Moving to Medium

Hello Dear Readers,

Today I'm making the move over to Medium.
https://medium.com/@PrimeViridian

This will be the last post I make on blogger, but all the old links will stay active.

Cheers
-Robert

Mom's Icosahedrons

Mom is a nurse with the Red Cross. Recently she took up the hobby of stringing together syringe-cover tubes to make icosahedrons (30 tubes and 1 piece of string):

I couldn't figure out how she managed each of the tubes being run through exactly twice, so I had her show me when I was out for a visit. I made mental notes so later I could put this assembly guide together: https://drive.google.com/open?id=0B2eEHrHZqClKdlZiOG1wRGVPOVk

When the lesson concluded, she insisted I take enough tubes home so I could make a "Mom mobile." While the icosahedrons have been assembled pretty much since I got home, today I finally hung them up with cotton string and disposable bamboo chopsticks:

This was a fun project. I'm ecstatic to find Mom has the same affinity for 3D geometry that I first started exploring when I was 7. Apparently all she needed to get going was a combination of Pinterest and a preponderance of disposable tubes.

Rabbits in my Garden

Earlier today I was watering my garden and in the back corner I noticed something running around. It’s not uncommon to see rabbits chilling out in my back yard, but the fence I put up has always kept them out of the actual garden. I expected the things running around to run away, but not today… No, today there was a baby bunny INSIDE the garden. It was trying to get out the corner, which made it easy to pick up by the scruff of its neck. I started talking to it:

     “You are in so much trouble.”
     “You’re so small you can get in the fence holes I didn’t bother to patch, didn’t you?”
     “Are you why my peppers didn’t sprout?”
     “You know I eat creatures like you? You’re called ‘meat.’ You know that, right?”
     “You’re a pest! I should exterminate you!”

This went on for a few more minutes and (swearing) I let the thing go.

I went back to watering my garden and saw a second baby bunny in there. I had to sneak up on this one because it was hiding under the Chinese cabbage. This time I got a picture:

Unlike the first baby bunny, this one made a lot of noise; so much noise that this full-grown bunny ran out in response:

I realized this was probably the mother. I yelled at it for making things more complicated.

Out of the corner of my eye I saw a third baby bunny running around; it was probably just excited mama was there:

I did not pick this one up.

The mom remained still, just sitting there:

I realized this animal must be too big to get through the holes her babies used, and now it wants me to get her third baby out. 

A prey animal had made itself vulnerable to me. I eat meat. I had a desire to kill this animal and eat it, but the neighbors (who definitely heard all the swearing when I let the first bunny go), and in particular the lady next door, would NOT be okay with me slaughtering wild/feral rabbits. Particularly heavy on my mind is my cat (indoor/screenporch/leash only). Callisto will always eat meat. This would be the best free-range grass-fed cat food ever! I would adore the opportunity to slaughter such a fine meal for my pet.

I was standing there so long (just staring at this rabbit) that I called my dad. I expected him to tell me I was crazy and that I should just let the third bunny go and find something else to worry about. My dad said he couldn't tell me what I want to hear, though we did talk about it for a while. I have a habit of pacing while I'm on the phone. At one point I was within 3 ft of the mother. While she looked scared, she didn't move at all.

Still on the phone, I got worn out to the point where I just left. A half-hour later, the mother was still sitting on the lawn. I took this photo from my rear window.

A cat eventually scared the mother away. The cat just wandered back the way it came.

It's dark now, and as far as I know, the third baby bunny is still in the garden. It certainly has enough food in there, and the mother will probably stay nearby. Tomorrow when I water the garden I'll probably just do what I did today: I'll pick the third bunny up, it'll holler, it's mama will come running, I'll let it go, and then I'll just leave this wild/feral warren alone.

I'll also fix the fence.

As an aside, if the city and my apartment-management company are okay with with it, I'm now considering raising a few rabbits for meat, only I won't catch wild/feral bunnies from my garden. I figure domesticated rabbits would be a better suited to cages and my neighbor wouldn't be on my case about cruelty to the native wildlife. We have plenty of basement space we could use in the winter too. I'll file rabbits on the "someday" list, and for now accept some level in hypocrisy in eating meat and letting the garden pests go. I never expected gardening to challenge me morally. Send help.

