title: COVID-19 pandemic in Singapore

text: As the pandemic spread throughout the world, the Singapore government had organised several repatriation efforts to bring Singaporeans back from various overseas locations. MFA and other governmental agencies subsequently brought back at least 1,000 Singaporeans, permanent residents, and family members stranded at other locations where there were similar lockdowns and suspension of flights: Cambodia, Egypt, Fiji, India, Iran, Nepal, Saudi Arabia, Slovenia, and the UK. Most of the evacuees were brought back on direct flights, whilst some in Fiji, Iran, Nepal, and Slovenia saw some assistance from other countries. All evacuees had to serve a 14-day Stay Home Notice or be quarantined at designated locations, such as hotels with cost borne by the government, government quarantine faculties, or at home.


title: COVID-19 pandemic in Singapore

text: With flights to Wuhan suspended due to the Hubei lockdown, Scoot had offered two one-way flights back to Wuhan for tourists stuck in Singapore. As the travel restrictions began to grow and accumulated into an ongoing ban on short-term visitors arriving or transiting through Singapore which started from 23 March, Singapore had allowed visitors to transit through Singapore if they are being repatriated by various governments. India had repatriated some of its citizens from Singapore in May 2020 on two separate flights. In the same flight which Singapore Airlines brought back Singaporeans from Cambodia on 12 April, it also carried Australians heading back to Australia with Singapore being a transit point. It was reported that Singapore and Bangladesh were in discussions to repatriate their respective citizens if necessary.


title: COVID-19 pandemic in Singapore

text: Both the Singapore government and private sector sent support packages to Indonesia, including test kits and personal protective equipment. The Singapore government sent swabs and other supplies to Malaysia to help with sample collection and testing. Temasek Foundation donated 30,000 test kits to India and this was affirmed and thanked by High Commissioner of India to Singapore, Jawed Ashraf. Indian low-cost airline SpiceJet operated flights carrying the test kits and other medical equipment from Singapore to both Bangalore and Chennai.


title: COVID-19 pandemic in Singapore

text: In December 2020, Singapore pledged US$5 million towards COVID-19 Vaccine Global Access (COVAX), a WHO-led global initiative aimed at securing COVID-19 vaccines for low- and middle-income countries. Together with Switzerland, Singapore co-chairs "Friends of the COVID-19 Vaccine Global Access (COVAX) Facility", which comprises the European Union and the following 14 nations: Australia, Canada, Iceland, Israel, Japan, the Kingdom of Saudi Arabia, New Zealand, Norway, Qatar, the Republic of Korea, Singapore, Switzerland, the United Arab Emirates, and the United Kingdom.


title: COVID-19 pandemic in Singapore

text: On 26 April 2021, the Singapore Indian Chamber of Commerce and Industry (SICCI) and Little India Shopkeepers Association (LISHA) initiated and launched a relief fund to raise money and support India in its fight against COVID-19.


title: COVID-19 pandemic in Singapore

text: On 31 August 2021, Singapore agreed to support Australia by sending 500,000 doses of Pfizer-BioNTech to aid Australia's coronavirus battle exacerbated by Delta variant. Based on the agreement, Australia will return the doses by December 2021.


title: COVID-19 pandemic in Singapore

text: On 28 September 2021, Singapore sent more than 120,000 doses of the AstraZeneca vaccine to Batam and Riau islands of Indonesia. Similarly, Singapore sent 100,000 doses of Moderna to Brunei on 29 September to help them tackle the pandemic.


title: COVID-19 pandemic in Singapore

text: With the impact of COVID-19 becoming greater, it was becoming clear to analysts that Singapore would need to respond with large scale government spending. In the Budget 2020, the Government provided help in four consecutive budget periods, under Unity Budget, Resilience Budget, Solidarity Budget and Fortitude Budget. All Singaporeans had $600 deposited into the bank account in 2020. In the Budget 2021, the $800 million support package was rolled out, where JSS was further increased to 50 per cent in hardest hit sectors. The government announced on 4 June 2020 that they would be keeping Central Provident Fund contribution rates unchanged as they believed the Jobs Support Scheme would help reduce the burden on employers.


title: COVID-19 pandemic in Singapore

text: It was announced that the government would draw an additional S$32 billion from past reserves, bringing the total used to S$52 billion. In addition, another S$13 billion will be set aside for contingencies due to the pandemic. Extending and enhancing the JSS, to include higher tiers of wage subsidies and lasting until August, would cost around S$2.9 billion, bringing the total cost to S$23.5 billion. The government introduced a one-off Solidarity Utilities Credit of S$100 for all households since many people were remote working. Food and beverage (F&B) and retail companies would receive up to S$10,000 as part of efforts to digitally transform their businesses, as they are likely to be seriously affected by safe distancing rules. S$2 billion was committed to SGUnited Jobs and Skills Package was announced to create more jobs and traineeships.


title: COVID-19 pandemic in Singapore

text: To help financial institutions and FinTech companies tide over the COVID-19 pandemic, the Monetary Authority of Singapore (MAS) released a $125 million package to help financial institutions and FinTech companies to strengthen long-term capabilities.


title: COVID-19 pandemic in Singapore

text: On 14 April, the Infocomm Media Development Authority announced that they will launch Public Service Content worth S$8 million and fund 90% of the course fees for Self-Employed Persons under Talent Assistance (T-Assist) Programme. To reduce operating costs, the Film Exhibition and Distribution Licence Fees will be waived from 17 April.


title: COVID-19 pandemic in Singapore

text: Every Singaporean who would be aged 18 and above in 2020 would receive S$100 worth of SingapoRediscovers vouchers, which can be used for a variety activities to support the local tourism industry. They were planned to be usable between December 2020 and June 2021 and would be paid via SingPass. Additionally, Trip.com has a "Pay It Forward" campaign throughout this usable period, where people can use their vouchers to pay for tourism products for various beneficiary groups. When the vouchers were launched, there were some complaints from Singaporeans about the complex way the vouchers were to be redeemed. On 30 April 2021, the Singapore Tourism Board (STB) announced that the expiry date of the SingapoRediscovers vouchers will be extended to 31 December 2021, and that improvements to the redemption process will be rolled out.


title: COVID-19 pandemic in Singapore

text: In February 2020, incidents were reported of healthcare workers such as nurses and ambulance drivers being ostracised, as members of the public who feared that their contact with coronavirus patients would pose an infection risk. Such acts were condemned by the authorities, who exhorted the public to show support and appreciation for their work. Ride-hailing service Grab announced the launch of GrabCare, a dedicated service for healthcare workers to obtain transport to and from their work places, in order to alleviate their difficulty of finding rides.


title: COVID-19 pandemic in Singapore

text: On 12 April 2020, CNA reported that some staff members of Lee Ah Mooi nursing home were evicted by landlords after a cluster of cases was reported at the home.


title: COVID-19 pandemic in Singapore

text: Despite effective handling of initial waves of infection, several serious outbreaks in April have brought the situation in Singapore out of control; many analysts points to poor conditions at foreign workers dormitories as a major factor of the failure.


title: COVID-19 pandemic in Singapore

text: The pandemic brought the living conditions at foreign worker dormitories to media attention. Dormitories were reported to be unsanitary and crowded, making preventive measures like social distancing difficult. Retired diplomat Tommy Koh criticised the living conditions, calling it "third world" and "a time bomb waiting to explode". Amnesty International called the situation a "recipe for disaster". Minister for Manpower Josephine Teo vowed to improve the living conditions of foreign workers after the quarantine was handled.


title: COVID-19 pandemic in Singapore

text: While churches in the 1960s and 1970s historically championed social justice and advocacy for worker rights within the Singaporean civil society, this has declined over the years due to the rise of a middle class and a dependency on state-centric growth. During the pandemic, Singaporean churches sought ways to bring about "neighbourly love" and national prayer. But some criticised this as not being enough to address the inequalities experienced by foreign workers who were disproportionately affected by the virus.


title: COVID-19 pandemic in Singapore

text: On 9 April, MOM said in a press release that it will improve quality of meals of foreign workers during quarantine and formed a task force to improve the living conditions of foreign workers. As of 25 April 25 dormitories have been gazetted as isolation areas. Some healthy workers are also progressively being moved to numerous empty premises such as SAF camps, HDB blocks, floating hotels and Changi Exhibition Centre. On 16 April, Minister for Manpower Josephine Teo said that there will be a "three-pronged strategy"; containing the spread, imposing lockdowns and separating workers in essential services.


title: COVID-19 pandemic in Singapore

text: Minister of National Development Lawrence Wong has said that the living standards in dormitories have steadily improved over the years, and suggested that the issue was the dormitories being designed for communal living, where migrant workers ate, lived, and cooked together, and that the initial precautions and safeguards put in place to reduce some of the non-essential activities were not sufficient. The government has promised to build new dormitories that are designed with public health in mind and provide more amenities, while some workers will be temporarily moved to makeshift dormitories before the new dormitories are ready.


title: COVID-19 pandemic in Singapore

text: It is noted that a large majority of COVID-19 cases in Singapore are foreign workers. On 20 April, Singapore reported a peak of 1,426 new cases, of which only 16 were not migrant workers but citizens or permanent residents. Foreign workers had accounted for three-fourths of the total infections in Singapore by then.


title: COVID-19 pandemic in Singapore

text: Towards the end of July 2020, a spate of attempted and successful suicides among the foreign workers in dormitories raised concerns about the mental well-being of the workers on lockdown. Authorities are monitoring the situation and are working with partners and NGOs to enhance their mental health support programmes.


title: COVID-19 pandemic in Singapore

text: Outside the lockdowns, social distancing was not required when on board public transportation, and it was not practical due to the high volume of commuters. The Minister of National Development Lawrence Wong explained that resulting reduction of passenger capacity will involve major society disruptions, such as lateness for work, more unnecessary increase of transportation, more manpower strength and enforcement. Authorities instead urged employers to allow more telecommuting and flexible work arrangements to help reduce the demand for public transport. With that, social distancing measures are only implemented in the lockdown periods, with stickers pasted on alternate seats to indicate that they should not be used, and auxiliary officers and transport ambassadors enforcing the rules throughout this period. Public transportation has resumed pre-lockdown operations after the period of lockdown, social distancing stickers were removed on bus and train seats, allowing all seats to be used. Regulations for social distancing were no longer applicable except in fixed queuing area such as in MRT stations and bus interchanges. Mask wearing continues to be mandatory in public transport and most places in Singapore. Since 29 March 2022, mask wearing is optional but still encouraged in some outdoor places in Singapore.


