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The Quest to Find Rectangles in a Square

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The Quest to Find Rectangles in a Square

Lisanne Taams, a pupil at Radboud College within the Netherlands, is engaged on a Ph.D. about, in her phrases, “computing motives of moduli stacks of vector bundles on stacky curves.”

“It took me two years to even say that correctly,” Ms. Taams mentioned. However, she added, such heights of abstraction solely elevated her delight as she just lately hung out on a extra concrete contemplation: counting the ways in which a sq. may be divided into related rectangles.

She discovered this geometric puzzle on Mathstodon, a group throughout the social community Mastodon. Created within the spring of 2017 by two mathematicians in England, Christian Lawson-Excellent and Colin Wright, registered accounts on Mathstodon totaled about 3,000 in September. Since then, with the Twitter exodus, the quantity has since elevated to round 13,000.

The puzzle was posted in December by John Carlos Baez, a mathematical physicist on the College of California, Riverside.

“There are 3 ways to divide a sq. into three rectangles with the identical proportions!” Dr. Baez wrote.

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He illustrated the reply with three photographs that he borrowed from Wikipedia:

Within the picture on the left, the rectangles are 3 times so long as they’re vast, he defined in an e mail. Within the center picture, the rectangles are one-and-a-half occasions so long as they’re vast. “The third resolution is trickier,” Dr. Baez mentioned. The rectangles are “about 1.75487 occasions so long as they’re vast, although one rectangle is circled so it’s brief and squat,” he added.

Dr. Baez famous that the #1.75487 is of curiosity to mathematicians. “It’s the sq. of the ‘plastic ratio,’” he mentioned, “which is a quantity that has numerous properties much like the extra well-known ‘golden ratio.’”

Having laid that basis, Dr. Baez requested his Mathstodon followers: “What should you chop a sq. into 4 related rectangles? What proportions can they’ve?”

Among the many first to take the bait was Rahul Narain, a pc scientist on the Indian Institute of Expertise, Delhi. “I used to be on Mastodon earlier than it was cool,” he says in his bio (he joined in December 2017). “And the truth that all of the cool individuals are right here now’s a bit unsettling.”

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Dr. Narain sketched out a scientific technique for fixing the puzzle, although he hoped another person would carry it out. As he mentioned in his reply to Dr. Baez, “I actually produce other issues I must work on proper now, I can’t afford to get nerd-sniped any greater than this!”

There turned out to be 11 options — 11 methods a sq. may be divided into 4 equally proportioned rectangles. The options step by step amassed with essential enter from Ian Henderson, an unbiased software program developer within the Bay Space, and Daniel Piker in Bristol, England, who works as a design techniques analyst creating software program for architects at Foster + Companions.

And numerous different folks additionally helped, Dr. Baez mentioned. “That’s why it was enjoyable.”

Ms. Taams discovered 11 options by hand and shortly found that she had made a number of errors. She then determined to let the pc do the work. She wrote software program and generated some photographs. However when she checked the progress on-line, “I noticed different folks already had much more photos,” Ms. Taams mentioned.

Mr. Piker, who enjoys making geometric animations, had drawn all 11 choices:

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The simplicity of the issue is what drew him. “I believed it was sort of cool that there was one thing so easy that apparently hadn’t been checked out earlier than,” Mr. Piker mentioned.

Nonetheless, he added, “The maths shortly went past my understanding.”

He might make sense of a proof posted by Ms. Taams, although it was not one thing that he would have simply produced. She posted an 11-part thread — with technical passages composed with LaTeX, a scientific typesetting language — exhibiting that this humble geometry puzzle is linked to extra critical and formal arithmetic.

In different phrases, she got here up with a proof that the ratio of the lengthy sides to the brief sides are “algebraic numbers,” a significant subject in quantity idea.

“I don’t suppose we’ve gotten wherever close to the underside of this but,” Dr. Baez mentioned. “However it’s a great step.”

