### kottke.org posts about mathematics

Cistercian numerals were invented by the Cistercian order of monks in the 13th century. Giuseppe Frisella explains how the notation system works:

A vertical straight line acts as an axis dividing the plane into four quadrants, each one representing one of the four digits: the upper right quadrant for the units, the upper left quadrant for the tens, the lower right quadrant for the hundreds, and the lower left quadrant for the thousands.

What this does well, indeed better than the roman or Arabic numeral systems it’s related to, is to represent both small and large numbers (1 up to 9999) in a single glyph. What it doesn’t do well, compared to roman or Arabic systems, is allow you to reduce operations on large numbers to operations on smaller ones. There’s no long division, in other words — and even addition and multiplication aren’t very straightforward.

So you might think about this as a kind of mathematical compression system, optimizing for storage rather than operations. If you just need to record a number — say, a four-digit year — you can do it quickly and in a minimum amount of space in the Cistercian system. If you need to do bookkeeping, then the Arabic numerals are probably what you want.

But if you’ve read this far, you’re probably thinking what I usually think anytime I encounter something a little strange in the world of mathematical notation — *what about aliens? * One can imagine an alien species that can easily do simple arithmetic operations on what they would call *small numbers* (less than 10,000) in their heads (or has offloaded such tasks to machine), and which would correspondingly value the storage and computational efficiency of a system of numbers like this. Maybe Cistercian numerals, rather than the clumsy digits of our intellectual infancy, will be the best way to make ourselves understood when first contact begins.

(Via Clive Thompson)

**Update:** Shelby Wilson has created an easy-to-use Cistercian numeral generator. (Via Alex Miller)

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Nicholas Rougeux designed a series of posters to visualize all 143 prime numbers with three digits based on simple rules.

Each print contains all 143 prime numbers with 3 digits. Each is represented by an image composed of simple geometric shapes based on its digital root and colors based on its digits. Arranging these images sequentially creates colorful collages of prime numbers based on simple rules.

For each poster, a unique shape was assigned to the digital root of each prime number which is calculated by iteratively summing its digits until one remains. (All known prime numbers greater than 3 have digital roots of 1, 2, 4, 5, 7, or 8.)

There are nine posters in all that use a few different styles of geometric shape.

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After seeing an abstract US map where the borders between the states were preserved but the shapes of the individual states were not, Tom Comerford was inspired to design what he calls The Topologist’s Map of the World.

I describe this a a topologist’s map because topology is a branch of mathematics concerned with the way that space is connected. In topology it’s common to think of stretchy, distortable surfaces that can be moved around without being punctured or torn.

With many connections to other countries and bodies of water, countries like Russia, India, Brazil, and China are prominent on the map while the US, which only borders two other countries, is a tiny box in the corner. (via hacker news)

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Bill Tavis designed this lovely vintage-style map of the familiar fractal shape, the Mandelbrot set. He is selling a poster version of the map, starting at the very reasonable price of $24. I don’t usually highlight the price on this sort of thing, but an unauthorized seller on Amazon was selling poor-quality counterfeits of the map and even though it wasn’t his fault, Tavis offered to replace any of the crappy maps for free. Great map, and apparently a great human who made it.

I found Tavis’s map when I was searching for the creator of this similarish map that I found on Twitter (bigger here).

~~Anyone know who made this version?~~ Jonny Laser made the map in the second image for VSauce (scroll down a bit). (thx, kirsten)

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Because of their relevance to social media, the algorithm has become an everyday concept. Why do we see the posts we see on Instagram or TikTok? Oh, it’s the algorithm. This video from BBC Ideas explains that the term has its roots in the work of 9th century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī, who also gave the world the word “algebra”. (via the morning news)

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Mathematics isn’t the most obviously cinematic academic discipline out there, but it is one that the movies (and to a lesser extent television) have repeatedly tried to understand, or in some cases, used to goose up a vaguely science-y story. Unsurprisingly, mathematicians often become sticklers for detail in such high-profile depictions of what they do, and a good or bad portrayal can become famous or infamous.

