Starting with TheDailyBeast

Long time since an update! I started on Monday with the Tech+Health desk at TheDailyBeast.com

Two articles this week. So far so good.

Ants Are Just as Effective as Chemical Pesticide http://thebea.st/1KsVwZz

I Was Shaken Down by Wikipedia’s Blackmail Bandits http://thebea.st/1M1z4Kz

The thing about Euler's identity is it's impossible to derive the number e without some use of infinitesimals, be it the limit definition, integration, or infinite series. That makes Euler's Identity REALLY HARD TO TALK ABOUT IN LAYMAN'S TERMS. That all said, I'm really happy how this turned out.

Enjoy.

Euler’s Identity: 'The Most Beautiful Equation'

http://www.livescience.com/51399-eulers-identity.html

New article today:

What Is Topology?

http://www.livescience.com/51307-topology.html

I generated the wireframe models using MATLAB. Here's the code I used along with links to all the parametric equations.
[u,v]=meshgrid(linspace(0,2*pi,25));

%cylinder
%http://mathworld.wolfram.com/Cylinder.html
% x=1.1*cos(u);
% y=1.1*sin(u);
% z=v/(2*pi);

%mobius strip
%http://mathworld.wolfram.com/MoebiusStrip.html
% r=5;
% s=(u-pi)/pi*2;
% t=v;
% x=(r+s.*cos(t/2)).*cos(t);
% y=(r+s.*cos(t/2)).*sin(t);
% z=(s.*sin(t/2));

%sphere
%http://mathworld.wolfram.com/Sphere.html
% r=5;
% theta=u;
% phi=v/2;
% x=r*cos(theta).*sin(phi);
% y=r*sin(theta).*sin(phi);
% z=r*cos(phi);

%torus
%http://mathworld.wolfram.com/Torus.html
% a=5;
% c=10;
% x=(c+a*cos(v)).*cos(u);
% y=(c+a*cos(v)).*sin(u);
% z=a*sin(v);

%klein bottle
%http://paulbourke.net/geometry/klein/
% r=4*(1-cos(u)/2);
% x=(u<pi).*(6*cos(u).*(1+sin(u))+r.*cos(u).*cos(v)) + (u>=pi).*(6*cos(u).*(1+sin(u))+r.*cos(v+pi));
% y=(u<pi).*(16*sin(u)+r.*sin(u).*cos(v)) + (u>=pi).*(16*sin(u));
% z=r.*sin(v);

%cross-cap disk
%https://en.wikipedia.org/wiki/Real_projective_plane#Cross-capped_disk
% r=5;
% x=r*(1+cos(v)).*(cos(u));
% y=r*(1+cos(v)).*(sin(u));
% z=-tanh(u-pi)*r.*sin(v);

hold on
mesh(x,y,z);
camlight left;
lighting phong;
alpha(0.4);
axis equal;
hold off

More math!

Properties of Pascal’s Triangle

http://www.livescience.com/51238-properties-of-pascals-triangle.html

My editor assured me he could find an image of a bean machine, so you can imagine my surprise when I discovered the inclusion of the following video:

I've been a fan of Numberphile videos for years, so seeing Matt Parker in was a real treat. I hope you all enjoy reading this as much as I enjoyed writing it.

Another math article!

What Is Symmetry?

http://www.livescience.com/51100-what-is-symmetry.html

It would have meant a lot to learn about a topic like this as a kid. That different kinds of symmetry had symbols associated with them was news to me as a college senior in chemical engineering.

It's a shame that the great diversity in 3-D patterns is rarely talked about outside of crystallography. What amazing sculptures are we missing out on by not exploring the topic more deeply in art?

Yet another excuse to write about the history of mathematics!

What Is Trigonometry?

http://www.livescience.com/51026-what-is-trigonometry.html

My editor left out the etymology of "sine" so I reproduce it here. From A History of Mathematics (3rd Edition) by Victor J. Katz (Section 8.7).

The English word ‘sine’ comes from a series of mistranslations of the Sanskrit ‘jyā-ardha’ meaning ‘chord-half.’ Aryabhata frequently abbreviated this term to ‘jyā’ or its synonym ‘jīvā.’ When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word ‘jiba.’ But since Arabic is written without vowels, later writers interpreted the consonants ‘jb’ as ‘jaib,’ which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word ‘sinus,’ which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf. This Latin word has now become our English ‘sine.’

That Malisha Dewalt over at MedievalPOC thought this was worth sharing tells me I'm doing something right in my representation of the history of math. I am happy.

Wow, I'm really behind on posting articles. Here's one from last month. I think the title scared everybody away:

What Are Logarithms?

http://www.livescience.com/50940-logarithms.html

I've been outdone on how to to explain logarithms simply to a rough approximation. From How Not to Be Wrong by Jordan Ellenberg (Chapter 11 I think).

The logarithm of a postiive number N, called log N, is the number of digits it has.
Wait, really? That's it?
No. That's not really it. We can call the number of digits the "fake logarithm," or flogarithm. It's close enough to the real thing to give the general idea of what the logarithm means in a context like this one. 

What is Calculus?

New article posted today!

What Is Calculus?

http://www.livescience.com/50777-calculus.html

No equations; just a bunch of pretty pictures. Enjoy!

Before and After the Scientific Revolution

It's easy to take for granted how quickly modern technology develops because the scientific revolution, which began in the 16th century, has been in full force for as long as anyone can remember. To offer some perspective on just how important this is, consider the differences in technological advancement before and after the scientific revolution was in effect:

• In a roughly 300 year span of the 13th-16th centuries, purely mechanical clocks (that is, ones that don't rely on the metering of water) went from chiming just hours to ticking seconds.
• In a roughly 300 year span of the 17th-20th centuries, human propulsion went from the horse-drawn carriages to rockets that took us to the moon.

Both of these advancements represent the pinnacle of technological and scientific thought at the time, but things have moved notably faster since the beginning of the Scientific Revolution. In my opinion the magic behind modern technological advancement is being able to use mathematics to characterize reality. This was the key to unleashing the floodgates, and so began the rush in the 16th century which has continued to this day

CafeSci Boston - April 2015 "Why 60 Minutes? 5000 years of tradition and science."

