 35 at this time, contracts influenza in the worldwide pandemic. She dies and the Bellamy household is devastated over her loss. Grief-stricken, James lets Georgina comfort him, but she stops him from any further closeness. Richard is elevated to the House of Lords as "Viscount Bellamy of Haversham" in the New Year Honours List of 1919.


title: Upstairs, Downstairs (1971 TV series)

text: At the end of the fourth series Meg Wynn Owen left the show, with her character Hazel dying of influenza in the Spanish flu pandemic. The character of Virginia Hamilton, played by Hannah Gordon, was introduced in the fourth series when she asked Richard for help in relation to her son.


title: Upstairs, Downstairs (1971 TV series)

text: According to Jean Marsh and Simon Williams, at the end of the fourth series they, along with other cast members, were eager to move on to other projects. However, with much persuasion by the producers and writers, they, along with the rest of the cast, agreed to do a fifth and final series showing the Bellamy family and staff entering the decade of the twenties and ending the series with the start of the Great Depression. During the final series, Marsh and Williams reduced their appearances on the show, both appearing in only 11 instalments out of the sixteen episodes produced. In addition, eight new cast members were added to the series, which included two servants, Frederick and Lily, played by Gareth Hunt and Karen Dotrice; Richard Bellamy's new wife, Virginia (played by Hannah Gordon); Virginia's two children Alice and William (Anne Yarker and Jonathan Seely); Shirley Cain as Miss Treadwell, the children's governess; Madeleine Cannon as Lady Dorothy (Dolly) Hale, Georgina's self-centred, catty, fair weather friend; and Anthony Andrews as Lord Robert Stockbridge, Georgina's fiancé and husband.


title: Upstairs, Downstairs (1971 TV series)

text: At the start of the fifth season, James is still grieving over Hazel's death. Richard has married a young widow named Virginia Hamilton, after the events of the fourth series when she asked Richard, along with his solicitor Sir Geoffrey Dillon (Raymond Huntley), to defend her son Michael, aged 17, who was facing a court martial charge for cowardice in the war. Michael was acquitted of the charges but was killed in the war shortly after. During that time, Richard and Virginia fell in love. In the first instalment of the fifth series, "On With The Dance", Richard and Virginia are just returning to London from their honeymoon. James is in a quandary because with only himself and Georgina living at Eaton Place, there is not enough for the servants to do and therefore, James feels he will have to dismiss them from their jobs and ultimately, sell the house. He tries to persuade Richard and Virginia to move in with him and Georgina, but Virginia is stubborn and adamant about finding a house of her own. Collaborating with Georgina, James invites Richard, Virginia, and her two children, Alice, aged 10 and William, aged 6, to tea. The house suddenly comes alive with their visit. James, Georgina, and the servants welcome them wholeheartedly to the point where Virginia is finally convinced that 165 Eaton Place is the perfect home for her family. As a result, she, Richard, and the children become residents of the house. Alice becomes very attached to Rose, who is soon given duties that keep her upstairs with the children and the new Lady Bellamy.


title: Upstairs, Downstairs (1971 TV series)

text: Downstairs, Edward and a pregnant Daisy have left Eaton Place and are replaced by Frederick (James's batman during the war) and Lily, respectively. Edward's job prospects are dim, and Daisy loses their baby due to malnutrition; both are re-hired at Eaton Place as chauffeur and house parlourmaid, respectively. Rose becomes children's maid and later lady's maid to Virginia.


title: Upstairs, Downstairs (1971 TV series)

text: In the fifth episode of that season, "Wanted: A Good Home", William is sent off to a boarding school while Alice resumes her schooling with Miss Treadwell. To compensate for William's absence, Rose gives Alice a dog - a cairn terrier named Thimble. While Lord and Lady Bellamy are away on a trip, Miss Treadwell quickly assumes the role as head of the house alienating the other servants as well as mentally and physically abusing Alice and the dog. Upon their return, Lord and Lady Bellamy dismiss Miss Treadwell and she leaves in disgust. Richard and Virginia then decide that Alice should go to a day school for her studies. After that instalment, Alice, William, and the dog are never seen again, although they are frequently referred to, and Alice returns for a brief appearance in the final episode.