title: COVID-19 pandemic in Singapore

text: On 4 January 2021, Minister of State for Home Affairs Desmond Tan announced in Parliament that Bluetooth proximity data from the TraceTogether app and token could be used by the police in criminal investigations, as per the Criminal Procedure Code. This resulted in significant backlash, as the announcement contradicted previous statements that the data would strictly be used for contact tracing; concerns were aired about the transparency of the government and how the personal data would be used, as well as possible infringement on the privacy of citizens. Some users even switched off their apps or left their tokens at home in protest. In response, Minister-in-charge of the Smart Nation Initiative Vivian Balakrishnan clarified that the police could only obtain TraceTogether data through a person involved in the investigation, adding that there were plans to introduce legislation that would restrict police use of TraceTogether data to serious offences such as terrorism or murder. The "COVID-19 (Temporary Measures) (Amendment) Bill" was passed on 2 February, which included a possible extension of the COVID-19 Control Order to next year and beyond if the situation does not improve or worsens, as well as the removal of SafeEntry/TraceTogether data at the end of the pandemic.


title: Malgudi Days (TV series)

text: Malgudi Days is an Indian television series that started in 1986, and was filmed in both English (first 13 episodes) and Hindi (all 54 episodes), based on the 1943 short story collection of the same name by R. K. Narayan. The series was directed by Kannada actor and director Shankar Nag. Carnatic musician L. Vaidyanathan composed the score, while R.K. Narayan's younger brother and acclaimed cartoonist R. K. Laxman was the sketch artist. The series was made by film producer T.S. Narasimhan. In 2006, the series was revived for an additional 15 episodes, which were directed by Kavitha Lankesh.


title: Malgudi Days (TV series)

text: The series is adapted from the collections of short stories written by R. K. Narayan in his books "Malgudi Days", "A Horse and Two Goats", "An Astrologer's Day and Other Short Stories", as well as the novels "Swami and Friends", and "The Vendor of Sweets."


title: Malgudi Days (TV series)

text: The series Malgudi Days, comprising thirty nine episodes, was first telecast on Doordarshan in 1986. Most of the series was shot in Agumbe village in Shimoga district, Karnataka. Some episodes were filmed in Bengaluru and some others in Devarayanadurga located in Tumakuru district in Karnataka. One episode, "Nitya", was shot entirely in Devarayanadurga.


title: Malgudi Days (TV series)

text: In 2004, the project was revived with filmmaker Kavitha Lankesh replacing Shankar Nag as director. The new series was telecast from 26 April 2006 on Doordarshan. Most of the stories in the series are one episode long and were derived from the books "A Horse and Two Goats", "Malgudi Days", "Swami and Friends" and "The Vendor of Sweets".


title: Malgudi Days (TV series)

text: Indian Railways decided to rename Arasalu Railway station of Shivamogga District, Karnataka, India to Malgudi Railway station as homage to the location of Malgudi Days serial.


title: Serena van der Woodsen

text: Serena Celia van der Woodsen is the fictional character and the protagonist in the "Gossip Girl" novel series and in its TV adaptation, in which she is portrayed by Blake Lively. Serena is featured on the blog of the series' mysterious narrator, "Gossip Girl". Serena is known as the 'it girl of Manhattan' and is a character that appears to easily get whatever she wants because of her captivating beauty and charismatic personality. She is the daughter of a successful doctor, and a well known socialite/heiress.


title: Serena van der Woodsen

text: Serena is a tall, blonde, slim, and beautiful teenage girl from the Upper East Side of Manhattan in New York City. Many of the girls at the Constance Billard school accuse her of using her good looks as a means to secure everything she wants. She is not particularly academically skilled, but she is often told that she is not meeting her full potential. Her father runs the same Dutch shipping firm his great-great-grandfather founded in the 18th century, and her mother, Lillian van der Woodsen, is a socialite, art collector, and philanthropist. Her mother and father are on the boards of all major charities and art organizations in the city. The van der Woodsens reside at 994 Fifth Avenue, a ritzy, white-gloved doorman building directly across the street from The Met and Central Park. Serena and her family own half the top floor in a 14-room penthouse.


title: Serena van der Woodsen

text: Serena has a nail-biting habit and also cuts her split ends off, mainly during her classes at Constance Billard. Though exposed and aware of the on-goings in the fashion world, she, unlike Blair, is not as meticulous about her fashion choices and wardrobe. She is described to have an effortless way about her. Serena is charismatic, charming, talented, funny, kind, care-free, and laid-back. These attributes help her to thrive and often, whether willingly or unwillingly, attract older men.


title: Serena van der Woodsen

text: Despite all of her attributes, Janet Malcolm of "The New Yorker" considers her "incandescently beautiful, exceptionally kind, and, in the end, it has to be said, somewhat boring."


title: Serena van der Woodsen

text: "People" compares her character with Josh Schwartz's original it-girl creation, Marissa Cooper, stating that Serena "seems to have it all, but in addition to a party girl reputation, dark family secrets and a disregard for high society, Serena one-ups the sulky Coop with a history of BFF betrayal." Jason Gay of "Rolling Stone" describes Serena as "the bad girl gone good(ish) who serves as Gossip Girl's wobbly moral compass." He added praise while describing her early strained relationship with Blair, stating, "Lively's Serena is a former queen bee who mysteriously disappeared from campus, only to return and find her spiteful ex-best friend, Blair (Meester), in charge." With regards to her character's direction in the show, actress Blake Lively commented on her character's adventurous storylines during an interview with "Nylon" saying, "I feel ridiculous at times with her, because I'm, you know, killing someone or marrying someone, but I look like me. I'm like, 'Oh, this is absurd.'" "Vogue" magazine considers Lively's upbringing to be advantageous to her character background and finds her "dazzling and worldly and optimistic". Her mother, a former model from Georgia would dress her differently and her differing fashion choices drew the attention of her classmates from the L.A. private school she enrolled in. "It was the only school where people were just downright mean to me," Lively stated. "They would make fun of my clothes because I dressed differently than the other kids."


title: Serena van der Woodsen

text: In November 2012 concept music artist Marty McKay released a single called "Serena" featuring former Dr. Dre / Aftermath producer Focus... The song is based on Serena's character in Gossip Girl. Due to a lack of promotion, the song did not garner much attention.


title: Serena van der Woodsen

text: The first season introduces Serena as the beautiful, wealthy daughter of divorced parents who returns from boarding school. Her return sparks her old rivalry with her best friend, Blair. Serena's return is due to her younger brother, Eric van der Woodsen, who attempted suicide. She is considered unwelcome, most so by her old best friend Blair, who has always seen Serena as a threat to her reign as Queen Bee of Constance. After Serena successfully reconciles with her, Chuck Bass reveals to Serena that he knows the cause of her sudden departure prior to the first season - taking Nate Archibald's (Blair's boyfriend) virginity during a wedding, and he attempts to kiss her. Serena manages to escape and runs into Dan, a St. Jude's student from Brooklyn, who often expresses cynicism about the lifestyle his wealthier classmates lead. Serena also finds a friend in Jenny Humphrey, Dan's younger sister. Nate eventually reveals his tryst with Serena to Blair, and Serena is ostracized in the first few episodes of the show. Blair and Serena consistently fight and reconcile throughout the show's subsequent episodes, often dealing with Serena's tendency to overshadow Blair. Serena and Blair reconcile after a heartfelt confrontation that prompted Serena to admit her mistake with Nate and leaving Blair in her time of need.


title: Serena van der Woodsen

text: Serena encounters a myriad of problems as her mother becomes engaged to Bart Bass, who forces her family to adjust to living with Chuck, then she discovers her mother's relationship with Dan's father, Rufus Humphrey, and the complications of embarking on a relationship with Dan that is met with disapproval of her mother and her peers. Her relationship with Dan further reaches complications with the arrival of Georgina Sparks, a longtime friend who slowly reveals to viewers the real reason why Serena left Manhattan. Georgina drives her to commit the same mistakes she did before leaving Manhattan, ranging from partying excessively to drinking heavily, causing Serena to miss the SATs. Serena confronts Georgina, only to draw her anger, prompting Georgina to destroy her relationship with Dan and blackmail her with knowledge of a particular event in the past: she accidentally killed a man. Serena's constant lying in order to keep Dan from discovering her past eventually takes its toll on their relationship and Dan breaks up with her.


title: Serena van der Woodsen

text: Blair eventually retaliates against Georgina with the help of Chuck and Nate and the intervention of Vanessa, successfully driving Georgina out of Manhattan. Serena finally tells Dan the truth, only to discover during Lily's wedding, that he wants to end their relationship. A disappointed Serena attempts to move on throughout the summer.


title: Serena van der Woodsen

text: The second season shows Serena spending the summer with her family and Blair in the Hamptons, attempting to move on from her failed relationship with Dan. During a dramatic party, she and Dan briefly reconcile their relationship and return to New York. Their relationship becomes public again but ends sooner than expected during a blackout as both are unable to move past the problems that previously ruined their relationship. After a disastrous double date with Dan and a new transfer, Amanda, that ends with two of Blair's mean girls throwing Nair at Amanda and Dan accusing Serena of being behind it, Serena takes matters into her own hands and easily re-establishes herself as Queen, thus reigniting her rivalry with Blair. She has Dan publicly ostracized throughout the school. Exposing herself to New York's high society, she meets Poppy Lifton, a socialite, and further strains her relationship with Blair during Eleanor Waldorf's fashion show when she and Poppy participate in it. Serena and Blair eventually reconcile during a trip to Yale, where she didn't originally want to apply to but then took the opportunity to do so after Blair told her she couldn't possibly get in.


