Why California Keeps Making Homelessness Worse

Why California Keeps Making Homelessness Worse

On Tuesday, fifteen officials from the White House toured Skid Row in Los Angeles with the head of a local homeless shelter. “Four or five of them were from the Environmental Protection Agency,” Rev. Andy Bales of Union Mission church told me. “That’s because human waste flows into storm sewers.” California is home to some of the world’s toughest environmental and public health laws, but skyrocketing homelessness has created an environmental and public health disaster. The 44,000 people living, eating, and defecating on the streets of L.A. have brought rats and medieval diseases including typhus. Garbage is everywhere. Experts fear the return of cholera and leprosy.
And homelessness is making people violent . “We are seeing behaviors from our guests that I’ve never seen in 33 years,” said Bales. “They are so bizarre and different that I don’t even feel right describing the behaviors. It’s extreme violence of an extreme sexual nature. I have been doing this for 33 years and never seen anything like it.”
Bales says he was one of the people who urged the US Government’s Federal Emergency Management Administration (FEMA) to intervene. “We’ve been crying out for a National Guard-like response,” said Bales, whose church provides food, showers, and shelter to 1,350 people camped nearby. In 2016 Bales lost the lower half of his leg to a flesh-eating bacteria from contamination on Skid Row.
How did things get so bad in California? The state has long prided itself on being humanistic and innovative. It is home to some of the world’s largest public health philanthropies, best hospitals, and most progressive policies on mental health and drug addiction. The Democrats have a supermajority. What went wrong?
According to Bales and other experts, California made homelessness worse by making perfect housing the enemy of good housing, by liberalizing drug laws, and by opposing mandatory treatment for mental illness and drug addiction.
Other states have done a better job despite spending less money. “This isn’t rocket science,” said John Snook, who runs the Treatment Advocacy Center , which advises states on mental health and homelessness policy around the country. “Arizona is a red state that doesn’t spend a ton on its services but is the best scenario in every aspect. World-class coordination with law enforcement. Strong oversight. They don’t let people fall apart and then return to jail in 30 days like California does.”
What happened in California isn’t the first time that we progressives let our idealism get the better of us. To understand how the current disaster unfolded, we have to go back in time, back to the post-World War II era when progressive reformers convinced themselves and others that they could destroy the country’s system for dealing with the mentally ill and replace it with a radically different and wholly unproven alternative.
A Mania for Reform
People considered the creation of state mental institutions in the 1800s to be a major progressive reform because they took the mentally ill out of prisons and hospitals and put them into a safer and kinder environment, notes the psychiatrist E. Fuller Torrey, M.D., in his devastating and critically-acclaimed 2014 history, American Psychosis: How the Federal Government Destroyed the Mental Illness System.
In many respects, the mental institutions were a step in the right direction, but by the middle of the 20th Century, their reputation was in tatters. They were understaffed and overcrowded. Some patients were poorly treated, even abused. Others were neglected. During World War II, Mennonites and Quakers worked in the institutions as an alternative to military service. After the war, they drew attention to the deplorable conditions.
Reformers felt they could do better. In 1945 they proposed community-based clinics not just to treat but also to prevent mental illness. They called for a federal takeover. Congressional advocates frequently invoked the US government’s Manhattan Project as inspiration. If America could build a nuclear bomb in a few years, why couldn’t we prevent and cure mental illness?
As Congress debated mental health reform in 1946, some were suspicious. “Men get strange ideas,” said Republican congressman Clarence J. Brown of Ohio. “They decide the only way in the world they are going to solve all the problems of mankind is to do a certain thing and that their field is the most important.” Many reformers believed mental illness was created by poverty and inequality and argued that solving it required creating “mentally healthy” environments, organizing tenants, and fighting landlords.
These reformers viewed mental illnesses like schizophrenia and bipolar disorder as socially constructed and not the result of biology, as most doctors believe today. They sought clinics that would “promote health” and “the development of a resilient character.” They wanted clinics to treat the “totality of [a patient’s] being in the totality of his relationships.” The psychiatrist played a special role, the reformers said. “One might even say,” wrote Francis Braceland, an influential psychiatrist who had studied with Carl Jung, “ the ideal goal of the psychiatrist is to achieve wisdom.”
The reformers were so confident in their convictions that they smashed the state mental institutions before creating an alternative. The reformers hyped new psychiatric drugs, which reduced the symptoms of schizophrenia, as a bridge to the new system. There was little resistance to the radical changes by existing mental institutions, whose leadership had been demoralized and discredited. And yet there was no evidence that community-based treatment would work. Between 1948 and 1962, the test that clinic reformers pointed to as the model had not prevented a single case of mental illness or treated a single individual with schizophrenia.
But attacking mental institutions had become hugely popular. In two hugely influential 1961 books, a psychiatrist argued that mental illnesses didn’t exist and a sociologist argued that the institutions themselves created mental illness. One year later, One Flew Over the Cuckoo’s Nest , a novel about a sane but socially maladjusted man who was drugged, electro-shocked, and lobotomized by a mental institution, became a best-seller. In 1967, the film “King of Hearts” depicted psychiatric inmates after World War II as living happily once freed from their asylum. In 1975, the year “One Flew Over the Cuckoo’s Nest” became a hit film, Michel Foucault argued in Discipline and Punish that mentally ill people had been better off in the Middle Ages when they could roam the streets without being shamed as deviant.
Over the next two decades, state mental hospitals would empty out. But the vast majority of released patients ended up homeless on the street. Congress had “encouraged the closing of state mental hospitals without any realistic plan regarding what would happen to the discharged patients,” notes Dr. Torrey, “especially those who refused to take medication they needed to remain well.”
And yet the reformers were becoming only more radical. “The changes I am talking about,” said a leader at the new National Institute of Mental Health, ” involves a redistribution of wealth and resources… society for the urban poor of such beauty and richness… nothing less than a privilege to be called poor.”
But when the community mental health clinics did start operating, they tended to treat the easiest-to-treat, not the hardest. It was a trend that worsened the longer the clinics were in existence. The clinic saw “very few individuals with serious mental illnesses,” reported a young psychiatrist working in Santa Monica near LA. “Instead, the patients were people from the community with various personal crises.”
In the end, no more than 5% of the federally-funded clinics “made any significant contributions to the care of patients being released from state mental hospitals,” finds Torrey. Financial abuses were rife, with clinics building tennis courts, swimming pools, and rooms for fads like “inhalation therapy” that did nothing for people with schizophrenia.
When critics faulted the clinics for their abuses, reformers defended themselves behind a wall of political correctness. One reformer-aligned task force that investigated the situation concluded in 1976 that “to criticize the [mental health] centers themselves for many (but not all) of their failings is to ‘blame the victim!’” The Carter Administration recommended making federal support permanent and included new money to prevent mental illness by reducing “societal stresses produced by racism, poverty, sexism, ageism, and urban blight.”
Republicans who had initially supported deinstitutionalization as a cost-savings measure became increasingly resentful of what they viewed as an anarchistic approach and sought to cut the budget for mental illness. But as federal support for the clinics declined, the state institutions were no longer in place to care for the homeless evicted to the streets. Everybody was in charge and nobody was in charge. The reformers grew depressed. “The deformed creature that has developed from the original community mental health center movement does not arouse much enthusiasm in any of us who had some more grandiose visions,” said one.
The problem, Torrey and other advocates for the mentally ill say, wasn’t de-institutionalization but rather the failure to provide new forms of treatment. “The majority of lives were little different than they had had while hospitalized,” he concludes, “and a significant number were considerably worse off.” Many didn’t even realize they were mentally ill, similar to some Alzheimer’s patients. For decades, radical reformers sought de-institutionalization in even the most extreme situations. In 1985, a public defender got a mentally ill client released from jail even though he had been found eating his feces.
Importantly, reformers never had evidence that community-based clinics would work better than big institutions. They just assumed it in a way that is eerily similar to the way that 1960s environmentalists in California, including Governor Jerry Brown, assumed “small-is-beautiful” policies would be better for the environment. Out of hubris, the reformers sought to smash the old institution before creating a new one. Intriguingly, that’s exactly what reformers would do again in California, 50 years later.
When Dogmatism Is Deadly
For decades, many progressives have claimed that homelessness is really just a kind of poverty, a manifestation of social inequality. In 1986, celebrity comedians Whoppi Goldberg, Robin Williams, and Billy Crystal held “Comic Relief,” a telethon for homelessness. Throughout it, they emphasized that the homeless were just like you and me, just poorer. Today, many of California’s leading homelessness advocates insist that the current crisis is due mostly to the housing shortage.
Homelessness experts and advocates disagree. “I’ve rarely seen a normal able-bodied able-minded non-drug-using homeless person who’s just down on their luck,” L.A. street doctor Susan Partovi told me. “Of the thousands of people I’ve worked with over 16 years, it’s like one or two people a year. And they’re the easiest to deal with.” Rev. Bales agrees. “One hundred percent of the people on the streets are mentally impacted, on drugs, or both,” he said.
Most of the time what people mean by the homelessness problem is really a drug problem and a mental illness problem. ”The problem is we don’t know if you’re psychotic or just on meth,” said Dr. Partovi. “And giving it up is very difficult. I worked in the local jail, and half of the inmates in the women’s jail were Latinas in their 20s, and all were in there for something related to meth.”
The people who work directly with the homeless say things worsened after California abandoned the “carrot and stick” approach toward treating the severely mentally ill and drug addicts who are repeat offenders. “The ACLU will come after me if I say the mentally ill need to be taken off the street,” said Dr. Partovi, “so let me be clear that they need to be taken care of, too.”
Bales says things worsened ten years ago when L.A. and other California cities rejected drug recovery (treatment) as a condition of housing. “When the ‘Housing First’ with a harm reduction model people came in they said ‘Recovery doesn’t work,’” said Bales. “But it was after that when homelessness exploded exponentially.”
Bales says people have little incentive to do treatment when there is no threat of jail time. “[The Housing First harm reduction advocates] talked about new services, but they were all voluntary.” Things went further in this direction with the passage of Proposition 47 in 2016, which decriminalized hard drugs and released nonviolent offenders from prison without providing after-care support. “Our guests went from 12 – 17% addicted to 50% or higher,” Bales says. “Policymakers need to understand that if you allow the use, you also allow the sales, and if you allow the sales, then you allow the big guys to break your legs when you owe them money,” says Bales.
Snook says that California is so unwilling to require non-voluntary mental health care that it is only now considering more extensive “conservatorship” — where a health official is given the authority to make decisions for a mentally incapacitated individual — and only after nine acts of violence against themselves or others.
“Imagine having a sick child and hoping he attacks someone once a month so somebody can do something!” said Snook. “That is so out of sync with the rest of the country, and with what mental health care looks like, that it is laughable.”