Last night I spoke at CafeSci Boston for the Cambridge Science Festival. I had a great time and the audience was lovely. The title of my talk was "Why 60 Minutes? 5000 years of tradition and science."

Unfortunately official recordings don't start until next month, but my wife managed decent capture herself:

You can follow CafeSci Boston on Twitter at @CafeSciBoston

An earlier article version of this talk can be found at:
http://www.livescience.com/44964-why-60-minutes-in-an-hour.html

The talk is based on a passing mention in the following book:

http://www.amazon.com/Introduction-History-Algebra-Mesopotamian-Mathematical/dp/0821844733/

Page 4:

Several centuries of continuous [Mesopotamian] observations provided an invaluable body of data for the computation of planetary periods… As it would have been an overwhelming task to convert all these data into the decimal system… the Greeks maintained the sexagesimal system for astronomical measurements… This was also adopted by the Indians as early as antiquity… Then… it reached the Muslims who in turn transmitted this notation to medieval Christian Europe. The sexagesimal division… still used today is thus a living witness to the sexagesimal base once used by the Sumerians… in prehistoric times.

Here's some books that helped me stitch the narrative together:

http://www.amazon.com/Universal-History-Numbers-Prehistory-Invention/dp/0471375683/

http://www.amazon.com/Euclids-Window-Geometry-Parallel-Hyperspace/dp/0684865246/

http://www.amazon.com/Zero-The-Biography-Dangerous-Idea/dp/0140296476

And more generally about the history of mathematics:

http://www.amazon.com/History-Mathematics-Carl-B-Boyer/dp/0470525487/

http://www.amazon.com/History-Mathematics-3rd-Victor-Katz/dp/0321387007/

ACS Chemistry Champions

Last week I made a video for the ACS Chemistry Champions contest.

You can see the rest of the entries here.

I had fun making this, and wish all the entrants the best of luck.

The History of "Using Equations and Stuff"

As a kindergartner falling madly in love with science and science fiction, the chalkboard had a symbolic mysticism to it. It was a place where brilliant people made discoveries and had eureka moments. A person could walk up to a chalkboard knowing only a little bit about something and after writing some symbols down, “A-HA!”, walk away knowing a great deal more about that something. I wished that someday I too could wield the power of this symbolic magic, and that I too would have eureka moments. It took more than a decade for me to get my kindergartner's wish, but it did come true.

Today I want to share the history of humans wielding such magic. Let’s explore an example:

What happens to the liquid level when a floating freshwater cube melts in a glass of heavily salted water?

Your instinct might be to run to the kitchen and give it a try, but don’t; we can get an answer much faster with only a pencil and paper. I didn't know the answer to this question when I first stumbled upon it, so did what any trained scientist would do: I sat down, drew some diagrams, did some algebra, and got an answer in a few minutes.

DISCLAIMER: It’s not at all important that you understand the following bits of math. Today I'm only interested in the historical and philosophical significance.

The water level will go up. To folks trained as scientists this will be trivial, but to most folks this will be indistinguishable from sorcery.

Briefly, Δh is the height of the water level. If it’s positive, the level went up; if it’s negative, the level went down. My goal was to figure out whether Δh is greater or less than zero based on the other things I could hypothetically determine if I were actually doing the experiment. In my final evaluation, all I needed to know was that salt water is denser than freshwater and that densities, masses, and areas can’t be negative.

Let’s take a step back to think about this. I didn't know the answer when I sat down… yet I was able to figure it out with nothing more than pencil and paper. Who was the first person to solve a problem in this way? Did they live in Ancient Greece? Rome? Maybe Classical India? Perhaps the Golden Age of Islam? Or Medieval Europe? Nope, nope, nope, nope, and… nope. This achievement certainly came after 1687. Why? Because I used two of Newton’s (1642–1727) three laws of motion, and 1687 is the year these laws were published. This fashion of solving physics problems came even later because Newton actually published his work in the tradition of Ancient Greek geometry; algebraic versions came later.

Newton’s forerunners were Galileo (1564–1642) and Kepler (1571-1630). Galileo’s model was a truly (perhaps the first) theoretical model because he showed that a simple rule like “constant gravitational acceleration” manifests in an observable phenomenon; namely, the parabolic motions of projectiles. Kepler's work on the other hand, while is often referred to as his “Laws of Planetary Motion” is more correctly called an empirical model; namely that planets follow elliptical trajectories around the sun.

An empirical model falls short of a theoretical model because it only proposes that data follow some mathematical shape without offering any reason why. Seventy years after Kepler published his model, Newton determined a theoretical basis in a new theoretical model called the "Law of Universal Gravitation" (which states that the gravitational force between two bodies is directly proportional to the masses of those bodies and inversely proportional to the square of the distance between them). Galileo’s work turned out to be a subset of this law. As for why Newton’s Law of Gravitation works, scientists are still trying to figure that out.

There was no such thing as a theoretical or empirical model (at least not relating to motion) before Galileo and Kepler. Before this there were variations of Aristotlean motion such as the theory of impetus. These ideas, in addition to being outright wrong, had no mathematical basis whatsoever. There is, however, something to be said about models relating to geometry: A theoretical example is Eratosthenes’s 240 BCE determination of the circumference of the earth, and an empirical example is Bhāskara I's 629 CE method of approximating sines; neither of these models characterize motion.

That mathematical models could be used to describe reality (and motion in particular) was the key to unleashing the floodgates of the Scientific Revolution, which began in 16th-century Europe and continues to this day.

The discovery of being able to use symbolic math to characterize reality is, in my opinion, the greatest intellectual achievement ever made by humans. Virtually every technological advancement of the last 300 years owes thanks to the practice of applying algebra to the physical world.

The Wrong Answer From a Little Bit of Right

Example 1:

"I checked my odometer; it's a shorter distance to go over that big hill than around it. Going over will save gas. Don't worry about the gas us use getting to the top of the hill; you'll get all that energy back when you start going down again."

No. Unless you intend on not using your brakes all the way down the hill, the vast majority of that hill-climbing energy is converted to heat (IE wasted) as the brake pads rub against the wheels. You're right only if you're okay with driving 90 mph when you get to the bottom.

Example 2:

"Fizz keepers don't work. Solubility of CO2 is only influenced by the partial pressure of CO2 in the container. Since air has negligible amounts of CO2, pumping air into the container won't do anything at all."