My friend Jordan Ellenberg, a math professor at the University of Wisconsin-Madison, is also an expert in translating math to popular audiences, in his books and sometimes on screen. In this video, he takes a look at some popular representations of math in TV and the movies, and tries to explain what’s going on, including what the filmmakers do well or not so well.

*Good Will Hunting*’s use of math is famously bad, and Ellenberg unsurprisingly agrees (although, surprisingly, he had never before seen the movie or even the math scenes in question). Portrayals that get a perhaps-surprisingly high score include *The Simpsons* (which includes several former mathematicians among its writers) and *Jurassic Park* — Jeff Goldblum pulls off a passable explanation of chaos theory while also eerily accurately capturing the slightly-creepy vibe of a neurotic academic asked to describe what he studies to a layperson. “He was the one who I most felt might have spent a long time studying mathematicians and truly trying to give off a mathematician vibe,” says Ellenberg.

One thing I love is Ellenberg’s attention to how each of the on-screen mathematicians write (if they do any writing themselves at all, rather than ponder something that’s already been written by a character offscreen) — the connection between math and writing is so powerful, and math is one of the great remaining repositories of manuscript culture (even as it’s also taken on computers and machines, like everything else).

Ellenberg also adds that the most important thing a movie about mathematics can do is to convey to the audience that being a mathematician is something real, ordinary people still do, rather than being just a bunch of old dead men wearing robes.

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The father-son duo of Martin and Erik Demaine make typefaces that are algorithmic, mathematical, or puzzle-like in nature. For instance, here’s their Tetris font, where each letter is made up of the seven possible Tetris pieces:

Or their newest one, Everything, where each letter can be folded into any other letter:

Everything to everything. This typeface illustrates how to fold any letter into any other letter, or more precisely, how to fold a piece of paper in the shape of any letter into the shape of any other letter. This lets you write one message inside another in a couple of ways. On the one hand, you could present the 6x6 crease patterns whose silhouettes look like one message (first text), and folding them reveals another message (second text). On the other hand, you could present the folded forms (as physical objects) whose silhouettes look like one message (second text), and unfolding them reveals another message (first text).

From a recent-ish profile of the Demaines and their typefaces in the NY Times:

In a 2015 paper, “Fun With Fonts: Algorithmic Typography,” the Demaines explained their motivations: “Scientists use fonts every day to express their research through the written word. But what if the font itself communicated (the spirit of) the research? What if the way text is written, and not just the text itself, engages the reader in the science?”

Inspired by theorems or open problems, the fonts — and the messages they compose — can usually be read only after solving the related puzzle or series of puzzles.

You can check out the rest of their typefaces on their website — they include fonts with infinitely tiling letters, Sudoku puzzle fonts, and a font whose letters are made up of shapes that can be packed into a 6x6 square. So fun!

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Even for mathematically minded folks, statistics can be hard to grasp. Take statistical paradoxes for example: Simpson’s Paradox is a real mind-boggler. Ryan Anderson explains this paradox in a recent issue of Why is this interesting?

It’s simple to describe, yet it still stops me in my tracks when I see it in the wild. The paradox is that a measurable effect on a large population disappears, or even reverses when that population is split into subgroups. The cause of these results is almost always a material change in the denominators from one period to the next.

Showing is easier than telling with paradoxes, so here is a classic example: In 1995 and 1996, David Justice had a higher batting average than Derek Jeter in each year. However, Jeter had a higher cumulative batting average over those two years.

It’s true; look:

Anderson continues:

How does this work? Jeter’s 1996 stats accounted for over 92% of his total performance over the two years, as he was 20 years old and only called up to the major league for a few games in 1995. Meanwhile, Justice’s 1996 stats were only 25% of his total performance due to a separated shoulder he suffered barely two months into the season. So while Justice performed better on smaller sample size, Jeter’s 183 hits in 1996 were the strongest signal for overall performance.

Read the rest of the piece; he goes on to connect statistical paradoxes to efforts to mislead people about the pandemic and vaccine effectiveness.