Lack of shelter and leadership are factors alongside extreme progressive idealism. “It’s the impact of not having a stick and not having shelter,” says Bales. Snook agrees. “There’s a provision that says Medicaid will now pay for beds in psychiatric hospitals,” said Snook. “It’s a no-brainer, but California is hemming and hawing. They don’t want to involuntarily incarcerate, but it’s self-defeating because you end up with mentally ill in jail because a bed isn’t available.”
Is the problem a lack of money? “California spends more than most places,” said Snook, whose organization researches and advocates solutions for mentally ill homeless people nationally. What happened to the money from Proposition 63, the successful 2016 ballot initiative that taxed millionaires for mental health? “A Hoover Foundation audit found funds that were supposed to go to seriously mentally ill were used for yoga and trauma and other laudable things, but none for the seriously mentally ill,” said Snook.
“When you look at the amount of money being spent, and then you hear the argument that we need more money? You have to ask, ‘How much more?’” said Snook. “Right now it’s just good money after bad. There’s no oversight and no accountability.”
Liberal idealism also wasted much of the $1.2 billion that L.A. voters raised in 2016 when they voted to tax themselves to build housing for the homeless. “It was supposed to build 10,000 units but in truth will create half that because each one costs $527,000 to $700,000,” said Bales. “They will take ten years to build, at which point 44,000 lives will have been destroyed by living on the street.”
Why did progressive housing activists in L.A. insist on building such expensive apartments for so few people, so slowly, rather than quickly building cheaper units faster for 44,000 people?
“[Housing First] is a dogmatic philosophy,” said Bales. “I’ve lost friends. One of my closest friends is attacking me for pushing for housing that costs $11,000 instead of $527,000 per person. He can’t get that we can’t provide a $527,000 to $700,000 apartment for each person on the street. I’ve been in planning meetings where people said, ‘Everybody deserves a granite countertop,’ but that isn’t going to work for 44,000 people.”
L.A.’s woke housing advocates have intimidated the city’s mayor. “I think the mayor’s unwilling to put out bridge shelters because of backlash from some homeless advocates,” said Bales, “and is concerned about NIMBYs, and may be concerned about union workers because the shelters may not be built by the unions.”
Beyond Right and Left
Despite the emergency, and in some ways because of it, homeless reformers are hopeful today that California will seek a more moderate path toward treating mental illness and drug addiction and providing shelter. “The problem is so bad and so significant that there’s an opportunity now,” said Snook.
Bales agrees. “In New York where they put a roof over the heads of 95% of the homeless, it took a law to change things,” he said. “I thought it would take a law here, too, but maybe we can get there without a law if people continue to show political courage.”
California recently awarded L.A. $124 million for the homelessness emergency, of which L.A.’s mayor has spent $66 million on 27 cheap, quick-to-assemble temporary shelters that can quickly get 13,000 people off the street. “I think the mayor has been fighting an uphill battle against the powers that be, and his political courage is growing,” said Bales.
One sign of trouble was the resistance by some in California to receiving help from the federal government. “[Trump’s] budget has proposed slashing public housing … and eliminating community development block grant dollars,” LA’s mayor said . “It’s totally out of step with the idea that he’s here to help.”
But others were more conciliatory “I am wary of any such offer from an administration that consistently demonizes vulnerable people,” the governor’s top homelessness advisor, Sacramento Mayor Darrell Steinberg, told me “And yet, if the federal government wants to offer resources to help bring people indoors and to offer federal facilities to shelter and house people, we should readily listen. We cannot afford to politicize an issue which needs real thought and real commitment.”
Everyone I spoke to hopes that the current crisis makes all sides less dogmatic. Something like that happened at the federal level after a mentally ill man killed 20 elementary school children in 2012. Democrats and Republicans found common ground on sweeping mental health reform legislation in 2015. “It wasn’t a Democrat or Republican thing,” says Snook.
Steinberg, for his part, would like to see legislation codifying his “right/obligation to shelter” framework, which blends the traditional liberal emphasis on rights and care with the traditional conservative emphasis on order. “I’m open to more carrots and sticks and would be for an obligation to seek shelter. I don’t think living outside is a civil right,” he told me. “We back, philosophically, ‘Housing First’… but if all we did was permanent housing, it would be until 2037 before we housed every person on the streets in the Bay Area.”
It’s hard to see any of what needs to happen as particularly partisan. “The key is focusing services on the seriously ill,” argues Snook. “You provide extended care. You open up beds for when they are stable. You provide care to people to stay out of the system.” What about the homeless who are not mentally ill? Focus on the hardest population first, he urged. “Once you get that population addressed, you can move on to the others. This is what New York City did. Once you get that population serviced, you’re not in crisis mode anymore, and you free up money for everyone else.”
I left the reporting for this column surprised by how stuck California’s leaders remain in 1960s ideology and how slow they’ve been to react to the crisis. “It’s better late than never, but still we are not treating it in the urgent manner we should,” said Bales. “We’re not there yet where people are really taking it seriously as an emergency.” For Torrey, it all comes down to leadership. “It is not clear where the leadership for change will come from,” he wrote seven years ago, “but until it emerges, change is unlikely.”

40 Companies Changing Plus-Size Women’s Clothing Industry

3. Luvmemore
A brand focused on inclusivity and body positivity in designer plus size clothing, Luvmemore is setting the bar for service and style with their impressive lineup of clothing from size 2-26. Luvmemore believes that you don’t have to be a size zero to be beautiful, and their clothing lineup is designed to bring out the best features of any size. From tops to cover-ups, tie waist pants to skirts, and everything in between, plus size fashion has never been more simple or loving. Show off your curves with Luvmemore’s casual lineup, or opt for a more professional look with clothes for work, school, or that important meeting you have next week.
4. Maree Pour Toi
For those ladies in between sizes 12 and 24, there’s a unique plus size women’s clothing option for you that boasts both attractive styles and reasonable prices. Maree Pour Toi has everything you need, from blazers to skirts to summer dresses and comfortable pants. The company believes that every woman, regardless of shape or size, should feel comfortable in her own clothing. With this simple motto, the brand is working hard to change the way plus size fashion is viewed and how plus size ladies feel in their own clothes.
5. Baacal
Designer Cynthia Vincent has made her debut in the world of designer plus size clothing with the launch of her brand, Baacal . With contemporary pieces and a desire to add sustainability to the industry’s material acquisition process, Cynthia Vincent is truly taking a new approach to the business. Baacal has something for everyone, whether you’re a size 10 or a size 22. From work apparel to casual attire, Baacal has everything the plus-size lady needs in her wardrobe.
6. Loft Plus
With a high focus on body positivity and comfort, Loft has a great lineup of plus size clothing for the ladies of all sizes. The designs are simple yet sleek and attractive, made to bring out your best features and make you feel like a goddess in your clothes. Loft is one of the few retailers that sell clothes of all sizes but hasn’t forgotten to include stylish and comfortable designs in their plus-size lineup.
7. Apiece Apart
With a lineup for sizes between 14-22, Apiece Apart was founded on one simple concept: to make women feel free in their clothing to go anywhere and do anything. Designed by Laura Cramer and Starr Hout in 2008, this plus size fashion line has been going strong for over a decade. With attractive designs and even more attractive prices, combined with a company mission that anyone could get behind, it’s easy to see why.
8. Milly
For sizes 10-20, there’s Milly ; a stylish plus size clothing brand with a lot to offer. Milly was founded way back in 2000 by designer Michelle Smith, and has been making waves ever since with its lineup of stunning casual wear, swimsuits, and even children’s clothing. If you’re looking for plus size women’s clothing, Milly is available on
9. Cushnie
Fitting sizes 12-22, Cushnie , which used to be Cushnie et Ochs, sports a lineup of impressive plus size fashion statements. Crisp lines, bold colors, and sophistication that can hardly be matched, Cushnie is the go-to clothing line for plus size girls looking to make a statement. With four collections to choose from, there’s no shortage of attractive styles for the plus-size lady to choose from.
10. Good American
Good American started out as a denim line in 2016, slowly evolving into a full-tilt clothing line due to its successful reception. Designed by Khloe Kardashian, the lineup caters to the tastes of curvier ladies (like Khloe herself) and supports style and comfort while offering prices anyone can manage. Plus size clothing is Good American’s specialty, and you’re sure to find exactly what you need to fit our personal tastes with the brand’s versatile lineup.
11. Chantelle
Finding plus size bras and undergarments is even more difficult than finding comfortable jeans. The designers at Chantelle understand this better than anyone, and that’s why the company offers a great lineup of SoftStretch Plus underwear. Plus-size fashion designers often forget the underwear side of things, but not so with Chantelle. Their Softstretch panties fit 1X-4X sizes and offer a full-coverage look for sizes XS-XL.
12. Chromat
For fashionable and comfortable sport and swimwear, plus size ladies choose Chromat . Sportswear is often overlooked in the world of plus size women’s clothing, but Chromat hasn’t forgotten that plus size ladies like to be swim too! Form attractive swimsuits to form-fitting sportswear, this designer offers clothing in sizes XS-4X.
13. Parker NY
Sporting clothing from size 00 to XXL, Parker NY has something for everyone. With sleek, sexy styles, a reputation for quality and comfort, and a designer who truly cares for the needs of plus size ladies, Parker NY is one of those brands you won’t want to miss. You can find their attractive lineup of plus size women’s clothing on their website, .
14. ModCloth
ModCloth’s plus size clothing line offers elegant dresses, comfortable rompers, sweatshirts and jeans, and much more. For casual or semi-formal wear, ModCloth shines with its simple and rustic approach to plus size fashion. With an affordable price range, and supporting up to size 28 and 4X, there are plenty of options for ladies of all sizes available. Plus size ladies shouldn’t have to sacrifice style or comfort, and ModCloth understands this simple fact.
15. City Chic
For exclusively plus size clothing options, City Chic is your go-to for incredible fashion and affordability. With over ten collections available, including everything from swimwear to casual and formal wear to lingerie and denim, City Chich truly is your one-stop-shop for all things designer plus size clothing. Right now, the company is running a sale on new season clothing, formal dresses, and swimwear, with deals of up to 50% off. With such an incredible lineup, it’s no wonder City Chich is becoming the premier online destination for plus size ladies everywhere.
16. Monif C.
Finding sexy and comfortable lingerie as a plus-size lady can be quite the challenge. Most lingerie stores don’t even stock in anything bigger than a medium or large, and online outlets can be just as picky. Enter Monif C . , the premier plus size lingerie destination for ladies of all sizes. At Monif C., you’ll find everything from stylish corsets to show off your curves to lace bodysuits and everything in between.