title: Upstairs, Downstairs (1971 TV series)

text: Downstairs, romance blooms between Hudson and Lily, but she spurns him and Lily leaves her position in the seventh episode "Disillusion". Frederick leaves service at the Bellamy house in the eleventh instalment "Alberto" when he discovers the allure of escorting young upper-class women to balls and other functions. Also during this season, the roles played by actresses Jenny Tomasin and Joan Benham were expanded to show more character depth. For Tomasin, the role of the childlike, simple-minded kitchen maid Ruby Finch was showcased in two episodes - the eighth instalment "Such A Lovely Man" where Ruby meets a man named Herbert through magazine correspondence, but rejects him because "he was no Rudolph Valentino"; and the fourteenth chapter of the series, "Noblesse Oblige" in which Ruby, fed up and disgusted with Mrs. Bridges's treatment of her, temporarily leaves her position at Eaton Place after they have a bitter quarrel. Benham's character of Lady Prudence Fairfax (or "Pru" as she was affectionately referred to) had been featured throughout the previous four series as a long time close friend of the Bellamy family who was a delightful social butterfly always welcome at family parties and events, as well as a harmless gossipy woman. However, in the fifth series, Benham was able to show another side of Lady Prudence - as a wise, kind, loyal, loving, and devoted friend to the Bellamys in two instalments - in the fourth episode "Joy Ride" and the eleventh episode "Alberto".


title: Upstairs, Downstairs (1971 TV series)

text: In the roaring 1920s, Georgina and her friends rebel against the depression and hard times of the post-war period, but her frivolity and merriment quickly end — firstly after the suicide of a friend who professed his love for her and threatened to kill himself unless she married him, and then killed himself during a Roaring 20s party at the Bellamy house. Secondly, Georgina accidentally kills a working class man after taking Richard's car without permission and driving it herself to compete in a scavenger hunt. Her friends, part of the moneyed, bored and pleasure-seeking "bright young things", desert her at the inquest. Only the rich but shallow Lady Dolly Hale and also the seemingly dull Lord Robert Stockbridge, heir to a dukedom, defend her. Georgina and Robert quickly fall in love. James suffers long-term emotional stress from his war experiences and Hazel's death. Restless, he stands for Parliament, but loses the election.


title: Upstairs, Downstairs (1971 TV series)

text: Lord Stockbridge's parents send their son on a trip around the world to try to wean him from Georgina, under the guise of helping him discover whether his feelings are true. James returns from America in October 1929, where he has visited Elizabeth and become rich through speculation on Wall Street. Rose allows James to invest the money Gregory left her when he died in the war, but the market crashes and James loses everything. James had also "borrowed a fair bit" that he now is unable to repay. He has disgraced his family and taken advantage of a member of staff who trusted him. Depressed and ashamed, he goes to a hotel in Maidenhead and commits suicide.


title: Upstairs, Downstairs (1971 TV series)

text: The final episode, set in 1930, finds things looking up at Eaton Place as Georgina marries Lord Stockbridge on 12 June 1930. Mr Hudson and Mrs Bridges also finally marry, and take the uneducated but surprisingly shrewd kitchen maid, Ruby Finch, to the seaside with them, to run a guest house. When asked by Rose how she feels about becoming part of the Hudsons' household, Ruby says, "They'll not last long, and I'll have guest house all to meself." Lord Bellamy has delivered his retirement speech to the House of Lords. He and Virginia retreat to a small villa in Dorset, keeping Rose Buck in their employ. Young Edward and his wife, Daisy, are elevated to the posts of butler and head house parlourmaid in the country household of the Marquess and new Marchioness of Stockbridge.


title: Upstairs, Downstairs (1971 TV series)

text: The last scene shows Rose taking a final walk through the now empty rooms at 165 Eaton Place, which is for sale. She 'hears' in her head the voices of Lady Marjorie, Mr. Hudson, and others, as well as the many incidents she witnessed over the years. On hearing James reassuring her about Gregory's honourable death in the war echoing in her ears, she becomes spooked and hurriedly leaves through the front door.


title: Upstairs, Downstairs (1971 TV series)

text: "Upstairs, Downstairs" ran for five series from 10 December 1971 to 21 December 1975. The first four series had thirteen episodes each, while the final series had sixteen. Due to an industrial dispute over extra payments for using newly introduced colour equipment, during which broadcasting unions refused to allow their members to use colour cameras, the first six episodes of the first series were shot in black-and-white, and when colour production resumed, the first episode was remade in colour. Two endings were made, which could be shown depending on whether the black and white episodes were broadcast by the channel. The original black-and-white version of the first episode is believed to have been wiped.