**Update:** A pair of videos on Simpson’s Paradox, in case you need some more explanation or examples.

(thx, @JunieGrrl)

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This video from Veritasium is a nice explanation of the mathematician David Hilbert’s paradox of the Grand Hotel, which illustrates that a hotel with an infinite number of rooms can still accommodate new guests even when it’s full. Until it can’t, that is. See also Steven Strogatz’s explanation of Hilbert’s infinite hotel and how Georg Cantor’s discovery of different types of infinities complicated the hotel’s hospitality. (via digg)

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What do swaying bridges, flashing fireflies, clapping audiences, the far side of the Moon, and beating hearts have in common? Their behavior all has something to do with synchronization. In this video, Veritasium explains why and how spontaneous synchronization appears all the time in the physical world.

I was really into the instability of the Millennium Bridge back when it was first opened (and then rapidly closed), so it was great to hear Steven Strogatz’s explanation of the bridge’s failure.

Oh, and do go play with Nicky Case’s firefly visualization to see how synchronization can arise from really simple rules.

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The Bank of England unveiled the final design of the new £50 banknote honoring mathematician and computer scientist Alan Turing.

Commenting on the new note, Governor Andrew Bailey said: “There’s something of the character of a nation in its money, and we are right to consider and celebrate the people on our banknotes. So I’m delighted that our new £50 features one of Britain’s most important scientists, Alan Turing. Turing is best known for his codebreaking work at Bletchley Park, which helped end the Second World War. However in addition he was a leading mathematician, developmental biologist, and a pioneer in the field of computer science. He was also gay, and was treated appallingly as a result. By placing him on our new polymer £50 banknote, we are celebrating his achievements, and the values he symbolises”.

The note will be placed into circulation beginning June 23, 2021. As part of the introduction of the note, GCHQ (the successor agency to the one Turing worked for) has created a series of 12 puzzles for folks to decipher. Good luck!

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In popular press and social media, there’s been a misunderstanding of what is actually meant when scientists say that the Pfizer and Moderna Covid-19 vaccines have an efficacy of 94-95%. It *does not* mean that 95% of vaccinated people are protected from infection — these vaccines are better than that. Dr. Piero Olliaro explains in a letter to The Lancet:

The mRNA-based Pfizer and Moderna vaccines were shown to have 94-95% efficacy in preventing symptomatic COVID-19, calculated as 100 x (1 minus the attack rate with vaccine divided by the attack rate with placebo). It means that in a population such as the one enrolled in the trials, with a cumulated COVID-19 attack rate over a period of 3 months of about 1% without a vaccine, we would expect roughly 0.05% of vaccinated people would get diseased.

Another way to put it: **you’re 20 times less likely to get Covid-19 with a vaccine than without**. (And again, data indicates these are safe vaccines.) Olliaro explains with some simple math:

If we vaccinated a population of 100,000 and protected 95% of them, that would leave 5000 individuals diseased over 3 months, which is almost the current overall COVID-19 case rate in the UK. Rather, a 95% vaccine efficacy means that instead of 1000 COVID-19 cases in a population of 100,000 without vaccine (from the placebo arm of the abovementioned trials, approximately 1% would be ill with COVID-19 and 99% would not) we would expect 50 cases (99.95% of the population is disease-free, at least for 3 months).

And of course if you vaccinate widely, it becomes a compounding situation because the virus just runs out of people to infect.

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Web performance and security company Cloudflare uses a wall of lava lamps to generate random numbers to help keep the internet secure. Random numbers generated by computers are often not exactly random, so what Cloudflare does is take photos of the lamps’ activities and uses the uncertainty of the lava blooping up and down to generate truly random numbers. Here’s a look at how the process works:

At Cloudflare, we have thousands of computers in data centers all around the world, and each one of these computers needs cryptographic randomness. Historically, they got that randomness using the default mechanism made available by the operating system that we run on them, Linux.

But being good cryptographers, we’re always trying to hedge our bets. We wanted a system to ensure that even if the default mechanism for acquiring randomness was flawed, we’d still be secure. That’s how we came up with LavaRand.

LavaRand is a system that uses lava lamps as a secondary source of randomness for our production servers. A wall of lava lamps in the lobby of our San Francisco office provides an unpredictable input to a camera aimed at the wall. A video feed from the camera is fed into a CSPRNG [cryptographically-secure pseudorandom number generator], and that CSPRNG provides a stream of random values that can be used as an extra source of randomness by our production servers. Since the flow of the “lava” in a lava lamp is very unpredictable, “measuring” the lamps by taking footage of them is a good way to obtain unpredictable randomness. Computers store images as very large numbers, so we can use them as the input to a CSPRNG just like any other number.