title: Serena van der Woodsen

text: An exhibit in Rufus' gallery has Serena finding new love with a childhood friend and artist, Aaron Rose. She eventually reconciles with Dan but their friendship soon threatens her new love with Aaron, prompting him to escort her to a ball where Serena receives news from her mother of Bart's death. During Bart's funeral, Aaron notices her continually drawn towards Dan and invites her to come with him to Argentina, which she accepts. Upon their return, Serena breaks up with Aaron, and she is free to pursue a relationship with Dan. The two reunite, only to face the awkward discovery that they share a sibling. Dan and Serena's relationship further suffers when she realizes their diverging college choices, her continuous mistrust in him, and Dan's affair with a teacher that drives her to prematurely end their relationship.


title: Serena van der Woodsen

text: After throwing a disastrous Sweet Sixteen party for Jenny, Serena retreats to Spain with Poppy and Poppy's boyfriend, Gabriel Edwards. She returns to New York pursued by Gabriel after he broke up with Poppy and finds herself in a new relationship. During a party held by her mother, Serena discovers that Gabriel and Poppy have conned her mother's guests and calls on Blair's aid to obtain their stolen money, breaking up with Gabriel as she pursues Poppy. Blair reluctantly teams up with Chuck and a reformed Georgina, ignoring Lily's wishes that they let her handle it herself, ending with the plan completely back-firing: Serena is arrested, Poppy escapes and Georgina, becoming the scapegoat, eases back into her old ways.


title: Serena van der Woodsen

text: A stubborn Serena chooses to stay in prison, until Dan bails her out and the two go to prom together. Her arrest soon adds to her fame as she becomes a local celebrity. On the day of graduation, Gossip Girl sends a blast that further ruins Serena and her friends' reputations. Declaring war on Gossip Girl, Serena finds herself the center of blame when a ticked-off Gossip Girl decides to drop every unreleased piece of gossip she had in store, creating further divide within the group. Sending a text to Gossip Girl, Serena calls a bluff that she knows Gossip Girl's identity, only for Gossip Girl to escape and warning everyone that she'll be pursuing them in college. She spends the summer leaving New York with Carter in pursuit of her father.


title: Serena van der Woodsen

text: Serena returns from her European adventure and a complicated relationship with Carter that indicates a possible romance. "The Los Angeles Times" noted her character's penchant for secrecy but stated "It initially seems as though Bad Serena is back, but we all know her highway of crazy behavior is always paved with good intentions." Serena's relationship with Carter earns the disapproval of her peers but discovers redeeming qualities in him. Lily first assumes that she's preparing to go to Brown but defers from entering, preferring to stay in the city and find herself. During Lily's wedding, she discovers Carter's previous transgressions and she loses him when he chooses to make amends for his mistakes, breaking her heart in the process.


title: Serena van der Woodsen

text: Serena eventually makes her time in New York useful by taking a job as a publicist, one that is short-lived as her relationships with both her friends and her family begin to erode. Her friendship with Blair and Nate crumble due to her job but finds even more trouble when she falls for Nate's married congressman cousin, Tripp van der Bilt (Aaron Tveit). Her relationship with him is tumultuous as both Nate and Blair fail in persuading her from continuing the affair. The affair ends when an offer to become a mistress from Tripp's wife makes her come to her senses. As Serena and Tripp return to New York, their car crashes, leaving her hospitalized. She survives and Nate proves his love for her and the two begin a relationship.


title: Serena van der Woodsen

text: When Chuck's mother returns, Serena's resolve to find her father is reignited. Her decision slowly damages her relationship with Nate as she reconnects with Carter who is back to his old habits. Knowing that he has information regarding her father, the two work together to find him but is no longer interested in a relationship with Carter. Jenny's growing feelings for Nate also threaten her relationship with him when she makes every effort to steal Nate for herself. One incident occurs during Nate's "assassin" birthday bash. During the party, Nate and Jenny happen to be the last two players. After being stuck in a supply closet, she kisses Nate, despite him being in love with Serena, straining their relationship further. Following a night drinking with Chuck and Nate, Nate and Jenny go back to his apartment. Just as soon as Serena gets off the elevator, she sees Jenny trying to kiss Nate. Nate kicks Jenny out of the apartment to reconcile with Serena, spurning Jenny's advances.


title: Serena van der Woodsen

text: During Dorota's wedding, Serena leaves with Carter and gives Jenny a message telling Nate that she has found her father. Serena soon finds out that Carter has been delaying her in an attempt to get close to her and leaves him, flies alone. Upon arriving, she is shocked to find her mother answering the hotel door, assuming that she had been in Canyon Ranch the entire time.


title: Serena van der Woodsen

text: William van der Woodsen (William Baldwin) returns, making Serena happy but is unknowingly dragged into her father's plot to win back his ex-wife. The plot nearly works until Jenny, Blair, Nate and Chuck scheme against William, who had been trying to frame Rufus for adultery and intentionally misdiagnosing Lily, and William leaves. Her father's departure leads to Dan comforting her and reignites her feelings for him. Serena sleeps over at Dan's and wake up to find themselves in a compromising position, one that Jenny photographs and sends to Gossip Girl. As the photo spreads, Nate and Serena temporarily end their relationship and she joins Blair for a summer in Paris.


title: Serena van der Woodsen

text: Serena is seen spending her summer in Paris with Blair, reluctant to tell Blair that she has enrolled in Columbia, something that Blair fears will have them return to their high school pettiness, when she receives news that Chuck might be dead. After an unfortunate double date involving a handsome royal, Blair eventually forgives her. Chuck's arrival and a visit to a Parisian morgue confirms that he is alive but Serena pursues him in hopes of convincing him to return to New York when he tries to run away to London. She returns to New York having made her decision between Dan and Nate but her decision becomes irrelevant when she sees them with their new respective significant others, Vanessa and Juliet Sharp (Katie Cassidy), a girl with a personal vendetta against Serena.


title: Serena van der Woodsen

text: Serena's enrollment in Columbia has her face Juliet's many attempts at getting rid of her, ranging from excluding her from an exclusive society along with driving Blair against her, exposing her relationship with her professor, Colin Forrester (Sam Page), who happens to be Juliet's cousin to get her kicked out, and spreading various rumors about Serena, who manages to evade these attempts. Juliet then recruits Jenny and Vanessa and succeeds in gaslighting her into toying with Dan and Nate's feelings, attempting to take a foundation position from Blair, and returning to her old partying habits. Serena is rehabilitated while Dan and Blair subsequently discover Juliet's reasons for ruining Serena. Serena was supposedly in an illicit, but not ever consummated, relationship with Juliet's half-brother and Serena's boarding school English teacher, Ben Donovan (David Call) and was arrested when Lily forged an affidavit that falsely accused Ben of statutory rape, out of concern for Serena.


title: Serena van der Woodsen

text: The holidays have Serena trying to figure out how to release Ben from prison and her relationship with Lily is yet again strained. Serena and Dan put a possible return to their relationship on hold, knowing that they might not have the same chance of loving and trusting each other. Meanwhile, Ben's release from prison reignites Serena's feelings for him and they pursue a loving, yet troubled relationship. Serena's relationship with Ben slowly erodes when she receives news from Vanessa that he is responsible for hurting Nate's father in prison and when Ben's mother, Cynthia, raises Ben's personal troubles while at the same time seeking to exonerate him. Serena becomes proud of her mother when Lily decides to face the consequences of sending an innocent Ben to prison. Serena and Ben's relationship eventually ends due to family complications.


title: Serena van der Woodsen

text: Serena's cousin, Charlie Rhodes, comes to stay with Serena's family for a while. Charlie helps Serena find out that Dan and Blair have had a secret romance, causing a fight between Serena and Blair. Serena continues to use Charlie as a spy until noticing that Charlie is starting to act exactly as Serena and steals Serena's dress. In the season finale, Serena gets Charlie to help with a medical condition, though it is later revealed that Charlie is really Ivy, and Ivy is just acting. Later, while walking the streets of Los Angeles, Serena gets offered a job in Hollywood where she decides to move to.


title: Serena van der Woodsen

text: In Season 5, Serena is working as a film producer's assistant in Los Angeles and spending her free time with Nate and Chuck. Although Serena's boss, Marshall, is difficult to please, his boss, Jane, promotes Serena after she succeeds at a task that Marshall failed. Later, she runs into Charlie and convinces her to move with her back to New York.


title: Serena van der Woodsen

text: When Dan tells Serena, Nate, Blair, Chuck, Rufus and Lily that he wrote a book about his life and their roles in it, Serena is offended when he portrays her as a self-centered, promiscuous party-girl and is even more upset when Blair tells her that's exactly who she's been. After this revelation, she stops speaking to Dan. This cannot last, however, when her boss, Jane, requires Serena to get the movie rights to Dan's book. He agrees eventually saying that he trusts her and she gives him her word to protect him and his work. However, when Serena's boss tries to smear Dan's character and his real life image in the movie, she kills the deal to protect him, but when Dan finds out, he accuses her of killing the movie because of the negative way she is portrayed. Heartbroken over his accusation, she reveals that her real reason for killing his movie was to protect him, not herself, then walks away.


title: Serena van der Woodsen

text: After being fired by Jane, Serena is hired by Nate's boss, Diana Payne, to start a blog based on her life in order to take down Gossip Girl. Later, Nate and Serena discover a list of sources for Gossip Girl, which would be the silver bullet in Diana's campaign against Gossip Girl. They struggle over whether or not to use it and ultimately decide it would be too damaging to release. The list is leaked anyway and the two accuse each other before Blair discovers that it was Louis who leaked Gossip Girl's list of sources. Charlie encourages Serena to start dating in order to motivate her for her blog and she begins dating a guy named Max that she met on the street, not knowing that Max is actually Charlie's ex-boyfriend. Diana arranges for Max to meet Serena at the opening of Chuck's premiere of "Sleep No More" in order to make it look like Max has stood her up. Serena forgives Max for standing her up, and the two begin dating.