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Sarah Hart, a mathematician at Birkbeck, College of London — whose ebook, “As soon as Upon a Prime,” which explores connections between arithmetic and literature, comes out in April — referred to as this rectangulation recreation “superior” and “pretty.” (She had not partaken within the Mathstodon pursuit.)

“What makes an issue pretty?” she mentioned. “That’s a difficult one.” For Dr. Hart, it helps if the issue is simple to explain, and simple to play with — “you will get your palms soiled immediately with easy examples.” And if it “turns into deliciously complicated and difficult.”

Dr. Hart additionally famous that “numerous essentially the most fascinating issues come from recreations like this.”

Ms. Taams discovered her proof computationally, after which contemplated it additional. The computations produced a set of equations, she mentioned. “And you then surprise, ‘Oh, are these all of the equations? Sure or no?’” She satisfied herself the reply was “Sure” by simply three examples. “It’s slightly bit arduous to argue why. For those who stare on the photos, you kind of see it.”

(Extra formally, Ms. Taams proved that when conducting an identical rectangulation of a sq. — that’s, dividing it right into a quantity, N, of comparable rectangles — the ratio is an algebraic variety of diploma at most N.)

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The net dialogue at one level turned to an identical investigation into “squaring the sq.” by William Tutte and his collaborators within the Thirties, which is expounded to electrical circuit idea.

“It seems you’ll be able to consider the peak and width of every sq. as associated to voltage and present in {an electrical} circuit — and utilizing this, yow will discover methods to ‘sq. the sq.’ utilizing electrical circuit idea,” Dr. Baez mentioned by e mail. “One thing like that is additionally true for the rectangle dissection downside, however we haven’t exploited it but.”

David Eppstein, a pc scientist on the College of California, Irvine, commented that the time period “guillotine partition” is the usual option to describe the method of recursively slicing off, vertically or horizontally, rectangular items from a sq.. Jules Hedges, a pc scientist on the College of Strathclyde, Glasgow, instructed that “Mondrian” — after the Dutch summary artist Piet Mondrian — may additionally be a becoming identify for this course of. This prompted Stefano Gogioso, a pc scientist on the College of Oxford, to level out that, “in machine studying, Mondrian tree/forests is used to indicate a sure sort of classification carried out by guillotine cuts.”

Returning to the matter of the 11 four-way rectangulations: This end result was confirmed with two batches of code, one by Dr. Narain, enumerating and testing all of the guillotine partitions, and one by Mr. Henderson utilizing a extra concerned strategy.

They didn’t cease there.

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“The query appeared in my thoughts, ‘Oh, what about 5? What about six?’” Mr. Henderson mentioned.

Each he and Dr. Narain discovered 51 options when it got here to dividing a sq. into 5 related rectangles:

Mr. Henderson discovered 245 doable rectangular proportions that divide a sq. into six related rectangles, and 1,371 choices for seven related rectangles. Initially, he gave up on eight rectangles — he tried, however this system simply saved operating. Ultimately, it ran out of reminiscence.

“I obtained it out of my system,” he mentioned.

However then he circled again and went troubleshooting and realized that there was one thing amiss with the code for eight rectangles. “In any case, with this mounted, it does truly end operating,” Mr. Henderson mentioned in his e mail. “There are (in line with the code, at the least) 8,506 totally different facet ratios for eight rectangles.” He might attempt for 9.

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Judge halts ban on syringe programs as El Dorado County legal battle continues

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Judge halts ban on syringe programs as El Dorado County legal battle continues

El Dorado County cannot enforce its ban on programs that hand out clean syringes as a legal battle continues between the county and the California Department of Public Health, a Superior Court judge has ruled.

Judge Gary S. Slossberg granted a preliminary injunction to prevent El Dorado County from enforcing an ordinance that makes it unlawful to operate syringe programs in its unincorporated areas.

The judge said he was not weighing in on the heated arguments for or against syringe programs, which provide sterile needles to people who use drugs, but whether the Department of Public Health had a “reasonable probability” of prevailing in its argument that the county ordinance clashes with state law.