17. ASOS Curve
ASOS is a popular clothing brand with a plus size women’s clothing lineup to fit all of your needs. From skirts and shorts to sweaters and cardigans, jumpsuits and rompers to jeans and dresses, Curve is an all-inclusive fashion line that caters to the needs of plus-size ladies. Style, comfort, and class blend perfectly in this attractive lineup, making it one of the best plus size clothing lines available online.
18. Navabi
With a huge range of plus size women’s clothing options, including options for sizes 12-32, the stylish lineup has something to offer ladies of all shapes and sizes. Feel comfortable and beautiful in your clothes with Navabi’s unique trends and prices that won’t break the bank. Whether you’re looking for jackets, dresses, denim, or knitwear, Navabi has one of the biggest arrays of plus size women’s clothing in the world. Shop today and learn why so many ladies trust Navabi for their plus size clothing needs!
19. Boohoo
Despite the brand’s name, there’s nothing to cry about with their impressive lineup of plus size clothing. Whether you’re looking for a formal dress to wear to a wedding or some comfy leggings for everyday wear, Boohoo has everything you need and more. Covering sizes 16-28, Boohoo includes everyone in their fashion statement with incredible styles and prices that won’t leave you feeling down. Leggings can be especially hard to find in plus sizes, so take a look at Boohoo’s collection today and find the right fit for you!
20. River Island
While River Island isn’t strictly a plus size clothing brand, it’s still an all-inclusive company that supports sizes up to 38. Sleek, trendy, and affordable, River Island’s lineup contains everything a curvy girl might need; from nightwear to denim and sweaters, to jumpers, cardigans, and even boots and shoes. It’s rare for retailers to include sizes up to River Island’s range, so we’re pretty excited to see a major outlet supporting the comfort and style of plus size ladies as well as their smaller counterparts.
21. Nicholas
With vivid colors, bold prints, and body-inclusive designs, Nicholas offers a plus-size women’s clothing line with beautiful floral patterns to mark the new season, simple colors that showcase style and comfort, and a reputation for excellence in design and service, Nicholas is a clothing lineup you won’t want to miss in the world of plus size fashion.
22. Simply Be
Ranging from sizes 12-34, Simply Be is perhaps one of the most size-inclusive brands on the market; providing stylish and comfortable plus size fashion designs that will leave you feeling satisfied and beautiful in your new clothes. Simply Be’s approach is simple; every woman is beautiful, no matter what her size, and her clothes should reflect that. From boots and shoes to casual and formal wear and even underwear, Simply Be has whatever plus size clothing you’re looking for at affordable prices.
23. Chi Chi London
Fashion is all about making a statement, and with Chi Chi London’s Curve lineup offers stylish dresses for any occasion that will leave the competition in envy. These sleek, form-fitting dresses will show off your best curvature while making you feel comfortable and content in your own skin. With size ranges from 18-26, the Curve lineup is all-inclusive and ready to make your dress collection stand out from the crowd.
24. Isolated Heroes
With a strong belief that fashion should be applicable to all walks of life, despite gender, race, size, or age, Isolated Heroes makes clothes for everyone. With plenty of glitter and glam, this bold clothing company doesn’t shy from plus size clothing; in fact, it embraces it and encourages plus size ladies to be comfortable in their own skin. From bold skirts and tops to sequin jackets, this clothing brand certainly offers a unique fashion statement to those plus size ladies who don’t want to shy away from their size.
25. Zelie for She
“An unapologetic expression of authenticity” as the brand’s website puts it, Zelie for She is one of the more unique and proud clothing lines for ladies between sizes 14 and 24. This plus size women’s clothing company doesn’t hide behind fashion trends or society’s expectations. Zelie truly sets the bar with their own unique lineup and an attitude that says “I’m beautiful no matter what you say.” For those plus size ladies that like to make a statement with their clothing, Zelie is the place for you.
26. Jibri
The plus-size fashion company Jibri specializes in clothing in sizes 10-28, and designer Jasmine Elder is not holding back when it comes to expressing beauty in plus sizes. Formal wear is the line’s most impressive addition; from wedding dresses to ball-gowns, you’ll be able to find that perfect fashion statement to go with whatever the occasion is. From poolside wear to formal dresses and beyond, Jibri has something for plus size gals of all sizes.
27. Lane Bryant
The popular clothing brand Lane Bryant is actually quite a bit more size-inclusive than one might think. With sizes ranging from 10-32, the brand has undergone some changes to make it not only contemporary and more accessible, but also as size-inclusive as possible. Lane Bryant offers plenty of options for plus size ladies, from jeans to dresses and everything in between. The website even offers free shipping on orders over $100!
28. Anthropologie
Supporting sizes 16-26, the A Plus lineup by Anthropologie blends casual style and format fittings for a unique look that plus size ladies can enjoy no matter what shape or size they are. The brand offers shoes, dresses, jackets, and more, and the prices are something to consider when choosing between high-end retailers. Show off your curves and your confidence with Anthropologie’s A Plus lineup.
29. Reformation
While Reformation is brand-new to the plus size women’s clothing industry, their new lineup of plus size clothing is nothing to balk at. The company only just released its plus-size lineup recently, but so far it’s been making waves with its stylish dresses and comfortable, sleek denim collection. With a press release that went something like “ Sorry it took us so long ” Reformation entered the world of plus size fashion and forever changed their public image and their commitment to empowering women of all sizes.
30. Swimsuits for All
Let’s be honest; as plus-sized ladies, buying swimsuits in a typical retail store isn’t exactly the best option available. That’s why Swimsuits for All offers swimwear for ladies of every size; whether you’re a 4 or a 40. Swimwear can be one of the most difficult things to shop for when you’re a plus-size lady, so head on over to Swimsuits for All and take a look at their impressive lineup of swimsuits today. With summer ending, you could save up to 40% on already reduced prices.
31. SmartGlamour
For a more tailored experience in plus size women’s clothing, you can try out the wonderful designs from SmartGlamour . Each piece is hand-produced with sustainable materials by designer and one-woman shop-owner Mallorie Dunn. Items are produced to order, and will match your exact measurements for the ultimate in comfort and flexibility. Supporting sizes from 2XS to 15X (and beyond) you’re certain to find the exact piece you’re looking for in the perfect size for your body shape.
32. Dear Kate
From size 00 to 26W, this all-inclusive clothing brand includes plus size fashion in its incredible lineup. Underwear and comfortable leggings should be something that plus size ladies have equal access to, but the fact is, it’s rather difficult to find the right fit at certain sizes. Dear Kate ensures that you’re getting the right underwear and/or leggings for your body shape and size; something other retailers simply can’t match.
33. Torrid
For trendy designs and a size-inclusive lineup, Torrid is a plus-size women’s clothing company that supports sizes from 10-30. Whether you’re looking to fill up your wardrobe with more fashionable items, or simply want to feel comfortable and beautiful in your own skin, Torrid has something for you. From intimates to shoes and accessories and more, Torrid has everything a plus-size lady needs to feel her best without breaking the bank.
34. A’Beautiful Soul
For the best in plus size dresses, visit A’Beautiful Soul . This trendy retailer believes that plus size ladies should look and feel good in dresses or casual wear just as much as the next girl; creating stunning designs that bring out both your confidence and your beauty all at once. Designer plus size clothing shouldn’t exclude formal wear like dresses, and A’Beautiful Soul certainly doesn’t.
35. Gwynnie Bee
If you’re looking for something a bit more unique, try Gwynnie Bee’s plus size subscription box. For sizes 10-32, Gwynnie Bee provides trendy and attractive clothing designs for girls of all shapes and sizes, and with the subscription box, you can have a constant supply of trendy clothing delivered right to your door every few months!
36. Premme
Sassy, proud, and full of style, Premme provides an attractive and stylish lineup of plus size women’s clothing from size 12-30. This truly inclusive clothing line contains everything you need; with multiple collections to meet the tastes of plus size ladies all over the world. If you’ve never tried Premme before, head over to their website and learn why they’re one of the most trusted plus size clothing outlets available today.
37. Fat Mermaids
Perhaps no company on the market today is quite as unapologetic as Fat Mermaids . With an attitude that says “shove it” to industry standards, Fat Mermaids sets itself apart by providing bold, unapologetic fashion to plus size ladies of all shapes and sizes. Don’t let your plus size hold you back from looking absolutely fabulous with Fat Mermaid’s diverse lineup of clothing and accessories.
Perhaps the gold standard of designer plus size clothing Eloquii has been turning heads for a number of years with its incredible lineup of dresses, jackets, denim, and more. You shouldn’t have to sacrifice comfort for style, and you’ll find none of that with Eloquii. The brand is trusted worldwide as one of the best providers of plus size clothing on the market.
39. American Eagle
Believe it or not, American Eagle has actually improved their clothing lineup to include sizes up to 20. Known for the comfort of their jeans, American Eagle has often been viewed as a “small-size” only brand, but this simply isn’t the case. The plus-size lineup contains American Eagle’s signature style in sizes made to fit ladies of all sizes.
40. Chubby Cartwheels
With over 12 collections to choose from, including intimates, Chubby Cartwheels is as unapologetic about their fashion as they are stylish. The brand focuses on body positivity and the idea that no matter what size a lady is, she should feel comfortable wearing any clothing; from sexy lingerie to the most formal of dresses. Visit Chubby Cartwheels online and discover their full lineup of plus size women’s clothing.

