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Baruch Samuel Blumberg

Baruch Samuel Blumberg (July 28, 1925 – April 5, 2011), known as Barry Blumberg, was an American physician, geneticist, and co-recipient of the 1976 Nobel Prize in Physiology or Medicine (with Daniel Carleton Gajdusek), for his work on the hepatitis B virus while an investigator at the NIH and at the Fox Chase Cancer Center. He was president of the American Philosophical Society from 2005 until his death.

Blumberg and Gajdusek received the Nobel Prize for discovering "new mechanisms for the origin and dissemination of infectious diseases." Blumberg identified the hepatitis B virus, and later developed its diagnostic test and vaccine.

Early life and education
Blumberg was born in Brooklyn, New York, into a Jewish family, the son of Ida (Simonoff) and Meyer Blumberg, a lawyer. He first attended the Orthodox Yeshivah of Flatbush for elementary school, where–in addition to all regular school subjects–he learned to read and write in Hebrew, and to study the Bible and Jewish texts in their original language. (That school also had among its students a contemporary of Blumberg, Eric Kandel, who is another recipient of the Nobel Prize in medicine.) Blumberg then attended Brooklyn's James Madison High School, a school that Blumberg described as having high academic standards, including many teachers with Ph.D.s. After moving to Far Rockaway, Queens, he transferred to Far Rockaway High School in the early 1940s, a school that also produced fellow laureates Burton Richter and Richard Feynman. Blumberg served as a U.S. Navy deck officer during World War II. He then attended Union College in Schenectady, New York, and graduated from there with honors in 1946.

Originally entering the graduate program in mathematics at Columbia University, Blumberg switched to medicine and enrolled at Columbia's College of Physicians and Surgeons, from which he received his MD in 1951. He remained at Columbia Presbyterian Medical Center for the next four years, first as an intern and then as a resident. He then moved to the University of Oxford and began graduate work in biochemistry at Balliol College, Oxford, and earned his DPhil there in 1957. He later became the first American to be master at Balliol College, Oxford.

Scientific career
Throughout the 1950s, Blumberg traveled the world taking human blood samples, to study the genetic variations in human beings, focusing on the question of why some people contract a disease in a given environment, while others do not. In 1964, while studying "yellow jaundice" (hepatitis), he discovered a surface antigen for hepatitis B in the blood of an Australian aborigine, hence initially called the 'Australian antigen'. His work later demonstrated that the virus could cause liver cancer. Blumberg and his team were able to develop a screening test for the hepatitis B virus, to prevent its spread in blood donations, and developed a vaccine. Blumberg later freely distributed his vaccine patent in order to promote its distribution by drug companies. Deployment of the vaccine reduced the infection rate of hepatitis B in children in China from 15% to 1% in 10 years.

In 1964, Blumberg became a member of the Institute of Cancer Research (ICR) of the Lankenau Hospital Research Institute in Philadelphia, known today as the Lankenau Institute for Medical Research (LIMR), which later joined the Fox Chase Cancer Center in 1974, and he held the rank of University Professor of Medicine and Anthropology at the University of Pennsylvania starting in 1977. Concurrently, he was Master of Balliol College from 1989 to 1994. He was elected a Fellow of the American Academy of Arts and Sciences in 1994. From 1999 to 2002, he was also director of the NASA Astrobiology Institute at the Ames Research Center in Moffett Field, California.

In 1992, Blumberg participated in the founding of the Hepatitis B Foundation (HBF), a nonprofit organization dedicated to finding a cure for hepatitis B and improving the lives of those affected by hepatitis B worldwide. He served on the Scientific and Medical Advisory Board, and as its distinguished scholar from 1992 until his passing in 2011. Blumberg was a regular and inspirational presence at the Hepatitis Foundation, maintaining an office at the foundation in Doylestown, Pennsylvania.

In 2000, Blumberg received the Golden Plate Award of the American Academy of Achievement. In 2001, Blumberg was named to the Library of Congress Scholars Council, a body of distinguished scholars that advises the Librarian of Congress. Blumberg served on the council until his death.

In November 2004, Blumberg was named chairman of the scientific advisory board of United Therapeutics Corporation, a position he held until his death. As chairman, he convened three "Conference[s] on Nanomedical and Telemedical Technology", as well as guiding the biotechnology company in the development of a broad-spectrum anti-viral medicine.

Beginning in 2005, Blumberg also served as the president of the American Philosophical Society. He had first been elected to membership in the society in 1986.

In October 2010, Blumberg participated in the USA Science and Engineering Festival's Lunch with a Laureate program, in which middle and high school students of the Greater Washington, D.C., Northern Virginia and Maryland area got to engage in an informal conversation with a Nobel Prize–winning scientist over a brown-bag lunch.

In discussing the factors that influenced his life, Blumberg always gave credit to the mental discipline of the Jewish Talmud, and as often as possible, he attended weekly Talmud discussion classes until his death.

Bioethics
Blumberg devoted his 1976 Nobel lecture to the subject of bioethics. He predicted the discovery of the Hepatitis B chronic carrier state would lead to calls for exclusion and quarantine of chronic carriers and the denial of health care. Blumberg came down solidly on the side of liberty and stated it was better not to test for the condition in medical practice. The following year, the teacher's union in New York City moved to exclude chronic carriers from the New York school system. At the time, a number of developmentally disabled children who had been institutionalized at Willowbrook were being mainstreamed into the public schools. As part of the Willowbrook hepatitis experiments, most children had been involuntarily tested and over 50 chronic carriers had been identified. The New York Public Health department convened a panel to decide policy led by Saul Krugman; however, Blumberg with his open views, was notably excluded. The panel and school system decided to exclude all known Hepatitis B carriers from school attendance and impose compulsory blood testing on all their classmates without informed consent about the nature of the blood tests. Litigation on behalf of the excluded children reversed the policy, and Blumberg advised the excluded children's lawyers. This set important precedent for the AIDS era.

In an interview with The New York Times in 2002 he stated that "[Saving lives] is what drew me to medicine. There is, in Jewish thought, this idea that if you save a single life, you save the whole world".

Death
Blumberg died on April 5, 2011, shortly after giving the keynote speech at the International Lunar Research Park Exploratory Workshop held at NASA Ames Research Center. At the time of his death Blumberg was a distinguished scientist at the NASA Lunar Science Institute, located at the NASA Ames Research Center in Moffett Field, California.

Jonathan Chernoff, the scientific director at the Fox Chase Cancer Center where Blumberg spent most of his working life said, "I think it's fair to say that Barry prevented more cancer deaths than any person who's ever lived." In reference to Blumberg's discovery of the Hepatitis B vaccine, former NASA administrator Daniel Goldin said, "Our planet is an improved place as a result of Barry's few short days in residence."

In 2011, the Library of Congress and National Aeronautics and Space Administration (NASA) announced the establishment of the Baruch S. Blumberg NASA/Library of Congress Chair in Astrobiology, a research position housed within the library's John W. Kluge Center, which explores the effects of astrobiology research on society. The chair was named for Blumberg in recognition of his service to the Library of Congress Scholars Council, and his commitment to "research and dialogue between disciplines."

In 2011, in recognition of Blumberg's long professional and personal association with the department of biochemistry and the Glycobiology Institute, Oxford University established the Baruch Blumberg Professorship in Virology.

Manuscript collection
The Baruch S. Blumberg papers are held at the American Philosophical Society in Philadelphia, PA. The collection contains 458 linear feet of materials documenting his life and career.

title: Baruch Samuel Blumberg


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The Raven (1963 film)

The Raven is a 1963 American comedy gothic horror film produced and directed by Roger Corman. The film stars Vincent Price, Peter Lorre, and Boris Karloff as a trio of rival sorcerers. The supporting cast includes Jack Nicholson as the son of Lorre's character.

It was the fifth in the so-called Corman-Poe cycle of eight films largely featuring adaptations of Edgar Allan Poe stories produced by Roger Corman and released by American International Pictures (AIP). The film was written by Richard Matheson, based on references to Poe's 1845 poem "The Raven". AIP released the film as a double feature with Night Tide.

Three decades earlier, Karloff had appeared in another film with the same title, Lew Landers's 1935 horror film The Raven with Bela Lugosi.

Plot
In the year 1506, the sorcerer Dr. Erasmus Craven has been mourning the death of his wife Lenore for over two years, much to the dismay of his daughter (Lenore's stepdaughter), Estelle. One night he is visited by a raven, the wizard Dr. Bedlo. Together they brew a potion that restores Bedlo to human form.

Bedlo explains that he was transformed by Dr. Scarabus in an unfair duel. Both decide to see Scarabus, Bedlo to exact revenge and Craven to look for his wife, who Bedlo saw alive at Scarabus' castle. After fighting off an attack by Craven's coachman, who acted under the influence of Scarabus, they are joined by Estelle and Bedlo's son Rexford, and set out to the castle. Estelle and Rexford become fond of each other over the journey.

At the castle, Scarabus greets his guests with false friendship, and Bedlo is apparently killed as he conjures a storm in an act of defiance. At night, however, Rexford finds Bedlo alive and well, hiding in the castle. Bedlo conspired with Scarabus to deliver Craven to him, so that Scarabus could acquire his magical secrets. Craven, meanwhile, is visited and tormented by Lenore, who is revealed to have faked her death to become Scarabus' mistress.

As Craven, Estelle, Rexford and Bedlo try to escape from the castle, Scarabus stops them, and they are imprisoned. Bedlo panics and begs Scarabus to turn him back into a raven rather than torture him. He then flees the dungeon by flying away. Craven is forced to choose between surrendering his magical secrets to Scarabus or watching his daughter be tortured. Having only pretended to desert his friends and son, Bedlo secretly returns and frees Rexford, and together they aid Craven.