title: Serena van der Woodsen

text: Charlie finds out the two are dating and convinces her to slow down the relationship. Serena's aunt Carol returns and Serena overhears an argument between the two about Carol forcing Charlie to live under the name 'Ivy Dickens'. Serena makes the connection between Max and Charlie when she remembers that Max's ex-girlfriend's name was Ivy. She demands the truth from Charlie and Max. Ivy as Charlie and Carol manage to convince Serena and the family that Max is lying and Serena tells Max to leave before she calls the police.


title: Serena van der Woodsen

text: As Serena and Lily prepare Charlie for her debut into New York society, Dan tells Serena that he is going to tell Blair how he feels about her. In order to make Dan see that Blair's happy ending will be with Louis, she sends Louis to the Humphrey loft. When she finds Dan with Blair and Chuck later, she realizes that he gave up his desire for Blair for Blair's own happiness with Chuck and realizes that Blair is still in love with Chuck. Later that night, after Charlie's debutante party, Serena receives a call from Nate that Blair and Chuck have been in an accident. This causes Serena to declare war on Gossip Girl, using Nate's power at the Spectator, and Dan.


title: Serena van der Woodsen

text: With Gossip Girl gone, Serena finds that Gossip Girl's tipsters have begun sending their blasts to her via her blog. Nate encourages her to use the blasts productively, but Serena is reluctant to become the next Gossip Girl. Meanwhile, Serena is concerned for Blair's well-being and confronts her at Nate's New Year's Eve party where Blair tells her that the accident was God's way of telling her that she and Chuck are not meant to be together and that she made a promise with God that if Chuck lived, she would marry Louis and that Dan has been going to church with her since. In order to cover Blair, Serena pretends that she has rekindled her romance with Dan. This "romance" becomes the focus of her soon-to-be-launched blog which causes problems for Dan, as while Dan thinks that their romance is fake, Serena is shown to have genuine feelings for Dan again.


title: Serena van der Woodsen

text: Serena is humiliated at the press premiere of her new blog, being photographed holding a tablet that shows "Site not found." Upset, she is told the time wasn't right for her blog, but she ultimately discovers that Nate had it shut down as a favour to Gossip Girl. In return, Gossip Girl would help him find out the truth about Blair and Chuck's car accident. Serena doesn't agree with Nate's suspicions that Tripp was responsible, and confronts the latter over at the Waldorf residence. This is revealed to be a set-up brought about by Nate, Serena and Nate's grandfather as a way for Tripp to admit that he was behind the accident thinking that Nate was going to be in the car instead.


title: Serena van der Woodsen

text: Following this, Serena finds out that Blair was arrested at her own bachelorette party, and rushes to her aide. Blair tells her that she's alright, and that since she's going to go through with her wedding, Serena doesn't have to pretend to be together with Dan anymore. Serena, however, decides to keep this truth from Dan a little while longer.


title: Serena van der Woodsen

text: On Blair's wedding day, Serena tries to convince Blair not to marry Louis. Blair goes through with her wedding anyway, even after Gossip Girl launches a video of Blair professing her true feelings to Chuck. After the vows are said, Serena, Dan and Nate confront Georgina and she blames each of them in a vague manner before leaving. When Blair goes missing after her wedding, Serena and Chuck search for her, and find her in a motel in Queens with Dan, which upsets Serena as she still has feelings for him. When Georgina arrives, Serena is revealed to have Georgina's video camera that was used to record Blair's confession and she takes the fall for the video.


title: Serena van der Woodsen

text: Following this, Blair and Serena are in yet another rut, but forgive each other and Serena is unknowingly set up on a blind date with Dan, after she told him her feelings and he rejected her. At a party later, she and Georgina find Dan and Blair kissing, which causes another rift between her and Blair. After her grandmother's death and shocking news of her will, Serena starts a vendetta against Ivy (who received the major part of CeCe's estate), trying to win back her family fortunes. In a surprising turn of events, she gets a package from former Gossip Girl, Georgina Sparks, containing a computer and the means necessary to become the next Gossip Girl. Serena accepts.


title: Serena van der Woodsen

text: After losing the reins to Gossip Girl and her unintentional role in exposing Blair's diary to the Upper East side, Serena attempts to deceive Blair by having sex with Dan. The plan almost works only for Dan to realize what has happened. Serena once again tells Dan that she loves him but Dan tells her that he does not feel anything for her. A depressed Serena is last seen leaving the city with one of Damien Dalgaard's drug dealers, a flashback to her days before she returned to the Upper East Side in season one.


title: Serena van der Woodsen

text: In season 6, Serena is seen together with a new boyfriend, Steven Spence (who is significantly older than her). She is seen leading a happy, peaceful life away from the UES. Confronted by Blair, she simply says that she doesn't see how they could ever be friends again. Serena ultimately moves back to the UES with Steven, and his jealous daughter Sage. Sage uses every chance she gets to sabotage her father's relationship with Serena, and proves herself to be on par, scheming-wise, with Georgina and Blair. After some shocking revelations, including the fact that Steven has slept with Lily when her last name was Müller, they decide to end things.


title: Serena van der Woodsen

text: Serena is re-united with Dan again in a last attempt to make things work. Dan, on his hand, has been writing a scathingly honest sequel to "Inside" with the help of Georgina. One of his last and most poignant chapters is The Serena Chapter, which is released during the Thanksgiving party. Serena is, in the chapter, described as a "Golden girl" who lives for attention and love, and dramatically ends things with Dan. She decides to move away from New York, but is intercepted by Dan, who tries to convince her that he has always loved her and always will. Hurt and upset, she doesn't believe him, and leaves. Before she leaves, Dan slides a file into her bag, which she opens on the plane before take-off. It is revealed that Dan wrote two chapters on Serena, and that the second one was the "Good Serena Chapter". She gets off the plane and goes to talk things through thoroughly with Dan.


title: Serena van der Woodsen

text: In the finale, Dan reveals to her how they first met at a party at Blair's, and how he fell in love at first sight. He had then realised that he could never pull her out of her world and into his, so he found a way to enter her world by creating a UES blog. This blog, launching Serena's rise to legend, became known as Gossip Girl. Serena realises that Gossip Girl was only ever a manifestation of Dan's love for her, and that he practically wrote them all a seven-year-long love letter.


title: Serena van der Woodsen

text: A scene set in the future shows everyone reunited at the Bass-Waldorf residence, witnessing the marriage of Dan and Serena.


title: Serena van der Woodsen

text: Serena's style has earned noteworthy praise from periodicals such as "InStyle" and "NYLON". "EW"<nowiki>'</nowiki>s Meeta Agrawal places Blake Lively's Serena van der Woodsen amongst the wearers of the 20 Knockout Dresses of the '00s. Her Tory Burch designed dress in the pilot earning the 2nd spot on the list. "People" magazine references an issue of "NYLON", reporting that even the actress herself admits to liking her character's style and wardrobe and "InStyle"<nowiki>'</nowiki>s Joyann King praises "Serena’s vixen-like style and Lively’s smoldering off-screen choices". In an interview for "Vanity Fair", costume designers Eric Daman and Meredith Markworth-Pollack considered fashion model Kate Moss and New York socialites Tinsley Mortimer and Arden Wohl as a muse when dressing Lively and when asked if they were influenced by New York socialites, respectively. TV Guide named her the fifth most fashionable TV character. "Glamour" listed her as one of the 12 Most Stylish TV Characters.


title: Serena van der Woodsen

text: Blair and Serena's friendship was praised as "it offered a relationship whose depth and complexity approached Rory and Paris' [from "Gilmore Girls"]." "Vanity Fair" considered Serena's murder storyline to be "unrealistic" and "an obvious ratings ploy," going on to compare it to a dramatic scene in "The O.C.", in which Marissa shot Ryan Atwood's brother, stating that the show may have possibly jumped the shark.


title: Serena van der Woodsen

text: For her portrayal of Serena van der Woodsen, Blake Lively won the 2008 Teen Choice Award for Choice TV Actress Drama and was nominated for the same award in 2009 and 2010, losing to her co-star, Leighton Meester, who portrays Blair Waldorf. She, however, won the award again in 2011. Lively also earned a People's Choice Award nomination for Favorite TV Drama Actress in 2011 and 2012.


title: Animal Farm (1999 film)

text: Animal Farm is a 1999 political comedy-drama television film directed by John Stephenson and written by Alan Janes. Based on the 1945 novel of the same name by George Orwell and serving as an allegory of the Russian Revolution and its aftermath, it features Kelsey Grammer, Sir Ian Holm, Julia Louis-Dreyfus, Sir Patrick Stewart, Julia Ormond, Paul Scofield, Pete Postlethwaite and Sir Peter Ustinov. In the film, a group of anthropomorphic animals revolt successfully against their own human owner, only to slide into a more brutal tyranny among themselves.


title: Animal Farm (1999 film)

text: Mr. Jones, the cruel and rarely sober owner of Manor Farm, invites the Pilkingtons to a gathering he is hosting, wishing to talk about the debts he owes with Pilkington. During the gathering, the animals gather in a meeting where Old Major, the prize Middle White boar, explains to the animals that humanity is their enemy, and while the animals sing a song, Jones accidentally shoots Major while investigating.


title: Animal Farm (1999 film)

text: Later, when Jones goes into town for a drink without feeding the animals, Boxer, a strong and kind-hearted shire horse, leads the animals to break into the food shed to help themselves. When Jones and his cronies investigate, the animals rebel against him, causing Jones to flee with his wife and men to the Red Lion Inn for refuge.


title: Animal Farm (1999 film)

text: A boar named Snowball renames the place "Animal Farm" and puts down the Seven Commandments of "Animalism" which embody Old Major's feelings and ideas. Meanwhile, Napoleon, a Berkshire boar, calls for a secret meeting in which he has Pincher, one of the farm's dogs, swear loyalty to him and become part of the animal guard before ordering him to sneak out Jessie the border collie's newborn puppies, claiming that it is best for them to receive an education from him, despite Jessie's reluctance.