The 100 best films of the 21st century | Film

100 Once Upon a Time in Hollywood (2019) Quentin Tarantino’s latest jaw-dropper bumps Kill Bill: Vol 1 off the list in gloriously irreverent fashion. Leonardo DiCaprio and Brad Pitt star as a fading western star and his mutt-loving stunt double in this relaxed and loving roast of bygone Tinseltown. CS
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99 Bright Star (2009) An early lead for Ben Whishaw as the ailing John Keats romancing Fanny Brawne (Abbie Cornish) is the tremulous soul of this underappreciated Jane Campion drama. The butterflies are too tropical for Hampstead, but the rest is spot-on. CS
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98 The Dark Knight (2008) The only comic book movie to make the cut is Christopher Nolan’s genre masterpiece: fatalist, bracing and forever the legacy of Heath Ledger, posthumously awarded an Oscar for his terrifying performance. CS
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97 Fahrenheit 9/11 (2004) Michael Moore’s finest hour: a blazing juggernaut with George W Bush, the Iraq war, the media, democracy and us, the gullible masses, in its crosshairs. Agitprop, and essential. CS
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Facebook Twitter Pinterest Paul Giamatti and Kathryn Hahn in Private Life. Photograph: Jojo Whilden/Netflix 96 Private Life (2018)
Kathryn Hahn and Paul Giamatti struggle to start a family, and to keep their marriage together, in this subtle, funny and often wondrously uncomfortable Netflix comedy written and directed by Tamara Jenkins. CS
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95 Call Me By Your Name (2017) Rarely has summer lust been so headily captured as in Luca Guadagnino’s breakout Italian romance. Transformative leads from Timothée Chalamet and Armie Hammer captured the collective imagination; Michael Stuhlbarg gently grounded realities. CS
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Facebook Twitter Pinterest Maximus attitude … Russell Crowe in Gladiator. Photograph: Dreamworks/Sportsphoto/Allstar 94 Gladiator (2000)
Ridley Scott’s deluxe Roman blockbuster is toga soap turned up to the absolute maximus. Russell Crowe bellows and glowers opposite hyper-evil Joaquin Phoenix and lugubrious Oliver Reed (who died during production). Yet there are many grace notes under the fire and fury. CS
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93 You, the Living (2007) The second in Roy Andersson’s trilogy of wackily incisive Swedish vignettes comes at you thick and fast – about 50 micro-sketches, sometimes loosely linked – yet sticks like plasticine beneath your fingernails. CS
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92 The Hurt Locker (2008) Kathryn Bigelow’s extraordinary story of a controlled explosions team – headed by a never-better Jeremy Renner – is intense, immersive and impossible to shake. CS
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91 Etre et Avoir (2002) Events soured after the shoot but Nicolas Philbert’s sole big hit remains a disarmingly funny study of a graceful and kind schoolteacher caring for a motley crew of under-11s in rural France. CS
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Facebook Twitter Pinterest Vital tale of love and clubbing … Eden. Photograph: Barbara Kinney/Eden 90 Eden (2012)
Even non-ravers can’t fail to be shaken by Mia Hansen-Løve’s vital tale of love and clubbing, vaguely based on the rise of Daft Punk . Giddy yet gripping. CS
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89 The Selfish Giant (2013) Clio Barnard’s second feature doesn’t have the shock of innovation of her verbatim cinema debut, The Arbor, but the story of two lads scrapping around junkyards to escape their homes is a masterpiece of lyrical social realism. CS
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Facebook Twitter Pinterest Savagery … Gomorrah. Photograph: Mario Spada/AP 88 Gomorrah (2008)
Director Matteo Garrone announced himself big-time with this blazing screen treatment of Roberto Saviano’s fearless account of the contemporary activities of Neapolitan mobsters: a thoroughly chilling chronicle of corruption and savagery rendered in tremendous style. AP
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87 The Wind that Shakes the Barley (2006) When Ken Loach won the Palme d’Or at Cannes for his film about the Irish rebellion against British rule, the tabloids went on the attack (Daily Mail: “Why Does Ken Loach hate his country so much?”). None of them had actually seen the film, a powerful, compassionate drama starring Cillian Murphy and Padraic Delaney as Republican brothers split by the civil war that followed independence in 1922. CC
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Facebook Twitter Pinterest Javier Barden in No Country for Old Men. Photograph: Allstar/Paramount Vantage 86 No Country for Old Men (2007)
The Coens’ Cormac McCarthy adaptation is a scorching study of benevolence and evil with rich and weathered turns from Tommy Lee Jones, Josh Brolin and a glossily horrible one from Javier Bardem. CS
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85 Burning (2018) One of the recent stream of fine dramas issuing from South Korea, Lee Chang-dong’s adaptation of a Haruki Murakami story is an elusive, unsettling thriller, in which a young writer reconnects with a former schoolfriend, only to find she mysteriously disappears after a trip away. AP
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Facebook Twitter Pinterest Hypnotic experimentalism … Tropical Malady. Photograph: Kick the Machine Films 84 Tropical Malady (2005)
A young solider and a feral boy fall in love, dance to the Clash then trek to the jungle searching for a shaman dressed up as a tiger. Thai master Apichatpong Weerasethakul ’s hypnotic experimentalism has never been bettered; sorry, Uncle Boonmee. CS
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83 The Son’s Room (2001) Nanni Moretti’s Palme d’Or-winning drama about a father crippled by grief after the accidental death of his child is not for the faint-hearted. Yet the Italian writer/director/star performs miracles making a movie so wrenching also so hopeful. CS
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82 Stories We Tell (2012) Sarah Polley followed Away From Her and Take This Waltz by turning the camera on her own family secrets in this tricksy and compassionate documentary uncovering the true identity of her father. CS
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Facebook Twitter Pinterest Upset … Katie Jarvis and Michael Fassbender in Fish Tank. Photograph: Holly Horner 81 Fish Tank (2009)
Andrea Arnold ’s bad mum high-rise dance tragedy is singular, sensuous and alive with everyday upset. Actor Katie Jarvis took six years off after shooting; roughly the same as audiences needed to recover from the shake it gave, and the sight of Michael Fassbender. CS
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80 Requiem for a Dream (2000) Hubert Selby Jr’s lacerating novel that lasers in on the exhilaration and tragedy of addiction is given expansive, stylish treatment by the then-emerging director Darren Aronofsky. Incredibly glamorous and miserably heartbreaking, this film gave notice of Aronofsky’s brilliance. AP
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Facebook Twitter Pinterest It still looks unique … Persepolis. Photograph: Marjane Satrapi/Vincent Paron 79 Persepolis (2007)
Iranian-French director Marjane Satrapi adapted her own graphic novel in this animated fantasy-memoir about a 10-year-old girl growing up in Tehran after the 1979 revolution. A real original, and it still looks unique. AP
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78 Ocean’s Eleven (2001) Steven Soderbergh is the Renaissance man of American cinema, and this intricately crafted heist movie – remade from the old Frank Sinatra chestnut – shows him on never-bettered, commercially minded form. George Clooney is at his most Cary Grant-ish as the leader of the crack team of robbers. AP
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Facebook Twitter Pinterest Wildly romantic … Lost in Translation. Photograph: The Ronald Grant Archive 77 Lost in Translation (2003)
Sofia Coppola’s second feature stands up: utterly distinctive, wildly romantic and fleetingly queasy. Scarlett Johannson and Bill Murray are impeccable casting as the unlikely soulmates thrown together in high-rise Tokyo. CS
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76 Ten (2002) Iranian director Abbas Kiarostami had already proved himself a master in the late 20th century; this simple but effective piece – featuring a woman driving different people around Tehran – proved he could do it in the 21st. Kiarostami and his star Mania Akbari conjure knotty drama out of a series of conversations about marriage, family, religion and sex. AP
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75 Philomena (2013) Stephen Frears brings tonal tact and unobtrusive genius to this wonderfully funny and touching real-life tale of an Irish natterer (Judi Dench) and cynical reporter (Steve Coogan) who demolish red tape and challenge evil nuns to try to find her long-lost son. CS
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Facebook Twitter Pinterest Supercharged … Tahar Rahim in A Prophet. 74 A Prophet (2009)
French film-maker Jacques Audiard ’s blistering arthouse prison thriller begins with a 19-year-old rookie prisoner ( Tahar Rahim ) being made an offer he can’t refuse by the mob: execute a police informant or be killed. The murder, a brutal struggle with a razor blade in a six by eight cell, is unforgettable. It’s the start of the kid’s prison education. A film supercharged with edginess. CC
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73 Love & Friendship (2016) Whit Stillman, Kate Beckinsale and Chloë Sevigny reunite 20 years after The Last Days of Disco for the most blindingly funny – and faithful – Jane Austen adaptation yet. Spun from her first novel, Lady Susan , this is the tale of an epically bitchy and ambitious widow upending her nearest and dearest. Beckinsale has never been better; Tom Bennett steals the show as the fantastically dim lord lined up for her daughter. CS
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72 Waltz With Bashir (2008) Israeli soldier-turned-film-maker Ari Folman’s film is a kind of animated companion to Apocalypse Now, a hallucinatory statement about the trauma of conflict and the madness of war. It’s an autobiographical documentary, Folman interviewing the men he fought alongside, aged 19, in the first Lebanon war of 1982. He has repressed his memories of the time. The film’s climax is the massacre of Palestinian refugees by Christian Phalangists at the Sabra and Shatila refugee camps . CC
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Facebook Twitter Pinterest Juvenile emancipation … Capernaum. Photograph: Lifestyle pictures/Alamy 71 Capernaum (2018)
A sprawling drama that functions both as an excoriating treatise on the nature of poverty in Lebanon, and an idiosyncratic drama in which a child takes his parents to court for their ill-treatment of him. We tend to think of the latter type of juvenile emancipation as the province of overprivileged westerners, but director Nadine Labaki makes it work in the toughest of social circumstances: a 12-year-old, living in the Beirut slums, takes steps to deal with his parents’ neglect. A highly original and affecting film. AP
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70 Anchorman: The Legend of Ron Burgundy (2004) Such was the glut of Judd Apatow-ish comedies to come our way about 10 years ago that it’s easy to forget what a gem this is; how deep and weird the performances (stand up, Steve Carell), how fast the laughs and rich the detail. CS
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Facebook Twitter Pinterest Inventive … Paddington 2 69 Paddington 2 (2017)
Hugh Grant recently called this the best film in which he’s ever been involved – and he might well be right. Paul King did the unthinkable and made a sequel to his insta-classic yet more charming, inventive and across-the-generations entertaining. CS
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68 Mr Turner (2014) Passed over by the British and American film academies – though Timothy Spall’s glorious grunting lead was rightly recognised by Cannes – Mike Leigh’s painter biopic is meticulous, moving and still underappreciated. CS
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67 Dogtooth (2009) Yorgos Lanthimos ’s debut film was the only one, in the end, to make our list; its tonal idiosyncrasy and battily unsettling story and performances just edging out Alps, The Lobster and The Favourite. CS