Craven and Scarabus engage in a magic duel. After a series of attacks, counterattacks and insults, during which Scarabus sets the castle on fire, Craven defeats Scarabus. Lenore tries to reconcile with him, claiming that she was bewitched by Scarabus, but Craven disbelieves and rejects her. Craven, Bedlo, Estelle and Rexford escape as the castle collapses on Scarabus and Lenore; both survive, but Scarabus's magic is gone.

Bedlo, still a raven, tries to convince Craven to restore him to human form. Still bitter over Bedlo's earlier betrayal, Craven casts a spell to render him mute.

Cast

 * Vincent Price as Dr. Erasmus Craven
 * Peter Lorre as Dr. Adolphus Bedlo
 * Boris Karloff as Dr. Scarabus
 * Hazel Court as Lenore Craven
 * Olive Sturgess as Estelle Craven
 * Jack Nicholson as Rexford Bedlo
 * Connie Wallace as Maid
 * William Baskin as Grimes
 * Aaron Saxon as Gort

Script
Roger Corman and Richard Matheson had both enjoyed making the comic "The Black Cat" episode of Tales of Terror and wanted to try an entirely comic Poe feature.

"After I heard they wanted to make a movie out of a poem, I felt that was an utter joke, so comedy was really the only way to go with it", said Matheson.

Shooting
The film was shot in 15 days.

Roger Corman said that although they kept closely to the structure and story script, "We did more improvisation on that film than any of the others." During shooting, Peter Lorre ad-libbed a number of lines in the film including: Roger Corman said that Lorre's improvisations confused both Vincent Price and Boris Karloff, but Price adapted to it well while Karloff struggled: "Overall I would say we had as good a spirit on The Raven as any film I've ever worked on, except for a couple of moments with Boris. There was a slight edge to it, because Boris came in with a carefully worked out preparation, so when Peter started improvising lines, it really threw Boris off from his preparation."
 * "How the hell should I know?", after Vincent Price asks "shall I ever see the rare and radiant Lenore again?"
 * "Where else?" after Vincent Price says "I keep her here." (referring to the body of his lost love Lenore, kept in a coffin in the hall)
 * "Hard place to keep clean."

Corman said the tension between Peter Lorre and Jack Nicholson as father and son came from the actors rather than the script, as the two did not get along well.

Vincent Price later recalled about the final duel: "Boris hated being strung up in the air on those chairs. He was terribly crippled, and we were both floating in the air on these wires. It wasn't a pleasant feeling! And I hated having that snake wrapped around my neck for two hours... I hate snakes." Boris Karloff later said he was annoyed at having to wear the heavy cape.

The footage of the burning interior of the castle was reused from Corman's 1960 film House of Usher.

Release
The Raven opened on January 23, 1963 in 100 theaters in the Greater New York area before opening the following week in 50 theaters in Southern California.

Critical reception
Bosley Crowther of The New York Times panned the film as "comic-book nonsense ... Strickly (sic) a picture for the kiddies and the bird-brained, quote the critic." Variety wrote that while Poe "might turn over in his crypt at this nonsensical adaptation of his immortal poem", Corman nevertheless "takes this premise and develops it expertly as a horror-comedy." The Chicago Tribune called it "fairly thin fare, made up mostly of camera tricks, and some very obviously false scenery, but Peter Lorre's performance is mildly entertaining. Youngsters may find it fun." A generally positive review in The Monthly Film Bulletin wrote that the film "starts off with the inestimable advantage of a script which not only makes it amply clear from the outset that [Corman] is cheerfully and wholeheartedly sending himself up, but manages to do it wittily." Its main criticism was a "long central section" of the film that drags until things pick up again for the final duel. Peter John Dyer of Sight & Sound wrote, "Richard Matheson's script, a good deal more tenuous than its predecessors in the Corman-Poe canon, at least treats its actors generously to props, incantations and quotable lines ... A pity the equation doesn't always add up; there's too much slack, due perhaps to an imbalance between the comedy, which runs riot, and the horror, which trails behind in the wake of previous Corman films."

Colin Greenland reviewed The Raven for Imagine magazine, and stated that "Not so much a raven, more a bit of a lark."

On Rotten Tomatoes the film has an approval rating of 88% based on reviews from 17 critics, with an average rating of 6.8 out of 10.

Shown on the MeTV show Svengoolie on January 7, 2022, and also Dec. 9, 2023.

Box office
The film was popular at the box office.

Novelization
A novelization of the film was written by Eunice Sudak adapted from Richard Matheson's screenplay and published by Lancer Books in paperback. This novel was republished by Bear Manor Media in 2012.

Comic book adaptation

 * Dell Movie Classic: The Raven (September 1963)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: The Raven (1963 film)


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MediaWiki talk:Uctop

"Top" seems rather cryptic to me, I had no idea what it meant and had to ask. Elsewhere the current revision is referred to as "current" or "cur", would it not make sense to do so here too? Or how about "latest", "unchanged" or "unedited"? -- Tom- 01:47, 11 Mar 2004 (UTC)

 * It sure is an issue with Wikipedia, but not an important one. I only found out what "top" meant a few months ago; it never really troubled me up until then... - Mark 01:49, 11 Mar 2004 (UTC)

 * I do care, so I was happy to see that someone had changed the message and I took the new suggestion to make it default on MediaWiki: 58049, still to be approved. --Nemo 05:26, 8 April 2013 (UTC)

title: MediaWiki talk:Uctop


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A Sunday Afternoon on the Island of La Grande Jatte

A Sunday Afternoon on the Island of La Grande Jatte (Un dimanche après-midi à l'Île de la Grande Jatte) was painted from 1884 to 1886 and is Georges Seurat's most famous work. A leading example of pointillist technique, executed on a large canvas, it is a founding work of the neo-impressionist movement. Seurat's composition includes a number of Parisians at a park on the banks of the River Seine. It is held in the collection of the Art Institute of Chicago.

Background
Georges Seurat painted A Sunday Afternoon between May 1884 and March 1885, and from October 1885 to May 1886, focusing meticulously on the landscape of the park and concentrating on issues of colour, light, and form. The painting is approximately 2 x in size. Seurat completed numerous preliminary drawings and oil sketches before completing his masterpiece. One complete painting, the study featured to the right, measures 27 3/4 x 41 in. (70.5 x 104.1 cm) and is on display in the Metropolitan Museum of Art.

Inspired by optical effects and perception inherent in the color theories of Michel Eugène Chevreul, Ogden Rood and others, Seurat adapted this scientific research to his painting. Seurat contrasted miniature dots or small brushstrokes of colors that when unified optically in the human eye were perceived as a single shade or hue. He believed that this form of painting, called Divisionism at the time (a term he preferred) but now known as Pointillism, would make the colors more brilliant and powerful than standard brushstrokes. The use of dots of almost uniform size came in the second year of his work on the painting, 1885–86. To make the experience of the painting even more vivid, at the paintings edge, he surrounded it with a frame of painted dots, which in turn he enclosed with a pure white, wooden frame, which is how the painting is exhibited at the Art Institute of Chicago. The Island of la Grande Jatte is located at the very gates of Paris, lying in the Seine between Neuilly and Levallois-Perret, a short distance from where La Défense business district currently stands. Although for many years it was an industrial site, it has become the site of a public garden and a housing development. When Seurat began the painting in 1884, the island was a bucolic retreat far from the urban center.

The painting was first exhibited at the eighth (and last) Impressionist exhibition in May 1886, then in August 1886, dominating the second Salon of the Société des Artistes Indépendants, of which Seurat had been a founder in 1884. Seurat was extremely disciplined, always serious, and private to the point of secretiveness—for the most part, steering his own steady course. As a painter, he wanted to make a difference in the history of art and with La Grande Jatte, Seurat was immediately acknowledged as the leader of a new and rebellious form of Impressionism called Neo-Impressionism.

Interpretation
Seurat's painting was a mirror impression of his own painting, Bathers at Asnières, completed shortly before, in 1884. Whereas the bathers in that earlier painting are doused in light, almost every figure on La Grande Jatte appears to be cast in shadow, either under trees or an umbrella, or from another person. For Parisians, Sunday was the day to escape the heat of the city and head for the shade of the trees and the cool breezes that came off the river. And at first glance, the viewer sees many different people relaxing in a park by the river. On the right, a fashionable couple, the woman with the sunshade and the man in his top hat, are on a stroll. On the left, another woman who is also well dressed extends her fishing pole over the water. There is a small man with the black hat and thin cane looking at the river, and a white dog with a brown head, a woman knitting, a man playing a trumpet, two soldiers standing at attention as the musician plays, and a woman hunched under an orange umbrella. Seurat also painted a man with a pipe, a woman under a parasol in a boat filled with rowers, and a couple admiring their infant child.

Some of the characters are doing curious things. The lady on the right side has a pet monkey on a leash. A lady on the left near the river bank is fishing. The area was known at the time as being a place to procure prostitutes among the bourgeoisie, a likely allusion of the otherwise odd "fishing" rod. In the painting's center stands a little girl dressed in white (who is not in a shadow), who stares directly at the viewer of the painting. This may be interpreted as someone who is silently questioning the audience.

