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Tuesday 25th of September 2018
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One Weird Trick to Keep Female Employees Happy — The Cut

We all know the narrative about why women leave the workforce. By now — decades after “the second shift,” five years after Sheryl Sandberg asked us to lean in, four years after we learned why “Why Women Still Can’t Have It All” — conventional wisdom says that employers lose women in their 30s because those women are starting families, and need more flexibility than the workplace is designed to grant them.

Well, here’s a shocker: A new global study of women in their 30s found they don’t leave jobs because they’re worried about family obligations. They leave because employers won’t pay and promote them. “Surprisingly,” reads the report, “young women identified finding a higher paying job, a lack of learning and development, and a shortage of interesting and meaningful work as the primary reasons why they may leave.”

This is only surprising if you have never spoken to a woman in her 30s. Most women don’t have to be exhorted to care more about work or apply themselves more vigorously. They are all in — no lean about it. The problem is that, all too often, their efforts are not recognized, cultivated, and compensated in the way their male colleagues’ are. This is often spun into a complex issue that some of corporate America’s brightest minds have struggled to solve — the stuff of Supreme Court cases and contentious legislation.

But the reality is that, if you’re an employer, you can retain your female employees longer and keep them happier by taking just three simple steps:

1. PAY WOMEN MORE. Pay us what you pay our male co-workers who do similar jobs. Pay us enough that if you were to accidentally email the entire office a spreadsheet containing everyone’s salary, you wouldn’t be ashamed. Pay us what you know we deserve, even if we haven’t demanded it. Pay us what we’ve earned.

2. PAY WOMEN MORE. Don’t assume we want to become mothers. And if we already are mothers, don’t assume that we’d rather have fewer hours or responsibilities. Assume, in all cases, that we work hard and we want money. More money. As long as we keep showing up and doing the job well, and until we tell you that we need different hours or a new role, just pay us more. And keep paying us.

3. PAY WOMEN MORE. Do it. Now.

So now that you’re paying your female employees on par with the men, let’s take a look at what else you might be able to do to retain women. You could work with them to develop their skills and use their talents in interesting, meaningful ways. If that seems too time-consuming and “managerial” for you, don’t worry. You can default to paying them even more money, and you just might get lucky and have them stick around a while longer. All you have to do is recognize that women’s potential is equal to that of men. Yes, even if the women in question are mothers.

The brilliant thing about this three-step plan is that, even though it’s not explicitly about mothers, it accounts for the opt-out question. If your primary concern is enabling mothers to stay in the workforce, paying women more money solves that problem, too. For privileged hetero couples I know, these days the decision about which partner stays home with the kids is fundamentally financial. If his position pays more, which it usually does, they tend to decide that she’ll be the one to stay home when child care is too expensive. It’s gendered, but less because of roles at home and more because of pay at work. Pay inequity and caregiving obligations are actually not two separate workplace issues facing women: They are one and the same.

The opt-out problem is, in other words, a money problem. Of course biology plays a role — women are more likely to physically need time off after the birth of a child. But if they were fairly compensated at every stage of their careers, they’d be on equal footing with their male partners and co-workers even after becoming parents. Guaranteed paid family leave for all workers and flexible work environments would help, too. But paying women more, from their first job onward, is a game-changer.

In a slightly older study of Harvard Business School graduates, men and women were equally career-driven in their first years in the workforce, and both men and women slowly grew more invested in their lives outside of work as they aged. The only difference was that women tended to rate “opportunities for career growth and development” as slightly more important to them than men did — probably because those opportunities proved more elusive to women. The study also found that only 40 to 50 percent of women were satisfied with their professional accomplishments and opportunities for career growth.

The great thing about paying women more is that it doesn’t just help business-school grads. For years, conversations about working women have centered on those who are college-educated and pursuing prestigious jobs. The great thing about simply paying women — all women — more money is that it benefits those who are usually left out of these debates, too. This innovative strategy works for hourly wage-earners as well as it does for corporate executives, and every woman in-between. The formula is simple. Is she a woman? Are you in charge of paying her? Pay her more.

Pay them as much as men!

Source: One Weird Trick to Keep Female Employees Happy — The Cut

the sierpinski triangle page to end most sierpinski triangle pages

Constructing the Sierpinski triangle
Throughout my years playing around with fractals, the Sierpinski triangle has been a consistent staple. The triangle is named after Wacław Sierpiński and as fractals are wont the pattern appears in many places, so there are many different ways of constructing the triangle on a computer.