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Facebook Twitter Pinterest Doomed … Jake Gyllenhaal, left, and Heath Ledger in Brokeback Mountain. Photograph: Kimberly French/AP 66 Brokeback Mountain (2005)
Ang Lee’s romance missed out to Crash for the best picture Oscar, but its legacy as a five-hankie ode to doomed romance lives on. Jake Gyllenhaal and Heath Ledger star as the farmhands whose love survives marriages, years of separation – and even death. CS
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65 Happy as Lazzaro (2018) A beautiful, strange dream of a film, Italian director Alice Rohrwacher ’s drama looks at first as if it’s set sometime in the dim and distant, a portrait of villagers exploited by feudal oppression. But no, there’s a mobile phone. OK, a flip-phone, but this is modern rural Italy. Well, the first half, anyway. After that, it’s complicated, with a flight into magic realism or perhaps even reincarnation. CC
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64 The Incredibles (2004) The Guardian’s Peter Bradshaw ranked The Incredibles as Pixar’s best ever film, the jewel in the crown. And only Pixar could make a superhero movie for kids about a midlife crisis. Mr Incredible is living in suburbia with his family after one lawsuit too many. Edna Mode, fashion designer to the supers, is an utter delight: “This is a hobo suit, dahlin, you can’t be seen in it!” CC
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Facebook Twitter Pinterest Impossible to turn away … Ezra Miller and Tilda Swinton in We Need to Talk About Kevin. Photograph: Cannes Film Festival/EPA 63 We Need to Talk about Kevin (2011)
Lynne Ramsay didn’t soften the blows adapting Lionel Shriver’s bestselling novel about a Columbine-style high-school massacre. A what-if feminist parable, this is a movie that thinks the unthinkable: what if a mother doesn’t like her child, or even love him? And the casting is killer, with Ezra Miller as Kevin and Tilda Swinton playing the mother. It’s a bruising watch, but Ramsay makes it’s impossible to turn away. CC
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62 Waiting for Happiness (2002) Malian director Abderrahmane Sissako won a lot of admirers for this slow-burn study of life in a west African town. Returning to Mauritania, his country of birth, Sissako puts together a string of vignettes and encounters, linked together by a returning, westernised student who can barely remember the local language. AP
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61 The Souvenir (2019) Joanna Hogg’s belated international breakthrough is a story of extraordinary specificity – young Hogg has disastrous affair while living in Knightsbridge and studying as a film student in the early 1980s – with rare cut-through and relatability. Honor Swinton Byrne is astonishing in her first film; Tom Burke inch-perfect as the charming but parasitic older man. CS
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Facebook Twitter Pinterest Foul-mouthed … Ted. Photograph: Allstar/Universal Pictures 60 Ted (2012)
Seth MacFarlane ’s brief ascent to the Hollywood firmament was down to this scabrously funny talking-bear farce, which helped reinvent the grossout comedy. Mark Wahlberg is great as the straight man to the foul-mouthed toy of the title, with Mila Kunis as his censorious fiancee. MacFarlane’s creation was simultaneously endearing and outrageous. AP
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59 Gangs of Wasseypur (2012) Mammoth two-part Indian crime film that’s a long, long way from Bollywood. Directed by Anurag Kashyap, this is conceived on a giant scale, as generations of three gangster families fight for supremacy over the course of half a century. Stylish, visceral film-making, violent and hard-hitting, it’s got a valid claim to be India’s answer to The Godfather. AP
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58 Wuthering Heights (2011) Andrea Arnold tossed out the costume drama rulebook with her raw, passionate retelling of Emily Brontë’s novel. I’d argue the case for Wuthering Heights as one of the most criminally underrated movies of recent years – though it’s been influential, blazing a trail for stripped-back period movies such as Lady Macbeth . Arnold was an early adopter of inclusive casting, too, giving the role of Heathcliff to black actor James Howson. CC
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57 Leave No Trace (2018) It took Winter’s Bone’s Debra Granik eight years to get this off the ground, but was worth the wait: a brilliantly moving eco drama with Ben Foster and Thomasin McKenzie as a father and daughter living off grid in an Oregon forest, but whose relationship and priorities are changed as the child begins the transition to adulthood. AP
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56 Behind the Candelabra (2013) Relegated to telly in the US, Steven Soderbergh’s wondrously funny and lavish Liberace biopic had a cinema release in the UK. Michael Douglas cast vanity aside and caution to the wind with virtuosic results as the promiscuous ivories-tickler; Matt Damon was terrific against type as his lover, Rob Lowe pinched and uproarious as their much-employed cosmetic surgeon. CS
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Facebook Twitter Pinterest An elaborately choreographed procession of tableaux and set pieces … Russian Ark. Photograph: Allstar/Cinetext/ARTIFICIAL EYE 55 Russian Ark (2002)
Groundbreaking single-shot paean to the Winter Palace in St Petersburg from Russian director Alexander Sokurov. Exploiting then newly developed video technology, Sokurov crafted an elaborately choreographed procession of tableaux and set pieces that explored three centuries of Russian history and culture, from the imperial era to the wartime siege of Leningrad. AP
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54 The Social Network (2010) After the Cambridge Analytica revelations , the treachery and backstabbery in Aaron Sorkin and David Fincher’s Facebook origin tale looks positively quaint – the geeks and nerds fighting over who had the idea for Facebook first. Nevertheless, this is still an outrageously watchable hatchet job. Jesse Eisenberg is a knockout Mark Zuckerberg, the smartest guy in the room (though not sartorially, in flip-flops and a hoodie). CC
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Facebook Twitter Pinterest A documentary with evocative lyricism… Fire at Sea. Photograph: PR 53 Fire at Sea (2016)
A beautifully shot observational documentary about the continuing humanitarian crisis in the Mediterranean: the lethally dangerous boats that carry refugees from Africa and end up on the Italian island of Lampedusa . Shot by director Gianfranco Rosi with an evocative lyricism that sits in counterpoint to the blazing anger at the film’s heart. AP
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52 Amores Perros (2000) A film that grabs you by the neck and shakes hard, this brutal crime drama announced the Mexican director Alejandro González Iñárritu as a major new talent in 2000. (And lumbered him for the while with the label “Mexico’s Tarantino”.) A film of mayhem and fury, three stories intersect around a car crash in which one of the passengers is a champion fighting dog. CC
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Facebook Twitter Pinterest Flying fighters … Crouching Tiger, Hidden Dragon. Photograph: Columbia TriStar Films 51 Crouching Tiger, Hidden Dragon (2000)
Western audiences unfamiliar with the wuxia martial arts genre had never seen anything like Ang Lee’s dazzling 18th-century-set epic in 2000 – fighters flying through the air with balletic grace. In the most exhilarating scene, the daughter of a regional governor (Ziyi Zhang) goes sword-to-sword with a famous warrior (Chow Yun-fat) in the branches of bamboo trees swaying high above the ground. CC
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50 Before Sunset (2004) In Richard Linklater’s gorgeous, romantic Before Sunrise , a pair of twentysomethings (Julie Delpy and Ethan Hawke) spent the day together in Vienna. Here in the second movie when they meet again in Paris for another brief encounter, they are in their 30s. So the questions are for grownups. Am I with the right person? Where did my life go? It also has the best line ever about being in a couple with small children: “I feel like I’m running a small nursery with someone I used to date.” CC
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49 24 Hour Party People (2002) Michael Winterbottom and Steve Coogan’s truth-tickling hit a high note with this joyful sorta-biopic of the record label boss and broadcaster Tony Wilson. Playful, ingenious and prodigiously informative, it’s a triumph of vision over verite. It’s also a total blast. CS
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Facebook Twitter Pinterest Elegant … Gillian Anderson and Eric Stoltz in The House of Mirth. Photograph: Cine Text/Sportsphoto/Allstar 48 The House of Mirth (2000)
Terence Davies utilised Gillian Anderson’s poised elegance to good advantage in this brilliantly controlled adaptation of the Edith Wharton novel. Anderson plays Lily Bart, the woman whose reputation and standing are gradually sullied until she becomes an unmarriageable outcast in end of 19th-century America. AP
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47 Margaret (2011) Here’s another dark American tale from Manchester by the Sea writer-director Kenneth Lonergan (Margaret was completed in 2007 but only released in 2011 after a wrangle with the studio). Set in post 9/11 New York, Anna Paquin is an overentitled teenager partly responsible for a tragic accident. As in Manchester by the Sea, the effect is shattering; it is like watching actual lives fall apart. CC
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Facebook Twitter Pinterest Murder, family strife and supernatural shenanigans … Volver. Photograph: United Archives GmbH/Alamy Stock Photo 46 Volver (2006)
Arguably Penélope Cruz’s finest performance, in one of Pedro Almodóvar’s key films: a heady stew of murder, family strife and supernatural shenanigans. Cruz plays a woman forced to kill and bury her ex-husband, while her dead mother appears to be haunting her hairdresser sister. All is resolved after various traumas are lanced. AP
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45 13th (2016) Intense, anger-driven documentary from Ava DuVernay on the racialisation of the US’s justice system, positing the idea that the massively disproportionate incarceration of African-American men is simply slavery by another name. With a title referring to the constitutional amendment that abolished slavery, DuVernay suggests that privatised prisons, cheap labour and light-touch capitalism are all in it together. Tough stuff. AP
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44 Toni Erdmann (2016) A knockout blow for the lazy, patronising stereotype that Germans don’t have a sense of humour, Maren Ade’s Toni Erdmann is one of the funniest films to hit (arthouse) cinemas in years. It’s the story of a workaholic management consultant (Sandra Hüller) whose embarrassing dad turns up unannounced for the weekend wearing joke-shop false teeth. A genuine one-off, the film is partly a satire on Europe, globalisation and workplace misogyny, as well as being a prickly sweet father-daughter movie. CC
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Facebook Twitter Pinterest Outrageously entertaining … The Wolf of Wall Street. Photograph: Paramount/Everett/Rex 43 The Wolf of Wall Street (2013)
Possibly the most fun anyone’s had at the cinema so far this century, Martin Scorsese’s The Wolf of Wall Street ought to be a cautionary tale. It’s based on the autobiography of crooked stockbroker Jordan Belfort, convicted in 1999 for fraud and money-laundering. But why focus on regret, when there are hookers, drugs and fast cars? Leonardo DiCaprio is outrageously entertaining as Belfort. CC
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42 4 Months, 3 Weeks and 2 Days (2007) The film whose Palme d’Or win heralded the arrival of a new wave of Romanian cinema. A young woman, helped by her friend, arranges an illegal abortion in the late 80s; the squalid events that follow parallel the decay and chaos of the country as the communist dictatorship began to collapse. Harrowing but clear-sighted. AP
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Facebook Twitter Pinterest The most twisted of love stories … The Handmaiden. Photograph: Allstar/Amazon Studios 41 The Handmaiden (2016)
The Handmaiden is one of cinema’s great literary adaptations: Park Chan-wook transposes Sarah Waters’s crime novel Fingersmith from Victorian London to Korea in the 1930s. In this most twisted of love stories, a pickpocket (Kim Tae-ri) poses as maid to a wealthy heiress (Kim Min-hee). But who is the double-crosser? Depending on your tastes, a candidate for sexiest film of the century. CC