No. It turns out adding pressure from an inert gas slows the speed at which CO2 leaves the soda by lowering diffusion in the gas phase. Fizz keepers work, just not indefinitely.

Example 3:

"Because of the second law of thermodynamics, larger temperature differences between fridge's interior and exterior translate to greater efficiency. Therefore, your refrigerator uses less energy when it's hot outside."

No. The rate of heat transfer across the refrigerator's walls is proportional to the temperature difference between the fridge's interior and exterior. While your refrigerator might use less energy to cool down the air inside, it has to do so more often, so more energy gets used.

---

You'll notice a theme here: each statement starts off by outlining a bit of knowledge, but still gets the wrong answer. I can only tear into these lines of logic because hindsight is 20/20. I, at one point, thought all of these were correct. 

What kills me is how much sense such statements seem to make until you know better. The real world is complicated. The only way to get good at explaining it accurately it is to make mistakes, but also recognize and correct prior mistakes as more knowledge is acquired. A little knowledge goes a long way... but if it stays little it's likely to cause damage or waste you money.

Two Kinds of "Coincidence"

Two kinds of "coincidence" that at first glance, might seem kind of similar.

1. The moon and sun are about the same size when viewed up in the sky. This makes solar eclipses really neat to look at.
2. The time it takes for the moon to rotate and go around the earth are the same. From this, the same side of the moon always faces the earth.

There is a huge (HUGE) difference between the nature of these two coincidences. To find out, all you have to do for each example is ask "Why?".

For the 1st example, it's just dumb luck it worked out that way. The moon can orbit the earth at any distance; there's no reason why it should appear the same size as the sun. For the 2nd example, tidal forces lock the moon's period of rotation to be the same as it's period of revolution. If you're interested in learning more about this you can watch my video here: http://www.youtube.com/watch?v=Spz_IHYeoLY

The point I want to make is the 2nd example has a much more satisfying answer than the first. "It just is" is an answer that has never satisfied anybody, but it turns out most good questions have answers that have a lot more substance. I didn't understand the significance of this difference until late in high school. Coming to understand this difference was life-changing for me at the time; it made me care about science in completely new ways.

There are real reasons for say, why a ball's trajectory traces the path of a parabola, or that a plant gets four times as much light when it's half the distance from a bulb, or that exponential functions perfectly model the decay of radioactive atoms, or a cue ball stays still when it smacks another ball straight on... none of these are dumb-luck coincidences. There are perfectly satisfying answers for "Why?" in all these cases.

I was lucky to find satisfying answers before I stopped asking "Why?". Tragically, I think most people don't. There's a lot to be gained in re-invoking people's curiosity.

The Quest for the Oracular Chalkboard

One evening I was writing a video script about ice-melt. I had to take pause because I wasn't sure if water level goes up or down when a floating freshwater cube melts in saltwater. I did what any practiced scientist would do: I made a sketch, assigned some variables, did some algebra and got an answer in a few minutes: it goes up. I've done this in my career as a scientist more times than I can count, but because I was trying to communicate this to a general audience, something hit me:“I wanted this…” For years as a child I wanted to be able to do this very thing, wondering if it was real.

As a kindergartener falling madly in love with science and science fiction, the chalkboard had a symbolic mysticism to it. It was a place where brilliant people made discoveries and had eureka moments. A person could walk up to a chalkboard knowing only a little bit about something, and after writing some symbols down, “A-HA!”, walk away knowing a great deal more about that something. I wished that someday I too could wield the power of this symbolic magic, and that I too would have eureka moments. It took more than a decade for me to get my kindergartener’s wish… but it did come true.

I remember thinking when I was eight, even though I’d been doing math for as long as I could remember, it looked nothing like the kind that scientists used. First and foremost, my math didn't have any letters in it. The matter was simple; once my math had letters, I’d start having eureka moments. To this day I can recall my sheer disappointment when the first letters came in the form of “Line PQ” and “Triangle ABC”; ten-year-old me would have to wait a while longer. The next year my teacher let us in on a secret after a week of doing the sort of problem where you have 6 oranges on one side of a balance and 2 oranges and a mystery box on the other. “This is algebra,” she explained, “just label the box as x.” I remember being excited. To say this feeling was premature would be an understatement. The wait continued.

The start of middle school set the stage for more disappointment. Letters (variables, rather) were used in formulas like the area of a triangle, the period of the wave, the quadratic equation, etc. I felt cheated that formulas "seemed to be about more math” rather than the real world. Things like finding the heights of a tree with a protractor were neat, but hardly what I was after. The other source of constant torture was my calculator. Even though it was the kind the school had told me to buy, it had all these mysterious buttons on it. Nobody could tell me what the “cosh” or the “Σ” button was for. They looked science-y and there they were… taunting me. All I could get from teachers was "just wait." Mind you, this was before Google so I was on my own.

The chapter introductions to the school math book had these interviews with people who use that chapter’s math in their line of work. In one example a policeman explained a formula that used a square root to determine a car’s cruising speed from the length of skid marks. If I’d had the words to ask at the time, they’d been along the lines of “So, is there a real reason for using a square root, or did somebody just notice that the graphs for skid marks and square roots kinda match?” Other formulas would show up in science class (such as the period of a pendulum) but since nobody would show me where these formulas came from I began to suspect that scientists created formulas just by trying different arrangements of variables until some formula just happened to match data and/or units worked out. It did occur to me that the graph of a quadratic looks exactly like a ball flying through the air, but I didn't know how to ask if this was by coincidence or for a real reason.

With high school starting I began to believe the oracular chalkboard was a myth. I remember thinking that the chalkboard scenes from “The Day the Earth Stood Still” must have been glamorized. I continued to be unimpressed by the now common formulas in geometry, algebra, trigonometry and chemistry.

When I was 16 I placed into advanced physics. I went in not expecting much, but I was in for a surprise. Just in the first week it was clear just how wrong I’d had it; the oracular chalkboard was REAL. For the first time it was clear that math describes real-world phenomena for real reasons, and because of this, anyone who knows algebra can use math to learn about the real world. I remember feeling silly for suspecting otherwise, but considering how long I waited…

My last year of public school I took advanced chemistry and calculus. My experience was much the same as that in physics, and this would continue on as a college student of chemical engineering. It was plain as day now: because math describes the natural world in profoundly effective ways, to learn from the natural world we do math.