Friday’s decision does not end the courtroom dispute over whether the ban passed by the El Dorado County Board of Supervisors was preempted by state law, as public health officials have argued, or opposing claims by county officials that the syringe program was improperly approved by the state. Slossberg said Friday that the preliminary injunction is meant to remain in place pending a later trial.

The Department of Public Health filed suit against El Dorado County and its county seat of Placerville this year contending that their bans on syringe programs defied the state health and safety code.

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The state health department first authorized the nonprofit Sierra Harm Reduction Coalition to operate a syringe program in the county four years ago. State officials have long endorsed such programs as a proven way to prevent HIV and hepatitis C from running rampant as people share contaminated syringes.

California law gives the public health agency the power to approve syringe programs anywhere that deadly or disabling infections might spread through used needles, “notwithstanding any other law.”

Local bans on syringe programs have nonetheless sprung up across California as city and county officials argue that handing out free syringes does more harm than good. El Dorado County leaders passed their rule in December, which was followed in February by a similar ordinance in Placerville.

The lawsuit lodged by the California Department of Public Health drew objections from El Dorado County leaders: Earlier this year, Dist. Atty. Vern Pierson called it “madness” and argued that California officials were “seeking to impose the normalization of hardcore drug use.”

In a cross-complaint filed against the Department of Public Health, the county said that the syringe program approved by the state had caused “profound nuisance and public safety impacts,” including a “drastic increase in discarded needles,” and that overdoses had risen since it started.

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The county said in a legal filing that since the ban went into effect, “there has been a reduction of syringe waste, decreased incidents of public nuisance, and a resulting reduction of the burdens on law enforcement.”

It also accused the public health department of failing to follow state requirements when it approved the syringe program.

The judge did not weigh in on the cross-complaint lodged by El Dorado County at the Friday hearing. In a court filing, California officials said studies show that syringe programs provide important resources for needle disposal and play a crucial role in preventing overdoses. They credited the Sierra Harm Reduction Coalition with handing out thousands of boxes of Narcan, a brand of naloxone, a medication that reverses the effects of an opioid overdose.

The Department of Public Health argued in a legal filing that stopping the syringe program would be likely to ramp up HIV and hepatitis C infections among people who use drugs, increasing state costs for their care; lead to more deaths from drug overdoses; and reduce access to options for syringe disposal, among other harmful effects.

Because of the bans, “our most vulnerable, stigmatized, and marginalized community members are actively being denied lifesaving interventions,” Sierra Harm Reduction Coalition interim executive director Shilo Jama said in a court filing.

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Slossberg said that although he was preventing the El Dorado County ordinance from being enforced, the county might have other mechanisms to address nuisance issues that were not addressed by the decision.

Pierson, the district attorney, said in a statement Friday that “we will propose narrowing the ordinance” in response to comments made by the judge.

The California Department of Public Health said in a statement that it was “pleased with the court’s decision that upholds that state’s role in protecting the public’s health while this case proceeds.”

The Friday ruling applies only to the ordinance passed by El Dorado County. Attorney Mona Ebrahimi, who represents the city of Placerville, said a hearing involving the city had been postponed.

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This one thing may derail your shot at healthy aging, scientists say

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This one thing may derail your shot at healthy aging, scientists say

Before you settle in to binge the new season of “The Bear” or watch Team USA go for the gold at the Paris Olympics, think twice about the amount of time you spend on the couch in front of the TV. Your future self may thank you.

A new study by Harvard researchers links the popular pastime of sitting and watching television to the likelihood of reaching one’s senior years in a state of good health: the more time spent doing the former, the lower the odds of achieving the latter.

The problem doesn’t seem to be with sitting in general. After controlling for a variety of risk factors such as diet quality and smoking history, the researchers found no relationship between time spent in a chair at work and the chances of aging well. Ditto for sitting in cars or at home doing something besides watching TV, such as reading, eating meals or paying bills.