In the 1950s, historian and Marxist philosopher Ernst Bloch drew social and political significance from Seurat's La Grande Jatte. The historian's focal point was what he saw as Seurat's mechanical use of the figures and what their static nature said about French society at the time. Afterward, critique of the work often centered on the artist's presumed mathematical and robotic interpretation of the meaning of modernity in Paris. According to historian of Modernism William R. Everdell: "Seurat himself told a sympathetic critic, Gustave Kahn, that his model was the Panathenaic procession in the Parthenon frieze. But Seurat didn't want to paint ancient Athenians. He wanted 'to make the moderns file past ... in their essential form.' By 'moderns' he meant nothing very complicated. He wanted ordinary people as his subject, and ordinary life. He was a bit of a democrat—a "Communard", as one of his friends remarked, referring to the left-wing revolutionaries of 1871; and he was fascinated by the way things distinct and different encountered each other: the city and the country, the farm and the factory, the bourgeois and the proletarian meeting at their edges in a sort of harmony of opposites."

The border of the painting is, unusually, in inverted color, as if the world around them is also slowly inverting from the way of life they have known. Seen in this context, the boy who bathes on the other side of the river bank at Asnières appears to be calling out to them, as if to say, "We are the future. Come and join us".

Painting materials
Seurat painted the La Grande Jatte in three distinct stages. In the first stage, which was started in 1884, he mixed his paints from several individual pigments and was still using dull earth pigments such as ochre or burnt sienna. In the second stage, during 1885 and 1886, Seurat dispensed with the earth pigments and also limited the number of individual pigments in his paints. This change in his palette was due to his application of the advanced color theories of his time. His intention was to paint small dots or strokes of pure color that would then mix on the retina of the beholder to achieve the desired color impression instead of the usual practice of mixing individual pigments.

Seurat's palette consisted of the usual pigments of his time such as cobalt blue, emerald green and vermilion. Additionally, he used the then new pigment zinc yellow (zinc chromate), predominantly for yellow highlights in the sunlit grass in the middle of the painting but also in mixtures with orange and blue pigments. In the century and more since the painting's completion, the zinc yellow has darkened to brown—a color degeneration that was already showing in the painting in Seurat's lifetime. The discoloration of the originally bright yellow zinc yellow (zinc chromate) to a brownish color is due to the chemical reaction of the chromate ions to orange-colored dichromate ions. In the third stage during 1888–89 Seurat added the colored borders to his composition.

The results of investigation into the discoloration of this painting have been combined with further research into natural aging of paints to digitally rejuvenate the painting.

Acquisition by the Art Institute of Chicago
In 1923, Frederic Bartlett was appointed trustee of the Art Institute of Chicago. He and his second wife, Helen Birch Bartlett, loaned their collection of French Post-Impressionist and Modernist art to the museum. It was Mrs. Bartlett who had an interest in French and avant-garde artists and influenced her husband's collecting tastes. Sunday Afternoon on the Island of La Grande Jatte was purchased on the advice of the Art Institute of Chicago's curatorial staff in 1924.

In conceptual artist Don Celender's 1974–75 book Observation and Scholarship Examination for Art Historians, Museum Directors, Artists, Dealers and Collectors, it is claimed that the institute paid $24,000 for the work (over $354,000 in 2018 dollars ).

In 1958, the painting was loaned for the only time, to the Museum of Modern Art in New York. On 15 April 1958, a fire there, which killed one person on the second floor of the museum, forced the evacuation of the painting, which had been on a floor above the fire, to the Whitney Museum, which adjoined MoMA at the time.

In popular culture
The painting is the basis for the 1984 Broadway musical Sunday in the Park with George by Stephen Sondheim and James Lapine, which tells a fictionalized story of the painting's creation. Subsequently, the painting is sometimes referred to by the misnomer "Sunday in the Park".

In Topiary Park (formerly Old Deaf School Park) in Columbus, Ohio, sculptor James T. Mason re-created the painting in topiary form; the installation was completed in 1989.

The 1986 John Hughes film Ferris Bueller's Day Off featured the painting during a scene at the Art Institute of Chicago.


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Faraday effect

The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the magnetic field along the direction of the light propagation. Formally, it is a special case of gyroelectromagnetism obtained when the dielectric permittivity tensor is diagonal. This effect occurs in most optically transparent dielectric materials (including liquids) under the influence of magnetic fields.

Discovered by Michael Faraday in 1845, the Faraday effect was the first experimental evidence that light and electromagnetism are related. The theoretical basis of electromagnetic radiation (which includes visible light) was completed by James Clerk Maxwell in the 1860s. Maxwell's equations were rewritten in their current form in the 1870s by Oliver Heaviside.

The Faraday effect is caused by left and right circularly polarized waves propagating at slightly different speeds, a property known as circular birefringence. Since a linear polarization can be decomposed into the superposition of two equal-amplitude circularly polarized components of opposite handedness and different phase, the effect of a relative phase shift, induced by the Faraday effect, is to rotate the orientation of a wave's linear polarization.

The Faraday effect has applications in measuring instruments. For instance, the Faraday effect has been used to measure optical rotatory power and for remote sensing of magnetic fields (such as fiber optic current sensors). The Faraday effect is used in spintronics research to study the polarization of electron spins in semiconductors. Faraday rotators can be used for amplitude modulation of light, and are the basis of optical isolators and optical circulators; such components are required in optical telecommunications and other laser applications.

History
By 1845, it was known through the work of Augustin-Jean Fresnel, Étienne-Louis Malus, and others that different materials are able to modify the direction of polarization of light when appropriately oriented, making polarized light a very powerful tool to investigate the properties of transparent materials. Faraday firmly believed that light was an electromagnetic phenomenon, and as such should be affected by electromagnetic forces. He spent considerable effort looking for evidence of electric forces affecting the polarization of light through what are now known as electro-optic effects, starting with decomposing electrolytes. However, his experimental methods were not sensitive enough, and the effect was only measured thirty years later by John Kerr.

Faraday then attempted to look for the effects of magnetic forces on light passing through various substances. After several unsuccessful trials, he happened to test a piece of "heavy" glass, containing equal proportions of silica, boracic acid and lead oxide, that he had made during his earlier work on glass manufacturing. Faraday observed that when a beam of polarized light passed through the glass in the direction of an applied magnetic force, the polarization of light rotated by an angle that was proportional to the strength of the force. He used a Nicol prism to measure the polarization. He was later able to reproduce the effect in several other solids, liquids, and gases by procuring stronger electromagnets.

The discovery is well documented in Faraday's daily notebook. On 13 Sept. 1845, in paragraph #7504, under the rubric Heavy Glass, he wrote: "BUT, when the contrary magnetic poles were on the same side, there was an effect produced on the polarized ray, and thus magnetic force and light were proved to have relation to each other."

He summarized the results of his experiments on 30 Sept. 1845, in paragraph #7718, famously writing: "Still, I have at last succeeded in illuminating a magnetic curve or line of force, and in magnetizing a ray of light."

Physical interpretation
The linear polarized light that is seen to rotate in the Faraday effect can be seen as consisting of the superposition of a right- and a left- circularly polarized beam (this superposition principle is fundamental in many branches of physics). We can look at the effects of each component (right- or left-polarized) separately, and see what effect this has on the result.

In circularly polarized light the direction of the electric field rotates at the frequency of the light, either clockwise or counter-clockwise. In a material, this electric field causes a force on the charged particles that compose the material (because of their large charge to mass ratio, the electrons are most heavily affected). The motion thus effected will be circular, and circularly moving charges will create their own (magnetic) field in addition to the external magnetic field. There will thus be two different cases: the created field will be parallel to the external field for one (circular) polarization, and in the opposing direction for the other polarization direction – thus the net B field is enhanced in one direction and diminished in the opposite direction. This changes the dynamics of the interaction for each beam and one of the beams will be slowed more than the other, causing a phase difference between the left- and right-polarized beam. When the two beams are added after this phase shift, the result is again a linearly polarized beam, but with a rotation of the polarization vector.

The direction of polarization rotation depends on the properties of the material through which the light is shone. A full treatment would have to take into account the effect of the external and radiation-induced fields on the wave function of the electrons, and then calculate the effect of this change on the refractive index of the material for each polarization, to see whether the right- or left-circular polarization is slowed more.

Mathematical formulation
Formally, the magnetic permeability is treated as a non-diagonal tensor as expressed by the equation:

 * $$\mathbf{B}(\omega) = \begin{bmatrix}

\mu_{1} & -i \mu_{2} &      0 \\ i \mu_{2} &   \mu_{1} &       0 \\ 0 &         0 & \mu_{z} \\ \end{bmatrix} \mathbf{H}(\omega)$$

The relation between the angle of rotation of the polarization and the magnetic field in a transparent material is:

 * $$ \beta = \mathcal{V}Bd $$

where

 * β is the angle of rotation (in radians)
 * B is the magnetic flux density in the direction of propagation (in teslas)
 * d is the length of the path (in meters) where the light and magnetic field interact
 * $$\scriptstyle \mathcal{V}$$ is the Verdet constant for the material. This empirical proportionality constant (in units of radians per tesla per meter) varies with wavelength and temperature  and is tabulated for various materials.

A positive Verdet constant corresponds to L-rotation (anticlockwise) when the direction of propagation is parallel to the magnetic field and to R-rotation (clockwise) when the direction of propagation is anti-parallel. Thus, if a ray of light is passed through a material and reflected back through it, the rotation doubles.