title: Upstairs, Downstairs (1971 TV series)

text: The opening credits of each episode featured a cartoon from the magazine "Punch", and the lettering was drawn by the graphic designer Terry Griffiths. The theme tune was composed by Alexander Faris and entitled "The Edwardians". It won an Ivor Novello Award. Part of this tune would be made into the song "What Are We Going to Do with Uncle Arthur?", sung by Sarah, with lyrics written by Alfred Shaughnessy. Pauline Collins released this as a single in 1973. The theme tune was also used as the processional march for the church wedding of Elizabeth and Lawrence in series 1, Episode 13: "For Love of Love".


title: Upstairs, Downstairs (1971 TV series)

text: Many writers wrote episodes throughout the five series, including Alfred Shaughnessy, John Hawkesworth, Fay Weldon, Terence Brady and Charlotte Bingham, John Harrison, Julian Bond, Raymond Bowers, Jeremy Paul, Rosemary Anne Sisson, Anthony Skene and Elizabeth Jane Howard.


title: Upstairs, Downstairs (1971 TV series)

text: Each episode of "Upstairs, Downstairs" was made in a fortnightly production schedule. The book "Inside Updown" has a detailed production history. The first week and a half would be spent rehearsing, with two days in the studio - the latter part of the second day being used for recording. Location footage was usually shot beforehand. The exterior shots of 165 Eaton Place were filmed at 65 Eaton Place with the "1" painted on. "Upstairs, Downstairs" was one of the first major colour productions to be made by LWT. Interior sequences were first recorded in LWT's first studio facility in Wembley in London, for all of series one and the episode "A Pair of Exiles" in series two. For the rest of series two and for the remaining three series the interior sequences were recorded in LWT's new studio complex called Kent House (later known as The London Studios) on London's Southbank.


title: Upstairs, Downstairs (1971 TV series)

text: Herne Bay features in series three, episode thirteen "The Sudden Storm", when the staff take a day out to the seaside.


title: Upstairs, Downstairs (1971 TV series)

text: "Upstairs, Downstairs" was nominated for and won many national and international awards, winning two BAFTA awards, two Royal Television Society awards, three Writers Guild Awards, eight Emmys, and a Golden Globe. It was nominated for a further seven BAFTAs, nine Emmys and four Golden Globes. It was nominated for the BAFTA Television Award for Best Drama Series in 1972, 1973, 1975, and 1976, winning in 1972 and 1974. Pauline Collins was nominated for Best Actress in 1973 for her portrayal of Sarah Moffat and Gordon Jackson was nominated for Best Actor for playing Mr. Hudson in 1975.


title: Upstairs, Downstairs (1971 TV series)

text: In the United States, "Upstairs, Downstairs" was honoured for both the Primetime Emmy Awards and the Golden Globe Awards. In 1974, 1975, and 1977 it won the Primetime Emmy Award for Outstanding Drama Series. In 1975, Jean Marsh won the Primetime Emmy Award for Outstanding Lead Actress in a Drama Series, and Bill Bain won the Primetime Emmy Award for Outstanding Directing for a Drama Series. The following year, it won for Outstanding Limited Series, and Gordon Jackson won for Outstanding Single Performance by a Supporting Actor in Comedy or Drama Series, and Angela Baddeley was nominated for Outstanding Continuing Performance by a Supporting Actress in a Drama Series. In 1977, Jacqueline Tong was nominated for Outstanding Continuing Performance by a Supporting Actress in a Drama Series. For the Golden Globes, "Upstairs, Downstairs" won the Golden Globe Award for Best TV Show - Drama in 1975 and was nominated again for the same award in 1978. Jean Marsh was nominated in 1975 and 1977 for Best TV Actress - Drama.


title: Upstairs, Downstairs (1971 TV series)

text: Alfred Shaughnessy, script editor and frequent writer, was nominated for a Primetime Emmy Award twice for the episodes "Miss Forrest" and "Another Year". John Hawkesworth, frequent writer and producer, was nominated for Outstanding Writing for a Drama Series for the episode "The Bolter". Fay Weldon won a Writers Guild of America Award for Best British TV Series Script of 1971 for "On Trial".