Have a nice weekend everyone.

Sustainable Fuels Question 1

I occasionally get questions about sustainable fuels. I'm happy to answer them, but I hate losing track of answers on Facebook, so from now on my answers go on my blog!

Today's question comes from an old high school jazz band friend:

Hey Logan. There’s layers to this question. Here goes.

I know for sure you’re not electrolyzing water since methane (CH₄) has a carbon atom in it, and water (H₂O) doesn’t. That methane's carbon has to come from somewhere, so it either (A) started off as in a fossil fuel (and ended up in the atmosphere when you burned it in the rocket), or else (B) started off in the atmosphere as CO₂ and got converted into methane (and thus the burning it to make CO₂ was part of a cycle that didn’t contribute to the total amount of atmospheric carbon).

As a brief aside, electrolysis is the process of zapping things like water or methane to get hydrogen gas (H₂) and leave all the carbon behind. This is useful for folks (with which I disagree) who envision a “hydrogen economy.” To use carbon (as opposed to just hydrogen) sustainably takes more steps, but it’s worth it since carbon-based fuels like gasoline are so much more energy dense and easier to transport than hydrogen. To get the hydrogen to have the same energy in the same space requires dangerously high pressures.

The rest of this blog entry discusses how to accomplish the sustainable option of making fuel from atmospheric CO₂. In order accomplish this it's absolutely necessary that we start with concentrating carbon from the atmosphere. This can either be done with plants like grass and trees (which concentrate atmospheric carbon in the form of a solid chemicals such as cellulose) or big solar-powered gizmos that filter out atmospheric CO₂ for storage in tanks (such gizmos probably won’t ever be as land-space efficient as a plants, but they could work in the desert where plants can’t grow).

After concentrating the carbon, we then need some method of converting it into fuel. Thermochemical conversion methods such as gasification/pyrolysis and several varieties of catalysis can all use some form of solar energy to convert both biomass and bottled CO_2­ into pretty much any carbon-based fuel (be it methane, gasoline, or diesel). Biochemical conversion methods are another set of techniques that all have the disadvantage of not converting CO₂ in tanks, meaning they only work on biomass. Many biochemical methods (such as fermentation to make ethanol) are really inefficient at producing fuel and only work on things we nominally consider food (like cane sugar).

Since you asked about methane, one biochemical method I like quite a bit is anaerobic digestion. Anaerobic digestion works on practically any kind of biomass (not just food) and is much more efficient at making fuel than other biochemical methods. The process makes gaseous methane which is both more convenient than hydrogen since it’s more energy dense, but less convenient than liquids like gasoline or ethanol. Here’s an article I wrote about how Washington D.C. is using anaerobic digestion to get a portion of their power from poop: http://www.thedailybeast.com/articles/2015/10/14/america-s-capital-powered-by-poop.html

Since all these methods take carbon from the atmosphere and turn it into fuel, the burning of that fuel in a rocket is just a part of a cycle that goes plant (or tank) -> fuel -> atmosphere -> and back to plant (or tank). Fuel gathered in this way is sustainable carbon fuel, and it doesn’t contribute to the total amount of carbon in the atmosphere.

Infinitely positive is neither infinite nor positive?

There's a recently publicized British biographical drama called "The Man Who Knew Infinity" about Srinivasa Ramanujan (1887-1920), a mathematician who, after growing up poor in Madras, India, earns admittance to Cambridge University and becomes a pioneer in mathematics. One of the things Ramanujan is particularly remembered for is adding an infinite number of (positive) things together, but getting something that is neither infinite, nor positive. For example: 1 + 2 + 3 + 4 + ... = -1/12

Every time this comes up, somebody asks me how adding an infinite number of progressively larger things can come out negative. Shouldn't the right answer be infinity?

It turns out (depending on what the dots mean) that both answers are correct. To understand how this is possible, suppose Achilles is out for a run and after spotting a wandering tortoise 100 meters ahead, he decides to catch up with it, only he plays a game of catching up to where the tortoise is now rather than where it will be. By the time Achilles catches up to where the tortoise was, the tortoise is 50 meters away. If Achilles again catches up to where the tortoise was, that sneaky tortoise is now 25 meters away. Achilles can keep playing this game indefinitely; to actually catch the tortoise would require catching up to where the tortoise was an infinite number of times.