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40 Unrelated (2007) At the age of 47, after a career directing TV soaps such as Casualty and EastEnders, Joanna Hogg reinvented herself as auteur of a new breed of cinema. In her feature debut, Unrelated, Kathryn Worth played a fortyish woman holidaying in Tuscany with two dysfunctional families and flirting outrageously with one of the lads (Tom Hiddleston in his first movie). A cinema of awkwardness, wielding a scalpel on the well-to-do middle classes, was born. CC
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39 Meek’s Cutoff (2010) Kelly Reichardt is a master of slow cinema, the maker of films about American outsiders, living without a safety net. Meek’s Cutoff is a western set in 1840s Oregon, following three families on the wagon train west. Their leader is show-offy Stephen Meek (Bruce Greenwood), but Reichardt’s focus, as is customary for her, is on the women – a trio played by Michelle Williams, Zoe Kazan and Shirley Henderson. CC
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Facebook Twitter Pinterest Despair … Yilmaz Erdogan and Firat Tanis in Once Upon A Time in Anatolia. Photograph: Everett Collection/Rex 38 Once Upon a Time in Anatolia (2011)
Here’s a police procedural with a difference by the Turkish director Nuri Bilge Ceylan – the whodunnit and why playing second fiddle to long stretches of silence. It’s set in rural Turkey where officials are spending the night driving a murder suspect around looking for a body. What they find, however, is mostly existential despair. Not exactly easy viewing, but it’s a masterpiece of slow cinema. CC
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37 Dogville (2003) Lars von Trier’ s Brechtian parable about coercive capitalism remains arguably the Danish provocateur’s masterpiece. Nicole Kidman and Paul Bettany both excel in this study of a woman on the run from gangsters who is offered shelter in a small town in return for undertaking chores. Von Trier’s use of stylised, floor-painted sets is the inspired final touch. AP
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36 A Separation (2011) The film begins with a couple in front of a judge asking for a divorce. She wants to leave Iran and take their daughter. He cannot go; his elderly father is sick. Everyone behaves badly in Asghar Farhadi ’s desperately painful family drama. Farhadi’s superpower is empathy, making the audience see all points of view. He lays depth charges in seemingly inconsequential moments with emotionally thrilling consequences. CC
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Facebook Twitter Pinterest Sensational … Tom Courtenay and Charlotte Rampling in 45 Years. Photograph: Agatha A. Nitecka/AP 35 45 Years (2015)
British director Andrew Haigh’s quietly devastating drama is a deeply moving portrait of marriage with the shiver of a ghost story. Charlotte Rampling and Tom Courtenay play a Norfolk couple planning their 45th wedding anniversary. A week before the party, a letter lands on their doormat like a hand grenade with news of his early lost love. Rampling is sensational. CC
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34 The Child (L’Enfant , 2002) The Dardenne brothers’ second Palme d’Or was bestowed on this stark portrait of underclass desperation, filmed in their characteristic hyper-naturalist manner. Jérémie Renier plays a petty criminal who sells his newborn baby in the adoption black market, but his devastated girlfriend’s response forces a kind of redemption. AP
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33 The Royal Tenenbaums (2001) Probably most Wes Anderson-y of Wes Anderson’s films and certainly his finest, with a to-die-for cast and the best fur coat in the history of cinema. Gwyneth Paltrow, Luke Wilson and Ben Stiller are the Tenenbaum siblings, all former child prodigies. The brilliance has faded. Who’s to blame? Enter paterfamilias Royal Tenenbaum (Gene Hackman), a man who consoles his grieving grandsons with: “I’m sorry for your loss. Your mother was a terribly attractive woman.” CC
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32 Gravity (2013) If we are living through a golden age of space movies, here’s where it started, Alfonso Cuarón’s spectacular thriller, shot with unbearable tension and Discovery Channel realism. Sandra Bullock is the rookie astronaut with George Clooney by her side, a living, breathing Buzz Lightyear. When a storm of debris hits the pair, a terrifying fight for survival ensues. Astoundingly good. CC
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Facebook Twitter Pinterest Stop-motion breakdown … Anomalisa. Photograph: Picasa/Paramount 31 Anomalisa (2015)
Charlie Kaufman’s existential breakdown with stop-motion puppets is a miniature masterpiece of concept and execution. David Thewlis voices the depressed motivational speaker to whom everyone sounds the same – except for Jennifer Jason Leigh ’s scarred sales agent. “What is it to be human?” asks Michael Stone (Thewlis). “To ache?” Few films try to answer: this Fabergé egg of a film does. CS
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30 Leviathan (2014) The hardest-drinking movie on our list – with some stiff competition – Andrey Zvyagintsev ’s anti-Putin polemic is brilliant, ballsy and completely sozzled. Our hero fisherman is forced from his home so that the corrupt local mayor can build his own palace on the site. A priest speaks of “reawakening the soul of the Russian people” as their spirits lie crushed at his feet. Corruption is so endemic, these people have even lost God. This is the most almighty achievement. CS
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29 Nebraska (2013) Bruce Dern discards his marbles on a windmill-tilting road trip with loving but frustrated son Will Forte. Alexander Payne’s black-and-white ode to small-town America is his best this century ( Sideways has not aged like a fine wine). It also features June Squibb being completely filthy. CS
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28 The Tree of Life (2011) Terrence Malick’s return to cinema six years after The New World has been vaguely tainted by the slew of woozy filmic xeroxes that have followed, but his first comeback – in which Brad Pitt and Jessica Chastain stand in for his parents in 1950s Texas – is a choking knockout. CS
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Facebook Twitter Pinterest Glorious camp … The Grand Budapest Hotel. Photograph: Fox Searchlight Pictures/Sportsphoto/Allstar 27 The Grand Budapest Hotel (2014)
Just nudging Gene Hackman’s Tenenbaum clan down the list, Wes Anderson’s glorious 1930s confection is a delight with a hard nugget of politics at its core. Ralph Fiennes’s central turn as charming concierge M Gustave, all beneficent sex and abashed camp, remains the man’s finest hour. CS
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26 A One and a Two (Yi Yi , 2000) Edward Yang ’s final film is a delicate domestic miracle: the story of one family seen through the perspectives of the father, the son and the daughter. A wedding begins proceedings, a funeral ends them. The stuff in the middle is the everyday, dissected with rare beauty and gravity. CS
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Facebook Twitter Pinterest Chilling … Daniel Kaluuya in Get Out. Photograph: Universal Pictures 25 Get Out (2017)
Jordan Peele ’s debut is a perfect, hard-polished gem of a film. A race satire that skewers beautifully, it’s also a chilling comedy, a proper horror and 104 minutes of complete entertainment. CS Read the review
24 Ida (2013) Brief as a dream and just as devastating, Paweł Pawlikowski ’s black and white story of a novice nun on a road trip with her aunt in 1962 Poland to discover the fate of her Jewish parents is spare, shocking and utterly unforgettable. CS
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23 Borat: Cultural Learnings of America for Make Benefit Glorious Nation of Kazakhstan (2006) Still Sacha Baron Cohen’s finest moment, a feature-length upscaling of his ludicrously hilarious TV character, whose purpose is to sucker the unsuspecting into condemning themselves out of their own mouths. Borat is on a trip in the US to try to marry Pamela Anderson; not everything works, but when it does it’s astounding: cruelly revelatory and hysterically funny at the same time. AP
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Facebook Twitter Pinterest Mysterious … Spirited Away. Photograph: Disney/Everett/Rex Features 22 Spirited Away (2001)
Hayao Miyazaki ’s wondrous animation, the greatest success of a spectacular run from Japan’s Studio Ghibli. A gentle, mysterious fable about a 10-year-old girl whose family stumble upon a haunted bathhouse After her parents are turned into pigs, she works to lift the curse, encountering a variety of spirit-world beings along the way. AP
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21 The White Ribbon (2009) Michael Haneke won his first Palme d’Or with this chilling, steel-hard parable set in Germany just before the first world war. The inhabitants of a small village are dogged by mysterious, violent incidents that serve mostly to exacerbate the dysfunctional social codes they all live by – and elliptically suggests the moral climate that evolved into Nazism. AP
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Facebook Twitter Pinterest Lyrical … Roma. Photograph: Carlos Somonte/AP 20 Roma (2018)
Despite lingering controversy over its adoption by Netflix in its war with the film industry, Roma still stands as an absolutely major work. Mexican auteur Alfonso Cuarón returned to the Mexico City of his childhood, telling the story of a middle-class family and their nanny-cum-maid in swooning, lyrical black and white. Part memoir, part elegaic fiction, Cuaron hit the heights with this. AP
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19 Lincoln (2012) Steven Spielberg’s portrait of the great US president looked at the time like a history lesson come to life: graced by a monumental, Oscar-winning performance by Daniel Day-Lewis , it detailed the arm-twisting and chicanery behind the passing of his slave-freeing constitutional amendment. These days, it looks like a fantasy: a president with principles: who’d have thought? AP
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18 A Serious Man (2009) The Coen brothers don’t really do personal, but this is as close as they’ve got (so far). Set in their home town of Minneapolis in the late 60s, A Serious Man stars Michael Stuhlbarg as an academic whose life is roiled by continuing uncertainty and self-doubt – triggering repeat visits to his rabbis, a marriage breakdown and extended interactions with his oddball brother. AP
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17 The Great Beauty (2013) The Young Pope director Paolo Sorrentino crafted this swooningly beautiful love letter to Rome – “la grande bellezza” – in its decadent, jaded glory. Toni Servillo, Sorrentino’s regular onscreen foil, plays journalist Jep Gambardella, a bon viveur beginning to sense the dying of his personal light, and hunting out meaning and substance in the world around him. AP
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Facebook Twitter Pinterest Almost unwatchable … An Act Of Killing. Photograph: Real/Novaya Ze/Rex/Shutterstock 16 The Act of Killing (2012)
An extraordinary documentary about the wave of barbaric killings that swept Indonesia in the mid-60s. Orchestrated by director Joshua Oppenheimer, this film revisits the perpetrators of some horrific events and asks them – with little need for encouragement – to re-enact them. The result is almost unwatchable: the murderers’ glee at performing, and the remorse they may or may not experience as a result. AP
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15 Shoplifters (2018) Japanese director Hirokazu Kore-eda won the Palme d’Or for this exquisitely turned drama that – like much of Kore-eda’s previous output – explores what it is to be a family in entirely non-conventional circumstances. A shoplifting gang take in a young girl who seems abandoned; how they hang together – or not – is the film’s key theme. AP
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14 White Material (2009) One of Isabelle Huppert ’s finest performances, and that’s saying something. Director Claire Denis drew on her own upbringing in colonial west Africa to give this study of a hard-as-nails plantation owner a pungent whiff of authenticity. Huppert is Maria, obsessed with getting in the coffee harvest as a violent civil conflict moves ever closer. Saddled with an untrustworthy husband and an erratic son, it’s all she can do to survive. AP