Some materials, such as terbium gallium garnet (TGG) have extremely high Verdet constants (≈ $-134 rad/(T·m)$ for 632 nm light). By placing a rod of this material in a strong magnetic field, Faraday rotation angles of over 0.78 rad (45°) can be achieved. This allows the construction of Faraday rotators, which are the principal component of Faraday isolators, devices which transmit light in only one direction. The Faraday effect can, however, be observed and measured in a Terbium-doped glass with Verdet constant as low as (≈ $-20 rad/(T·m)$ for 632 nm light). Similar isolators are constructed for microwave systems by using ferrite rods in a waveguide with a surrounding magnetic field. A thorough mathematical description can be found here.

Interstellar medium
The effect is imposed on light over the course of its propagation from its origin to the Earth, through the interstellar medium. Here, the effect is caused by free electrons and can be characterized as a difference in the refractive index seen by the two circularly polarized propagation modes. Hence, in contrast to the Faraday effect in solids or liquids, interstellar Faraday rotation (β) has a simple dependence on the wavelength of light (λ), namely:

 * $$ \beta = \mathrm{RM} \, \lambda^2 $$

where the overall strength of the effect is characterized by RM, the rotation measure. This in turn depends on the axial component of the interstellar magnetic field Bundefined, and the number density of electrons ne, both of which vary along the propagation path. In Gaussian cgs units the rotation measure is given by:
 * $$\mathrm{RM} = \frac{e^3}{2\pi m^2c^4}\int_0^d n_e(s) B_\parallel(s) \;\mathrm{d}s$$

or in SI units:
 * $$\mathrm{RM} = \frac{e^3}{8\pi^2 \varepsilon_0 m^2 c^3} \int_0^d n_e(s) B_{||}(s) \;\mathrm{d}s \approx

(2.62 \times 10^{-13}\, \mathrm T^{-1})\times\, \int_0^d n_e(s) B_\parallel(s)\; \mathrm{d}s $$

where
 * ne(s) is the density of electrons at each point s along the path
 * B‖(s) is the component of the interstellar magnetic field in the direction of propagation at each point s along the path
 * e is the charge of an electron;
 * c is the speed of light in vacuum;
 * m is the mass of an electron;
 * $$\scriptstyle\epsilon_0$$ is the vacuum permittivity;

The integral is taken over the entire path from the source to the observer.

Faraday rotation is an important tool in astronomy for the measurement of magnetic fields, which can be estimated from rotation measures given a knowledge of the electron number density. In the case of radio pulsars, the dispersion caused by these electrons results in a time delay between pulses received at different wavelengths, which can be measured in terms of the electron column density, or dispersion measure. A measurement of both the dispersion measure and the rotation measure therefore yields the weighted mean of the magnetic field along the line of sight. The same information can be obtained from objects other than pulsars, if the dispersion measure can be estimated based on reasonable guesses about the propagation path length and typical electron densities. In particular, Faraday rotation measurements of polarized radio signals from extragalactic radio sources occulted by the solar corona can be used to estimate both the electron density distribution and the direction and strength of the magnetic field in the coronal plasma.

Ionosphere
Radio waves passing through the Earth's ionosphere are likewise subject to the Faraday effect. The ionosphere consists of a plasma containing free electrons which contribute to Faraday rotation according to the above equation, whereas the positive ions are relatively massive and have little influence. In conjunction with the Earth's magnetic field, rotation of the polarization of radio waves thus occurs. Since the density of electrons in the ionosphere varies greatly on a daily basis, as well as over the sunspot cycle, the magnitude of the effect varies. However the effect is always proportional to the square of the wavelength, so even at the UHF television frequency of 500 MHz (λ = 60 cm), there can be more than a complete rotation of the axis of polarization. A consequence is that although most radio transmitting antennas are either vertically or horizontally polarized, the polarization of a medium or short wave signal after reflection by the ionosphere is rather unpredictable. However the Faraday effect due to free electrons diminishes rapidly at higher frequencies (shorter wavelengths) so that at microwave frequencies, used by satellite communications, the transmitted polarization is maintained between the satellite and the ground.

Semiconductors
Due to spin-orbit coupling, undoped GaAs single crystal exhibits much larger Faraday rotation than glass (SiO2). Considering the atomic arrangement is different along the (100) and (110) plane, one might think the Faraday rotation is polarization dependent. However, experimental work revealed an immeasurable anisotropy in the wavelength range from 880–1,600 nm. Based on the large Faraday rotation, one might be able to use GaAs to calibrate the B field of the terahertz electromagnetic wave which requires very fast response time. Around the band gap, the Faraday effect shows resonance behavior.

More generally, (ferromagnetic) semiconductors return both electro-gyration and a Faraday response in the high frequency domain. The combination of the two is described by gyroelectromagnetic media, for which gyroelectricity and gyromagnetism (Faraday effect) may occur at the same time.

Organic materials
In organic materials, Faraday rotation is typically small, with a Verdet constant in the visible wavelength region on the order of a few hundred degrees per Tesla per meter, decreasing proportional to $$ \lambda^{-2} $$ in this region. While the Verdet constant of organic materials does increase around electronic transitions in the molecule, the associated light absorption makes most organic materials bad candidates for applications. There are however also isolated reports of large Faraday rotation in organic liquid crystals without associated absorption.

Plasmonic and magnetic materials
In 2009 γ-Fe2O3-Au core-shell nanostructures were synthesized to integrate magnetic (γ-Fe2O3) and plasmonic (Au) properties into one composite. Faraday rotation with and without the plasmonic materials was tested and rotation enhancement under 530 nm light irradiation was observed. Researchers claim that the magnitude of the magneto-optical enhancement is governed primarily by the spectral overlap of the magneto-optical transition and the plasmon resonance.

The reported composite magnetic/plasmonic nanostructure can be visualized to be a magnetic particle embedded in a resonant optical cavity. Because of the large density of photon states in the cavity, the interaction between the electromagnetic field of the light and the electronic transitions of the magnetic material is enhanced, resulting in a larger difference between the velocities of the right- and left-hand circularized polarization, therefore enhancing Faraday rotation.

title: Faraday effect


text:
Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or $j$-faces (for all ${5}.$, where $n$ is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension ${5,3}.$.

Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians.

Classically, a regular polytope in $n$ dimensions may be defined as having regular facets (${5,3,3}.$-faces) and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes.

A regular polytope can be represented by a Schläfli symbol of the form ${4,3,4}.$ with regular facets as $0 ≤ j ≤ n$ and regular vertex figures as $j≤ n$

Classification and description
Regular polytopes are classified primarily according to their dimension.

Three classes of regular polytopes exist in every dimension: Any other regular polytope is said to be exceptional.
 * Regular simplex
 * Measure polytope (Hypercube)
 * Cross polytope (Orthoplex)

In one dimension, the line segment simultaneously serves as the 1-simplex, the 1-hypercube and the 1-orthoplex. In two dimensions, there are infinitely many regular polygons, namely the regular n-sided polygon for n ≥ 3. The triangle is the 2-simplex. The square is both the 2-hypercube and the 2-orthoplex. The n-sided polygons for n ≥ 5 are exceptional.

In three and four dimensions, there are several more exceptional regular polyhedra and 4-polytopes.

In five dimensions and above, the simplex, hypercube and orthoplex are the only regular polytopes. There are no exceptional regular polytopes in these dimensions.

See also the list of regular polytopes.

Regular polytopes can be further classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron. Two distinct regular polytopes with the same symmetry are dual to one another. Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries.

The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.

Schläfli symbols
A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.

 * A convex regular polygon having n sides is denoted by {n}. So an equilateral triangle is {3}, a square {4}, and so on indefinitely. A regular n-sided star polygon which winds m times around its centre is denoted by the fractional value {n/m}, where n and m are co-prime, so a regular pentagram is {5/2}.
 * A regular polyhedron having faces {n} with p faces joining around a vertex is denoted by {n, p}. The nine regular polyhedra are {3, 3} {3, 4} {4, 3} {3, 5} {5, 3} {3, 5/2} {5/2, 3} {5, 5/2} and {5/2, 5}. {p} is the vertex figure of the polyhedron.
 * A regular 4-polytope having cells {n, p} with q cells joining around an edge is denoted by {n, p, q}. The vertex figure of the 4-polytope is a {p, q}.
 * A regular 5-polytope is an {n, p, q, r}. And so on.

Duality of the regular polytopes
The dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} and so on.

The vertex figure of a regular polytope is the dual of the dual polytope's facet. For example, the vertex figure of {3, 3, 4} is {3, 4}, the dual of which is {4, 3} — a cell of {4, 3, 3}.

The measure and cross polytopes in any dimension are dual to each other.

If the Schläfli symbol is palindromic, i.e. reads the same forwards and backwards, then the polyhedron is self-dual. The self-dual regular polytopes are:
 * All regular polygons, {a}.
 * All regular n-simplexes, {3,3,...,3}
 * The regular 24-cell in 4 dimensions, {3,4,3}.
 * The great 120-cell ({5,5/2,5}) and grand stellated 120-cell ({5/2,5,5/2}) in 4 dimensions.
 * All regular n-dimensional cubic honeycombs, {4,3,...,3,4}. These may be treated as infinite polytopes.
 * Hyperbolic tilings and honeycombs (tilings {p,p} with p>4 in 2 dimensions, {4,4,4}, {5,3,5}. {3,5,3}, {6,3,6}, and {3,6,3} in 3 dimensions, {5,3,3,5} in 4 dimensions, and {3,3,4,3,3} in 5 dimensions).

Regular simplices
Begin with a point A. Mark point B at a distance r from it, and join to form a line segment. Mark point C in a second, orthogonal, dimension at a distance r from both, and join to A and B to form an equilateral triangle. Mark point D in a third, orthogonal, dimension a distance r from all three, and join to form a regular tetrahedron. And so on for higher dimensions.

