Monthly Archives: April 2014

Helicase Hijinks

Geneticist’s pick-up line: I wish I were DNA Helicase so I could unzip your genes.

Okay, it’s slightly off-color, but hopefully not too offensive. As an actual pick-up line, I give it about one chance in ten of succeeding; in the right setting, with the right person, you might do all right.

Being a living organism is hard work. You need a specific set of materials to keep you ticking. You must consume some of those materials from the environment, such as oxygen and food. Other materials are manufactured inside your cells, which requires specialized molecular machinery. Each of your cells has a full set of instructions telling it how to build and operate this machinery. These instructions are known as genes.

A human cell contains between 30,000 and 40,000 genes. Each gene is made of a strand of DNA (DeoxyriboNucleic Acid). You’ve probably seen a model of DNA before. It vaguely resembles a twisted ladder, like this:

The rungs of DNA are quite different from the rungs of a real ladder in that they are not continuous across the width of the ladder. The rungs are actually in two pieces, held together in the middle by a force called hydrogen bonding. When your cells divide, each gene must be copied so that both daughter cells receive the complete set of instructions. This requires “unzipping” the DNA molecule, which in turn requires breaking the hydrogen bonds that hold its halves together.

The process of unzipping DNA is achieved by a special enzyme called DNA helicase. An enzyme is a protein manufactured by your cells to perform specific functions. DNA helicases (there’s more than one type) “crawl” along the length of a DNA molecule, breaking its hydrogen bonds and separating it into two separate strands. Two new strands are then built from the templates of the original strands.

In this video, Hank Green talks about the structure, history, and replication of DNA. He describes the role of DNA helicase in unzipping DNA strands, and he tells a version of this joke. It’s all fascinating but if you want to jump straight to the replication discussion, it starts at 8:50.

DNA helicase is also used in a process called transcription, which is where the DNA code is transferred into single-stranded RNA (RiboNucleic Acid) segments. These RNA chains are then shipped to the cell’s manufacturing district for further processing. Just as in replication, the double-stranded DNA must be unwound and unzipped so the RNA can be built from its template.

So whether you’re replicating or transcribing, if you need help unzipping your genes DNA helicase is your enzyme.

i vs Pi

i vs pi

When we say that a person is being irrational, we mean that his actions aren’t consistent with reason or logic. For example, believing that you’re fit to make legislative decisions about public education when you have no teaching experience – that’s irrational. It completely flies in the face of logic or wisdom.

When we say that a number is rational or irrational, that’s quite a different story. A rational number is one that can be expressed as the ratio of two integers. To put it another way, all rational numbers can be written as a/b, where b is not equal to zero. Consider the number 2. Two is a rational number because you can express it as 2/1, or 4/2, or 56/28, etc. Another example of a rational number is 0.4, which can be expressed as 2/5, or 4/10, or 150/375, etc.

A defining characteristic of rational numbers is that when you write them out in decimal form, their digits either stop, or fall into an infinitely repeating loop. For example, the rational number 5/8 can be written as 0.625. There’s no more after that (unless you fancy writing out lots of unnecessary zeroes). The rational number 1/7 can be written as 0.142857… with the 142857 part repeating forever and ever.

An irrational number is any number that cannot be written as the ratio of two integers. There are infinitely many irrational numbers (just as there are infinitely many rational numbers), but a few well-known examples are the square root of 2 and, of course, pi.

Pi (or π) is what you get when you divide a circle’s circumference (the distance around) by its diameter (the distance across). No matter how large or small the circle is, the ratio between its circumference and its diameter is always equal to pi. Numerically, pi is approximately equal to 3.14159265, but it doesn’t stop there. The digits of pi go on forever without falling into a repeating pattern. Also, there is no fraction you can imagine that is exactly equal to pi (although 22/7 is often used as a decent approximation).

By the way, that’s not just blind conjecture: it has been mathematically proven that pi is quite irrational. Using computers and complicated algorithms, people have determined the value of pi to an astonishing ten trillion digits, and we’re still no closer to reaching the “end” of pi than we were when we started, nor will we ever be.