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Facebook Twitter Pinterest Acute … Dennis Quaid and Julianne Moore in Far From Heaven. Photograph: Allstar/Cinetext/USA Films 13 Far From Heaven (2002)
From director Todd Haynes, this is pastiche at its most brilliantly acute. Haynes takes the bold, vivid melodrama beloved of Douglas Sirk, and reconfigures it to fully reveal the social faultlines of race, sex and class that were considerably more latent in the original. A beautifully crafted act of ancestor worship. AP
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12 Son of Saul (2015) Brutally visceral fable that plunges the viewer headlong into the all-encompassing horror of a Nazi extermination camp. Shot in remorseless, unforgiving close-up by first time Hungarian director László Nemes, the story of a Jewish prison-camp worker whose job it is to help clear the gas chamber of corpses is cinema at its absolute rawest. AP
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11 Mulholland Drive (2001) A magisterial achievement from David Lynch, despite the difficulties he had getting it off the ground. Originally conceived as the pilot of a new TV series, this expertly fuses Lynch’s softcore pulp obsessions with his trademark creepy surrealism. Naomi Watts was the big discovery here: she plays a wannabe actor who midway seems to switch personalities with another, more jaded one. AP
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Facebook Twitter Pinterest Fearless … Team America: World Police. Photograph: Melinda Sue Gordon/AP 10 Team America: World Police (2004)
The most audacious slaughter of sacred cows seen on celluloid, Matt Stone and Trey Parker’s marionette action-musical is a gleeful hail of precision-aimed bullets. It’s totally fearless: pops are taken at Hollywood, Broadway, evil dictators, gung-ho superpowers, the intelligence service, bleeding heart liberals, actors – especially actors – before signing off with a devastating, if obscene, defence of US interventionism. Politically, it’s scattergun; satirically, it’s spot-on. Mostly, though, it’s just ferociously funny, even if most of the humour does, finally, come from the sight of the 2ft puppets tottering around, getting drunk, having wild sex, attempting to walk through doorways and wrestling panthers played by kittens. CS
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9 Zama (2017) An official working for the Spanish crown descends into madness while waiting for a transfer out of his backwater post in 18th-century Paraguay in Lucrecia Martel’s fevered historical drama. Like a disorienting dream, it’s a film of fragments, moments that worm their way into your memory – a slave limping with flayed feet, a llama barging into frame during an uncomfortable meeting. Daniel Giménez Cacho is petty, wretched Zama, clinging to his white man’s sense of importance (and his ill-fitting periwig), a symptom of colonial rot. Martel has been called “the Malick of Latin American cinema” but this feels closer to Herzog. A strange masterpiece. CC
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Facebook Twitter Pinterest Revelatory … Moonlight. Photograph: David Bornfriend/AP 8 Moonlight (2016)
The triumph of Barry Jenkins’s coming-of-age tale over La La Land for the best picture Oscar was extraordinary in all sorts of ways, of which Faye Dunaway’s envelope mixup was maybe the least remarkable. It was the first film with an all-black cast as well as with an LGBTQ theme to scoop the prize – and it must also rank as one of the most visually and tonally ambitious: told in three parts, with three different leads, each showing the stages of repression and internalised loathing in the young life of a Miami man. It’s simply revelatory: innovative, wildly affecting, utterly beautiful. CS
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7 Synecdoche, New York (2008) After a string of brilliant, industry-transforming scripts, Charlie Kaufman made his directorial debut with this complex, convoluted drama, starring Philip Seymour Hoffman as theatre director Caden Cotard, who is swamped by personal crises as he works on his dream project: building an ever-expanding replica of the city streets and buildings inside a giant warehouse, and populating it with lookalikes; the blurred boundary between performance and reality is mirrored in Cotard’s own breakdown, with the title giving the big clue – this is all symbolic. AP
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6 Hidden (2005) Having made his name as one of the pioneers of ordeal arthouse with unflinching chronicles of trauma and cruelty, Austrian auteur Michael Haneke achieved a unlikely popular success with this film that connected with France’s deep well of unease about events of the relatively recent past. Daniel Auteuil plays a successful TV host whose contentment is disturbed by the arrival of mysterious surveillance tapes. This seems to be connected with a young Algerian boy whose parents were apparently killed in the infamous 1961 Paris massacre . Haneke ratchets up the tension with an unerring sense of dread and dismay. AP
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Facebook Twitter Pinterest Beautiful … Maggie Cheung and Tony Leung in In the Mood for Love. Photograph: Miramax/Everett/Rex Features 5 In the Mood for Love (2000)
Has there ever been a more beautiful couple in the history of cinema than Tony Leung and Maggie Cheung in Wong Kar-Wai’s smouldering love story In the Mood for Love? Not that they’re a couple, technically. It’s 1962. Chow (Tony Leung) and Li-zhen (Maggie Cheung) move in next door to each other in a cramped Hong Kong block of flats. His wife is having an affair – with her husband. The cheated-on pair become friends, but vow not to behave badly. Like Brief Encounter, the film aches with the understanding that impossible love makes for a more romantic movie. It’s gorgeously detailed, drenched in sensuality – a scene in which the two squeeze past each other in a narrow alleyway by night has a humid sexiness. CC Read the review
4 Under the Skin (2013) Jonathan Glazer’s first film in nearly a decade (and still his most recent) turned out to be an uncategorisable masterwork. Scarlett Johansson plays an alien in human form, trawling the streets of Glasgow for unsuspecting males to “take home” – in fact, using them as a food source. From its unnerving alien-POV sequences, to the empathetic scene with actor Adam Pearson (who has neurofibromatosis ), to the sheer coldness of the predatory logic of its central figure, Under the Skin achieves a mood and texture unlike anything else before or since. AP Read the review
Facebook Twitter Pinterest Gentle revolution … Ellar Coltrane in Boyhood. Photograph: Allstar/Universal Pictures 3 Boyhood (2014)
Twelve years in the shooting, Richard Linklater ’s story of a child’s life between six and 18 is a vindication of artistic ambition in an age of cinematic snacking. Its downside is to ruin almost every single other film for you – at least all those in which the actors are conspicuously aged up or down. In watching the bonafide progress of Ellar Coltrane – as well as Patricia Arquette and Ethan Hawke as his parents – Boyhood provides its audience with an intimacy and an investment like no other. This is cinema as gentle revolution. CS
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2 12 Years a Slave (2013) Steve McQueen’s real-life story of Solomon Northup, a free man kidnapped and sold into slavery in 19th-century Louisiana, exudes all of the dignity, impatience and artistic fidelity of its director. It is perfectly cast and paced, endlessly surprising, uncompromising and compassionate: a story purely and powefully told, yet full of the extraordinary visual grace notes. It never descends into cliche or even self-pity; it remains a film for adults, uninterested in anything but the truth. To read Northup’s 1853 memoir is to be astonished by the film’s fidelity. CS
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Facebook Twitter Pinterest Stark … Daniel Day-Lewis in There Will Be Blood. Photograph: Allstar/Miramax/Sportsphoto 1 There Will Be Blood (2007)
Paul Thomas Anderson’s strange masterpiece, freely adapted by him from Upton Sinclair’s 1927 novel Oil!, has a dark title that threatens a calamity now visible on the horizon: destruction of the Earth itself. And it is all inscribed in the story of the movie’s leading character, a man with the Bunyanesque name of Daniel Plainview. Daniel Day-Lewis gives perhaps the greatest, certainly the most exotic performance of his career as an oil prospector in the early 20th century, rewarded with colossal wealth that never gives him the smallest pleasure.
The movie perhaps looks even stranger, starker and more unforgiving now than it did in 2007 when it first came out. But from 2016, there has been a raging Plainview in plain sight in the White House: Trump, the eccentric property billionaire and spoilt baby whose cranky tweets are as crazy as Plainview’s deranged “milkshake” pronouncement. What a spectacle Anderson and Day-Lewis create: a portrait of male belligerence and fear, a Tutankhamun of misery, walled up in his own sarcophagus of wealth and prestige. PB
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Hard Math Problems | Hardest Math Problems With Answers

10 of the Toughest Math Problems Ever Solved
They’re guaranteed to make your head spin. Sep 11, 2019 Andrew Daniels
Earlier this week, a math puzzle that had stumped mathematicians for decades was finally solved. It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”: Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.
On the surface, it seems easy. Can you think of the integers for x, y, and z so that x³+y³+z³=8? Sure. One answer is x = 1, y = -1, and z = 2. But what about the integers for x, y, and z so that x³+y³+z³=42?
That turned out to be much harder—as in, no one was able to solve for those integers for 65 years until a supercomputer finally came up with the solution to 42. (For the record: x = -80538738812075974, y = 80435758145817515, and z = 12602123297335631. Obviously.)
That’s the beauty of math: There’s always an answer for everything, even if takes years, decades, or even centuries to find it. So here are nine more brutally difficult math problem s that once seemed impossible until mathematicians found a breakthrough. Advertisement – Continue Reading Below The Poincaré Conjecture Popular Science Monthly Volume 82 [Public domain] Wikimedia Commons
In 2000, the Clay Mathematics Institute, a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered $1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.
Henri Poincaré was a French mathematician who, around the turn of the 20th century, did foundational work in what we now call topology. Here’s the idea: Topologists want mathematical tools for distinguishing abstract shapes. For shapes in 3-D space, like a ball or a donut, it wasn’t very hard to classify them all . In some significant sense, a ball is the simplest of these shapes.
Poincaré then went up to 4-dimensional stuff, and asked an equivalent question. After some revisions and developments, the conjecture took the form of “Every simply-connected, closed 3-manifold is homeomorphic to S^3,” which essentially says “the simplest 4-D shape is the 4-D equivalent of a sphere.”
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A century later, in 2003, a Russian mathematician named Grigori Perelman posted a proof of Poincaré’s conjecture on the modern open math forum arXiv. Perelman’s proof had some small gaps, and drew directly from research by American mathematician Richard Hamilton. It was groundbreaking, yet modest.
After the math world spent a few years verifying the details of Perelman’s work, the awards began . Perelman was offered the million-dollar Millennium Prize, as well as the Fields Medal, often called the Nobel Prize of Math. Perelman rejected both. He said his work was for the benefit of mathematics, not personal gain, and also that Hamilton, who laid the foundations for his proof, was at least as deserving of the prizes. Fermat’s Last Theorem Wikimedia Commons
Pierre de Fermat was a 17th-century French lawyer and mathematician. Math was apparently more of a hobby for Fermat, and so one of history’s greatest math minds communicated many of his theorems through casual correspondence. He made claims without proving them, leaving them to be proven by other mathematicians decades, or even centuries, later. The most challenging of these has become known as Fermat’s Last Theorem.
It’s a simple one to write. There are many trios of integers (x,y,z) that satisfy x²+y²=z². These are known as the Pythagorean Triples, like (3,4,5) and (5,12,13). Now, do any trios (x,y,z) satisfy x³+y³=z³? The answer is no, and that’s Fermat’s Last Theorem.