Here’s what kills me: Most people don’t take algebra-based physics in high school or college. Most people don’t get to see firsthand how someone can walk up to a chalkboard and teach herself something without doing an experiment. If I hadn't had this opportunity before leaving high school I could have easily went the rest of my life thinking the oracular chalkboard was a myth. If the average person isn't given the tools to experience this first hand, I think they should at least be able to consider the implications. I think most people understand that algebra is a tool that scientists use all the time, but they have no idea how or why it’s used.

In my experience, the average person has some vague notion that algebra is somehow powerful and that somewhere scientists use it for… something. The average person doesn't know it can be used for everyday musings such as freshwater cubes melting in salt water. It’s as if we’re showing children the various contraptions of a helicopter year after year, but most have no idea the blasted thing can fly. Only a select few students get to the point where they understand “Yes, I just saw a machine take off into the air. Yes, this is real. Yes, I can learn how to do this myself. Yes, I can use this tool to explore the world in a whole new way.” Without this perspective, it’s no wonder many opt to abandon the machine on some rooftop to rust.

There are plenty of skills that are by no means required to succeed in life: playing music, drawing, dancing, cooking, or speaking a foreign language. To respond to xkcd’s question over why people are so proud to have not learned math: with these other skills it’s clear the sort of amazing things that can be accomplished with mastery of those skills. I think people who take pride in never having used algebra literally have no idea the kinds of amazing things that can be done with it.

My two cents. Thanks for reading.

New article posted on LiveScience.com today!

If you've ever wondered where the quadratic formula came from, now you can find out.

What Are Quadratic Equations?

Cheers All.

Mathematics Crossing the Lines of Civilzation

There's a number of reasons for which I find in Eurocentric teachings of the history of mathematics harmful, but today I'll focus on one facet in particular: incompleteness. When I say "incomplete" I'm not saying "this isn't a compendium of the histories of all human mathematical activity". I specifically mean leaving out pieces of the story of mathematics as it is practiced today, which stretches in an unbroken (though perhaps split) line from present times all the way back to the the first civilizations of Egypt, Mesopotamia, and the Indus Valley.

While a teaching that focuses on Europe isn't necessarily bad (certainly great things in the history of mathematics have happened there...), I do see intrinsic harm in failing to contextualize these contributions. As I've stated previously on this blog, the progress of algebra over the ages can be found in:

• Procedural Babylonian and Egyptian texts around the 19th-century BCE
• Procedural Chaldean texts from the seventh century BCE
• Procedural Greek texts from the fourth-century BCE
• Procedural Indian texts from the fifth-century CE
• Rhetorical and Syncopated Arabic texts from the ninth-century CE, and
• Syncopated and Symbolic European texts from the 11th-century CE

But Robert! What's making you say this is all one coherent narrative? Isn't it completely possible that some of these civilizations developed mathematics independent of one-another?

That is an excellent question. It is possible... likely even, but only for one of the civilizations listed here. It is demonstrably NOT the case for the rest. This blog entry will focus on the evidence that shows how mathematics was passed from civilization to civilization (and consequently, how when the mathematics of say, Europeans, is studied in great detail, it needs to be understood as a piece of a much bigger narrative).

As an aside, up until the beginning of the Scientific Revolution in Europe, significant advances in mathematics almost invariably had close ties to astronomy.

Babylonia → Chaldea

Okay, this isn't so hard to imagine. Same geographic region. They both wrote in cuneiform and used sexagesimal. It's pretty clear Chaldean tables have a lot in common with Babylonian star catalogs. Moving on.

Chaldea → Greece

The conquests of Alexander the Great were from 335 BCE to 324 BCE and established the reaches of the Hellenistic kingdoms as far as India.

The first known trigonometric table was compiled by Hipparchus of Nicaea (180 – 125 BCE) who divided the circle into 360 degrees and fractions of chord lengths in sexagesimal (base 60). Paraphrased from page 107 of "A History of Mathematics, 3rd Ed." by Uta C. Merzbach and Carl B. Boyer: Hipparchus likely adopted this convention from Hypsicles (190 – 120 BCE), who wrote an adaptation of a work explaining a Babylonian technique for computing the rise times (at the latitude of Alexandria) of the signs of the zodiac. Both Hypsicles's adaptation and the Babylonian text divided the ecliptic into 360 degrees.

Also, on page 3 of “An Introduction to the History of Algebra” by Jacques Sesiano, it reads: "Ptolemy (A.D. 150) mentions... that since the beginning of the reign of Nabonassar (747 B.C.), "the ancient (Mesopotamian) observations are, on the whole, preserved down to our own time."

Chaldea & Greece → India

This is the hardest one, since it's not clear how much Indian mathematics originated on the subcontinent from the time of the Indus Valley Civilization (flourished between 2600 – 1900 BC) onward, and how much of it came from the Hellenistic kingdoms or prior contact with Mesopotamia.

The Āryabhaṭīya (499 CE) describes degrees and sexagesimal in verses 3.1 and 3.2:

• 1 revolution = 12 signs
• 1 sign = 30 degrees
• 1 degree = 60 arc-minutes (kalās)
• 1 arc-minute (kalās) = 60 arc-seconds (vikalās)
• 1 arc-second (vikalās) = 60 arc-thirds (tatparās)

Based on what I've said so far, this might appear Babylonian in origin, but this is tricky because the degree (though apparently not sexagesimal) appear in the Rigveda (most likely between 1500 – 1200 BCE, though possibly between 1700 – 1100 BCE):

Twelve spokes, one wheel, navels three.
Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.
     — Dirghatamas , Rigveda 1.164.48

For what it's worth, the Egyptians and Mesopotamians (thus completing ancient trifecta of Egypt, Mesoptoamia, and the Indus Valley) were also both fond of cutting the sky into 12 sections (one for each lunar cycle or 'month') and within each section, taking particular note of 3 stars or groups of stars.

http://en.wikipedia.org/wiki/Babylonian_star_catalogues#Three_Stars_Each

http://en.wikipedia.org/wiki/Decans

Anyway, if there was any one civilization to develop mathematics independently of the others on this list, it would have been India.