All of the methods are fundamentally iterative. The most obvious method is probably the triangle-in-triangle approach. We start with one triangle, and at every step we replace each triangle with 3 sub triangles:

Source: oftenpaper.net/sierpinski.htm

Take It to the Limit – The New York Times

In middle school my friends and I enjoyed chewing on the classic conundrums.   What happens when an irresistible force meets an immovable object?  Easy — they both explode.  Philosophy’s trivial when you’re 13.

But one puzzle bothered us: if you keep moving halfway to the wall, will you ever get there?  Something about this one was deeply frustrating, the thought of getting closer and closer and yet never quite making it.  (There’s probably a metaphor for teenage angst in there somewhere.)  Another concern was the thinly veiled presence of infinity.  To reach the wall you’d need to take an infinite number of steps, and by the end they’d become infinitesimally small.  Whoa.

Questions like this have always caused headaches.  Around 500 B.C., Zeno of Elea posed a set of paradoxes about infinity that puzzled generations of philosophers, and that may have been partly to blame for its banishment from mathematics for centuries to come.  In Euclidean geometry, for example, the only constructions allowed were those that involved a finite number of steps.  The infinite was considered too ineffable, too unfathomable, and too hard to make logically rigorous.

But Archimedes, the greatest mathematician of antiquity, realized the power of the infinite.  He harnessed it to solve problems that were otherwise intractable, and in the process came close to inventing calculus — nearly 2,000 years before Newton and Leibniz.

In the coming weeks we’ll delve into the great ideas at the heart of calculus.  But for now I’d like to begin with the first beautiful hints of them, visible in ancient calculations about circles and pi.

Let’s recall what we mean by pi.  It’s a ratio of two distances.  One of them is the diameter, the distance across the circle through its center.  The other is the circumference, the distance around the circle.   Pi is defined as their ratio, the circumference divided by the diameter.

circle with diameter and circumference indicated 

If you’re a careful thinker, you might be worried about something already.  How do we know that pi is the same number for all circles?  Could it be different for big circles and little circles?  The answer is no, but the proof isn’t trivial.  Here’s an intuitive argument.

Imagine using a photocopier to reduce an image of a circle by, say, 50 percent.  Then all distances in the picture — including the circumference and the diameter — would shrink in proportion by 50 percent.  So when you divide the new circumference by the new diameter, that 50 percent change would cancel out, leaving the ratio between them unaltered.  That ratio is pi.

Of course, this doesn’t tell us how big pi is.  Simple experiments with strings and dishes are good enough to yield a value near 3, or if you’re more meticulous, 3 and 1/7th.  But suppose we want to find pi exactly or at least approximate it to any desired accuracy.  What then?  This was the problem that confounded the ancients.

Before turning to Archimedes’s brilliant solution, we should mention one other place where pi appears in connection with circles.  The area of a circle (the amount of space inside it) is given by the formula

formula for area of a circle

Here A is the area, π is the Greek letter pi, and ris the radius of the circle, defined as half the diameter.

Circle, with area filled in, and radius marked with letter r  

All of us memorized this formula in high school, but where does it come from?  It’s not usually proven in geometry class.  If you went on to take calculus, you probably saw a proof of it there, but is it really necessary to use calculus to obtain something so basic?

Yes, it is.

What makes the problem difficult is that circles are round.  If they were made of straight lines, there’d be no issue.  Finding the areas of triangles, squares and pentagons is easy.  But curved shapes like circles are hard.

The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works … as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small.  That’s the crucial idea behind all of calculus.

Here’s one way to use it to find the area of a circle.  Begin by chopping the area into four equal quarters, and rearrange them like so.

Four quarters of a circle on left, then rearranged on right 

The strange scalloped shape on the bottom has the same area as the circle, though that might seem pretty uninformative since we don’t know its area either.  But at least we know two important facts about it.  First, the two arcs along its bottom have a combined length of πr, exactly half the circumference of the original circle (because the other half of the circumference is accounted for by the two arcs on top).  Second, the straight sides of the slices have a length of r, since each of them was originally a radius of the circle.

Next, repeat the process, but this time with eight slices, stacked alternately as before.

Circle showing eight slices  

The scalloped shape looks a bit less bizarre now.  The arcs on the top and the bottom are still there, but they’re not as pronounced.  Another improvement is the left and right sides of the scalloped shape don’t tilt as much as they used to.  Despite these changes, the two facts above continue to hold: the arcs on the bottom still have a net length of πr, and each side still has a length of r.  And of course the scalloped shape still has the same area as before — the area of the circle we’re seeking — since it’s just a rearrangement of the circle’s eight slices.

As we take more and more slices, something marvelous happens: the scalloped shape approaches a rectangle.  The arcs become flatter and the sides become almost vertical.