title: Animal Farm (1999 film)

text: Pilkington leads an attack into Animal Farm with the aid of other local farm workers led by Frederick, only to lose. Though he is defeated, Pilkington considers working with the animals instead. During a meeting, Snowball's plans to build a windmill to better the animals' lives and improve their operations are opposed by Napoleon, who summons Jessie's puppies (who are now grown up dogs working as his henchmen) to chase Snowball out of the farm. Napoleon decrees the pigs will decide the future and the animals begin the hard work of building the windmill with Boxer's help. Meanwhile, Pilkington hears over a microphone planted in the barn that the pigs can speak English and begins to trade with Napoleon. Later, Jessie reveals she saw the pigs living in the abandoned farmhouse and sleeping in the beds, though the commandment of sleeping in a bed being altered to not sleeping in a bed with sheets.


title: Animal Farm (1999 film)

text: Jones conspires with his wife to sabotage Animal Farm by blowing up the almost-complete windmill with dynamite as revenge on the animals for taking his farm. Napoleon blames it on Snowball, pigs consume more food, and blame Snowball for the food shortage and that the hens will have to surrender their eggs to the market. When the hens oppose, Napoleon makes feeding a hen punishable by death. Squealer begins making propaganda films about Napoleon showing animals on trial for working with Snowball that are sentenced to death, as well as animals supposedly happy with Napoleon's rule. It is revealed that the alcohol and killing commandments were altered as well into "no animal shall drink alcohol to excess" and "no animal shall kill another animal without cause".


title: Animal Farm (1999 film)

text: During the rebuilding of the windmill, Boxer is injured, and Jessie and Benjamin, a wise donkey, realize that the van taking Boxer is from the glue factory, causing everyone to unsuccessfully attempt to save him. Napoleon is paid by Pilkington for selling Boxer to the glue factory in exchange for more whiskey, and Squealer's latest propaganda film claims the van was previously the glue factory. That night, Jessie watches through a warped glass window as Pilkington and his wife dine with the pigs in the farmhouse. Napoleon then changes the farm's name back to Manor Farm. The animals can no longer tell the difference between them. Muriel the goat and Benjamin notice that the final commandment, "All animals are equal", has been extended to include "but some animals are more equal than others". Now seeing Napoleon and Squealer's evil nature, Jessie, Muriel, Benjamin and a few other animals sneak out before things can get any worse, while Napoleon (who now "fully resembles a human") enslaves the rest of the farm by falsely declaring all animals free.


title: Animal Farm (1999 film)

text: Years later, during a disastrous rainstorm, the escapees return and investigate the remains of the now-destroyed Manor Farm. They find Napoleon and Squealer are long gone, having died from their own dictation, though a few animals have survived the fall, including Jessie's puppies (who all recognize her as their mother). Jessie finds out a new family has purchased the farm (although the whereabouts of Jones and his wife is unknown), and vows to not let them make the same mistakes as Jones and Napoleon.


title: Animal Farm (1999 film)

text: Filming began on 25 August 1998 and ended on 6 November. Because of the extensive CGI work and other post-production requirements, the film was not delivered to TNT and Hallmark Entertainment until June 1999.


title: Animal Farm (1999 film)

text: Fourteen animals were built to represent the animals of Animal Farm at Jim Henson's Creature Shop in London: four pigs (Old Major, Snowball, Napoleon, and Squealer), two horses (Boxer and Mollie), a sheepdog (Jessie), a donkey (Benjamin), a raven (Moses), a goat (Muriel), a sheep, a rat, a chicken, a duck, a cat, and a dove.


title: Animal Farm (1999 film)

text: Ten dogs were cast into the film from Fircroft Kennels. Their Border collie, Spice, played the role of Jessie.


title: Animal Farm (1999 film)

text: In early screenplays done by Martyn Burke for this film, Jessie was set to be a male character, rather than a female.


title: Animal Farm (1999 film)

text: The film received mixed reviews. It holds a 40% approval rating on Rotten Tomatoes, based on five reviews.


title: Animal Farm (1999 film)

text: It was criticized for its loose adaptation of the book, its simplicity and lack of subtlety, and for being too dark and political for children while being too familiar and simplistic for adults.


title: Animal Farm (1999 film)

text: The film won Best Special Effects and was nominated for best film in the 2000s Fantasporto International Fantasy Film Award.


title: Animal Farm (1999 film)

text: The film's director John Stephenson was nominated for Starboy Award in the 2000s Oulu International Children's and Youth Film Festival.


title: Grammatical person

text: In linguistics, grammatical person is the grammatical distinction between deictic references to participant(s) in an event; typically the distinction is between the speaker (first person), the addressee (second person), and others (third person). A language's set of "personal" pronouns are defined by grammatical person, but other pronouns would not. "First person" includes the speaker (English: "I", "we", "me", and "us"), "second person" is the person or people spoken to (English: "you"), and "third person" includes all that are not listed above (English: "he", "she", "it", "they", "him", "her", "them"). It also frequently affects verbs, and sometimes nouns or possessive relationships.


title: Grammatical person

text: In Indo-European languages, first-, second-, and third-person pronouns are typically also marked for singular and plural forms, and sometimes dual form as well (grammatical number).


title: Grammatical person

text: Some other languages use different classifying systems, especially in the plural pronouns. One frequently found difference not present in most Indo-European languages is a contrast between inclusive and exclusive "we": a distinction of first-person plural pronouns between including or excluding the addressee.


title: Grammatical person

text: Many languages express person with different morphemes in order to distinguish degrees of formality and informality. A simple honorific system common among European languages is the T–V distinction. Some other languages have much more elaborate systems of formality that go well beyond the T–V distinction, and use many different pronouns and verb forms that express the speaker's relationship with the people they are addressing. Many Malayo-Polynesian languages, such as Javanese and Balinese, are well known for their complex systems of honorifics; Japanese, Korean, and Chinese also have similar systems to a lesser extent.


title: Grammatical person

text: In many languages, the verb takes a form dependent on the person of the subject and whether it is singular or plural. In English, this happens with the verb "to be" as follows:


title: Grammatical person

text: Other verbs in English take the suffix "-s" to mark the present tense third person singular, excluding singular 'they'.


title: Grammatical person

text: In many languages, such as French, the verb in any given tense takes a different suffix for any of the various combinations of person and number of the subject.


title: Grammatical person

text: The grammar of some languages divide the semantic space into more than three persons. The extra categories may be termed fourth person, fifth person, etc. Such terms are not absolute but can refer depending on context to any of several phenomena.


title: Grammatical person

text: Some Algonquian languages and Salishan languages divide the category of third person into two parts: "proximate" for a more topical third person, and "obviative" for a less topical third person. The obviative is sometimes called the fourth person. In this manner, Hindi and Bangla may also categorize pronouns in the fourth, and with the latter a fifth person.


title: Grammatical person

text: The term fourth person is also sometimes used for the category of indefinite or generic referents, which work like "one" in English phrases such as "one should be prepared" or "people" in "people say that...", when the grammar treats them differently from ordinary third-person forms. The so-called "zero person" in Finnish and related languages, in addition to passive voice may serve to leave the subject-referent open. Zero person subjects are sometimes translated as "one," but the problem with that is that English language constructions involving "one", e.g. "One hopes that will not happen," are rare and could be considered expressive of an overly academic tone to the majority of people, while Finnish sentences like "Ei saa koskettaa" ("Not allowed to touch") are recognizable to and used by young children in both languages.


title: Kym Karath

text: Kym Karath is an American former actress, best known for her role as Gretl, the youngest of the Von Trapp children in "The Sound of Music".


title: Kym Karath

text: Kym Karath was born in Los Angeles, California, and started her career at the age of 4, when she appeared in "Spencer's Mountain" (1963) with Henry Fonda and Veronica Cartwright (she subsequently worked with Veronica's younger sister, Angela Cartwright, in "The Sound of Music"). Also in 1963, Karath was seen in "The Thrill of It All" with Doris Day and James Garner. In 1964, she was seen in "Good Neighbor Sam" with Jack Lemmon.


title: Kym Karath

text: Karath is best remembered for her role in "The Sound of Music", where she played the youngest child, Gretl Von Trapp.


title: Kym Karath

text: Karath graduated from the University of Southern California with a degree in humanities, and shortly after that she moved to Paris, France, where she studied art history and modeled. At the age of 26, she married Philippe L'Equibec; their son was born in 1991. Thereafter she left acting for a while and lived in Greenwich, Connecticut, before resuming acting in 2005.


title: Kym Karath

text: Karath has said she still is not fond of water after nearly drowning in the boat scene in "The Sound of Music", as she could not swim.


title: Mathematics and art

text: Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.


title: Mathematics and art

text: Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his "Canon", prescribing proportions conjectured to have been based on the ratio 1: for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise "De divina proportione" (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as "De Prospectiva Pingendi", and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work "Melencolia I". In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting.


title: Mathematics and art

text: Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of symmetry, and mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching. Mathematical concepts such as recursion and logical paradox can be seen in paintings by René Magritte and in engravings by M. C. Escher. Computer art often makes use of fractals including the Mandelbrot set, and sometimes explores other mathematical objects such as cellular automata. Controversially, the artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes; the architect Philip Steadman similarly argued that Vermeer used the camera obscura in his distinctively observed paintings.


title: Mathematics and art

text: Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditional batiks from different regions of Java have distinct fractal dimensions, and stimuli to mathematics research, especially Filippo Brunelleschi's theory of perspective, which eventually led to Girard Desargues's projective geometry. A persistent view, based ultimately on the Pythagorean notion of harmony in music, holds that everything was arranged by Number, that God is the geometer of the world, and that therefore the world's geometry is sacred.


title: Mathematics and art

text: Polykleitos the elder (c. 450–420 BC) was a Greek sculptor from the school of Argos, and a contemporary of Phidias. His works and statues consisted mainly of bronze and were of athletes. According to the philosopher and mathematician Xenocrates, Polykleitos is ranked as one of the most important sculptors of classical antiquity for his work on the "Doryphorus" and the statue of Hera in the Heraion of Argos. While his sculptures may not be as famous as those of Phidias, they are much admired. In the his "Canon", a treatise he wrote designed to document the "perfect" body proportions of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body.


title: Mathematics and art

text: The "Canon" itself has been lost but it is conjectured that Polykleitos used a sequence of proportions where each length is that of the diagonal of a square drawn on its predecessor, 1: (about 1:1.4142).