Greece & India → Arabia

The following is based almost entirely on page 205 of "A History of Mathematics, 3rd Ed." by Uta C. Merzbach and Carl B. Boyer:

The first century of Muslim Empire had been devoid of scientific achievement. Arab conquerors fought amongst themselves and with their enemies until about 750 CE when their warlike spirit subsided. Under the rule al-Mamum, the 7th caliph of the Abbasid Caliphate, a House of Wisdom comparable to the ancient Museum in Alexandria was built in Baghdad. This institution served as a focus of medieval scholarship until it was destroyed by the Mongols in the 13th century. 

In this time, Al-Mamun is said to have had a dream in which Aristotle appeared and became determined to have made Arabic versions of all of the Greek works he could lay his hands on. From the Byzantine Empire, with which the Arabs now maintained an uneasy peace, Greek manuscripts were obtained through treaties.

Delegations of scholars from India were also invited to share their knowledge of mathematics and astronomy. Probably one of the more significant examples of mathematics crossing lines of civilization is the use of Brahmangupta’s Khandakhadyaka by House of Wisdom scholar Al-Biruni (who amazingly, read it in the original Sanskrit).

Arabia → Europe

Since the fall of Western Rome in the 5th century CE, Medieval Christian Europe had been practically devoid of scientific achievement. This changed significantly in the 11th-13th centuries with the transmission of knowledge to Europe from the Muslim Empires, primarily through the Iberian Peninsula, which was known to the Arabs as Al-Andalus.

Particular points of transmission to Europe were the 1085 conquest of Toledo by Spanish Christians, the 1091 re-claiming of Sicily by the Normans (after the Islamic conquest in 965) and the Crusader battles in the Levant from 1096 to 1303. Additionally, a number of Christian scholars such as Constantine the African (1017-1087), Adelard of Bath (1080-1152) and Leonardo Fibonacci (1170-1250) traveled to Muslim lands to learn sciences.

http://en.wikipedia.org/wiki/Islamic_contributions_to_Medieval_Europe

Europe Onward

The reason the rest of the world continues this now 4000 year-long story is because Europe, from the 16th century onward, became obsessed with a little something called "colonialism". The Pan Afro-Eurasian heritage of mathematics was of course key (it still is) to the Scientific Revolution and the Industrial Revolution, both which have continued to this day.

So yeah... pretty good reasons to think it's all one narrative.

Crafting a prolate spheroid

It's time for a Mathematical Adventure!

http://i.cdn.turner.com/v5cache/CARTOON/site/Images/i70/adventure-time.png

Last year, this little guy here sent me on a journey:

https://www.youtube.com/watch?v=Lt9mjoaO2oc

As a birthday gift, my wife offered to make me a stuffed version:

I am so happy!

There was one condition: I needed to determine what fabric pattern cut so that the bee would come out as nice as possible.

I'd actually thought about a similar problem a few years prior. My college friend Warren had called me over winter break 2007 expressing frustration that a particular family tradition involved far too much guesswork and there had to be "some kind of math" that would fix it.

The tradition in question was these Christmas ornaments made from foam balls upholstered with petals of fabric; in my family we called them "tannenbaums" (though after a brief bit of Googling this might be something my brother made up).

https://craftsncoffee.files.wordpress.com/2014/11/diy-fabric-ornaments.jpg?w=640

It took some thought, but I figured the best shape of petal would be the spaced enclosed by two out-of-phase sine waves:

Made using Microsoft Math 4.0: http://www.microsoft.com/en-us/download/details.aspx?id=15702

Understanding this, I figured it couldn't be that much harder to figure out what "petal shape" would make a prolate spheroid. Wow, was I in for a surprise...

Looking at the drawing, I estimated the aspect ratio of the bee to be 1.25, meaning the bee was 1.25 times longer on one axis than it was on the other two. My first thought was just to stretch the petal shape along the horizontal direction, but immediately it was clear this wouldn't work. Since the overall shape needs to be perfectly round at the tips, the angle of each petal still needs to be the same: 360° divided by however many petals I decide to use. If I stretch the petal in the horizontal direction, this would change the angle at the ends of the petals. This would turn the ball not into a prolate spheroid, but a football.

I was not about to let my bee have the shape of a football! I'd have to find the shape of some weird curve I'd not heard of before. Hmm...

Imagine being small and standing on the surface of a proloate spheroid, marching from the equator to the north pole. The horizontal coordinate of our curve (u) should be how far we've traveled along the surface of the prolate spheroid. The vertical coordinate of our curve (v) should be a twelfth (since we're using six petals, and a petal is twice as tall as the curve we're looking for) of the circumference of the circle of latitude where you're at. Think of the cross section produced when slicing a lemon; you get circles. That's the circumference we're talking about.

I was pretty sure I'd end up using the formula for arc length.

http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx

Now I just needed to plug in the equation for an ellipse:

http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

DO NOT TRY THIS. IT WILL ONLY END IN TEARS.

Basically, it leads to a really mean integral. I cheated by looking at the answer, and I still have no idea what happened. Most students leave public school knowing the formula for a circle's circumference: C=2πr. Now, try to think back if you've ever heard the formula for an ellipse's circumference. I can almost guarantee nobody has ever even mentioned it, and here's why:

http://en.wikipedia.org/wiki/Ellipse#Circumference

Never-mind that this is the formula for just the circumference and I need pieces of this. I'm not even sure what coordinate system to use! Screw this. Bye bye. I need to start over.

When in doubt, solve the problem numerically

Lets define the spheroid as having semi-axes lengths of r_x=1.00, r_y=1.00, and r_z=1.25 (these are kind of like radii). Now we just need to numerically determine u(z) and v(z) and plot the curve of v vs. u. to form the petal pattern which my wife will use to make the bee.

It'll be convenient to define r(z). This is the radius of the cross section of each slice along the z axis. Since the side view of the prolate spheroid is also an ellipse, we can use the following equation:

Made using the equation editor in Power Point.

From here, v(z) = 2π*r(z)/12 .

Determining u(z) is harder. First we have to define n=100 or so points so that z₀=0 and z_n=1.25 . We then use the Pythagorean Theorem to figure out the distances between each (r_i, z_i) and (r_i+1, z_i+1), and cumulatively add everything together:

We get the following curves:

r(z) vs. z and v(z) vs. u(z)

Now it's a simple manner of making a printout my wife can use:

Yaaaay! It's done.