Circle with many slices 

In the limit of infinitely many slices, the shape is a rectangle.  Just as before, the two facts still hold, which means this rectangle has a bottom of width πr and a side of height r.


But now the problem is easy.  The area of a rectangle equals its width times its height, so multiplying πr times r yields an area of πr2 for the rectangle.  And since the rearranged shape always has the same area as the circle, that’s the answer for the circle too!

What’s so charming about this calculation is the way infinity comes to the rescue.  At every finite stage, the scalloped shape looks weird and unpromising.  But when you take it to the limit — when you finally “get to the wall” — it becomes simple and beautiful, and everything becomes clear.  That’s how calculus works at its best.

Archimedes used a similar strategy to approximate pi.  He replaced a circle by a polygon with many straight sides, and then kept doubling the number of sides to get closer to perfect roundness.  But rather than settling for an approximation of uncertain accuracy, he methodically bounded pi by sandwiching the circle between “inscribed” and “circumscribed” polygons, as shown below for 6-, 12- and 24-sided figures.

Circles inscribed in polygons 

Then he used the Pythagorean theorem to work out the perimeters of these inner and outer polygons, starting with the hexagon and bootstrapping his way up to 12, 24, 48 and ultimately 96 sides.  The results for the 96-gons enabled him to prove that

formula for 96-gons

In decimal notation (which Archimedes didn’t have), this means pi is between 3.1408 and 3.1429.

This approach is known as the “method of exhaustion” because of the way it traps the unknown number pi between two known numbers that squeeze it from either side.  The bounds tighten with each doubling, thus exhausting the wiggle room for pi.

In the limit of infinitely many sides, both the upper and lower bounds would converge to pi.  Unfortunately, this limit isn’t as simple as the earlier one, where the scalloped shape morphed into a rectangle.  So pi remains as elusive as ever.  We can discover more and more of its digits — the current record is over 2.7 trillion decimal places — but we’ll never know it completely.


Aside from laying the groundwork for calculus, Archimedes taught us the power of approximation and iteration.  He bootstrapped a good estimate into a better one, using more and more straight pieces to approximate a curved object with increasing accuracy.

More than two millennia later, this strategy matured into the modern field of “numerical analysis.”  When engineers use computers to design cars to be optimally streamlined, or when biophysicists simulate how a new chemotherapy drug latches onto a cancer cell, they are using numerical analysis.  The mathematicians and computer scientists who pioneered this field have created highly efficient, repetitive algorithms, running billions of times per second, that enable computers to solve problems in every aspect of modern life, from biotech to Wall Street to the Internet.  In each case, the strategy is to find a series of approximations that converge to the correct answer as a limit.

And there’s no limit to where that’ll take us.


  1. The history and intellectual legacy of Zeno’s paradoxes are discussed in: J. Mazur, Zeno’s Paradox (Plume, 2008).
  2. For a delightfully opinionated and witty history of pi, see: P. Beckmann, A History of Pi (St. Martin’s Press, 1976).
  3. Bill Willis at Worsley School OnLine has given a very clear explanation of how to find the area of the circle, using the same argument as above but fleshed out in more detail. The school website contains many other excellent math and science resourcesfor students, teachers and parents.
  4. The PBS television series “Nova” ran a wonderful episode about Archimedes, infinity and limits called “Infinite Secrets.” It originally aired on Sept. 30, 2003. The program website includes many online resources, including the program transcript and interactive demonstrations.
  5. For readers wishing to see the mathematical details of Archimedes’s method of exhaustion, Neal Carothers has used trigonometry (equivalent to the Pythagorean gymnastics that Archimedes relied on) to derive the perimeters of the inscribed and circumscribed polygons between which the circle is trapped. Peter Alfeld’s website features an interactive Java applet that lets you change the number of sides in the polygons.
  6. The individual steps in Archimedes’s original argument are of historical interest but you might find them disappointingly obscure.
  7. Anyone curious about the heroic computations of pi to enormous numbers of digits should enjoy Richard Preston’s profile of the Chudnovsky brothers. Entitled “The Mountains of Pi,” this affectionate and surprisingly comical piece appeared in the Mar. 2, 1992, issue of The New Yorker, and more recently as a chapter in: R. Preston, Panic in Level Four (Random House, 2008).
  8. For a textbook introduction to the basics of numerical analysis, see: W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University Press, 2007).

Thanks to Tim Novikoff and Carole Schiffman for their comments and suggestions, and to Margaret Nelson for preparing the illustrations.

Editor’s Note: A correction was made to an earlier version of this column, to fix a misspelling of the name of the publisher of Zeno’s Paradox.

Source: Take It to the Limit – The New York Times