These are the regular simplices or simplexes. Their names are, in order of dimension:

 * 0. Point
 * 1. Line segment
 * 2. Equilateral triangle (regular trigon)
 * 3. Regular tetrahedron
 * 4. Regular pentachoron or 4-simplex
 * 5. Regular hexateron or 5-simplex
 * ... An n-simplex has n+1 vertices.

Measure polytopes (hypercubes)
Begin with a point A. Extend a line to point B at distance r, and join to form a line segment. Extend a second line of length r, orthogonal to AB, from B to C, and likewise from A to D, to form a square ABCD. Extend lines of length r respectively from each corner, orthogonal to both AB and BC (i.e. upwards). Mark new points E,F,G,H to form the cube ABCDEFGH. And so on for higher dimensions.

These are the measure polytopes or hypercubes. Their names are, in order of dimension:

 * 0. Point
 * 1. Line segment
 * 2. Square (regular tetragon)
 * 3. Cube (regular hexahedron)
 * 4. Tesseract (regular octachoron) or 4-cube
 * 5. Penteract (regular decateron) or 5-cube
 * ... An n-cube has 2n vertices.

Cross polytopes (orthoplexes)
Begin with a point O. Extend a line in opposite directions to points A and B a distance r from O and 2r apart. Draw a line COD of length 2r, centred on O and orthogonal to AB. Join the ends to form a square ACBD. Draw a line EOF of the same length and centered on 'O', orthogonal to AB and CD (i.e. upwards and downwards). Join the ends to the square to form a regular octahedron. And so on for higher dimensions.

These are the cross polytopes or orthoplexes. Their names are, in order of dimensionality:

 * 0. Point
 * 1. Line segment
 * 2. Square (regular tetragon)
 * 3. Regular octahedron
 * 4. Regular hexadecachoron (16-cell) or 4-orthoplex
 * 5. Regular triacontakaiditeron (Pentacross) or 5-orthoplex
 * ... An n-orthoplex has 2n vertices.

Classification by Coxeter groups
Regular polytopes can be classified by their isometry group. These are finite Coxeter groups, but not every finite Coxeter groups may be realised as the isometry group of a regular polytope. Regular polytopes are in bijection with Coxeter groups with linear Coxeter-Dynkin diagram (without branch point) and an increasing numbering of the nodes. Reversing the numbering gives the dual polytope.

The classification of finite Coxeter groups, which goes back to, therefore implies the classification of regular polytopes:
 * Type $$A_n$$, the symmetric group, gives the regular simplex,
 * Type $$B_n$$, gives the measure polytope and the cross polytope (both can be distinguished by the increasing numbering of the nodes of the Coxeter-Dynkin diagram),
 * Exceptional types $$I_2(n)$$ give the regular polygons (with $$n = 3, 4, ...$$),
 * Exceptional type $$H_3$$ gives the regular dodecahedron and icosahedron (again the numbering allows to distinguish them),
 * Exceptional type $$H_4$$ gives the 120-cell and the 600-cell,
 * Exceptional type $$F_4$$ gives the 24-cell, which is self-dual.

The bijection between regular polytopes and Coxeter groups with linear Coxeter-Dynkin diagram can be understood as follows. Consider a regular polytope $$P$$ of dimension $$n$$ and take its barycentric subdivision. The fundamental domain of the isometry group action on $$P$$ is given by any simplex $$\Delta$$ in the barycentric subdivision. The simplex $$\Delta$$ has $$n+1$$ vertices which can be numbered from 0 to $$n$$ by the dimension of the corresponding face of $$P$$ (the face they are the barycenter of). The isometry group of $$P$$ is generated by the $$n$$ reflections around the hyperplanes of $$\Delta$$ containing the vertex number $$n$$ (since the barycenter of the whole polytope $$P$$ is fixed by any isometry). These $$n$$ hyperplanes can be numbered by the vertex of $$\Delta$$ they do not contain. The remaining thing to check is that any two hyperplanes with adjacent numbers cannot be orthogonal, whereas hyperplanes with non-adjacent numbers are orthogonal. This can be done using induction (since all facets of $$P$$ are again regular polytopes). Therefore, the Coxeter-Dynkin diagram of the isometry group of $$P$$ has $$n$$ vertices numbered from 0 to $$n-1$$ such that adjacent numbers are linked by at least one edge and non-adjacent numbers are not linked.

Convex polygons and polyhedra
The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from ancient Greek mathematicians. The five Platonic solids were known to them. Pythagoras knew of at least three of them and Theaetetus (c. 417 BC – 369 BC) described all five. Later, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids.

 * {| class="wikitable"

 * colspan=5 align=center|Platonic solids
 * align=center|[[Image:Tetrahedron.svg|75px]]
 * align=center|[[Image:Hexahedron.svg|75px]]
 * align=center|[[Image:Octahedron.svg|75px]]
 * align=center|[[Image:POV-Ray-Dodecahedron.svg|75px]]
 * align=center|[[Image:Icosahedron.svg|75px]]
 * Tetrahedron||Cube||Octahedron||Dodecahedron||Icosahedron
 * }
 * Tetrahedron||Cube||Octahedron||Dodecahedron||Icosahedron
 * }
 * }

Star polygons and polyhedra
Our understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number. Thomas Bradwardine (Bradwardinus) was the first to record a serious study of star polygons. Various star polyhedra appear in Renaissance art, but it was not until Johannes Kepler studied the small stellated dodecahedron and the great stellated dodecahedron in 1619 that he realised these two were regular. Louis Poinsot discovered the great dodecahedron and great icosahedron in 1809, and Augustin Cauchy proved the list complete in 1812. These polyhedra are known as collectively as the Kepler-Poinsot polyhedra.

Higher-dimensional polytopes
It was not until the 19th century that a Swiss mathematician, Ludwig Schläfli, examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in, six years posthumously, although parts of it were published in and. Between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians — see for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by Reinhold Hoppe, one of Schläfli's rediscoverers, in 1882, and first used in English by Alicia Boole Stott some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).

is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six regular convex polytopes in 4 dimensions. Five of them can be seen as analogous to the Platonic solids: the 4-simplex (or pentachoron) to the tetrahedron, the hypercube (or tesseract) to the cube, the 4-orthoplex (or hexadecachoron or 16-cell) to the octahedron, the 120-cell to the dodecahedron, and the 600-cell to the icosahedron. The sixth, the 24-cell, can be seen as a transitional form between the hypercube and 16-cell, analogous to the way that the cuboctahedron and the rhombic dodecahedron are transitional forms between the cube and the octahedron.

In five and more dimensions, there are exactly three regular polytopes, which correspond to the tetrahedron, cube and octahedron: these are the regular simplices, measure polytopes and cross polytopes. Descriptions of these may be found in the list of regular polytopes. Also of interest are the star regular 4-polytopes, partially discovered by Schläfli.

By the end of the 19th century, mathematicians such as Arthur Cayley and Ludwig Schläfli had developed the theory of regular polytopes in four and higher dimensions, such as the tesseract and the 24-cell.

The latter are difficult (though not impossible) to visualise through a process of dimensional analogy, since they retain the familiar symmetry of their lower-dimensional analogues. The tesseract contains 8 cubical cells. It consists of two cubes in parallel hyperplanes with corresponding vertices cross-connected in such a way that the 8 cross-edges are equal in length and orthogonal to the 12+12 edges situated on each cube. The corresponding faces of the two cubes are connected to form the remaining 6 cubical faces of the tesseract. The 24-cell can be derived from the tesseract by joining the 8 vertices of each of its cubical faces to an additional vertex to form the four-dimensional analogue of a pyramid. Both figures, as well as other 4-dimensional figures, can be directly visualised and depicted using 4-dimensional stereographs.

Harder still to imagine are the more modern abstract regular polytopes such as the 57-cell or the 11-cell. From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives.

At the start of the 20th century, the definition of a regular polytope was as follows.
 * A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
 * A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose vertex figures are all congruent and regular.
 * And so on, a regular n-polytope is an n-dimensional polytope whose (n − 1)-dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.

This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.

 * An n-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to n−1 dimensions, can be mapped to any other such set by a symmetry of the polytope.

So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, known as a flag, (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:
 * A regular polytope is one whose symmetry group is transitive on its flags.

In the 20th century, some important developments were made. The symmetry groups of the classical regular polytopes were generalised into what are now called Coxeter groups. Coxeter groups also include the symmetry groups of regular tessellations of space or of the plane. For example, the symmetry group of an infinite chessboard would be the Coxeter group [4,4].

Apeirotopes — infinite polytopes
In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges. Nowadays, they are called infinite polyhedra or apeirohedra. The regular tilings of the plane {4, 4}, {3, 6} and {6, 3} can also be regarded as infinite polyhedra.

In the 1960s Branko Grünbaum issued a call to the geometric community to consider more abstract types of regular polytopes that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes, that is, regular polytopes with infinitely many faces. A simple example of a skew apeirogon would be a zig-zag. It seems to satisfy the definition of a regular polygon — all the edges are the same length, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered.

Regular complex polytopes
A complex number has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A complex Hilbert space has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a complex polytope.

Abstract polytopes

Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face. The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.

This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners.

The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively curved (elliptic) hyperspace.

A few years after Grünbaum's discovery of the 11-cell, H. S. M. Coxeter independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984).

By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called abstract polytopes. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by containment. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes.

A geometric polytope is understood to be a realization of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the corresponding faces of the geometric realisation. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.