So pi truly is irrational, and despite i‘s wishes, it can never be anything but irrational.

So what’s up with i? Why isn’t it real?

Let’s back up and review the concept of squaring numbers and finding square roots. You square a number whenever you multiply it by itself. For example, 42 (read as “four squared” or “four to the second power”) is equal to 4 x 4, or 16. Easy so far, right?

Whenever you multiply a negative number by a negative number, the result is positive. So every negative number can be squared, but it will produce a positive result. (-4)2 is also equal to 16, not -16.

Taking the square root of a number is like working backwards; you’re asking what number can be multiplied by itself to produce a given value. If I ask you to tell me the square root of 16, you could correctly answer 4 (since 4 x 4 = 16) or you could say -4 (since -4 x -4 = 16). The most complete answer is ±4 (positive or negative four) although most calculators will only give you the positive square root.

So to review, all positive numbers have two square roots – one positive and one negative. Great…now, how do you find the square root of a negative number? What is the square root of -1?

If you’ve been following along, you’ll recognize the problem. Both positive and negative numbers, when squared, produce positive results. There are no real numbers that can be squared to produce negative results, and if there are no real numbers to do the job, mathematicians are all too happy to imagine a number that works.

So i is not a real number but an imaginary number. It is the square root of -1*. If you square i, the result is -1.

The square root of any other negative number is a multiple of i. For example, the square root of -16 is ±4i, and the square root of -100 is ±10i. There’s a whole number line of imaginary numbers, which mathematicians picture lying perpendicular to the real number line. Presumably the lines intersect each other at 0, or 0i if you prefer.

So no amount of bickering can ever make pi rational or i real…it’s just not in their natures. Of course, if the square root of -2 happens by, they can both turn their ire on him, since root-2 is neither real nor rational.

*Actually, negative numbers get two square roots as well, so the square root of -1 is ±i

Mathematician Madness

Infinite mathematicians walk into a bar. The first orders one drink. The second orders half a drink. The third orders a quarter of a drink. The fourth one orders an eighth of a drink.

The bartender pours two drinks and says “You mathematicians just don’t know your limits.

The bartenders in jokes are a strange and varied bunch; some of them have concentrated hydrogen peroxide on tap; others have an acute understanding of mathematical limits.

Since this is basically a joke about limits, let’s start there…but I do want to come back later and talk about some difficulties you might encounter when you try to serve progressively smaller drinks to infinite mathematicians.

When drinking in bars, know your limits.

When drinking in bars, know your limits.

Let’s look at the number of drinks each mathematician orders: 1, 1/2, 1/4, 1/8, and so on. This is a sequence. A sequence can be any set of numbers – it may be finite or endless, and it may or may not follow a pattern. Since there are infinite mathematicians, we may assume that this sequence is infinite. It also seems to follow a pattern; each mathematician asks for half of the amount requested by the fellow immediately prior to him.

Since you can continue the sequence ad infinitum, multiplying by the same number to get the next term in the sequence, this is called a geometric sequence. Here are some more examples of geometric sequences:

  • Start with 1 and multiply each term by 5 to get the next term: 1, 5, 25, 125, 625, 3125, etc
  • Start with 4 and multiply each term by -1/2 to get the next term: 4, -2, 1, -1/2, 1/4, -1/8, 1/16, etc

Geometric sequences can be further classified as convergent or divergent. Convergent sequences seem to hone in on some particular value. Divergent sequences, on the other hand, balloon out of control. The first sequence above is a divergent sequence; its terms will get farther and farther apart. The second sequence is convergent.

Now imagine that you decided to add the terms in a sequence. If the sequence is finite, that’s easy enough. To illustrate my point, let’s try adding the first ten orders made by the mathematicians:

1 drink + 1/2 drink + 1/4 drink + 1/8 drink + 1/16 drink + 1/32 drink + 1/64 drink + 1/128 drink + 1/256 drink + 1/512 drink = 1023/512 drinks

1023/512 is very close to two, but not quite there. What would happen if you could add infinitely many terms in the sequence?