Despite the infinite number of times Achilles catches up to the previous location of the tortoise, Achilles will, in fact, catch it. The trick is that these distances (as well as the times it takes for Achilles to cover them), get smaller and smaller. If Achilles runs at 1m/s and the tortoise at ½m/s, the infinite sum works out so the duo will meet after 100 seconds and Achilles has run 200 meters. (100m + 50m + 25m + ... = 200m)

This scenario, sometimes referred to as Zeno's Paradox, embodies what mathematicians call a "convergent series." The word "series" means all the terms are being added together and convergent means the sum approaches some number as terms are added. Usually, when terms get smaller and smaller, the series converges (though there are a few exceptions such as the harmonic series.) Convergent series are why, for example, a ball could theoretically bounce an infinite number of times before coming to rest a few seconds later.

Ramanujan's infinite sum is what mathematicians call a "divergent series" because the terms get bigger and bigger so the sum just blows up to infinity. Divergent series can apply to Achilles and the tortoise as well. Suppose Achilles is 100 meters ahead of the tortoise. How long until he catches the tortoise?

• A very good answer is "never." If the race were infinitely long, there's just no way for Achilles to catch the tortoise because Achilles already is ahead of it.
• Another very good answer is "in the past" since Achilles passed the tortoise 100 meters before the starting line 100 seconds before the race started. An equivalent way of saying this is that Achilles will catch the tortoise after negative one-hundred seconds (-100s) having traveled a distance of negative one-hundred meters (-100m).

So what does this have to do with Ramanujan and his infinite sum? Well, the tortoise can catch up to where Achilles was, and he can do this again and again... forever. It's just like before where Achilles closed in on the tortoise, only now the distances (as well as the time it takes to cover them) get bigger and bigger. Adding all these distances together makes a divergent series. Evaluating this sum, it makes sense to get an answer that's infinite... but it also makes sense to get an answer that's both non-infinite and negative. Similar logic can be applied to Ramanujan's sum.

Showing how Ramanujan's sum comes out to -1/12 takes a few more tricks. See below for a ad-hoc derivation:

Reflections on the Anthropocene

Humans are responsible for the spread of countless food/resource plants and animals, human parasites, and other invasive species across constants (particularly from the 16th century onward). We’re also the cause a sixth mass extinction, a measurable increase deforestation and desertification, the shifting of sedimentation patterns from roads and damns, a significant dent in the availability of fossil fuels, a 100-ppm (and counting) increase in atmospheric carbon, the detonating hundreds of atmospheric nuclear devices (thus ruining carbon-dating methods for anything past 1950), and the production of many new minerals such as ceramic, plastic, concrete, aluminum, titanium, and extremely radioactive substances such as corium (the product of nuclear meltdowns).

Anatomically modern humans have only existed for roughly 200 thousand years, and we’ve only been practicing agriculture for roughly the last 10 thousand. The longest-lived civilization ever to grace the earth only lasted 1.5 thousand years, and that’s being incredibly generous. How long into the next million years will we even last? What force will end the last human civilization, and how long after that will it take for the last human to die off?

The next time the evolutionary engine of Earth produces a species capable of discovering geology – be it the descendants of dolphins, bees, or slime molds – that species will be able to tell some interesting tales of an ancient ape who came down from the trees, developed stone harvesting and hunting tools, spread as far as the megafauna could lead them, plopped down to build civilizations around the domestication of specific plants and animals, and quickly developed technologies that altered the geology of the time.

Upon discovering our fossilized remains and our geologic impact, what will they think of us? Will they idolize us or hold us in contempt? Will there be lessons to be gleamed from our failures? With fossil fuels less available, will they be forced to develop renewable technologies? Will they be able to reverse engineer our ancient methods? Will they eventually be able to reach farther than us? Will they come to physically realize realities we never even dreamed of?

The story of humanity thus far is simultaneously amazing, tragic, and understandable. I hope that by the time the end of our story is written in the rocks, it will show that we learned, for a time, how to live sustainably, peacefully, equitably, and happily.

New article on Nautilus today!

This went up today. I've been wanting to write about this particular topic ever since I entered the world of science writing.