title: Mathematics and art

text: The influence of the "Canon" of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitos's prescription. While none of Polykleitos's original works survive, Roman copies demonstrate his ideal of physical perfection and mathematical precision. Some scholars argue that Pythagorean thought influenced the "Canon" of Polykleitos. The "Canon" applies the basic mathematical concepts of Greek geometry, such as the ratio, proportion, and "symmetria" (Greek for "harmonious proportions") and turns it into a system capable of describing the human form through a series of continuous geometric progressions.


title: Mathematics and art

text: In classical times, rather than making distant figures smaller with linear perspective, painters sized objects and figures according to their thematic importance. In the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen (Ibn al-Haytham) described a theory of optics in his "Book of Optics" in 1021, but never applied it to art. The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts. Two major motives drove artists in the late Middle Ages and the Renaissance towards mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas. Second, philosophers and artists alike were convinced that mathematics was the true essence of the physical world and that the entire universe, including the arts, could be explained in geometric terms.


title: Mathematics and art

text: The rudiments of perspective arrived with Giotto (1266/7 – 1337), who attempted to draw in perspective using an algebraic method to determine the placement of distant lines. In 1415, the Italian architect Filippo Brunelleschi and his friend Leon Battista Alberti demonstrated the geometrical method of applying perspective in Florence, using similar triangles as formulated by Euclid, to find the apparent height of distant objects. Brunelleschi's own perspective paintings are lost, but Masaccio's painting of the Holy Trinity shows his principles at work.


title: Mathematics and art

text: The Italian painter Paolo Uccello (1397–1475) was fascinated by perspective, as shown in his paintings of "The Battle of San Romano" (c. 1435–1460): broken lances lie conveniently along perspective lines.


title: Mathematics and art

text: The painter Piero della Francesca (c. 1415–1492) exemplified this new shift in Italian Renaissance thinking. He was an expert mathematician and geometer, writing books on solid geometry and perspective, including "De prospectiva pingendi (On Perspective for Painting)", "Trattato d'Abaco (Abacus Treatise)", and "De quinque corporibus regularibus (On the Five Regular Solids)". The historian Vasari in his "Lives of the Painters" calls Piero the "greatest geometer of his time, or perhaps of any time." Piero's interest in perspective can be seen in his paintings including the Polyptych of Perugia, the "San Agostino altarpiece" and "The Flagellation of Christ". His work on geometry influenced later mathematicians and artists including Luca Pacioli in his "De divina proportione" and Leonardo da Vinci. Piero studied classical mathematics and the works of Archimedes. He was taught commercial arithmetic in "abacus schools"; his writings are formatted like abacus school textbooks, perhaps including Leonardo Pisano (Fibonacci)'s 1202 "Liber Abaci". Linear perspective was just being introduced into the artistic world. Alberti explained in his 1435 "De pictura": "light rays travel in straight lines from points in the observed scene to the eye, forming a kind of pyramid with the eye as vertex." A painting constructed with linear perspective is a cross-section of that pyramid.


title: Mathematics and art

text: In "De Prospectiva Pingendi", Piero transforms his empirical observations of the way aspects of a figure change with point of view into mathematical proofs. His treatise starts in the vein of Euclid: he defines the point as "the tiniest thing that is possible for the eye to comprehend". He uses deductive logic to lead the reader to the perspective representation of a three-dimensional body.


title: Mathematics and art

text: The artist David Hockney argued in his book "" that artists started using a camera lucida from the 1420s, resulting in a sudden change in precision and realism, and that this practice was continued by major artists including Ingres, Van Eyck, and Caravaggio. Critics disagree on whether Hockney was correct. Similarly, the architect Philip Steadman argued controversially that Vermeer had used a different device, the camera obscura, to help him create his distinctively observed paintings.


title: Mathematics and art

text: In 1509, Luca Pacioli (c. 1447–1517) published "De divina proportione" on mathematical and artistic proportion, including in the human face. Leonardo da Vinci (1452–1519) illustrated the text with woodcuts of regular solids while he studied under Pacioli in the 1490s. Leonardo's drawings are probably the first illustrations of skeletonic solids. These, such as the rhombicuboctahedron, were among the first to be drawn to demonstrate perspective by being overlaid on top of each other. The work discusses perspective in the works of Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano. Leonardo studied Pacioli's "Summa", from which he copied tables of proportions. In "Mona Lisa" and "The Last Supper", Leonardo's work incorporated linear perspective with a vanishing point to provide apparent depth. "The Last Supper" is constructed in a tight ratio of 12:6:4:3, as is Raphael's "The School of Athens", which includes Pythagoras with a tablet of ideal ratios, sacred to the Pythagoreans. In "Vitruvian Man", Leonardo expressed the ideas of the Roman architect Vitruvius, innovatively showing the male figure twice, and centring him in both a circle and a square.


title: Mathematics and art

text: As early as the 15th century, curvilinear perspective found its way into paintings by artists interested in image distortions. Jan van Eyck's 1434 "Arnolfini Portrait" contains a convex mirror with reflections of the people in the scene, while Parmigianino's "Self-portrait in a Convex Mirror", c. 1523–1524, shows the artist's largely undistorted face at the centre, with a strongly curved background and artist's hand around the edge.


title: Mathematics and art

text: Three-dimensional space can be represented convincingly in art, as in technical drawing, by means other than perspective. Oblique projections, including cavalier perspective (used by French military artists to depict fortifications in the 18th century), were used continuously and ubiquitously by Chinese artists from the first or second centuries until the 18th century. The Chinese acquired the technique from India, which acquired it from Ancient Rome. Oblique projection is seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752–1815).


title: Mathematics and art

text: The golden ratio (roughly equal to 1.618) was known to Euclid. The golden ratio has persistently been claimed in modern times to have been used in art and architecture by the ancients in Egypt, Greece and elsewhere, without reliable evidence. The claim may derive from confusion with "golden mean", which to the Ancient Greeks meant "avoidance of excess in either direction", not a ratio. Pyramidologists since the 19th century have argued on dubious mathematical grounds for the golden ratio in pyramid design. The Parthenon, a 5th-century BC temple in Athens, has been claimed to use the golden ratio in its façade and floor plan, but these claims too are disproved by measurement. The Great Mosque of Kairouan in Tunisia has similarly been claimed to use the golden ratio in its design, but the ratio does not appear in the original parts of the mosque. The historian of architecture Frederik Macody Lund argued in 1919 that the Cathedral of Chartres (12th century), Notre-Dame of Laon (1157–1205) and Notre Dame de Paris (1160) are designed according to the golden ratio, drawing regulator lines to make his case. Other scholars argue that until Pacioli's work in 1509, the golden ratio was unknown to artists and architects. For example, the height and width of the front of Notre-Dame of Laon have the ratio 8/5 or 1.6, not 1.618. Such Fibonacci ratios quickly become hard to distinguish from the golden ratio. After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo's "Mona Lisa".


title: Mathematics and art

text: Another ratio, the only other morphic number, was named the plastic number in 1928 by the Dutch architect Hans van der Laan (originally named "le nombre radiant" in French). Its value is the solution of the cubic equation


title: Mathematics and art

text: an irrational number which is approximately 1.325. According to the architect Richard Padovan, this has characteristic ratios and , which govern the limits of human perception in relating one physical size to another. Van der Laan used these ratios when designing the 1967 St. Benedictusberg Abbey church in the Netherlands.


title: Mathematics and art

text: Planar symmetries have for millennia been exploited in artworks such as carpets, lattices, textiles and tilings.


title: Mathematics and art

text: Many traditional rugs, whether pile carpets or flatweave kilims, are divided into a central field and a framing border; both can have symmetries, though in handwoven carpets these are often slightly broken by small details, variations of pattern and shifts in colour introduced by the weaver. In kilims from Anatolia, the motifs used are themselves usually symmetrical. The general layout, too, is usually present, with arrangements such as stripes, stripes alternating with rows of motifs, and packed arrays of roughly hexagonal motifs. The field is commonly laid out as a wallpaper with a wallpaper group such as pmm, while the border may be laid out as a frieze of frieze group pm11, pmm2 or pma2. Turkish and Central Asian kilims often have three or more borders in different frieze groups. Weavers certainly had the intention of symmetry, without explicit knowledge of its mathematics.


title: Mathematics and art

text: The mathematician and architectural theorist Nikos Salingaros suggests that the "powerful presence" (aesthetic effect) of a "great carpet" such as the best Konya two-medallion carpets of the 17th century is created by mathematical techniques related to the theories of the architect Christopher Alexander. These techniques include making opposites couple; opposing colour values; differentiating areas geometrically, whether by using complementary shapes or balancing the directionality of sharp angles; providing small-scale complexity (from the knot level upwards) and both small- and large-scale symmetry; repeating elements at a hierarchy of different scales (with a ratio of about 2.7 from each level to the next). Salingaros argues that "all successful carpets satisfy at least nine of the above ten rules", and suggests that it might be possible to create a metric from these rules.


title: Mathematics and art

text: Elaborate lattices are found in Indian Jali work, carved in marble to adorn tombs and palaces. Chinese lattices, always with some symmetry, exist in 14 of the 17 wallpaper groups; they often have mirror, double mirror, or rotational symmetry. Some have a central medallion, and some have a border in a frieze group. Many Chinese lattices have been analysed mathematically by Daniel S. Dye; he identifies Sichuan as the centre of the craft.


title: Mathematics and art

text: Symmetries are prominent in textile arts including quilting, knitting, cross-stitch, crochet, embroidery and weaving, where they may be purely decorative or may be marks of status. Rotational symmetry is found in circular structures such as domes; these are sometimes elaborately decorated with symmetric patterns inside and out, as at the 1619 Sheikh Lotfollah Mosque in Isfahan. Items of embroidery and lace work such as tablecloths and table mats, made using bobbins or by tatting, can have a wide variety of reflectional and rotational symmetries which are being explored mathematically.