The theory has since been further developed, largely by, but other researchers have also made contributions.

Regularity of abstract polytopes
Regularity has a related, though different meaning for abstract polytopes, since angles and lengths of edges have no meaning.

The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes.

Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes do not retain information about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope.

All polygons are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world.

Vertex figure of abstract polytopes
The concept of vertex figure is also defined differently for an abstract polytope. The vertex figure of a given abstract n-polytope at a given vertex V is the set of all abstract faces which contain V, including V itself. More formally, it is the abstract section

 * Fn / V = {F | V ≤ F ≤ Fn}

where Fn is the maximal face, i.e. the notional n-face which contains all other faces. Note that each i-face, i ≥ 0 of the original polytope becomes an (i − 1)-face of the vertex figure.

Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures may or may not be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons.

The classical vertex figure will, however, be a realisation of the abstract one.

Polygons
The traditional way to construct a regular polygon, or indeed any other figure on the plane, is by compass and straightedge. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21,...

Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.

Polyhedra
Euclid's Elements gave what amount to ruler-and-compass constructions for the five Platonic solids. However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.) The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface.

If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron.

Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the Platonic solids (or the Archimedean solids), especially if given a little guidance from a knowledgeable adult.

In theory, almost any material may be used to construct regular polyhedra. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit.

Higher dimensions
In higher dimensions, it becomes harder to say what one means by "constructing" the objects. Clearly, in a 3-dimensional universe, it is impossible to build a physical model of an object having 4 or more dimensions. There are several approaches normally taken to overcome this matter.

The first approach, suitable for four dimensions, uses four-dimensional stereography. Depth in a third dimension is represented with horizontal relative displacement, depth in a fourth dimension with vertical relative displacement between the left and right images of the stereograph.

The second approach is to embed the higher-dimensional objects in three-dimensional space, using methods analogous to the ways in which three-dimensional objects are drawn on the plane. For example, the fold out nets mentioned in the previous section have higher-dimensional equivalents. One might even imagine building a model of this fold-out net, as one draws a polyhedron's fold-out net on a piece of paper. Sadly, we could never do the necessary folding of the 3-dimensional structure to obtain the 4-dimensional polytope because of the constraints of the physical universe. Another way to "draw" the higher-dimensional shapes in 3 dimensions is via some kind of projection, for example, the analogue of either orthographic or perspective projection. Coxeter's famous book on polytopes has some examples of such orthographic projections. Note that immersing even 4-dimensional polychora directly into two dimensions is quite confusing. Easier to understand are 3-d models of the projections. Such models are occasionally found in science museums or mathematics departments of universities (such as that of the Université Libre de Bruxelles).

The intersection of a four (or higher) dimensional regular polytope with a three-dimensional hyperplane will be a polytope (not necessarily regular). If the hyperplane is moved through the shape, the three-dimensional slices can be combined, animated into a kind of four dimensional object, where the fourth dimension is taken to be time. In this way, we can see (if not fully grasp) the full four-dimensional structure of the four-dimensional regular polytopes, via such cutaway cross sections. This is analogous to the way a CAT scan reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated hologram of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope.

Another way a three-dimensional viewer can comprehend the structure of a four-dimensional polytope is through being "immersed" in the object, perhaps via some form of virtual reality technology. To understand how this might work, imagine what one would see if space were filled with cubes. The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of himself. If one could travel in these directions, one could explore the array of cubes, and gain an understanding of its geometrical structure. An infinite array of cubes is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (Euclidean) space. However, a 4-polytope can be considered a tessellation of a 3-dimensional non-Euclidean space, namely, a tessellation of the surface of a four-dimensional sphere (a 4-dimensional spherical tiling).

Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at UIUC has a number of pictures of what one would see if embedded in a tessellation of hyperbolic space with dodecahedra. Such a tessellation forms an example of an infinite abstract regular polytope.

Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its symmetry group is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner.

Regular polytopes in nature
For examples of polygons in nature, see:

Each of the Platonic solids occurs naturally in one form or another:

title: Regular polytope


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Jennifer Hudson

Jennifer Kate Hudson (born September 12, 1981), also known by her nickname J.Hud, is an American singer, actress, and television personality. Having received numerous accolades for her work in music, film, television, and theater, Hudson became the youngest woman and third African-American recipient of all four major American entertainment awards: Emmy, Grammy, Oscar, and Tony (EGOT) in 2022. She was inducted into the Hollywood Walk of Fame in 2013, and Time named her one of the 100 most influential people in the world in 2020.

In her recording career, Hudson rose to fame in 2004 as a finalist on the third season of the reality series American Idol, in which she placed seventh. She signed with Arista Records to release her self-titled debut studio album (2008), which peaked at number two on the Billboard 200 and won Best R&B Album at the 51st Annual Grammy Awards. She made her film debut as Effie White in the musical Dreamgirls (2006), for which she won the Academy Award for Best Supporting Actress and became the youngest African-American recipient of the award.

Hudson's second and third studio albums, I Remember Me (2011) and JHUD (2014), both peaked within the top ten of the Billboard 200. Outside of her recording career, she was cast in further acting roles such as Sex and the City (2008), The Secret Life of Bees (2008), Winnie Mandela (2011), Black Nativity (2013), Sing (2016), Cats (2019) and Respect (2021), the television shows Smash (2012), Empire (2015) and Confirmation (2016). She made her Broadway acting debut with the 2015 iteration of the musical, The Color Purple. Hudson has also contributed as a coach on both the UK and the US versions of the reality series The Voice from 2017 to 2019, and became the first female coach to lead a winning team on the former. Since 2022, she has hosted her own talk show, The Jennifer Hudson Show.

Early life
Hudson was born on September 12, 1981 in Chicago, Illinois. She is the third and youngest child of Darnell Donerson and Samuel Simpson, a Greyhound Lines bus driver. She, her older sister Julia, a school bus driver, and her brother Jason Hudson, were raised primarily by their mother. Her father was absent from her childhood until, with the blessing of her mother, she went looking for him at age 14. When Hudson found her father, she learned that he had 26 other children. She has stated she has still not met all of her half-siblings.

She was raised as a Baptist in Englewood and attended Dunbar Vocational High School, from which she graduated in 1999. She cites Whitney Houston, Aretha Franklin, and Patti LaBelle as her overall biggest influences and inspiration. She has also credited Mariah Carey as being one of her musical "heroes". At age 7 she got her start in performing by singing with the church choir and doing community theater with the help of her late maternal grandmother, Julia. She enrolled at Langston University but she left after a semester due to homesickness and unhappiness with the weather, and registered at Kennedy–King College.

In January 2002, Hudson signed her first recording contract with Righteous Records, a Chicago-based independent record label. She was released from her five-year contract with Righteous Records so that she could appear on American Idol in 2004.

2004–2005: American Idol
Hudson auditioned for the third season of American Idol in Atlanta. After commenting that she had been singing on Disney Cruise Lines (aboard the Disney Wonder) for the past few months as one of the Muses from Hercules, judge Randy Jackson told her, "We're expecting more than a cruise ship performance from you." Hudson received the highest number of votes in the "Top 9" after her performance of Elton John's "Circle of Life" on April 6, 2004, but two weeks later, she was eliminated during the "Top 7" show after performing Barry Manilow's, "Weekend in New England". In May 2009, MTV listed Hudson as the sixth greatest contestant in American Idol history and noted her exit was the most shocking of all time. In May 2010, the Los Angeles Times claimed Hudson to be the third greatest Idol contestant in the history of the show, placing behind season one winner Kelly Clarkson and season four winner Carrie Underwood respectively.

2006–2007: Dreamgirls and breakthrough
In one of her first appearances on a record, Hudson is featured in a duet, "The Future Ain't What It Used to Be", on Meat Loaf's Bat Out of Hell III: The Monster Is Loose. In September 2006, Hudson performed the song, "Over It", live on Fox Chicago Morning News. In the interview she stated the song would be included on her debut album, to be released in early 2007; however, this was before she was signed to a record label. In November 2006, Hudson signed a record deal with Arista Records. Hudson also recorded a song she co-wrote with Bill Grainer and Earl Powell called "Stand Up", which was available for preview on her fan website. The track was produced by Chicago natives Powell and Herman Little III, who also arranged the song. The power-ballad would later become available on the deluxe edition of Hudson's self-titled album as a bonus track.

In November 2005, Hudson was cast in the role of Effie White for the film adaptation of the musical Dreamgirls, which also starred Jamie Foxx, Beyoncé Knowles and Eddie Murphy. The role marked Hudson's debut screen performance. She won the role over hundreds of professional singers and actresses, including Fantasia Barrino. Filming of Dreamgirls began on January 9, 2006, and the film went into limited release on December 25, 2006, and national release on January 12, 2007. Hudson has won particular praise for her show-stopping onscreen rendition of the hit song, "And I Am Telling You I'm Not Going", the signature song of the role, which had earlier been recorded, and had already reached the status of musical standard, because of the definitive performance of Jennifer Holliday. The New York Observer described Hudson's performance of the song as "five mellifluous, molto vibrato minutes that have suddenly catapulted Ms. Hudson... into the position of front-runner for the best supporting actress Oscar". Newsweek said that when moviegoers hear Hudson sing the song, she, "is going to raise goose bumps across the land." Variety magazine wrote that Hudson's performance "calls to mind debuts like Barbra Streisand in Funny Girl or Bette Midler in The Rose, with a voice like the young Aretha". "Love You I Do" was nominated for the 2007 Academy Award for Best Original Song and won the 2008 Grammy Award for Best Song Written for a Motion Picture, Television or Other Visual Media.