Fermat famously wrote the Last Theorem by hand in the margin of a textbook, along with the comment that he had a proof, but could not fit it in the margin. For centuries, the math world has been left wondering if Fermat really had a valid proof in mind.
Flash forward 330 years after Fermat’s death to 1995, when British mathematician Sir Andrew Wiles finally cracked one of history’s oldest open problems . For his efforts, Wiles was knighted by Queen Elizabeth II and was awarded a unique honorary plaque in lieu of the Fields Medal, since he was just above the official age cutoff to receive a Fields Medal.
Wiles managed to combine new research in very different branches of math in order to solve Fermat’s classic number theory question. One of these topics, Elliptic Curves, was completely undiscovered in Fermat’s time, leading many to believe Fermat never really had a proof of his Last Theorem. The Classification of Finite Simple Groups
From solving Rubik’s Cube to proving a fact about body-swapping on Futurama , abstract algebra has a wide range of applications. Algebraic groups are sets that follow a few basic properties, like having an “identity element,” which works like adding 0.
Groups can be finite or infinite, and if you want to know what groups of a particular size n look like, it can get very complicated depending on your choice of n .
If n is 2 or 3, there’s only one way that group can look. When n hits 4, there are two possibilities. Naturally, mathematicians wanted a comprehensive list of all possible groups for any given size.
The complete list took decades to finish conclusively, because of the difficulties in being sure that it was indeed complete. It’s one thing to describe what infinitely many groups look like, but it’s even harder to be sure the list covers everything. Arguably the greatest mathematical project of the 20th century, the classification of finite simple groups was orchestrated by Harvard mathematician Daniel Gorenstein, who in 1972 laid out the immensely complicated plan.
By 1985, the work was nearly done, but spanned so many pages and publications that it was unthinkable for one person to peer review. Part by part, the many facets of the proof were eventually checked and the completeness of the classification was confirmed.
By the 1990s, the proof was widely accepted. Subsequent efforts were made to streamline the titanic proof to more manageable levels, and that project is still ongoing today . The Four Color Theorem Inductiveload [CC BY-SA 3.0 (] Wikimedia Commons
This one is as easy to state as it is hard to prove.
Grab any map and four crayons. It’s possible to color each state (or country) on the map, following one rule: No states that share a border get the same color.
The fact that any map can be colored with five colors—the Five Color Theorem—was proven in the 19th century. But getting that down to four took until 1976.
Two mathematicians at the University of Illinois, Urbana-Champaign, Kenneth Appel and Wolfgang Hakan, found a way to reduce the proof to a large, finite number of cases . With computer assistance, they exhaustively checked the nearly 2,000 cases, and ended up with an unprecedented style of proof.
Arguably controversial since it was partially conceived in the mind of a machine, Appel and Hakan’s proof was eventually accepted by most mathematicians. It has since become far more common for proofs to have computer-verified parts, but Appel and Hakan blazed the trail. (The Independence of) The Continuum Hypothesis Wikimedia Commons
In the late 19th century, a German mathematician named Georg Cantor blew everyone’s minds by figuring out that infinities come in different sizes, called cardinalities. He proved the foundational theorems about cardinality, which modern day math majors tend to learn in their Discrete Math classes.
Cantor proved that the set of real numbers is larger than the set of natural numbers, which we write as |ℝ|>|ℕ|. It was easy to establish that the size of the natural numbers, |ℕ|, is the first infinite size; no infinite set is smaller than ℕ.
Now, the real numbers are larger, but are they the second infinite size? This turned out to be a much harder question, known as The Continuum Hypothesis (CH) .
If CH is true, then |ℝ| is the second infinite size, and no infinite sets are smaller than ℝ, yet larger than ℕ. And if CH is false, then there is at least one size in between.
So what’s the answer? This is where things take a turn.
CH has been proven independent, relative to the baseline axioms of math. It can be true, and no logical contradictions follow, but it can also be false, and no logical contradictions will follow.
It’s a weird state of affairs, but not completely uncommon in modern math. You may have heard of the Axiom of Choice, another independent statement. The proof of this outcome spanned decades and, naturally, split into two major parts: the proof that CH is consistent, and the proof that the negation of CH is consistent.
The first half is thanks to Kurt Gödel, the legendary Austro-Hungarian logician. His 1938 mathematical construction, known as Gödel’s Constructible Universe , proved CH compatible with the baseline axioms, and is still a cornerstone of Set Theory classes. The second half was pursued for two more decades until Paul Cohen, a mathematician at Stanford, solved it by inventing an entire method of proof in Model Theory known as “forcing.”
Gödel’s and Cohen’s halves of the proof each take a graduate level of Set Theory to approach, so it’s no wonder this unique story has been esoteric outside mathematical circles. Gödel’s Incompleteness Theorems Alejandro Mallea/From the book A Mathematical Introduction to Logic, by Herbert Enderton. Flickr
Gödel’s work in mathematical logic was totally next-level. On top of proving stuff, Gödel also liked to prove whether or not it was possible to prove stuff . His Incompleteness Theorems are often misunderstood, so here’s a perfect chance to clarify them.
Gödel’s First Incompleteness Theorem says that, in any proof language, there are always unprovable statements. There’s always something that’s true, that you can’t prove true. It’s possible to understand a (non-mathematically rigorous) version of Gödel’s argument, with some careful thinking. So buckle up, here it is: Consider the statement, “This statement cannot be proven true.”
Think through every case to see why this is an example of a true, but unprovable statement. If it’s false, then what it says is false, so then it can be proven true, which is contradictory, so this case is impossible. On the other extreme, if it did have a proof, then that proof would prove it true … making it true that it has no proof, which is contradictory, killing this case. So we’re logically left with the case that the statement is true, but has no proof. Yeah, our heads are spinning, too.
But follow that nearly-but-not-quite-paradoxical trick, and you’ve illustrated that Gödel’s First Incompleteness Theorem holds.
Gödel’s Second Incompleteness Theorem is similarly weird. It says that mathematical “formal systems” can’t prove themselves consistent. A consistent system is one that won’t give you any logical contradictions.
Here’s how you can think of that. Imagine Amanda and Bob each have a set of mathematical axioms—baseline math rules—in mind. If Amanda can use her axioms to prove that Bob’s axiom system is free of contradictions, then it’s impossible for Bob to use his axioms to prove Amanda’s system doesn’t yield contradictions.
So when mathematicians debate the best choices for the essential axioms of mathematics (it’s much more common than you might imagine) it’s crucial to be aware of this phenomenon. The Prime Number Theorem User:Dcoetzee [CC0]
There are plenty of theorems about prime numbers. One of the simplest facts—that there are infinitely many prime numbers—can even be adorably fit into haiku form .
The Prime Number Theorem is more subtle; it describes the distribution of prime numbers along the number line. More precisely, it says that, given a natural number N, the number of primes below N is approximately N/log(N) … with the usual statistical subtleties to the word “approximately” there.
Drawing on mid-19th-century ideas, two mathematicians, Jacques Hadamard and Charles Jean de la Vallée Poussin, independently proved the Prime Number Theorem in 1898. Since then, the proof has been a popular target for rewrites, enjoying many cosmetic revisions and simplifications. But the impact of the theorem has only grown.
The usefulness of the Prime Number Theorem is huge. Modern computer programs that deal with prime numbers rely on it. It’s fundamental to primality testing methods, and all the cryptology that goes with that. Solving Polynomials by Radicals Self [CC BY-SA 3.0 (] Wikimedia Commons
Remember the quadratic formula? Given ax²+bx+c=0, the solution is x=(-b±√(b^2-4ac))/(2a), which may have felt arduous to memorize in high school, but you have to admit is a conveniently closed-form solution.
Now, if we go up to ax³+bx²+cx+d=0, a closed form for “x=” is possible to find, although it’s much bulkier than the quadratic version. It’s also possible, yet ugly, to do this for degree 4 polynomials ax⁴+bx³+cx²+dx+f=0.
The goal of doing this for polynomials of any degree was noted as early as the 15th century. But from degree 5 on, a closed form is not possible. Writing the forms when they’re possible is one thing, but how did mathematicians prove it’s not possible from 5 up?
The world was only starting to comprehend the brilliance of French mathematician Evariste Galois when he died at the age of 20 in 1832. His life included months spent in prison, where he was punished for his political activism, writing ingenious, yet unrefined mathematics to scholars, and it ended in a fatal duel.
Galois’ ideas took decades after his death to be fully understood, but eventually they developed into an entire theory now called Galois Theory . A major theorem in this theory gives exact conditions for when a polynomial can be “solved by radicals,” meaning it has a closed form like the quadratic formula. All polynomials up to degree 4 satisfy these conditions, but starting at degree 5, some don’t, and so there’s no general form for a solution for any degree higher than 4. Trisecting an Angle Self [CC BY-SA 3.0 (] Wikimedia Commons
To finish, let’s go way back in history.
The Ancient Greeks wondered about constructing lines and shapes in various ratios, using the tools of an unmarked compass and straightedge. If someone draws an angle on some paper in front of you, and gives you an unmarked ruler, a basic compass, and a pen, it’s possible for you to draw the line that cuts that angle exactly in half. It’s a quick four steps, nicely illustrated like this , and the Greeks knew it two millennia ago.
What eluded them was cutting an angle in thirds. It stayed elusive for literally 15 centuries, with hundreds of attempts in vain to find a construction. It turns out such a construction is impossible.
Modern math students learn the angle trisection problem—and how to prove it’s not possible—in their Galois Theory classes. But, given the aforementioned period of time it took the math world to process Galois’ work, the first proof of the problem was due to another French mathematician, Pierre Wantzel . He published his work in 1837, 16 years after the death of Galois, but nine years before most of Galois’ work was published.
Either way, their insights are similar, casting the construction question into one about properties of certain representative polynomials. Many other ancient construction questions became approachable with these methods, closing off some of the oldest open math questions in history.
So if you ever time-travel to ancient Greece, you can tell them their attempts at the angle trisection problem are futile. Advertisement – Continue Reading Below

NDT Mounts for K31

Looking to add a NDT mount to my Swiss beauty. I’ve narrowed it down to the Swiss Products off-center mount and the BadAce Tactical center mount . I’ve seen the SP mount discussed and approved on various forums, but no mention of the BadAce; thoughts on it?
Swiss Products PROS NDT mount
Known reliable product with great reviews
Compatible with stripper clips
Works with full sized scopes
CONS Off-center (although this isn’t that big of a deal to me) BadAce Tactical PROS NDT mount
Compatible with stripper clips
Center mounted
CONS Not as many reviews
Only works with scout scopes (full sized scopes would interfere with brass ejection)