title: Mathematics and art

text: Islamic art exploits symmetries in many of its artforms, notably in girih tilings. These are formed using a set of five tile shapes, namely a regular decagon, an elongated hexagon, a bow tie, a rhombus, and a regular pentagon. All the sides of these tiles have the same length; and all their angles are multiples of 36° (π/5 radians), offering fivefold and tenfold symmetries. The tiles are decorated with strapwork lines (girih), generally more visible than the tile boundaries. In 2007, the physicists Peter Lu and Paul Steinhardt argued that girih resembled quasicrystalline Penrose tilings. Elaborate geometric zellige tilework is a distinctive element in Moroccan architecture. Muqarnas vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells.


title: Mathematics and art

text: The Platonic solids and other polyhedra are a recurring theme in Western art. They are found, for instance, in a marble mosaic featuring the small stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica in Venice; in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's 1509 book "The Divine Proportion"; as a glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495; in the truncated polyhedron (and various other mathematical objects) in Albrecht Dürer's engraving Melencolia I; and in Salvador Dalí's painting "The Last Supper" in which Christ and his disciples are pictured inside a giant dodecahedron.


title: Mathematics and art

text: Albrecht Dürer (1471–1528) was a German Renaissance printmaker who made important contributions to polyhedral literature in his 1525 book, "Underweysung der Messung (Education on Measurement)", meant to teach the subjects of linear perspective, geometry in architecture, Platonic solids, and regular polygons. Dürer was likely influenced by the works of Luca Pacioli and Piero della Francesca during his trips to Italy. While the examples of perspective in "Underweysung der Messung" are underdeveloped and contain inaccuracies, there is a detailed discussion of polyhedra. Dürer is also the first to introduce in text the idea of polyhedral nets, polyhedra unfolded to lie flat for printing. Dürer published another influential book on human proportions called "Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion)" in 1528.


title: Mathematics and art

text: Dürer's well-known engraving "Melencolia I" depicts a frustrated thinker sitting by a truncated triangular trapezohedron and a magic square. These two objects, and the engraving as a whole, have been the subject of more modern interpretation than the contents of almost any other print, including a two-volume book by Peter-Klaus Schuster, and an influential discussion in Erwin Panofsky's monograph of Dürer.


title: Mathematics and art

text: Salvador Dalí's 1954 painting "Corpus Hypercubus" uniquely depicts the cross of Christ as an unfolded three-dimensional net for a hypercube, also known as a tesseract: the unfolding of a tesseract into these eight cubes is analogous to unfolding the sides of a cube into a cross shape of six squares, here representing the divine perspective with a four-dimensional regular polyhedron. The painting shows the figure of Christ in front of the tessaract; he would normally be shown fixed with nails to the cross, but there are no nails in the painting. Instead, there are four small cubes in front of his body, at the corners of the frontmost of the eight tessaract cubes. The mathematician Thomas Banchoff states that Dalí was trying to go beyond the three-dimensional world, while the poet and art critic Kelly Grovier says that "The painting seems to have cracked the link between the spirituality of Christ's salvation and the materiality of geometric and physical forces. It appears to bridge the divide that many feel separates science from religion."


title: Mathematics and art

text: Traditional Indonesian wax-resist batik designs on cloth combine representational motifs (such as floral and vegetal elements) with abstract and somewhat chaotic elements, including imprecision in applying the wax resist, and random variation introduced by cracking of the wax. Batik designs have a fractal dimension between 1 and 2, varying in different regional styles. For example, the batik of Cirebon has a fractal dimension of 1.1; the batiks of Yogyakarta and Surakarta (Solo) in Central Java have a fractal dimension of 1.2 to 1.5; and the batiks of Lasem on the north coast of Java and of Tasikmalaya in West Java have a fractal dimension between 1.5 and 1.7.


title: Mathematics and art

text: The drip painting works of the modern artist Jackson Pollock are similarly distinctive in their fractal dimension. His 1948 "Number 14" has a coastline-like dimension of 1.45, while his later paintings had successively higher fractal dimensions and accordingly more elaborate patterns. One of his last works, "Blue Poles", took six months to create, and has the fractal dimension of 1.72.


title: Mathematics and art

text: The astronomer Galileo Galilei in his "Il Saggiatore" wrote that "[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures." Artists who strive and seek to study nature must first, in Galileo's view, fully understand mathematics. Mathematicians, conversely, have sought to interpret and analyse art through the lens of geometry and rationality. The mathematician Felipe Cucker suggests that mathematics, and especially geometry, is a source of rules for "rule-driven artistic creation", though not the only one. Some of the many strands of the resulting complex relationship are described below.


title: Mathematics and art

text: The mathematician Jerry P. King describes mathematics as an art, stating that "the keys to mathematics are beauty and elegance and not dullness and technicality", and that beauty is the motivating force for mathematical research. King cites the mathematician G. H. Hardy's 1940 essay "A Mathematician's Apology". In it, Hardy discusses why he finds two theorems of classical times as first rate, namely Euclid's proof there are infinitely many prime numbers, and the proof that the square root of 2 is irrational. King evaluates this last against Hardy's criteria for mathematical elegance: ""seriousness, depth, generality, unexpectedness, inevitability", and "economy"" (King's italics), and describes the proof as "aesthetically pleasing". The Hungarian mathematician Paul Erdős agreed that mathematics possessed beauty but considered the reasons beyond explanation: "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I "know" numbers are beautiful."


title: Mathematics and art

text: Mathematics can be discerned in many of the arts, such as music, dance, painting, architecture, and sculpture. Each of these is richly associated with mathematics. Among the connections to the visual arts, mathematics can provide tools for artists, such as the rules of linear perspective as described by Brook Taylor and Johann Lambert, or the methods of descriptive geometry, now applied in software modelling of solids, dating back to Albrecht Dürer and Gaspard Monge. Artists from Luca Pacioli in the Middle Ages and Leonardo da Vinci and Albrecht Dürer in the Renaissance have made use of and developed mathematical ideas in the pursuit of their artistic work. The use of perspective began, despite some embryonic usages in the architecture of Ancient Greece, with Italian painters such as Giotto in the 13th century; rules such as the vanishing point were first formulated by Brunelleschi in about 1413, his theory influencing Leonardo and Dürer. Isaac Newton's work on the optical spectrum influenced Goethe's "Theory of Colours" and in turn artists such as Philipp Otto Runge, J. M. W. Turner, the Pre-Raphaelites and Wassily Kandinsky. Artists may also choose to analyse the symmetry of a scene. Tools may be applied by mathematicians who are exploring art, or artists inspired by mathematics, such as M. C. Escher (inspired by H. S. M. Coxeter) and the architect Frank Gehry, who more tenuously argued that computer aided design enabled him to express himself in a wholly new way.


title: Mathematics and art

text: The artist Richard Wright argues that mathematical objects that can be constructed can be seen either "as processes to simulate phenomena" or as works of "computer art". He considers the nature of mathematical thought, observing that fractals were known to mathematicians for a century before they were recognised as such. Wright concludes by stating that it is appropriate to subject mathematical objects to any methods used to "come to terms with cultural artifacts like art, the tension between objectivity and subjectivity, their metaphorical meanings and the character of representational systems." He gives as instances an image from the Mandelbrot set, an image generated by a cellular automaton algorithm, and a computer-rendered image, and discusses, with reference to the Turing test, whether algorithmic products can be art. Sasho Kalajdzievski's "Math and Art: An Introduction to Visual Mathematics" takes a similar approach, looking at suitably visual mathematics topics such as tilings, fractals and hyperbolic geometry.


title: Mathematics and art

text: Some of the first works of computer art were created by Desmond Paul Henry's "Drawing Machine 1", an analogue machine based on a bombsight computer and exhibited in 1962. The machine was capable of creating complex, abstract, asymmetrical, curvilinear, but repetitive line drawings. More recently, Hamid Naderi Yeganeh has created shapes suggestive of real world objects such as fish and birds, using formulae that are successively varied to draw families of curves or angled lines. Artists such as Mikael Hvidtfeldt Christensen create works of generative or algorithmic art by writing scripts for a software system such as "Structure Synth": the artist effectively directs the system to apply a desired combination of mathematical operations to a chosen set of data.


title: Mathematics and art

text: The mathematician and theoretical physicist Henri Poincaré's "Science and Hypothesis" was widely read by the Cubists, including Pablo Picasso and Jean Metzinger. Being thoroughly familiar with Bernhard Riemann's work on non-Euclidean geometry, Poincaré was more than aware that Euclidean geometry is just one of many possible geometric configurations, rather than as an absolute objective truth. The possible existence of a fourth dimension inspired artists to question classical : non-Euclidean geometry became a valid alternative. The concept that painting could be expressed mathematically, in colour and form, contributed to Cubism, the art movement that led to abstract art. Metzinger, in 1910, wrote that: "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematician Maurice Princet has deduced a whole geometry". Later, Metzinger wrote in his memoirs:


title: Mathematics and art

text: Maurice Princet joined us often ... it was as an artist that he conceptualized mathematics, as an aesthetician that he invoked "n"-dimensional continuums. He loved to get the artists interested in the new views on space that had been opened up by Schlegel and some others. He succeeded at that.


title: Mathematics and art

text: The impulse to make teaching or research models of mathematical forms naturally creates objects that have symmetries and surprising or pleasing shapes. Some of these have inspired artists such as the Dadaists Man Ray, Marcel Duchamp and Max Ernst, and following Man Ray, Hiroshi Sugimoto.


title: Mathematics and art

text: Man Ray photographed some of the mathematical models in the Institut Henri Poincaré in Paris, including "Objet mathematique" (Mathematical object). He noted that this represented Enneper surfaces with constant negative curvature, derived from the pseudo-sphere. This mathematical foundation was important to him, as it allowed him to deny that the object was "abstract", instead claiming that it was as real as the urinal that Duchamp made into a work of art. Man Ray admitted that the object's [Enneper surface] formula "meant nothing to me, but the forms themselves were as varied and authentic as any in nature." He used his photographs of the mathematical models as figures in his series he did on Shakespeare's plays, such as his 1934 painting "Antony and Cleopatra". The art reporter Jonathan Keats, writing in "ForbesLife", argues that Man Ray photographed "the elliptic paraboloids and conic points in the same sensual light as his pictures of Kiki de Montparnasse", and "ingeniously repurposes the cool calculations of mathematics to reveal the topology of desire". Twentieth century sculptors such as Henry Moore, Barbara Hepworth and Naum Gabo took inspiration from mathematical models. Moore wrote of his 1938 "Stringed Mother and Child": "Undoubtedly the source of my stringed figures was the Science Museum ... I was fascinated by the mathematical models I saw there ... It wasn't the scientific study of these models but the ability to look through the strings as with a bird cage and to see one form within another which excited me."