As Effie White, Hudson has garnered 29 awards from film critics as Best Supporting Actress and Breakthrough Performer of 2006. She won the Golden Globe Award as Best Actress in a Supporting Role. In addition, she has been named Best Supporting Actress by the Broadcast Film Critics Association and also by the Screen Actors Guild. After seeing Hudson's performance in Dreamgirls, Simon Cowell taped a congratulatory message to her, which aired on The Oprah Winfrey Show. With the March 2007 issue, Jennifer Hudson became the third African-American celebrity and the first African-American singer to grace the cover of Vogue magazine. On February 11, 2007, the 60th British Academy Film Awards were held in London, but Hudson was not there to accept her BAFTA Award for Best Actress in a Supporting Role. The BAFTA Award went missing and Hudson did not receive her award until April 22, 2011, when it was presented to her on the Graham Norton Show. On February 25, 2007, she won the Academy Award for Best Supporting Actress for her role in the film. Later in 2007, the mayor of her home town, Chicago, Richard M. Daley, declared March 6 as "Jennifer Hudson Day". She was invited to join the Academy of Motion Picture Arts and Sciences on June 18, along with 115 other individuals for 2007. Entertainment Weekly put her performance on its end-of-the-decade, "best-of" list, saying, "Sure, Beyoncé's performance was great. And Eddie Murphy's was impressive. But there was really only one reason we all rushed to see 2006's Dreamgirls: Jennifer Hudson's soul-to-the-rafters rendition of the classic 'And I'm Telling You I'm Not Going'. When she sang 'You're gonna love me,' it wasn't just a lyric — it was a fact."

2008–2009: Jennifer Hudson album and films
In May 2008, Hudson appeared in the feature film Sex and the City as Louise, Carrie Bradshaw's assistant. Filming finished in December 2007 and the movie was released on May 30, 2008. In October 2008, Hudson made her third film appearance in The Secret Life of Bees as Rosaleen, the mother figure of Lily Owens (Dakota Fanning). Also starring in the film are fellow Oscar nominees Queen Latifah and Sophie Okonedo, as well as Alicia Keys. The film opened on October 17, 2008, and grossed over $37 million at the box office. The film won two People's Choice Awards in January 2009 in the categories, Favorite Drama Movie and Favorite Independent Movie. The film also received nine Black Reel Awards nominations including three wins. Hudson was nominated for her performance in the film, but lost to Queen Latifah. The film also received eight nominations at the 2009 NAACP Image Awards, including a nomination for Hudson for her performance.

In 2009, Hudson starred as Kathy Archenault in Fragments, a film based on the novel by Roy Freirich, with Dakota Fanning and Forest Whitaker. It was released on DVD on August 4, 2009. In 2010, Hudson started filming the biopic film Winnie Mandela based on the South African politician Winnie Mandela starring alongside Terrence Howard and directed by Darrell J. Roodt. Andre Pieterse and Roodt and Paul L. Johnson based the film's script on Anne Marie du Preez Bezdrob's biography, Winnie Mandela: A Life. The Creative Workers Union of South Africa have opposed this choice, stating they will push for a moratorium on the film if the casting is not reversed. Winnie Mandela has threatened legal action over this film, claiming she was not "consulted" about the film. Winnie Mandela was originally scheduled to be released in December 2011; the trailer was released in 2010. Image Entertainment released the film on September 6, 2013.

In January 2008, Hudson returned to the studio to record new songs for her debut studio album. Her label was reportedly unhappy with the direction they were sending Hudson musically and decided it would be best to scrap the older songs and instead focus on new ones. Hudson worked with Ryan Tedder and Timbaland on a number of songs. Her debut single, "Spotlight" was released on June 10, 2008, and became Hudson's first top 40 hit peaking at number 24 on the Billboard Hot 100 and became a top 20 hit in the United Kingdom and Turkey. Her debut studio album, Jennifer Hudson, was released on September 30, 2008, on Arista and was written by Ne-Yo, who co-produced along with Stargate. Additional contributors on the album include Timbaland, Missy Elliott, Robin Thicke, Harvey Mason, Jr., Diane Warren, Earl Powell, and Christopher "Tricky" Stewart, among others. The album debuted at number two on the Billboard 200 with 217,000 copies in the US and opened to positive reviews. As of August 2009, the album has sold 739,000 copies in the US, receiving a Gold certification for surpassing sales of 500,000. She has performed the song, "All Dressed in Love" for the Sex and the City soundtrack, which was released on May 27, 2008. Hudson performed the national anthem at the 2008 Democratic National Convention.

The album's second single was to be released in October 2008, and was originally set to be "My Heart" before "If This Isn't Love" was chosen as the official second single. After three of her family members were brutally murdered, the single was rescheduled for a January 2009 release. However, in January 2009, her label decided to postpone the release of the second single once more, choosing a February 2009 release date. After the release of "If This Isn't Love", the single eventually peaked at number 63 on the Billboard Hot 100 and number 37 on the UK Singles Chart. The third single was "Pocketbook" featuring Ludacris in April 2009, for a June 2009 release, despite speculation that the third single would be "My Heart" after it was canceled as the second single. In May 2009, "Pocketbook" was canceled as the third single due to negative reception from fans and critics alike. The third single was then rescheduled as "Giving Myself" and was released on June 2, 2009, to the US. Her debut album garnered Hudson three nominations at the 2009 Grammy Awards: Best Female R&B Vocal Performance for "Spotlight", Best R&B Performance by a Duo or Group with Vocals for "I'm His Only Woman" with Fantasia Barrino, and Best R&B Album for Jennifer Hudson. She won the last of these and performed at the awards ceremony.

In February 2009, Hudson made her first public appearance since the murders when she sang the Star-Spangled Banner at Super Bowl XLIII. She toured the United States with Robin Thicke in 2009. In early May 2009, Hudson suffered "throat fatigue" and had to reschedule her May 16–19, 2009, tour dates. She sang "Will You Be There" at the Michael Jackson memorial service in July 2009. Hudson recorded "Neither One of Us (Wants to Be the First to Say Goodbye)", originally by Gladys Knight & the Pips, for American Idol Season 3: Greatest Soul Classics — the official American Idol album for the third season. She is featured on Ne-Yo's second album, Because of You, on the track, "Leaving Tonight". Hudson performed a Christmas special in December 2009 called Jennifer Hudson: I'll Be Home for Christmas. During the special, she relived her childhood Christmases with musical performances filmed at her favorite locations in her native Chicago. Hudson also visited family members, friends and other childhood influences during the broadcast. On January 22, 2010, Hudson appeared on the telethon "Hope for Haiti Now: A Global Benefit for Earthquake Relief", performing the Beatles' "Let it Be". The telecast generated $61 million in donations. Also, that year, Hudson recorded the song "One Shining Moment" for the 2010 NCAA Men's Division I Basketball Tournament.

2010–2012: Weight Watchers and I Remember Me
In 2010, Hudson became the spokesperson for Weight Watchers. That year, she had lost 56 lb and "[did] not want to lose any more weight." However, on February 10, 2011, she appeared on The Oprah Winfrey Show and revealed she had lost a total of 80 pounds. Her 2012 autobiography, I Got This: How I Changed My Ways and Lost What Weighed Me Down, detailed her weight loss.

Early in the album I Remember Me's development, Ne-Yo told E! Online that Hudson was ready to get personal on her second studio album and that he might be producing the album. He said "She's gone through a lot over the last year, so she has a lot to talk about", he continues. "She's definitely gotten stronger. The things that she's gone through and for her to still be upbeat and happy, it's amazing." Hudson released her second studio album I Remember Me on March 22, 2011. It debuted at number two on the Billboard 200 selling 165,000 copies in its first week of release.

On January 24, 2011, Hudson premiered the album's lead single "Where You At". It was written by R. Kelly and produced by Harvey Mason, Jr. The song went for radio adds that same week. On February 3, 2011, due to radio adds the single made its debut on the US Billboard Hot R&B/Hip-Hop Songs chart at number 53, having so far reached a peak of number 10. The music video premiered on BET's 106 & Park and through Vevo in February 2011. "No One Gonna Love You" impacted on Urban adult contemporary radio in the United States in May 2011. A remix bundle, "No One Gonna Love You-The Remixes", was released as a digital download in the US and the UK in July 2011. The song has reached number 23 on the Hot R&B/Hip-Hop Songs. The third US single will be "I Got This". The non-single track "Don't Look Down" reached number 70 on the US Hot R&B/Hip-Hop Songs chart.

2013–2014: JHUD
Hudson played a nun in the Farrelly brothers film The Three Stooges (2012). The day after the death of Whitney Houston, Hudson paid tribute to her by singing Houston's "I Will Always Love You" during the 54th Grammy Awards held on February 12, 2012, at the Staples Center in Los Angeles, California. Hudson guest-starred for a multi-episode arc on the second season of Smash. She sang "America the Beautiful" with the Sandy Hook Elementary School chorus at Super Bowl XLVII. Hudson sang at the 85th Academy Awards in a tribute to musical films. She performed "Same Love" with Macklemore, Ryan Lewis, and Mary Lambert at the 2013 MTV Video Music Awards.

On September 21, 2013, a new single from Hudson's forthcoming third studio album was released via her website. The 70s stylized uptempo ballad, "I Can't Describe (The Way I Feel)", features T.I. with production by Pharrell, she performed this song alongside Chaka Khan, Evelyn Champagne King, and T.I. at the 2013 Soul Train Awards. In October 2013, Hudson starred in the coming of age drama The Inevitable Defeat of Mister & Pete alongside Jordin Sparks as well as the drama/musical Black Nativity. Furthermore, she appears alongside Amy Adams in the drama Lullaby. On November 13, 2013, Hudson was awarded with the 2,512th star of the Hollywood Walk of Fame.