Yet for every additional two hours spent in front of the boob tube, a person’s chance of meeting the researchers’ definition of healthy aging declined by 12%, according to their study published this week in JAMA Network Open.

That does not bode well for the United States, where 62% of adults between the ages of 20 and 64 say they watch TV for at least two hours a day, as do 84% of senior citizens.

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The findings are based on data from more than 45,000 women who participated in the Nurses Heath Study. All of them were at least 50 years old and had no major chronic diseases back in 1992, when they answered a slew of questions about their health and what they did all day.

For instance, the nurses were asked how much time they spent standing or walking around at work or at home. They were asked about various types of exercise, including jogging, swimming laps, playing tennis and doing yoga. They were asked if they mowed their own lawns.

And they were asked how many hours they spent doing all kinds of sitting.

A couple watches a movie on TV at their home in Norwalk while sharing a bowl of popcorn.

(Francine Orr / Los Angeles Times)

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You might not be surprised to learn that the most popular type of sitting was sitting while watching television. More than half of the women — 53% — said they watched between six and 20 hours of TV a week. (The median among this group was around 15.4 hours per week.) Another 15% of the women said they watched between 21 and 40 hours of TV each week, and 2% watched even more.

The nurses were tracked for 20 years or until they died, whichever came first. By the end of the study period, 41% of them were still free of 11 major health conditions, including cancer, diabetes, heart failure, chronic obstructive pulmonary disease and multiple sclerosis. In addition, 44% of the nurses were in good mental health, 52% had no memory impairments and 16% had no physical impairments.

Only 8.6% of the women met all four of those criteria, which was what it took to achieve healthy aging.

On the whole, the women who watched more TV tended to be older, were more likely to be smokers or drinkers, consumed more calories and had higher body mass index scores than women who watched less TV. The more devoted TV watchers were also more likely to have high blood pressure and high cholesterol.

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Once the researchers accounted for these and a host of other differences, they found that the women who spent an hour or less each week sitting in front of the TV were the most likely to achieve healthy aging. Compared to them, women who watched TV for two to five hours per week were 9% less likely to be healthy agers; those who watched for six to 20 hours per week were 19% less likely; those who watched for 21 to 40 hours per week were 40% less likely; and those who watched for at least 41 hours a week were 45% less likely.

The researchers also found that replacing TV time with pretty much anything else — including sleep, for women who got no more than seven hours of shut eye per night — would increase their odds of healthy aging. The more vigorous the new activity, the bigger the boost.

Although the actual percentage of women who succeeded in healthy aging was low, the study authors estimated that another 61% of the women could have joined that rarefied group if they had done four things:

  • Spent at least three hours per day engaged in light physical activity at work.
  • Invested at least 30 minutes a day in moderate to vigorous physical activity.
  • Kept their weight in the normal range instead of being overweight or obese.
  • Limited their TV-watching time to less than three hours a day.

The study didn’t show that excess TV time caused any of the nurses to miss out on healthy aging, only that there was a significant inverse correlation between the two. Still, there’s good reason to suspect that their favorite sedentary behavior bore at least some of the responsibility.

Previous studies have linked prolonged sitting — especially while watching television — to a variety of health problems, including diseases like breast cancer, colorectal cancer, type 2 diabetes, cardiovascular disease and early death. (That particular study found that compared to sitting for less than three hours a day, sitting for at least twice that long was associated with a 17% increased risk of premature death for men and a 34% increased risk of premature death for women.)

But the researchers from Harvard’s T.H. Chan School of Public Health have taken things a step further, said Dr. I-Min Lee, an epidemiologist at Brigham & Women’s Hospital in Boston who studies how physical activity can prevent chronic diseases and extend life.

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“This study expands what we know because it looked at ‘healthy aging,’” said Lee, who was not involved in the study. “‘Health’ is not just the absence of disease; it includes dimensions of physical and mental health, function and well-being.”