title: Mathematics and art

text: The artists Theo van Doesburg and Piet Mondrian founded the De Stijl movement, which they wanted to "establish a visual vocabulary elementary geometrical forms comprehensible by all and adaptable to any discipline". Many of their artworks visibly consist of ruled squares and triangles, sometimes also with circles. De Stijl artists worked in painting, furniture, interior design and architecture. After the breakup of De Stijl, Van Doesburg founded the Avant-garde Art Concret movement, describing his 1929–1930 "Arithmetic Composition", a series of four black squares on the diagonal of a squared background, as "a structure that can be controlled, a "definite" surface without chance elements or individual caprice", yet "not lacking in spirit, not lacking the universal and not ... empty as there is "everything" which fits the internal rhythm". The art critic Gladys Fabre observes that two progressions are at work in the painting, namely the growing black squares and the alternating backgrounds.


title: Mathematics and art

text: The mathematics of tessellation, polyhedra, shaping of space, and self-reference provided the graphic artist M. C. Escher (1898—1972) with a lifetime's worth of materials for his woodcuts. In the "Alhambra Sketch", Escher showed that art can be created with polygons or regular shapes such as triangles, squares, and hexagons. Escher used irregular polygons when tiling the plane and often used reflections, glide reflections, and translations to obtain further patterns. Many of his works contain impossible constructions, made using geometrical objects which set up a contradiction between perspective projection and three dimensions, but are pleasant to the human sight. Escher's "Ascending and Descending" is based on the "impossible staircase" created by the medical scientist Lionel Penrose and his son the mathematician Roger Penrose.


title: Mathematics and art

text: Some of Escher's many tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter on hyperbolic geometry. Escher was especially interested in five specific polyhedra, which appear many times in his work. The Platonic solids—tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons—are especially prominent in "Order and Chaos" and "Four Regular Solids". These stellated figures often reside within another figure which further distorts the viewing angle and conformation of the polyhedrons and provides a multifaceted perspective artwork.


title: Mathematics and art

text: The visual intricacy of mathematical structures such as tessellations and polyhedra have inspired a variety of mathematical artworks. Stewart Coffin makes polyhedral puzzles in rare and beautiful woods; George W. Hart works on the theory of polyhedra and sculpts objects inspired by them; Magnus Wenninger makes "especially beautiful" models of complex stellated polyhedra.


title: Mathematics and art

text: The distorted perspectives of anamorphosis have been explored in art since the sixteenth century, when Hans Holbein the Younger incorporated a severely distorted skull in his 1533 painting "The Ambassadors". Many artists since then, including Escher, have make use of anamorphic tricks.


title: Mathematics and art

text: The mathematics of topology has inspired several artists in modern times. The sculptor John Robinson (1935–2007) created works such as "Gordian Knot" and "Bands of Friendship", displaying knot theory in polished bronze. Other works by Robinson explore the topology of toruses. "Genesis" is based on Borromean rings – a set of three circles, no two of which link but in which the whole structure cannot be taken apart without breaking. The sculptor Helaman Ferguson creates complex surfaces and other topological objects. His works are visual representations of mathematical objects; "The Eightfold Way" is based on the projective special linear group PSL(2,7), a finite group of 168 elements. The sculptor Bathsheba Grossman similarly bases her work on mathematical structures. The artist Nelson Saiers incorporates mathematical concepts and theorems in his art from toposes and schemes to the four color theorem and the irrationality of π.


title: Mathematics and art

text: A liberal arts inquiry project examines connections between mathematics and art through the Möbius strip, flexagons, origami and panorama photography.


title: Mathematics and art

text: Mathematical objects including the Lorenz manifold and the hyperbolic plane have been crafted using fiber arts including crochet. The American weaver Ada Dietz wrote a 1949 monograph "Algebraic Expressions in Handwoven Textiles", defining weaving patterns based on the expansion of multivariate polynomials. The mathematician Daina Taimiņa demonstrated features of the hyperbolic plane by crocheting in 2001. This led Margaret and Christine Wertheim to crochet a coral reef, consisting of many marine animals such as nudibranchs whose shapes are based on hyperbolic planes.


title: Mathematics and art

text: The mathematician J. C. P. Miller used the Rule 90 cellular automaton to design tapestries depicting both trees and abstract patterns of triangles. The "mathekniticians" Pat Ashforth and Steve Plummer use knitted versions of mathematical objects such as hexaflexagons in their teaching, though their Menger sponge proved too troublesome to knit and was made of plastic canvas instead. Their "mathghans" (Afghans for Schools) project introduced knitting into the British mathematics and technology curriculum.


title: Mathematics and art

text: Modelling is far from the only possible way to illustrate mathematical concepts. Giotto's "Stefaneschi Triptych", 1320, illustrates recursion in the form of "mise en abyme"; the central panel of the triptych contains, lower left, the kneeling figure of Cardinal Stefaneschi, holding up the triptych as an offering. Giorgio de Chirico's metaphysical paintings such as his 1917 "Great Metaphysical Interior" explore the question of levels of representation in art by depicting paintings within his paintings.


title: Mathematics and art

text: Art can exemplify logical paradoxes, as in some paintings by the surrealist René Magritte, which can be read as semiotic jokes about confusion between levels. In "La condition humaine" (1933), Magritte depicts an easel (on the real canvas), seamlessly supporting a view through a window which is framed by "real" curtains in the painting. Similarly, Escher's "Print Gallery" (1956) is a print which depicts a distorted city which contains a gallery which recursively contains the picture, and so "ad infinitum". Magritte made use of spheres and cuboids to distort reality in a different way, painting them alongside an assortment of houses in his 1931 "Mental Arithmetic" as if they were children's building blocks, but house-sized. "The Guardian" observed that the "eerie toytown image" prophesied Modernism's usurpation of "cosy traditional forms", but also plays with the human tendency to seek patterns in nature.


title: Mathematics and art

text: Salvador Dalí's last painting, "The Swallow's Tail" (1983), was part of a series inspired by René Thom's catastrophe theory. The Spanish painter and sculptor Pablo Palazuelo (1916–2007) focused on the investigation of form. He developed a style that he described as the geometry of life and the geometry of all nature. Consisting of simple geometric shapes with detailed patterning and coloring, in works such as "Angular I" and "Automnes", Palazuelo expressed himself in geometric transformations.


title: Mathematics and art

text: The artist Adrian Gray practises stone balancing, exploiting friction and the centre of gravity to create striking and seemingly impossible compositions.


title: Mathematics and art

text: Artists, however, do not necessarily take geometry literally. As Douglas Hofstadter writes in his 1980 reflection on human thought, "Gödel, Escher, Bach", by way of (among other things) the mathematics of art: "The difference between an Escher drawing and non-Euclidean geometry is that in the latter, comprehensible interpretations can be found for the undefined terms, resulting in a comprehensible total system, whereas for the former, the end result is not reconcilable with one's conception of the world, no matter how long one stares at the pictures." Hofstadter discusses the seemingly paradoxical lithograph "Print Gallery" by M. C. Escher; it depicts a seaside town containing an art gallery which seems to contain a painting of the seaside town, there being a "strange loop, or tangled hierarchy" to the levels of reality in the image. The artist himself, Hofstadter observes, is not seen; his reality and his relation to the lithograph are not paradoxical. The image's central void has also attracted the interest of mathematicians Bart de Smit and Hendrik Lenstra, who propose that it could contain a Droste effect copy of itself, rotated and shrunk; this would be a further illustration of recursion beyond that noted by Hofstadter.


title: Mathematics and art

text: Algorithmic analysis of images of artworks, for example using X-ray fluorescence spectroscopy, can reveal information about art. Such techniques can uncover images in layers of paint later covered over by an artist; help art historians to visualize an artwork before it cracked or faded; help to tell a copy from an original, or distinguish the brushstroke style of a master from those of his apprentices.


title: Mathematics and art

text: Jackson Pollock's drip painting style has a definite fractal dimension; among the artists who may have influenced Pollock's controlled chaos, Max Ernst painted Lissajous figures directly by swinging a punctured bucket of paint over a canvas.


title: Mathematics and art

text: The computer scientist Neil Dodgson investigated whether Bridget Riley's stripe paintings could be characterised mathematically, concluding that while separation distance could "provide some characterisation" and global entropy worked on some paintings, autocorrelation failed as Riley's patterns were irregular. Local entropy worked best, and correlated well with the description given by the art critic Robert Kudielka.


title: Mathematics and art

text: The American mathematician George Birkhoff's 1933 "Aesthetic Measure" proposes a quantitative metric of the aesthetic quality of an artwork. It does not attempt to measure the connotations of a work, such as what a painting might mean, but is limited to the "elements of order" of a polygonal figure. Birkhoff first combines (as a sum) five such elements: whether there is a vertical axis of symmetry; whether there is optical equilibrium; how many rotational symmetries it has; how wallpaper-like the figure is; and whether there are unsatisfactory features such as having two vertices too close together. This metric, "O", takes a value between −3 and 7. The second metric, "C", counts elements of the figure, which for a polygon is the number of different straight lines containing at least one of its sides. Birkhoff then defines his aesthetic measure of an object's beauty as "O/C". This can be interpreted as a balance between the pleasure looking at the object gives, and the amount of effort needed to take it in. Birkhoff's proposal has been criticized in various ways, not least for trying to put beauty in a formula, but he never claimed to have done that.


title: Mathematics and art

text: Art has sometimes stimulated the development of mathematics, as when Brunelleschi's theory of perspective in architecture and painting started a cycle of research that led to the work of Brook Taylor and Johann Heinrich Lambert on the mathematical foundations of perspective drawing, and ultimately to the mathematics of projective geometry of Girard Desargues and Jean-Victor Poncelet.