(via open culture)

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DJ Earworm has made a chronological mix of songs, one from each year from 1970 to 2020.^{1} The Jackson 5 flows into Rod Stewart, Def Leppard into Milli Vanilli, Eric Clapton into Chumbawamba into The Verve, Shakira into Rihanna, and Ed Sheeran into Justin Bieber. Go on then, take a ride.

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How many holes does a donut have? That’s pretty easy: one. What about a straw? Two (one at each end) or just one? (Isn’t a straw just an elongated donut?) Does a coffee mug have one hole or two? Does a bowl have a hole? If no, then what about a hole in the ground or a hole in a wall that doesn’t pass all the way through? Does a basketball have a hole? A Reddit user asked 1600 people how many holes were in various objects and the results are fantastically all over the place.

This is a trivial question, but it reveals something interesting about people’s perceptions. The dictionary definition of “hole” includes two main meanings for the purposes of this question: “an opening through something” and “a hollowed-out place”. Mathematics offers another possible meaning:

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a disconnectivity is interpreted as a hole in the space. Examples of holes are things like the “donut hole” in the center of the torus, a domain removed from a plane, and the portion missing from Euclidean space after cutting a knot out from it.

But a hole isn’t clearly defined in math or topology. From What We Talk about When We Talk about Holes in Scientific American:

Here’s my short answer that is also the reason I’m not an algebraic topologist. If you can put it on a necklace, it has a one-dimensional hole. If you can fill it with toothpaste, it has a two-dimensional hole. For holes of higher dimensions, you’re on your own.

That answer isn’t very satisfying. Is there a better way to describe holes? I talked with some of my topologist friends and discovered two things: topologists don’t all agree on what a hole is, and it’s fun and interesting to think about different interpretations of a word whose mathematical definition isn’t completely settled. I think my larger conclusion, in the spirit of the season, is that holes are like Santa Claus: the true meaning is in your heart.

No wonder those poll results are all over the place. But at the same time, it’s interesting that many more people say that donuts have a hole than washers or rubber bands. I guess donut holes have better marketing? As for straws — reason tells me they only have one hole but I know in my heart they have two. (via the whippet)

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The director of the National Institute of Allergy and Infectious Diseases, Anthony Fauci, told a Senate committee today that the US could be heading towards *100,000* new reported cases of Covid-19 *per day*. 100,000 cases per day. Yesterday the US recorded about 40,000 new cases.

“It is going to be very disturbing, I will guarantee you that,” he said.

“What was thought to be unimaginable turns out to be the reality we’re facing right now,” Fauci said, adding that “outbreaks happen, and you have to deal with them in a very aggressive, proactive way.”

Fewer than 20 countries have recorded more than 100,000 cases in total. Canada, for instance, has confirmed about 106,000 Covid-19 cases since the outbreak began.

Public health and infectious diseases experts, who have been gravely concerned about the way the U.S. response has unfolded, concurred with Fauci’s assessment.

Bars and restaurants are reopening around the country without any serious effort to test/trace/isolate/support. In the absence of strident guidance from the federal government, people are worrying less about social distancing and wearing masks to protect others. As this guy says, it’s just a matter of math:

“It’s unfortunately just a simple consequence of math plus a lack of action,” said Marm Kilpatrick, an infectious diseases dynamics researcher at the University of California, Santa Cruz. “On the one hand it comes across as ‘Oh my God, 100,000 cases per day!’ But then if you actually look at the current case counts and trends, how would you not get that?”

Absolutely nothing has changed about the virus, so its spread is determined by pretty simple exponential growth.

Limiting person-to-person exposure and decreasing the probability of exposures becoming infections can have a huge effect on the total number of people infected because the growth is exponential. If large numbers of people start doing things like limiting travel, cancelling large gatherings, social distancing, and washing their hands frequently, the total number of infections could fall by several orders of magnitude, making the exponential work for us, not against us. Small efforts have huge results.