Hudson's third studio album, JHUD, was released on September 23, 2014. Hudson described the album as more "upbeat" than her previous material. She worked with Timbaland, Pharrell Williams, RedOne, and R. Kelly among others. RedOne said about the material: "It's not dance dance, but it's soulful dance. It's very '70s kinda style". Hudson stated: "What we're doing feels so right that I never want to go home (from the studio). My sister and my manager will be sleeping on the couch while we work until 4 or 5am".

Hudson confirmed that a song on the album will be "It's Your World" and features R. Kelly, in an interview with Jonathan Ross on The Jonathan Ross Show. The track went on to be nominated for Best R&B Vocal Performance at the 2015 Grammy Awards. On June 20, she released the music video for "Walk it Out" featuring Timbaland, which is the first single off her upcoming album. Her third album, JHUD, would include "different expressions of [Hudson] as an artist, a girl, [and] a music lover".

2015–2016: The Color Purple, Hairspray Live, and Sing
At the 87th Academy Awards, Hudson performed her song (originally from NBC's Smash) "I Can't Let Go" by Marc Shaiman and Scott Wittman as a tribute to the individuals Hollywood lost in the previous year.

Later in 2015, Hudson guest-starred as Michelle White, a minor semi-recurring character in the first season of Empire. She sang three songs on the show, "Remember the Music", "Whatever Makes You Happy", and "For My God".

Hudson helped welcome Pope Francis on his first visit to the United States by singing her rendition of Leonard Cohen's "Hallelujah".

In fall 2015, Hudson made her Broadway debut as Shug Avery in the Broadway revival of The Color Purple, alongside Cynthia Erivo and Danielle Brooks. New York Times critic Ben Brantley hailed her performance saying, "Ms. Hudson radiates a lush, supple stage presence that is echoed by her velvet voice." Hudson later picked up her second career Grammy for Best Musical Theater Album (2017) for her work on The Color Purple."

In June 2016, Hudson performed Prince's "Purple Rain", on the BET Awards, which she had previously sung onstage with the cast of The Color Purple, the day Prince died.

On June 28, 2016, Hudson signed with Epic Records. L.A. Reid and Clive Davis are joining forces for the first time since Whitney Houston's The Bodyguard soundtrack 25 years ago. "Jennifer Hudson is the defining voice of this generation", said Reid. "Her award-winning presence on records, in film and on stage establishes her as one of the premiere artists to emerge in this century." Davis says: "I signed Jennifer in 2007 believing her to be the strongest candidate as the next generation's Aretha Franklin and Whitney. She has become that, as LA and I excitingly reunite to capture Jennifer at her all-time best."

Meanwhile, Hudson picked up a few minor film roles, including in Spike Lee's musical Chi-Raq, and HBO's Confirmation, alongside Kerry Washington.

In December 2016, Hudson played scene-stealer Motormouth Maybelle in NBC's Hairspray Live!. Her version of the Civil Rights-themed ballad "I Know Where I've Been", blew audiences away, garnering Emmy buzz from fans and critics alike. During the credit sequence, Hudson duets with costar Ariana Grande on Shaiman and Wittman's "Come So Far".

Hudson lent her voice to the younger version of the character Nana Noodleman in the Illumination Entertainment film, Sing, which was released December 21, 2016. In the film, she sang The Beatles' classic, "Golden Slumbers/Carry That Weight", which was the recurring musical theme of the film. On the deluxe soundtrack, Hudson sings with Tori Kelly on a duet version of Leonard Cohen's "Hallelujah".

2017–2020: Sandy Wexler, The Voice, Cats, and new music
Hudson starred as singer Courtney Clarke in the film, Sandy Wexler, which premiered in April 2017. In early 2017, Hudson debuted in The Voice franchise as a coach on the sixth season of The Voice UK. She won the season with her act Mo Adeniran and became the first female coach to win the show. Hudson had joined the American version of The Voice as a coach for its thirteenth season.

Hudson released "Remember Me" on March 3, 2017. On December 12, 2017, she released another new song, "Burden Down". The same day, Hudson performed "Remember Me" live for the first time on the semi-final result night of The Voice. Later, a music video for the single was also released.

Hudson returned for her second series as a coach to The Voice UK; her second series premiered on January 6, 2018, and concluded on April 7, where Hudson's finalist Belle Voci finished third. On January 27, 2018, Clive Davis disclosed that Hudson had been handpicked by Aretha Franklin to portray her in an eventual biopic with a prospective release date in 2021.

Jennifer Hudson performed with the D.C. Choir at the March for Our Lives anti-gun violence rally in Washington, D.C., on March 24, 2018. Hudson returned as a coach for the fifteenth season of the American version of The Voice after taking a one-season hiatus. Her final act, Kennedy Holmes, finished in fourth place overall. Hudson returned for The Voice UK's eighth series. In 2018, Hudson was cast as the Cats character Grizabella in the film adaptation, which was released in 2019.

In March 2020, Hudson collaborated with Bono, Will.i.am, and Yoshiki to create "#SING4LIFE", a song written and compiled remotely by the four musicians to lift spirits during the COVID-19 crisis.

2021–present: Respect, EGOT, and talk show
On July 20, 2021, Hudson was interviewed by Oprah Winfrey about her role as Aretha Franklin in the film, Respect. The Aretha Franklin bio-pic, Respect, was directed by Liesl Tommy and was released in August 2021. Hudson's performance earned her the Chairman's Award at the Palm Springs International Film Festival as well as a nomination for the Screen Actors Guild Award for Outstanding Performance by a Female Actor in a Leading Role. The film was accompanied by a soundtrack sung entirely by Hudson featuring covers of several of Franklin's songs with one original song for Hudson, "Here I Am (Singing My Way Home)", being released as the soundtracks lead single in June 2021. The song earned Hudson nominations for the Golden Globe Award for Best Original Song and the Grammy Award for Best Song Written for Visual Media. The film's soundtrack received a nomination for the Grammy Award for Best Compilation Soundtrack for Visual Media.

On August 19, 2021, Hudson performed a show at the Apollo Theater called Jennifer Hudson - Live at the Apollo: A Night of Soul.

On August 21, Hudson performed "Nessun Dorma" at We Love NYC: The Homecoming Concert backed by New York Philharmonic in Central Park.

In September 2021, Hudson won a Daytime Emmy Award for Outstanding Interactive Media for a Daytime Program for being an executive producer for Baba Yaga, a 27-minute interactive fairytale story made for Oculus Quest.

In February 2022, Jennifer Hudson was one of the night's major winners at the 2022 NAACP Image Awards. She won the Entertainer of the Year and Outstanding Actress in a Motion Picture for her portrayal of Aretha Franklin in the film Respect. In May 2022, Hudson received a nomination for the Tony Award for Best Musical for serving as a producer for the Broadway musical, A Strange Loop. On June 12, 2022, Hudson became one of seventeen people to have won an Oscar, an Emmy, a Grammy, and a Tony ("an EGOT") at the 75th Tony Awards. She appeared in the 2022 film Tell It Like a Woman and will also star in the upcoming film Breathe.

Hudson's syndicated talk show, The Jennifer Hudson Show, launched on September 12, 2022.

Personal life
Hudson began dating James Payton in 1999, when she was 18. They separated in late 2007. Hudson met David Otunga, a professional wrestler in the WWE and a Harvard Law graduate, and the couple became engaged in September 2008. In August 2009, Hudson gave birth to her first child, her son David Daniel Otunga, Jr. In November 2017, Hudson and Otunga had split.

On October 24, 2008, Hudson's 57-year-old mother Darnell Donerson and 29-year-old brother Jason were found shot to death inside the Chicago home Donerson shared with Hudson's older sister, Julia. An AMBER Alert was issued for her 7-year-old nephew, Julian King, when Julia reported him missing. Three days later the Federal Bureau of Investigation confirmed a body found on Chicago's West Side was the nephew; an autopsy indicated he had died from "multiple gunshot wounds". Police charged William Balfour, Julia's estranged 27-year-old husband, with three counts of first-degree murder and one count of home invasion. Convicted on all seven counts against him, in July 2012, he was sentenced to three life sentences without the possibility of parole (to be served consecutively), followed by an additional 120 years for his other convictions. Hudson's family created The Hudson-King Foundation for Families of Slain Victims, in honor of the three victims. Hudson and her sister created the Julian D. King Gift Foundation in honor of her nephew. It provides Christmas presents and school supplies to families in need in the Chicago area.

Hudson has been described as a friend of former President Barack Obama, who invited her to appear with him at a fundraiser in Beverly Hills during his first term in May 2009. She also performed at the White House at the "Celebration of Music from the Civil Rights Movement" event.

, Hudson resides in Burr Ridge, Illinois, a western suburb of Chicago.

In 2022, while working together on their film Breathe, Hudson began dating fellow Chicago native, rapper and actor, Common. The pair confirmed their relationship with an appearance on The Jennifer Hudson Show in January 2024. When discussing their romance, Common stated "I'm in a relationship that is with one of the most beautiful people I've met in life," he admitted. "She's smart, she loves God, she has something real down to earth about her. She's talented."

Discography

 * Jennifer Hudson (2008)
 * I Remember Me (2011)
 * JHUD (2014)

title: Jennifer Hudson


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