The idea is that the sum gets forever closer to two. That’s what we mean when we say that the limit of this sum is two.

So our bartender, being well-versed in mathematics, quickly realized that the infinite mathematicians were ordering drinks in a convergent geometric series, and that the sum of the infinite orders would be two drinks. Rather than listen to drink orders for the rest of eternity, the bartender took a shortcut. Let’s hope that every mathematician after the first one doesn’t mind sharing a glass.

Now that we understand the bartender’s response, let’s examine the logistics of serving infinite mathematicians.

Is it even possible for there to be infinite mathematicians?

Maybe, but only if there is infinite mass in the universe. Nobody knows if that is the case or not. Even if the universe is infinitely massive, and there are infinite mathematicians, you wouldn’t necessarily have all of them together in one drinking establishment.

Why not?

At some point the collected mass of the assembled mathematicians will pass a critical limit and they will collapse into a black hole, pulling the bartender, the pub, and perhaps Earth itself along with them. Even before that point it would get uncomfortably stuffy in the building.

Assuming the mathematicians used their advanced mathematical knowledge to avoid collapsing into a singularity (mathematicians can do that, right?), how many of them would actually get something to drink?

An excellent point! (Thank you!) Even though numbers may be divided infinitely, liquid refreshment cannot. Eventually one of the mathematicians’ orders will constitute only a single molecule, and then what? Presumably any mathematician after that is drinking in theory only.

Let’s assume that each drink is one liter, and that they’re all asking for water (just to make things easier for me). One liter of water contains something like 33,420,000,000,000,000,000,000,000 (about 33.4 septillion) water molecules, and that’s how many molecules the first mathematician would consume. The second mathematician would get half as many; the third mathematician would get a fourth as many, and so on. How many times do you have to divide a liter of water in half until there is only one molecule remaining?

Not as many as you might think. (WARNING: Math ahead!) We can write a little equation to calculate how many molecules the nth mathematician consumes.

Start with the assumption that 1 liter of water contains 3.3418×1025 water molecules. The mathematician in position n consumes (3.342×1025)(1/2)n-1 molecules. So which mathematician only gets one molecule?

(3.342×1025)(1/2)n-1 = 1

Divide both sides by 3.342×1025, which gives…

(1/2)n-1 = 2.992×10-26

Break out the natural logarithms.

ln(1/2)n-1 = ln(2.992×10-26)

(n-1)ln(1/2) = ln(2.992×10-26)

n-1 = ln(2.992×10-26) / ln(1/2)

n-1 = 84.79

Finish up by adding one to both sides…

n = 85.79

So the 85th mathematician’s order amounts to little more than a single water molecule. Everybody else will have to be thirsty.

But you forgot to account for the fact that in the short span of time between when the bartender pours the drinks and serves them, some of the water molecules will have evaporated from the surface, which means that the 85th mathematician won’t even get his order (and neither will the 84th, 83rd, 82nd, and a few others, depending on conditions like ambient temperature and humidity.

Yes, and I also neglected to account for the fact that the combined body heat of infinite mathematicians (mathematicians are warm-blooded, right?) in one place would probably flash boil the water anyway, making the whole thought experiment moot.

You may be thinking too much about this joke.

Indeed I am, but that’s the stated purpose of this blog. Good night!

Logical Laffs

Three logicians walk into a pub. The bartender asks “Do all of you want a drink?”

The first logician says “I don’t know.”

The second logician says “I don’t know.”

The third logician says “Yes!”

Prior to entering the pub, the logicians have apparently agreed to answer all questions with “Yes”, “No”, or “I don’t know”, but for some reason neglected to discuss whether they wanted drinks or not.