All of the study subjects were women, but the biological mechanisms are likely to apply to men as well, Lee said. Even so, it would be good to actually test this relationship in men, as well as in people from a wider range of racial and ethnic backgrounds, she said. (The group of women in the original Nurses Health Study was overwhelmingly white.)

The youngest of the Baby Boomers are now turning 60, and the proportion of the U.S. population that’s at least 65 is projected to increase from roughly 17% today to nearly 21% in 2050, according to the U.S. Census Bureau.

“Population aging is an important public health issue,” the study authors wrote, and strategies to promote healthy aging “are urgently needed.”

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Column: The abortion pill is safe. Is your uterus?

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Column: The abortion pill is safe. Is your uterus?

Huge sigh of relief.

In a ruling that I happily admit surprised me, the Supreme Court on Thursday affirmed the obvious: Women should have the right to safe medication abortions.

But ladies, our uteri are not safe yet.

For now, in a unanimous decision, the justices have tossed a case that would have prevented the drug mifepristone from being used by women seeking to end pregnancies.

So mify, as the drug is commonly called, is safe. But this is far from the end of the MAGA war on women.

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Let’s be clear on this: The ruling wasn’t actually about the drug. It was about the folks who brought the suit, a bunch of doctors who really didn’t have much of a reason to keep millions of women from accessing care other than they didn’t like the care those women wanted.

That’s not actually a reason to sue, even by Samuel Alito and Clarence Thomas standards.

So this ruling is about “standing” and the fact that these docs didn’t have it. Already, antiabortion activists are lining up other cases with defendants whose legal footing is much more solid.

And the Supreme Court is hardly the only front in this war for women’s rights. Here’s three other ways the far-right wants to control female bodies:

First, “fetal personhood” has bubbled up as a scary push by the religious right.

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Alito hinted at this concept in the Dobbs ruling, which knocked out Roe vs. Wade, when he referred to an embryo as an “unborn human being.”

In Alabama, we saw this take greater life recently when state Supreme Court judges ruled that embryos created during in vitro fertilization should be considered protected human life (though the state Legislature for now has protected the procedure).

And this week, the Southern Baptist Conference, which speaks for more than 10 million Protestant Americans, announced it would now opposes IVF on those embryos-are-life grounds.

If courts do recognize the idea of fetal personhood, it would pave the way for abortion to be considered not just illegal, but murder. It would also give a state the right to police pregnant women in any way it deems necessary to protect the “unborn child.”

We are already seeing some states attempting to prosecute women for abortions under strict new abortion laws and dozens of states (such as Kansas) either have outright legal language broadly giving fetuses rights or language that edges right up to it. We are closer to this than you think.

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The second front on the war on women is contraception.

Though it seems insane and inane to most of us to forbid women from taking the pill or an emergency medication in the immediate aftermath of intercourse to prevent pregnancy, some folks do want to ban it as a form of abortion.

There is a logic to it. If all abortion is illegal, then anything affecting the embryo after conception is off limits.

Finally, there’s former President Trump.

I’ve written before about the Comstock Act, an obscure and angry old law that many speculated the Supreme Court justices might dredge up in this mifepristone case.

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That law (which is on the books, but not enforced) theoretically makes it illegal to mail anything that could be used in an abortion — so not just the medication. Hard-liners could argue that anything shipped to an abortion clinic to help it operate could be verboten, even latex gloves.

MAGA types are already floating the frightening notion that if Trump were elected, he could simply bypass courts and Congress and order his Department of Justice to enforce the Comstock Act — ending abortion access without technically ending abortion access.

This week, Trump send a recorded message to the Danbury Institute, an ultraconservative organization that has advocated for abortion to be prosecuted as homicide and called it “child sacrifice.”

He didn’t mention abortion, but there’s this:

“These are gonna be your years, because you’re gonna make a comeback like just about no other group,” he said. “I know what’s happening, I know where you’re coming from and where you’re going, and I’ll be with you side by side.”

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So while Thursday’s ruling is a welcome win in the fight to keep women equal, it’s a victorious battle.

The war continues.

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