title: Upstairs, Downstairs (1971 TV series)

text: Following the final episode of "Upstairs, Downstairs" many ideas for spin-offs evolved. These included having the new Lord and Lady Stockbridge buying back 165 Eaton Place and an American company wanted to make a programme based around Hudson and Rose immigrating to the United States. Jack Webb was interested in a series of Marsh and Gordon Jackson reprising their roles as head of a Los Angeles employment agency. Another idea, called "You Live or You Die", was to have Frederick Norton seeking his fortune in the US. A further idea would have followed Hudson, Mrs Bridges and Ruby running their seaside boarding house, and this probably would have been made had it not been for the death of Angela Baddeley on 22 February 1976. The only spin-off to make it onscreen was "Thomas & Sarah", which broadcast in 1979, and this followed the adventures of Thomas and Sarah after they left Eaton Place.


title: Upstairs, Downstairs (1971 TV series)

text: A short-lived CBS series entitled "Beacon Hill", which aired in the fall of 1975, was loosely based upon "Upstairs, Downstairs"; its executive producer, Beryl Vertue, was Jean Marsh's literary agent and had been responsible for helping sell the original "Upstairs, Downstairs" to LWT.


title: Upstairs, Downstairs (1971 TV series)

text: In 2011, the BBC ran a series entitled "Royal Upstairs Downstairs" in which Tim Wonnacott and Rosemary Shrager tour country houses visited by Queen Victoria. Tim tours the Upstairs concentrating on the architecture and events of Victoria's visit, while Rosemary concentrates on the downstairs, demonstrating recipes cooked for the Queen.


title: Upstairs, Downstairs (1971 TV series)

text: Each series of "Upstairs, Downstairs" was accompanied by a novelisation, with additional detail in each, but also with some episodes missing. All books were published by Sphere Books. The novelisation of the first series, "Upstairs, Downstairs or the secrets of an Edwardian household", was written by John Hawkesworth and published in 1972. Hawkesworth also wrote the series two novelisation, "In My Lady's Chamber" and this was published in 1973. The following year, Mollie Hardwick's novelisation of the third series, "The Years of Change", was published and she also wrote the 1975 "The War to End Wars", the fourth series novelisation. The fifth series, which was longer than the others, was novelised in two books, both by Michael Hardwick and published in 1975. They were called "On With The Dance" and "Endings and Beginnings".


title: Upstairs, Downstairs (1971 TV series)

text: As well as these novelisations, five books were separately published, again by Sphere Books, with each being the biography of a main character before the series started. "Rose's Story" was written by Terence Brady & Charlotte Bingham and published in 1972. The following year, Mollie Hardwick's "Sarah's Story" and Michael Hardwick's "Mr Hudson's Diaries" were both published. "Mr Bellamy's Story", by Michael Hardwick, was published in 1974 and "Mrs Bridge's Story" by Mollie Hardwick was published in 1975. Also in 1975, "The Upstairs, Downstairs Omnibus", featuring all five slightly edited stories, was published.


title: Upstairs, Downstairs (1971 TV series)

text: The BBC series "The Duchess of Duke Street" is widely seen as the BBC's answer to "Upstairs, Downstairs", not least because some of the same producers and writers worked on it, and it also has a theme tune by Faris. The 1990 BBC sitcom "You Rang, M'Lord?" also featured a similar situation. In 1975, an American version, entitled "Beacon Hill", debuted but due to low ratings it was soon cancelled, running for just thirteen episodes. Tom Wolfe called the series a plutography, i.e. a "graphic depiction of the lives of the rich".


title: Upstairs, Downstairs (1971 TV series)

text: A "Monsterpiece Theater" sketch on "Sesame Street", introduced/narrated by Cookie Monster (as Alistair Cookie – a play on Alistair Cooke, who was at that time the host of "Masterpiece Theater") – was entitled "Upstairs, Downstairs" and featured Grover running up and down a staircase until collapsing from exhaustion.


title: Upstairs, Downstairs (1971 TV series)

text: In 2000, a stop-motion animated series called "Upstairs, Downstairs Bears", co-produced by Canada's Cinar and Scottish Television for the CITV block, was based upon "Upstairs, Downstairs". Ironically, it happened to air on the last day of London Weekend Television's broadcasting under that brand (as part of the CITV block) on 27 October 2002; the next day LWT was subsumed into ITV London.


title: Upstairs, Downstairs (1971 TV series)

text: Company Pictures' 2008 television series "The Palace" has been described as a "modern "Upstairs, Downstairs"" as it features the points of view of both a fictional royal family and their servants.