How a Mathematical Superstition Stultified Algebra for Over a Thousand Years

The original topic I pitched was to actually show examples of Ancient-Greek and Medieval-Islamic mathematics, but after some back-and-forth with my editor we finally cut them. I don't know who reads this blog, but if you're here you'll probably find these examples interesting.

The Nautilus article mentions the Golden Ratio because it's an irrational value/length that Ancient Greek scholars were able to determine geometrically. Because of the phobia surrounding irrational numbers an algebraic determination didn't come until more than a thousand years later. For the algebraic solution, we can thank Medieval Islamic scholars.

Ancient Greek Geometry:

A rectangle with the golden ratio has the following property:

When you cut a square off a Golden Rectangle, the rectangle remaining has the same aspect ratio as the original, rotated at a right angle, like so:

So how do we construct a Golden Rectangle? And how can we be sure the construction actually works?

The following construction procedure is found in Eudlid's Elements (3rd-century BCE). The cult of the Pythagoreans (6th-century BCE) probably proved this particular construction produced a Golden Rectangle; Euclid just included it in his famous geometry textbook:

1.       Start with a square (shown in red).

2.       Draw a line from the midpoint of the base of the red square to its upper right corner.

3.       Swing this length down to the red square's base.

4.       Using the end point of the swung-down line, complete the rectangle.

To prove this construction produces a Golden Rectangle, certain areas within the rectangle had to be proven equal. To determine which areas, the Pythagoreans started with the one thing they knew about a Golden Rectangle: matching aspect ratios. While what follows is today called "cross multiplying", Euclid refers to it as Book 6 Prop. 17.

Thus, to show the construction produces a Golden Rectangle, the red and blue areas have to be shown to be equal. If the aspect ratio is off, one of these ares will be bigger:

Book 2 Prop. 11 shows that for the construction described above, the red and blue areas are indeed equal:

The first part on the left uses a rule about tacking on lengths to a bisected line (Book 2 Prop. 6). The second part on the right uses the Pythagorean Theorem (Book 1 Prop. 47).

This shows that Ancient Greek scholars figured out how to solve this problem geometrically. The algebraic solution came more than a millennium later in Medieval Baghdad.

Medieval Islamic Algebra:

Algebra began in Ancient Babylonia. The conquests of Alexander the Great in the 4th-century BCE, which stretched from Greece and Egypt to India, likely brought this knowledge to Hellenistic Greece. While Greek scholars did measurably advance the topic, Medieval Islamic scholars took it much further.

In 19th-century BCE Babylonia, a problem like “x² = x + 870” looked like this:

British Museum Tablet 13901

Here's a translation from Babylonian into English (and sexagesimal into decimal).

Problem: I added 870 to the side of my square to get its area. What is the side length of my square?

Solution: You divide 1 (multiplied by the side length) by two, it gives ½. You multiply it by itself, it gives ¼. You add it to 870, it gives 870+¼. It is the square of 29+½. You add ½ (which you multiplied) to 29+½, it gives 30 (the side length). (Adapted from this book)

And here it is again translated into modern notation:

Problem: x² = x + 870

Solution: 

More generally, all problems of this form can be solved using the following formula:

Problem: x² = px + q

Solution:

This "rhetorical form" of algebra was pretty much constant across its advancement under the Ancient Babylonians, Hellenistic Greeks, Classical Indians, and Medieval Muslims. What 9th-century CE Islamic scholars did that nobody else had ever thought to do before was apply these procedures to the quadratic irrational lengths found in Greek Geometry.

Since the aspect ratios are equal, the problem can be represented algebraically. Keep in mind, Medieval Islamic scholars would have written their equations out as sentences, but the steps taken would have been pretty much the same.

x/1 = 1/(x-1)

Cross multiply:

x·(x-1) = 1²

Distribute x across (x-1):

x² - x = 1

Add x to both sides:

x² = x + 1

Since the problem is now in the form of x² = px + q, we can apply the ancient Babylonian solution procedure.

Problem: x² = x + 1

Solution: 

A commentary on Euclid’s Elements by the 9th-century Islamic mathematician Al-Mahani is the first known work that algebraically explores quadratic (and cubic) irrational numbers. Thus Medieval Islamic Scholars were able to algebraically solve problems that Ancient Greek Scholars had geometrically solved more than a millennium earlier.

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.

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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.