We’ve known for months (and epidemiologists and infectious disease experts have known for their entire careers) what works and yet the federal government and many state governments have not listened and, in some cases, have actively suppressed use of such measures. So the pandemic will continue to escalate in the United States until proper measures are put in place by governments and people follow them. The virus will not change, the mathematics will not change, so we must.

Graph at the top of the post via Rishi Desai.

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This lovely short film by Cristóbal Vila shows how the simple Fibonacci sequence manifests itself in natural forms like sunflowers, nautilus shells, and dragonfly wings.

See also Arthur Benjamin’s TED Talk on the Fibonacci numbers and the golden ratio and the Fibonacci Shelf. (via @stevenstrogatz)

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As someone who suspects I may have had a mild case of Covid-19 a couple of months ago, I’ve been thinking about getting tested for antibodies. But as this video from ProPublica shows, even really accurate tests may not actually tell you all that much.

And the thing is, the “do I have Covid-19 right now” tests are plagued by the same issue.

For patients getting tested, the main concern is how to interpret the outcome: If I test negative with an RT-PCR genetic test, what are the chances I actually have the virus? Or if I test positive with an antibody test, does it actually mean I have the antibodies?

It turns out that the answers to these questions don’t just hinge on the accuracy of the test. “Mathematically, the way that works out, that actually depends on how many people in your area have Covid,” Eleanor Murray, an assistant professor of epidemiology at the Boston University School of Public Health, said.

The rarer the disease in the population, the less you’ll learn by testing.

Let’s say we have a hypothetical Covid-19 test for antibodies that is both 99 percent sensitive — meaning almost all people with antibodies will test positive — and 99 percent specific, meaning almost all people who were never infected will yield a negative result.

If you test a group of 100 uninfected people, odds are one of them will still test positive even though they don’t have the virus. Conversely, if you test 100 people who were infected, it’s likely one of them will still test negative.

Now let’s presume the virus has a prevalence rate of 1 percent, so one person in 100 carries antibodies to it. If you test 100 random people and get a positive result, what is the chance that this person was truly infected?

Deborah Birx, the White House Covid-19 response coordinator, explained the answer at a press conference on April 20: “So if you have 1 percent of your population infected and you have a test that’s only 99 percent specific, that means that when you find a positive, 50 percent of the time will be a real positive and 50 percent of the time it won’t be.”

So even if I test positive for antibodies and I assume that confers immunity, given that the number of confirmed infections in Vermont is so low (~900 statewide), it doesn’t seem like I would be justified in changing my behavior at all. I would still have to act as though I’ve never had the virus, both for my own health and the health of those around me. Maybe if I had two or three corroborating tests could I be more certain…

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Back when the COVID-19 pandemic was beginning to be taken seriously by the American public, 3blue1brown’s Grant Sanderson released a video about epidemics and exponential growth. (It’s excellent — I recommend watching it if you’re still a little unclear on how things are got so out of hand so quickly in Italy and, very soon, in NYC.) In his latest video, Sanderson digs a bit deeper into simulating epidemics using a variety of scenarios.

Like, if people stay away from each other I get how that will slow the spread, but what if despite mostly staying away from each other people still occasionally go to a central location like a grocery store or a school?

Also, what if you are able to identify and isolate the cases? And if you can, what if a few slip through, say because they show no symptoms and aren’t tested?

How does travel between separate communities affect things? And what if people avoid contact with others for a while, but then they kind of get tired of it and stop?

These simulations are fascinating to watch. Many of the takeaways boil down to: early & aggressive actions have a huge effect in the number of people infected, how long an epidemic lasts, and (in the case of a disease like COVID-19 that causes fatalities) the number of deaths. This is what all the epidemiologists have been telling us — because the math, while complex when you’re dealing with many factors (as in a real-world scenario), is actually pretty straightforward and unambiguous.

The biggest takeaway? That the effective identification and isolation of cases has the largest effect on cutting down the infection rate. Testing and isolation, done as quickly and efficiently as possible.

See also these other epidemic simulations: Washington Post and Kevin Simler.