title: Upstairs, Downstairs (1971 TV series)

text: From November 2008 to January 2010 variations (played in different styles, e.g., a fugue, jazz, calypso, death metal) of the theme music were played on BBC Radio 4's "PM" programme to introduce a segment entitled "Upshares, Downshares", in which Nils Blythe ran through the day's business news. In November 2010, with the composer Alexander Faris's blessing, a special CD of collected versions was released to raise money for the charity Children in Need.


title: Upstairs, Downstairs (1971 TV series)

text: The ITV series "Downton Abbey", which ran from 2010 to 2015, has many similarities with "Upstairs, Downstairs." Both series depict a noble family and their servants in the context of some of the same historical events; certain storylines and characterisations also appear in both series. In 2019, "Downton Abbey's" executive producer Gareth Neame acknowledged the inspiration, stating that upon watching "Upstairs, Downstairs" for the first time as an adult, he realised that "there's two generations now who have got something new to be exposed to” because they had never seen "Upstairs, Downstairs".


title: Upstairs, Downstairs (1971 TV series)

text: The phrase "upstairs-downstairs coalition", an allusion to "Upstairs, Downstairs", has been used by political writers to describe political coalitions (such as the Democratic Party in the United States) that appeal to both socially moderate white suburbanites and minorities.


title: Upstairs, Downstairs (1971 TV series)

text: In 2009, the BBC announced it was to broadcast a revival of the series, with Jean Marsh reprising her role as Rose alongside a new cast in the same Eaton Place household. The new series was created and written by Heidi Thomas. The first series aired from 26 December 2010. Eileen Atkins appeared in the first series, but left her role before production on the second series began.


title: Upstairs, Downstairs (1971 TV series)

text: "Upstairs, Downstairs" was first released to Region One DVD in December 2001 by A&E Home Entertainment. During 2002, it released the remaining series and then released "Thomas and Sarah" on DVD in 2004. The individual releases have also been collected together into two boxed sets, the second of which, "The Collector's Edition Megaset", also includes "Thomas and Sarah". These are all out of print. In 2011 Acorn Media released "Upstairs Downstairs: The Ultimate Collection" which includes "Thomas and Sarah" and 25 hours of special features in a 40th Anniversary 26-disc set.


title: Upstairs, Downstairs (1971 TV series)

text: "Upstairs, Downstairs" was originally released on DVD by VCI in Region 2 (UK). The colour episodes of the first series were released in 2001 followed by the other series finishing in 2003. In 2004, the black-and-white episodes and the first episode with the original ending were released. "Thomas & Sarah" was released in matching packaging in 2004. In 2005, VCI stopped making these DVDs, none of which had included any extras.


title: Upstairs, Downstairs (1971 TV series)

text: Network Video released the entire programme series-by-series from 2005 to 2006. The episodes were remastered and the black-and-white episodes were put in chronological order in the first series. Some episodes also featured audio commentaries, the LWT logo, commercial bumpers and the original preceding countdowns. In addition, each series was accompanied by a special one-hour documentary relating to that series featuring new and archive interviews. The fifth series release also featured the 1975 documentary "Russell Harty goes... "Upstairs, Downstairs"". In 2006, a boxed set featuring all the DVDs was released.


title: Upstairs, Downstairs (1971 TV series)

text: DVDs of the series have also been released in Denmark, Finland, Germany, the Netherlands, Portugal, Spain, and Sweden.


title: Upstairs, Downstairs (1971 TV series)

text: Universal DVD released all five series to DVD in Australia and New Zealand. These were later deleted. Timelife issued the series as a mail-order collection. ITV began re-releasing the series in Australia in January 2013.


title: Curvature

text: In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.


title: Curvature

text: For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature "at a point" of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.


title: Curvature

text: For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of "maximal curvature", "minimal curvature", and "mean curvature".


title: Curvature

text: For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature "intrinsically", that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.


title: Curvature

text: In "Tractatus de configurationibus qualitatum et motuum," the 14th-century philosopher and mathematician Nicole Oresme introduces the concept of curvature as a measure of departure from straightness; for circles he has the curvature as being inversely proportional to the radius; and he attempts to extend this idea to other curves as a continuously varying magnitude.


title: Curvature

text: The curvature of a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve.


title: Curvature

text: Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle in ), so it is a measure of the instantaneous rate of change of "direction" of a point that moves on the curve: the larger the curvature, the larger this rate of change. In other words, the curvature measures how fast the unit tangent vector to the curve rotates (fast in terms of curve position). In fact, it can be proved that this instantaneous rate of change is exactly the curvature. More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point is a function of the parameter , which may be thought as the time or as the arc length from a given origin. Let be a unit tangent vector of the curve at , which is also the derivative of with respect to . Then, the derivative of with respect to is a vector that is normal to the curve and whose length is the curvature.