So... that was interesting or as Finn would say:

http://media.giphy.com/media/9lMoyThpKynde/giphy.gif

New article posted on LiveScience.com today!

What Is Algebra?

http://www.livescience.com/50258-algebra.html

I've been wanting to write about the history of Algebra since I first started with LS. In my first draft I made the mistake of trying to get an explanation out of the way so I could talk about the history. I have to thank my editor for his insistence that it didn't connect, and that we really had to pander to the average Googleing passerby. I kind of gave up on the topic, but to my surprise, after some time away, my editor expressed interest in picking it back up.

Somehow I'd forgotten how effective the technique of using a topic's history to explain it is. (See my articles on the transistor or quantum mechanics). I'm really happy with how this came out.

Why I tell the story of why an hour has 60 minutes

Next month I’m giving a talk on why an hour has 60 minutes. (Thanks @cafesciboston!) This story encompasses the progress of science from before the first writing, exploring contributions from Ancient Sumer, Babylonia, Greece, India, the Golden Age of Islam, and the early days of the Scientific Revolution in Europe. You can find the original article here:

http://www.livescience.com/44964-why-60-minutes-in-an-hour.html

I was told to be ready for questions on how I discovered this story and what drove me to tell it. What follows is a result of that prompt.

---------------------

This started with a series of dinner debates with a theologian friend. As a disclaimer on my atheism, let me say: whatever drives you to go out and look at the world, kudos. If it’s wanting to know the mind of God or wanting to understand God’s creation, I can’t criticize whatever motivation you have for driving the wheel of science forward.

I didn't always feel this way, and as a counter to my then opinion that religion makes people unscientific, my friend vehemently argued that the Christian Enlightenment was responsible for the Scientific Revolution in Europe. I was convinced there had to be more to it than this, and I was also sickened by the undercurrent that, when boiled down, amounted to little more than "because our civilization has the right god, we were the ones who got the bomb."

This was when I started researching the history of algebra (something I was curious about anyway). On page 4 of “An Introduction to the History of Algebra” by Jacques Sesiano, it reads:

"Several centuries of continuous [Mesopotamian] observations provided an invaluable body of data for the computation of planetary periods… As it would have been an overwhelming task to convert all these data into the decimal system… the Greeks maintained the sexagesimal system for astronomical measurements… This was also adopted by the Indians as early as antiquity… Then… it reached the Muslims who in turn transmitted this notation to medieval Christian Europe. The sexagesimal division… still used today is thus a living witness to the sexagesimal base once used by the Sumerians… in prehistoric times."

After picking my jaw up off the floor, I started looking at maps and reading about Babylonia, Greece, India, and Arabia. Later I took a course in science writing and decided to use the topic for my final paper.

Setting aside that this story is amazing… it is also incredibly useful! Firstly, because it encompasses the progress of science from before writing up until the Scientific Revolution. Knowing this story, I now have context when thinking about anything else from history.

The second reason concerns representation of ethnic minorities. This is a sensitive topic, so let me start with an image I saw the other day on Facebook.

Link here

Some commenters wanted to know about the quadratic formula. Since the ancient Babylonians, Greeks, Indians, and medieval Muslims all used some progressively advanced version of the quadratic formula, it’s hard to summarize in a list, and is thus not so great for going up on Facebook. Yet, something very dangerous has happened here!

This list goes from the 6^th century BCE Europe to the 17^th century CE Europe. Inadvertently (I hope), this creates the impression that mathematics is from Europe, and thus Europeans are somehow superior or representative of “true civilization.” If you don’t see this as a problem, allow me to point at (with a 10 foot pole) the unfathomable number of YouTube commenters who insist something along the lines of “white people invented science and other races should be thankful.”

An amazing set of circumstances triggered the scientific revolution in Europe, and it’s been fascinating looking to look at scholarly reasons why people think it happened there rather than India, Arabia, or China.

I view Newton as a linchpin for the following flood of applying mathematics to motion and the physical world. His Law of Gravitation has two halves: the terrestrial from Galileo for describing the parabolic motions of projectiles, and the celestial from Kepler and Brahe for describing the elliptical motions of the planets around the sun. Here are my favorite explanations for why these two halves happened in Europe:

The celestial half concerns mechanical clocks. According to David Landes, author of “Revolution in Time”, It gets cold in Europe, which is why Europeans went to the trouble to develop mechanical clocks that didn't rely on water. Clock chimes also set the rhythm of life for medieval Europeans, but this wasn't culturally important to the Chinese. The first mechanical clocks couldn't compete with water clocks in terms of accuracy, but their potential to do so was staggering. Tycho Brahe absolutely needed clocks accurate to within 4 seconds of the Earth's time to take the data upon which Johannes Kepler based his laws of planetary motion.

The terrestrial half concerns Renaissance art. According to Joseph W. Dauben, Professor of History at CUNY, it is because Renaissance artists became obsessed with accurately portraying reality in art that Galileo similarly became obsessed with accurately portraying reality using mathematics. I've not been able to answer if traditions in the medieval art of India, Arabia, or China would have been prohibitive of similarly inspiring great minds from those parts of the world… but it’s an interesting idea.

As a reply to my theologian friend (who has since lost interest in discussing this with me): you’d be correct to argue that Christianity is responsible for the culture of clock chimes and sponsoring great works of art… but this is far cry from the philosophy of the enlightenment being responsible for the scientific revolution.

Nationally, I’m American (US-ian, as my Canadian brother-in-law says), as were my mother and father. Ethnically, I’m half mixed European and half Chinese. While I do occasionally encounter blatant, personal racism, lately I've become more aware of the kind of insidious, subtle racism that comes with inadequately representing people of color. I think one of the best countermeasures to this problem is to tell stories like this. Such narratives eviscerate the idea that “white people invented science,” and they give context for other discoveries throughout history. It is my hope you’ll fall in love with the story too, and tell it to other people.