*Note: Please keep in mind that these are simulations to help us better understand how epidemics work in general — it’s not about how the COVID-19 pandemic is proceeding or will proceed in the future.*

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By manipulating values like R0, incubation time, and hospitalization rate with this this epidemic graphing calculator, you really get a sense of how effective early intervention and aggressive measures can be in curbing infection & saving lives in an exponential crisis like the COVID-19 pandemic.

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Over the past week or so, echoing public health officials & epidemiologists, I’ve been trying to illustrate the often counterintuitive concept of exponential growth that you see in an epidemic and how flattening the curve can help keep people healthy and alive. But I think people have a hard time grasping what that means, personally, to them. Like, what’s one person in the face of a pandemic?

Well, epidemiologist Britta Jewell had a similar thought and came up with this brilliantly simple graph, one of the best I’ve seen in illustrating the power of exponential growth and how we as individuals can affect change:

Jewell explains a bit more about what we’re looking at:

The graph illustrates the results of a thought experiment. It assumes constant 30 percent growth throughout the next month in an epidemic like the one in the U.S. right now, and compares the results of stopping one infection today — by actions such as shifting to online classes, canceling of large events and imposing travel restrictions — versus taking the same action one week from today.

The difference is stark. If you act today, you will have averted four times as many infections in the next month: roughly 2,400 averted infections, versus just 600 if you wait one week. That’s the power of averting just one infection, and obviously we would like to avert more than one.

So that’s 1800 infections averted from the actions of *just one person*. Assuming a somewhat conservative death rate of 1% for COVID-19, that’s 18 deaths averted. Think about that before you head out to the bar tonight or convene your book group as usual. Your actions have a lot of power in this moment; take care in how you wield it.

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From 3blue1brown’s Grant Sanderson, this is an excellent quick explanation of exponential growth and how we should think about it in relation to epidemics like COVID-19. Depending on how rusty your high school math is, you might need to rewind a couple of times to fully grasp the explanation, but you should persevere and watch the whole thing.

The most important bit is at the end, right around the 7:45 mark, when he talks about how limiting person-to-person exposure and decreasing the probability of exposures becoming infections can have a huge effect on the total number of people infected because the growth is exponential. If large numbers of people start doing things like limiting travel, cancelling large gatherings, social distancing, and washing their hands frequently, the total number of infections could fall by several orders of magnitude, making the exponential work for us, not against us. Small efforts have huge results. If, as in the video, you’re talking about 100 million infected in two months (at the current transmission rate) vs. 400,000 (at the lowered rate) and if the death rate of COVID-19 is between 1-3%, you’re looking at *1-3 million dead vs. 4-12,000 dead*.

So let’s start flattening that exponential curve. South Korea and China both seem to have done it, so there’s no reason the rest of the world can’t through aggressive action. (thx, david)

**Update:** Vox has a nice explainer on what epidemiologists refer to as “flattening the curve”.

Yet the speed at which the outbreak plays out matters hugely for its consequences. What epidemiologists fear most is the health care system becoming overwhelmed by a sudden explosion of illness that requires more people to be hospitalized than it can handle. In that scenario, more people will die because there won’t be enough hospital beds or ventilators to keep them alive.

A disastrous inundation of hospitals can likely be averted with protective measures we’re now seeing more of — closing schools, canceling mass gatherings, working from home, self-quarantine, avoiding crowds - to keep the virus from spreading fast.

Epidemiologists call this strategy of preventing a huge spike in cases “flattening the curve”.

Here’s the relevant graphic explanation from Our World in Data’s COVID-19 package:

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If you’re a computer, it turns out that the fastest way to multiply two numbers, especially two very large numbers, is not by the grade school method of stacking the two numbers and then multiplying each digit in the top number by each digit in the bottom number and adding the results. Since 1960, mathematicians have been discovering ever faster methods to multiply and recently, a pair of mathematicians discovered a method that is perhaps the fastest way possible.

Their method is a refinement of the major work that came before them. It splits up digits, uses an improved version of the fast Fourier transform, and takes advantage of other advances made over the past forty years. “We use [the fast Fourier transform] in a much more violent way, use it several times instead of a single time, and replace even more multiplications with additions and subtractions,” van der Hoeven said.