title: Curvature

text: For being meaningful, the definition of the curvature and its different characterizations require that the curve is continuously differentiable near , for having a tangent that varies continuously; it requires also that the curve is twice differentiable at , for insuring the existence of the involved limits, and of the derivative of .


title: Curvature

text: The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use in kinematics, this characterization is often given as a definition of the curvature.


title: Curvature

text: Historically, the curvature of a differentiable curve was defined through the osculating circle, which is the circle that best approximates the curve at a point. More precisely, given a point on a curve, every other point of the curve defines a circle (or sometimes a line) passing through and tangent to the curve at . The osculating circle is the limit, if it exists, of this circle when tends to . Then the "center" and the "radius of curvature" of the curve at are the center and the radius of the osculating circle. The curvature is the reciprocal of radius of curvature. That is, the curvature is


title: Curvature

text: where is the radius of curvature (the whole circle has this curvature, it can be read as turn over the length ).


title: Curvature

text: This definition is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced.


title: Curvature

text: Every differentiable curve can be parametrized with respect to arc length. In the case of a plane curve, this means the existence of a parametrization , where and are real-valued differentiable functions whose derivatives satisfy


title: Curvature

text: If the curve is twice differentiable, that is, if the second derivatives of and exist, then the derivative of exists. This vector is normal to the curve, its norm is the curvature , and it is oriented toward the center of curvature. That is,


title: Curvature

text: with . The real number is called the oriented curvature or signed curvature. It depends on both the orientation of the plane (definition of counterclockwise), and the orientation of the curve provided by the parametrization. In fact, the change of variable provides another arc-length parametrization, and changes the sign of .


title: Curvature

text: Let be a proper parametric representation of a twice differentiable plane curve. Here "proper" means that on the domain of definition of the parametrization, the derivative


title: Curvature

text: These formulas can be derived from the special case of arc-length parametrization in the following way. The above condition on the parametrisation imply that the arc length is a differentiable monotonic function of the parameter , and conversely that is a monotonic function of . Moreover, by changing, if needed, to , one may suppose that these functions are increasing and have a positive derivative. Using notation of the preceding section and the chain rule, one has


title: Curvature

text: By using the above formula and the chain rule this derivative and its norm can be expressed in terms of and only, with the arc-length parameter completely eliminated, giving the above formulas for the curvature.


title: Curvature

text: In the general case of a curve, the sign of the signed curvature is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values of . This makes significant the sign of the signed curvature.


title: Curvature

text: The sign of the signed curvature is the same as the sign of the second derivative of . If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. It is zero, then one has an inflection point or an undulation point.


title: Curvature

text: When the slope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, using big O notation, one has


title: Curvature

text: It is common in physics and engineering to approximate the curvature with the second derivative, for example, in beam theory or for deriving the wave equation of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwise nonlinear to be treated approximately as linear.


title: Curvature

text: If a curve is defined in polar coordinates by the radius expressed as a function of the polar angle, that is is a function of , then its curvature is


title: Curvature

text: The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Also, changing into does not change the curve, but changes the sign of the numerator if the absolute value is omitted in the preceding formula.


title: Curvature

text: A point of the curve where is a singular point, which means that the curve is not differentiable at this point, and thus that the curvature is not defined (most often, the point is either a crossing point or a cusp).


title: Curvature

text: Above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has


title: Curvature

text: It can be useful to verify on simple examples that the different formulas given in the preceding sections give the same result.


title: Curvature

text: It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle.


title: Curvature

text: is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by in both the numerator and the denominator in the preceding formula.


title: Curvature

text: The same circle can also be defined by the implicit equation with . Then, the formula for the curvature in this case gives


title: Curvature

text: It has the sign of for all values of . This means that, if , the concavity is upward directed everywhere; if , the concavity is downward directed; for , the curvature is zero everywhere, confirming that the parabola degenerates into a line in this case.


title: Curvature

text: The (unsigned) curvature is maximal for , that is at the stationary point (zero derivative) of the function, which is the vertex of the parabola.


title: Curvature

text: Consider the parametrization . The first derivative of is , and the second derivative is zero. Substituting into the formula for general parametrizations gives exactly the same result as above, with replaced by . If we use primes for derivatives with respect to the parameter .