Ancient Babylonian Mathematics: Tablet VAT 8389

If the history of mathematics interests you, I highly recommend Jacques Sesiano's book “An Introduction to the History of Algebra“ (AMS, 2009). I particularly enjoyed the first chapter on ancient Babylonian algebra. There's one thing that I was really missing though... images of the original text. Digging just one of these images up sent me on something of a journey and sadly I can only use them for educational, non-profit purposes (which is why this is on my blog). The word "text" might give readers the wrong idea, so let me just throw this out there:

Photo credit: © Staatliche Museen zu Berlin – Museum of the Ancient Near East | Photo: Olaf M. Teßmer, May 2010 | USED WITH PERMISSION

What you're looking at is tablet VAT 8389 from the Museum of the Ancient Near East in Berlin, Germany. The tablet contains the solution to system of linear equations that today we'd write like this:

x + y = 1800

⅔∙x - ½∙y = 500

Such representation is anachronistic to say the least.

Today I'm going to pull out everything I know about this tablet: transliteration, literal and adapted translation, mathematical interpretation, and comparison with modern mathematics.

The Babylonians didn't have lined paper, so I've corrected that here:

Photo credit: © Staatliche Museen zu Berlin – Museum of the Ancient Near East | Photo: Olaf M. Teßmer, May 2010 | USED WITH PERMISSION | Edited by me in PowerPoint

Transliteration.

The following transliteration appears in Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Springer, 2013) by Jens Høyrup. I am no expert in ancient pronunciation of ancient Babylonian cuneiform (my background is in chemical engineering...). This was hard to type and I am not positive I got all the characters right.

I-1           i-na bùr 4 še.gur am-ku-us

I-2           i-na bùr ša-ni-im 3 še.gur am-ku-us

I-3           še-um ugu še-in 8,20 i-ter

I-4           garim ĝar.ĝar-ma 30

I-5           garim en.nam

I-6           30 bu-ra-am ĝar.ra 20 še-am ša im-ku-sú ĝar.ra

I-7           30 bu-ra-am ša-ni-am ĝar.ra

I-8           15 še-am ša im-ku-sú

I-9           8,20 ša še-um ugu še-im i-te-ru ĝar.ra

I-10        ù 30 ku-mur-ri a.šà garim.mes ĝar.ra-ma

I-11        30 ku-mur-ri a.šà garim.meš

I-12        a-na ši-na ḫe-pé-ma 15

I-13        15 ù 15 a-di si-ni-šu ĝar.ra-ma

I-14        igi 30 bu-ri-im pu-tur-ma 2

I-15        2 a-na 20 še ša im-ku-su

I-16        íl 40 še-um lul a-na 15 ša a-di ši-ni-šu

I-16a      ta-aš-ku-nu

I-17        íl 10 re-eš-ka li-ki-il

I-18        igi 30 bu-ri-im ša-ni-im pu-tur-ma 2

I-19        2 a-na 15 še-im ša im-ku-sú

I-20        íl 30 š-um lul a-na 15 ša a-di ši-ni-šu

I-20a      ta-aš-ku-nu íl 7,30

I-21        10 ša re-eš-ka ú-ka-lu

I-22        ugu 7,30 mi-nam i-ter 2,30 i-ter

I-23        2,30 ša i-te-ru i-na 8,20

I-24        ša še-um ugu še-im i-te-ru

II-1         ú-sú-uḫ-ma 5,50 te-zi-ib

II-2         5,50 ša te-zi-bu

II-3         re-eš-ka li-ki-il

II-4         40 ta-ki-ir-tam ù 30 ta-ki-ir-tam

II-5         ĝar.ĝar-ma 1,10 i-gi-a-am ú-ul i-de

II-6         mi-nam a-na 1,10 lu-uš-ku-un

II-7         ša 5,50 ša re-eš-ka ú-ka-lu i-na-di-nam

II-8         5 ĝar.ra 5 a-na 1,10 íl

II-9         5,50 it-ta-di-kum

II-10       5 ša ta-aš-ku-nu i-na 15 ša a-di ši-ni-šu

II-11       ta-aš-ku-nu i-na iš-te-en ú-sú-uḫ

II-12       a-na iš-te-en sí-im-ma

II-13       iš-te-en 20 ša-nu-um 10

II-14       20 a.šà garim iš-te-at 10 a.ša garim ša-ni-tim

II-15       šum-ma 20 a.šà garim iš-te-at

II-16       10 a.šà garim ša-ni-tim še-ú-ši-na en.nam

II-17       igi 30 bu-ri-im pu-tur-ma 2

II-18       2 a-na 20 še-im ša im-ku-sú

II-19       íl 40 a-na 20 a.šà garim iš-te-at

II-20       íl 13,20 še-um ša 20 a.šà garim

II-21       igi 30 bu-ri-im ša-ni-im pu-ṭur-ma 2

II-22       2 a-na 15 še-im ša im-ku-sú íl 30

II-23       30 a-na 10 a.šà garim ša-ni-tim

II-24       íl 5 še-um ša 10 a.šà garim ša-ni-tim

II-25       13,20 še-um a.šà garim iš-te-at

II-26       ugu 5 še-im a.šà garim ša-ni-tim

II-27       mi-nam i-ter 8,20 i-ter

Literal Translation

Next is a word-for-word translation. It's worth mentioning that Babylonian numbers were written in sexagesimal (base 60). For clarity I've included the place value of numerals which the Babylonians inferred by context.

               

I-1           From a bur 04. kur of grain I have collected.

I-2           from a second bur 03. kur of grain I have collected.

I-3           grain over grain, 08,20. it went beyond

I-4           My plots I have accumulated: 30,00.

I-5           My plots what?

I-6           30,00. the bur, posit. 20,00. , the grain which he has collected, posit.

I-7           30,00. , the second bur, posit.

I-8           15,00. , the grain which he has collected

I-9           08,20. which the grain over the grain went beyond

I-10        and 30,00. the accumulation of the surfaces of the plots posit:

I-11        30,00. the accumulation of the surfaces of the plots

I-12        to two break: 15,00.

I-13        15,00. and 15,00. until twice posit:

I-14        Inverse 30,00. , of the bur, detach 00.00,02

I-15        00.00,02 to 20,00. , the grain which he has collected

I-16        raise, 00.40 the false grain; to 15,00. which until twice

I-16a      you have posited,

I-17        raise, 10,00. may your head hold!