What’s interesting is that independently of these discoveries, computers have become a lot better at multiplication:

In addition, the design of computer hardware has changed. Two decades ago, computers performed addition much faster than multiplication. The speed gap between multiplication and addition has narrowed considerably over the past 20 years to the point where multiplication can be even faster than addition in some chip architectures.

(via @macgbrown, who passed this along after I posted this video on Russian multiplication)

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I’ve loved math since I was a kid. One of the big reasons for this is that there’s always more than one way to solve a particular problem and in discovering those solutions, you learn something about mathematics and the nature of numbers.^{1}

In this video, math fan Johnny Ball shows us a different method of multiplication. In Russian multiplication (also called Ethiopian multiplication and related to ancient Egyptian multiplication), you can multiply any two numbers together through simple addition and doubling & halving numbers. The technique works by converting the numbers to binary and turning it into an addition problem.

I loved learning about this so much that I scribbled an explanation out on a napkin at brunch yesterday to show a friend how it worked. We’re friends because she was just as excited as I was about it. (via the kid should see this)

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All art is bounded by one constraint or another. Mathematician Robert Bosch makes what he calls “optimization art”, which is best embodied by these images produced as solutions to the travelling salesman problem. Each image is made up of a continuous line that is the shortest possible route through a series of points without revisiting any single point, much like the optimal route of a travelling salesperson visiting cities. The rendition of a van Gogh self-portrait uses a solution for 120,000 “cities” while the single line forming the Girl with the Pearl Earring visits 200,000 cities.

I would love to see an Observable notebook where you could upload any photo to make images like these. (via @Ianmurren)

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Jessica Wynne has been taking photos of mathematicians’ blackboards for the past year or so, some of which were featured recently in the NY Times. I love the variety in density, style, color, and tidiness.

“I am also fascinated by the process of working on the chalkboard. Despite technological advances, and the creation of computers, this is how the masters choose to work.”

In their love of blackboards and chalk, mathematicians are among the last holdouts. In many fields of science and investigation, blackboards have been replaced with whiteboards or slide show presentations. But chalk is cheaper and biodegradable. It smells better than whiteboard markers and is easier to clean up, mathematicians say. It is also more fun to write with.

A book of Wynne’s chalkboard photos called Do Not Erase will be released next year.

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The folks behind the Nevertheless podcast commissioned a set of seven posters of STEM role models, people who have made significant contributions to science, technology, engineering, and mathematics. The posters are free to download and print out in eight different languages (including English, Spanish, and Simplified Chinese).

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The story goes that modern chaos theory was birthed by Edward Lorenz’s paper about his experiments with weather simulation on a computer. The computing power helped Lorenz nail down hidden patterns that had been hinted at by computer-less researchers for decades. But the early tenets of chaos theory were not the only things that were hidden. The women who wrote the programs that enabled Lorenz’s breakthroughs haven’t received their proper due.

But in fact, Lorenz was not the one running the machine. There’s another story, one that has gone untold for half a century. A year and a half ago, an MIT scientist happened across a name he had never heard before and started to investigate. The trail he ended up following took him into the MIT archives, through the stacks of the Library of Congress, and across three states and five decades to find information about the women who, today, would have been listed as co-authors on that seminal paper. And that material, shared with Quanta, provides a fuller, fairer account of the birth of chaos.

The two women who programmed the computer for Lorenz were Ellen Gille (née Fetter) and Margaret Hamilton. Yes, that Margaret Hamilton, whose already impressive career starts to look downright bonkers when you add in her contributions to chaos theory.

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If you could somehow fold a piece of paper in half 103 times, the paper would be as thick as the observable universe.

Such is the power (*cough*) of exponential growth, but of course you’d never get anywhere close to that many folds. The theoretical limit for folding paper was long thought to be seven or eight folds. You can see why watching this hydraulic press attempt the 7th fold…the paper basically turns to dust.

But in 2002, high school student Britney Gallivan proved that you could fold a piece of paper 12 times. Here’s Gallivan explaining the math involved and where the limits come in when folding:

(thx, porter)

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Nidhal Selmi combined the fractal Sierpinski triangle with the impossible Penrose triangle to create the M.C. Escher-like the Selmi triangle.

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