title: Curvature

text: The same parabola can also be defined by the implicit equation with . As , and , one obtains exactly the same value for the (unsigned) curvature. However, the signed curvature is meaningless here, as is a valid implicit equation for the same parabola, which gives the opposite sign for the curvature.


title: Curvature

text: The expression of the curvature In terms of arc-length parametrization is essentially the first Frenet–Serret formula


title: Curvature

text: where the primes refer to the derivatives with respect to the arc length , and is the normal unit vector in the direction of .


title: Curvature

text: For a general parametrization by a parameter , one needs expressions involving derivatives with respect to . As these are obtained by multiplying by the derivatives with respect to , one has, for any proper parametrization


title: Curvature

text: A "curvature comb" can be used to represent graphically the curvature of every point on a curve. If formula_29 is a parametrised curve its comb is defined as the parametrized curve


title: Curvature

text: where formula_31 are the curvature and normal vector and formula_32 is a scaling factor (to be chosen as to enhance the graphical representation).


title: Curvature

text: As in the case of curves in two dimensions, the curvature of a regular space curve in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if is the arc-length parametrization of then the unit tangent vector is given by


title: Curvature

text: The plane containing the two vectors and is the osculating plane to the curve at . The curvature has the following geometrical interpretation. There exists a circle in the osculating plane tangent to whose Taylor series to second order at the point of contact agrees with that of . This is the osculating circle to the curve. The radius of the circle is called the radius of curvature, and the curvature is the reciprocal of the radius of curvature:


title: Curvature

text: The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point. In three dimensions, the third-order behavior of a curve is described by a related notion of torsion, which measures the extent to which a curve tends to move as a helical path in space. The torsion and curvature are related by the Frenet–Serret formulas (in three dimensions) and their generalization (in higher dimensions).


title: Curvature

text: For a parametrically-defined space curve in three dimensions given in Cartesian coordinates by , the curvature is


title: Curvature

text: where the prime denotes differentiation with respect to the parameter . This can be expressed independently of the coordinate system by means of the formula


title: Curvature

text: where × denotes the vector cross product. This last formula is valid for the curvature of curves in a Euclidean space of any dimension:


title: Curvature

text: Given two points and on , let be the arc length of the portion of the curve between and and let denote the length of the line segment from to . The curvature of at is given by the limit


title: Curvature

text: where the limit is taken as the point approaches on . The denominator can equally well be taken to be . The formula is valid in any dimension. Furthermore, by considering the limit independently on either side of , this definition of the curvature can sometimes accommodate a singularity at . The formula follows by verifying it for the osculating circle.


title: Curvature

text: The curvature of curves drawn on a surface is the main tool for the defining and studying the curvature of the surface.


title: Curvature

text: For a curve drawn on a surface (embedded in three-dimensional Euclidean space), several curvatures are defined, which relates the direction of curvature to the surface's unit normal vector, including the:


title: Curvature

text: Any non-singular curve on a smooth surface has its tangent vector contained in the tangent plane of the surface. The normal curvature, , is the curvature of the curve projected onto the plane containing the curve's tangent and the surface normal ; the geodesic curvature, , is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), , measures the rate of change of the surface normal around the curve's tangent.


title: Curvature

text: Let the curve be arc-length parametrized, and let so that form an orthonormal basis, called the Darboux frame. The above quantities are related by:


title: Curvature

text: All curves on the surface with the same tangent vector at a given point will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing and . Taking all possible tangent vectors, the maximum and minimum values of the normal curvature at a point are called the principal curvatures, and , and the directions of the corresponding tangent vectors are called principal normal directions.


title: Curvature

text: Curvature can be evaluated along surface normal sections, similar to above (see for example the Earth radius of curvature).


title: Curvature

text: Some curved surfaces, such as those made from a smooth sheet of paper, can be flattened down into the plane without distorting their intrinsic features in any way. Such developable surfaces have zero Gaussian curvature (see below).


title: Curvature

text: In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, . It has a dimension of length and is positive for spheres, negative for one-sheet hyperboloids and zero for planes and cylinders. It determines whether a surface is locally convex (when it is positive) or locally saddle-shaped (when it is negative).


title: Curvature

text: Gaussian curvature is an "intrinsic" property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.


title: Curvature

text: Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.