The bartender asks if all of the logicians want a drink, which is sort of a bizarre thing to ask; maybe the bar was about to close and the bartender was annoyed that he might have to fix three drinks. Let’s think about what each logician knows and see how that affects their answers.

Last call is no time for logic puzzles.

Last call is no time for logic puzzles.

The first logician knows that he wants a drink. If he didn’t want a drink, then he would simply have said “No” to the bartender’s question, since he would have known that not all of them wanted a drink. He doesn’t know whether his companions want a drink, so he replies truthfully “I don’t know.” He could have said “I want a drink, but I don’t know about them” but I suppose that wouldn’t have been a concise enough answer.

The second logician also wants a drink; if he did not want a drink, he would have said “No” to the bartender’s question. He can deduce that the first logician wants a drink, but he still doesn’t know whether the third logician wants a drink or not. “I don’t know” is the most concise truthful answer he can give.

The third logician can only answer “Yes” or “No”, since by now he’s figured out that his two associates do want drinks. He correctly answers “Yes!”; he obviously wants a drink as well.

Now of course this joke ignores a certain amount of natural human ambivalence. It’s possible that one of the logicians truly didn’t know whether he wanted a drink or not. He might have wanted to see what the bartender had on tap before making his decision. He might have been pondering whether it was a good idea to drink when he had an early meeting to attend. But the joke also acknowledges that basic logic has a certain black-and-whiteness to it. Like the good little logic gates in a computer, each logician had only two states: wanting to drink or not wanting to drink.

Interestingly (to me, anyway), if the bartender had asked “Do any of you want a drink?” the logicians’ responses would have changed. If the first logician wanted a drink, he would be compelled to say “Yes” regardless of whether the second and third logician wanted a drink. After that, both logicians would have to say “Yes” as well; they already know that the first logician wants a drink, so the only truthful answer is “Yes”.

If you enjoy logic puzzles, and if you’re not too occupied with other tasks, it’s kind of fun to think about what each logicians’ response would have been based on the bartender’s phrasing of the question and their own particular drink desires. Feel free to construct your own charts showing all the possible permutations and responses. It’s what the logicians would have wanted you to do.

A Superluminal Limerick

There was a young woman named Bright,
Who traveled much faster than light.
She set out one day
In a relative way
And returned on the previous night.

Light is fast…really, really fast. It travels at 299,792,458 meters per second, or about 186,000 miles per second. If you could run at the speed of light, you could run around Earth almost 7 and a half times in one second.

The only thing that seems to be able to travel at the speed of light is light* itself. That’s because light has no mass. Anything with mass – including you and me – must forever travel slower than the speed of light. That might seem counter-intuitive; after all, the speed of light is just a number. Why can’t we keep accelerating until we eventually reach it?

Two very smart guys named Einstein and Lorentz showed that mega-strange stuff happens when you approach the speed of light. For one, the passage of time is altered. Whenever you move, time passes at a different rate for you than it does for somebody who isn’t moving. We don’t notice the difference in our everyday lives; however, if you could accelerate close to the speed of light the effect would be noticeable. The specifics are complicated (no surprise there), but suffice it to say that if you travel close to 299,792,458 meters per second, you’ll return to Earth to find that everybody else has aged more than you have.

Imagine a pair of twins: we’ll call them Alan and Bill. Alan boards a fantastically futuristic starship destined for Proxima Centauri, the nearest star to our Sun. Bill stays on Earth. Alan accelerates at one gee** for half the journey, then reverses his spaceship and spends the second half slowing down. When he reaches Proxima Centauri, he snaps a few pictures, then repeats the journey in reverse, finally returning to Earth.

The round trip takes nearly 12 years of Earth time. When Alan returns to Earth, Bill has aged 12 years. During the trip, however, Alan has reached about 95% of the speed of light. He has aged less rapidly than Bill. From his perspective, the round trip took only 7 years. The twins are now five years apart.