I-18        Inverse 30,00. , of the second bur, detach 00.00,02

I-19        00.00,02 to 15,00. , the grain which he has collected

I-20        raise, 00.30 the false grain; to 15,00. which until twice

I-20a      you have posited, raise 07,30. .

I-21        10,00. which your head holds

I-22        over 07,30. what goes beyond? 02,30. it goes beyond

I-23        02,30. which it goes beyond, from 08,20.

I-24        which the grain over the grain goes beyond

II-1         tear out: 05,50. you leave

II-2         05,50. which you have left

II-3         may your head hold!

II-4         00.40, the change, and 00.30 the change,

II-5         accumulate 01.10 . The inverse I do not know.

II-6         What to 01.10 may I posit?

II-7         which 05,50. which you head holds gives me?

II-8         05,00. posit. 05,00. to 01.10 raise.

II-9         05,50. it gives to you.

II-10       05,00. which you have posited, from 15,00. which until twice

II-11       you have posited, from one tear out,

II-12       to one append:

II-13       The first is 20,00. , the second is 10,00.

II-14       20,00. is the surface of the first plot, 10,00. is the surface of the second plot

II-15       If 20,00. is the surface of the first plot,

II-16       10,00. the surface of the second plot, their grains what?

II-17       Inverse 30,00. , of the bur, detach 00,00.02 .

II-18       00,00.02 to 20,00. , the grain which he has collected,

II-19       raise 00.40 to 20,00. . The surface of the first plot,

II-20       raise, 13,20. . The grain of 20,00. , the surface of the meadow

II-21       Inverse 30,00. , of  the second bur, detach 00.00,02

II-22       00.00,02 to 15,00. the grain which he has collect, raise, 00.30

II-23       00.30 to 10,00. , the surface of the second plot

II-24       raise, 05,00. the grain of the surface of the second plot.

II-25       13,20. the grain of the surface of the first plot

II-26       over 05,00. the grain of the surface of the second plot

II-27       what goes beyond? 08,20. it goes beyond.

Adapted Translation

Between the strange grammar, the use of sexagesimal, lack of units (which if written, would have been long outdated [says the guy from the US...]), and ancient methodology, I can't make much of this without differing expertise. The following adapted translation is based on that found in Jacques Sesiano's book: “An Introduction to the History of Algebra“ (AMS, 2009).

"Bur" and "sar" are units of field area. 1 bur = 1800 sar. A sar is about 36 square meters.

"Kur" and "sila" are units of grain volume. 1 kur = 300 sila. A sila is about 1 liter.

      

I-1           The 1st field gave 4 kur for each bur (2/3 sila for each sar)

I-2           The 2nd field gave 3 kur for each bur (1/2 sila for each sar)

I-3           The 1st field gave 500 more sila than the 2nd

I-4           The sum of the fields' areas is 1 bur (1800 sar)

I-5           What is the area of each field?

I-6           The 1st field gave 1200 sila for every 1800 sar

I-7           The 2nd field gave 900 sila for every 1800 sar

I-8           ↑

I-9           The 1st field gave 500 more sila than the 2nd

I-10        The total area of the two fields is 1800 sar

I-11        ↑

I-12        1800/2 = 900

I-13        You now have two values of 900

I-14        (1800)^(-1) = (1/1800)

I-15        (1/1800) × 1200 = (2/3) "false grain"

I-16        (2/3) × 900 = 600 "hold"

I-16a      ↑

I-17        ↑

I-18        (1800)^(-1) = (1/1800)

I-19        (1/1800) × 900 = (1/2) "false grain"

I-20        (1/2) × 900 = 450

I-20a      ↑

I-21        600 - 450 = 150

I-22        ↑

I-23        500 - 150 = 350 "hold"

I-24        ↑

II-1         ↑

II-2         ↑

II-3         ↑

II-4         (2/3) + (1/2) = (7/6)

II-5         (7/6)^(-1) = unknown

II-6         (7/6) × z = 350

II-7         ↑

II-8         300 × (7/6) = 350, z = 300

II-9         ↑

II-10       900 + 300 = 1200, 900 - 300 = 600

II-11       ↑

II-12       ↑

II-13       The 1st field has an area of 1200 sar, The 2nd field has an area of 600 sar

II-14       ↑

II-15       ↑

II-16       What are the yields of each field?

II-17       (1800)^(-1) = (1/1800)

II-18       (1/1800) × 1200 = (2/3)

II-19       (2/3) × 1200  = 800 "grain of the 1st field"

II-20       ↑

II-21       (1800)^(-1) = (1/1800)

II-22       (1/1800) × 900 = (1/2)

II-23       (1/2) × 600 = 300 "grain of 2nd field"

II-24       ↑

II-25       The 1st field produced 800 sila

II-26       The 2nd field produced 300 sila

II-27       800 - 300 = 500

Mathematical Interpretation

To make sense of this I've prepared the following graphic. I've done my best to identify the major events in each part of the procedure.

Image Credit: Self (Power Point)

The Babylonians solved this problem by figuring out how far the area of each field deviated (steps B-G) from half of the total area (step A). Step H has the answer, and steps I-K verify this answer is correct.

Analysis of Method

Working backward from the described method of verification, we can use modern methods to determine the ancient methods. Here we call the deviation from midpoint "z" and reverse engineer its method of determination. How did the Babylonians figure this out? I can speculate until the cows come home... but rigorous answers have been lost to the sands of time.

Image Credit: Self

Modern Method

Here's how the math looks today after filtering through 4000 years of advancement by civilization after civilization, specifically tenth-c. BCE Assyria, the seventh-c. BCE Chaldeans, sixth-c. BCE Persia, fourth-c. BCE Greece, first-c. CE Rome, fifth-c. CE India., ninth-c. CE Arabia, and 11th-c. CE Europe. By the beginning of the scientific revolution (17th-c. CE Europe), the notation and methodology would have been recognizable as something like this:

                x + y = 1800

                ⅔∙x - ½∙y = 500

                y = 1800 – x

                ⅔∙x - ½∙(1800 - x) = 500

                ⅔∙x - 900 + ½∙x = 500

                (7/6)∙x = 1400

                x = 1200

                (1200) + y = 1800

                y = 600

Is our modern method better? Yes. To be fair, it has the advantage of contributions of nearly every great civilization of Eurasia over the last 4000 years. Let us hope these methods survive even the fall of our own civilization.