title: Curvature

text: An intrinsic definition of the Gaussian curvature at a point is the following: imagine an ant which is tied to with a short thread of length . It runs around while the thread is completely stretched and measures the length of one complete trip around . If the surface were flat, the ant would find . On curved surfaces, the formula for will be different, and the Gaussian curvature at the point can be computed by the Bertrand–Diguet–Puiseux theorem as


title: Curvature

text: The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss–Bonnet theorem.


title: Curvature

text: The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss–Bonnet theorem is Descartes' theorem on total angular defect.


title: Curvature

text: Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher-dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.


title: Curvature

text: The mean curvature is an "extrinsic" measure of curvature equal to half the sum of the principal curvatures, . It has a dimension of length. Mean curvature is closely related to the first variation of surface area. In particular, a minimal surface such as a soap film has mean curvature zero and a soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.


title: Curvature

text: The intrinsic and extrinsic curvature of a surface can be combined in the second fundamental form. This is a quadratic form in the tangent plane to the surface at a point whose value at a particular tangent vector to the surface is the normal component of the acceleration of a curve along the surface tangent to ; that is, it is the normal curvature to a curve tangent to (see above). Symbolically,


title: Curvature

text: where is the unit normal to the surface. For unit tangent vectors , the second fundamental form assumes the maximum value and minimum value , which occur in the principal directions and , respectively. Thus, by the principal axis theorem, the second fundamental form is


title: Curvature

text: An encapsulation of surface curvature can be found in the shape operator, , which is a self-adjoint linear operator from the tangent plane to itself (specifically, the differential of the Gauss map).


title: Curvature

text: For a surface with tangent vectors and normal , the shape operator can be expressed compactly in index summation notation as


title: Curvature

text: The Weingarten equations give the value of in terms of the coefficients of the first and second fundamental forms as


title: Curvature

text: The principal curvatures are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gauss curvature is its determinant, and the mean curvature is half its trace.


title: Curvature

text: By extension of the former argument, a space of three or more dimensions can be intrinsically curved. The curvature is "intrinsic" in the sense that it is a property defined at every point in the space, rather than a property defined with respect to a larger space that contains it. In general, a curved space may or may not be conceived as being embedded in a higher-dimensional ambient space; if not then its curvature can only be defined intrinsically.


title: Curvature

text: After the discovery of the intrinsic definition of curvature, which is closely connected with non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory of general relativity, which describes gravity and cosmology, the idea is slightly generalised to the "curvature of spacetime"; in relativity theory spacetime is a pseudo-Riemannian manifold. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying spacetime curvature that is physically significant.


title: Curvature

text: Although an arbitrarily curved space is very complex to describe, the curvature of a space which is locally isotropic and homogeneous is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (all points and all directions are indistinguishable). A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. An example of negatively curved space is hyperbolic geometry. A space or space-time with zero curvature is called flat. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat spacetime. There are other examples of flat geometries in both settings, though. A torus or a cylinder can both be given flat metrics, but differ in their topology. Other topologies are also possible for curved space. See also shape of the universe.


title: Curvature

text: The mathematical notion of "curvature" is also defined in much more general contexts. Many of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions.


title: Curvature

text: One such generalization is kinematic. The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind of tidal force (this is one way of thinking of the sectional curvature). This generalization of curvature depends on how nearby test particles diverge or converge when they are allowed to move freely in the space; see Jacobi field.


title: Curvature

text: Another broad generalization of curvature comes from the study of parallel transport on a surface. For instance, if a vector is moved around a loop on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as holonomy. Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see curvature form. A closely related notion of curvature comes from gauge theory in physics, where the curvature represents a field and a vector potential for the field is a quantity that is in general path-dependent: it may change if an observer moves around a loop.


title: Curvature

text: Two more generalizations of curvature are the scalar curvature and Ricci curvature. In a curved surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the same radius in flat space. This difference (in a suitable limit) is measured by the scalar curvature. The difference in area of a sector of the disc is measured by the Ricci curvature. Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are particularly important in relativity theory, where they both appear on the side of Einstein's field equations that represents the geometry of spacetime (the other side of which represents the presence of matter and energy). These generalizations of curvature underlie, for instance, the notion that curvature can be a property of a measure; see curvature of a measure.


title: Curvature

text: Another generalization of curvature relies on the ability to compare a curved space with another space that has "constant" curvature. Often this is done with triangles in the spaces. The notion of a triangle makes senses in metric spaces, and this gives rise to spaces.