The faster Alan travels, the more dramatic the age difference will be. Supposing Alan could withstand a constant acceleration of 100 gees without turning into mush, he will reach 99.999% of the speed of light. From Bill’s perspective, Alan will return in about 8.6 years. To Alan, the trip will take only about 43 days. Alan will return in six weeks to find Bill (and everybody else on Earth) more than eight years older.

In a sense, traveling close to the speed of light is like traveling in time, but it’s a one-way trip into the future.

Mathematically speaking, if you could travel faster than light, the passage of time would be reversed. You would return to Earth before you left; in other words, you would have traveled (start reverb) BACK IN TIME! (end reverb)

It might be tempting to build a super-luminal back-in-time machine, but it just can’t be done. As I said before, no massive object can even reach the speed of light, let alone surpass it. Time dilation is not the only weird effect of traveling close to the speed of light. As you approach the light barrier, you require disproportionately more energy to make you go faster. You would require an infinite amount of energy to accelerate to the speed of light. Barring some undiscovered loophole in the laws of physics, humans will never, ever be able to reach the speed of light. Sorry to burst your hyper-dimensional bubble, Star Wars and Star Trek fans.

And so this humorous rhyme must fall into the category of Hypothetical Limericks With No Basis In Physical Reality. Still, it’s cute and it makes an important point about the bizarreness of relativity.

*The word light may refer only to the visible portion of the electromagnetic spectrum, but in this case I’m using it to refer to all forms of electromagnetic radiation: ultraviolet, infrared, microwaves, gamma rays, etc.

**One gee is the rate at which an object falls due to gravity near Earth’s surface (if you ignore complicating factors like air resistance). If Alan’s ship accelerates at one gee, he will feel a constant force equal to his own weight on Earth. In this way he could make the entire journey without suffering the ill effects of long-term exposure to weightlessness.

Cuckoo for Cross Products

Q: What do you get when you cross a mosquito with a mountain climber?

A: You can’t cross a vector and a scalar!

Man, that is one nerdy joke. To be honest, I had to do a little review so I could confidently discuss it.

See, it’s a pun! Actually, it’s a triple pun, because it relies on the multiple meanings of three words: vector, scalar, and cross. Prepare for MAXIMUM NERDAGE as we fearlessly leap across disciplines!

What is a vector?

Biologically speaking, a vector is an organism that transmits diseases from one animal or plant to another; a mosquito, e.g. Mosquitos can transmit malaria, West Nile virus, dengue fever, yellow fever, and a bunch of other germs you really don’t want.

In mathematics (and its beautiful child, physics) a vector is a quantity that has both a magnitude and a direction. If I tell you that I am driving east at 80 kilometers per hour, I have given you a vector: specifically, my velocity. Velocity is a vector quantity because it has both a magnitude (80 km/hr) and a direction (east). Other examples of vectors are acceleration, force, and momentum.

What is a scalar?

To scale a mountain is to climb it, so a mountain climber might be called a scalar (or would it be spelled scaler? I’m not sure, but the joke doesn’t work if you spell it with an e, so I’ll stick with scalar).

A scalar is also a quantity that has a magnitude but not a direction. Speed is an example of a scalar quantity. If I tell you that I am driving at 80 kilometers per hour – but neglect to mention which way I am driving – I have told you my speed. That’s the difference between speed and velocity: velocity has a direction; speed does not. Other examples of scalar quantities are: mass, distance, and energy.

What does it mean to cross two things?

There are lots of jokes that start with “What do you get when you cross X with Y?” I’ve always assumed that this crossing was some sort of ethically questionable breeding program being conducted in a secret laboratory deep beneath a mad scientist’s mansion. Think about that the next time somebody asks you a joke like this one. It’ll make you cringe a little and not feel so sure you want to know the answer.

In math/physics, crossing is a specific mathematical operation that can be performed on two vectors in three-dimensional space. That doesn’t make a bit of sense, I know, so let’s back up. Imagine that vectors are arrows…that’s how most scientists think of them anyway. Now imagine two arrows pointing outward from a single location, sort of like this:

Two vectors diverged in a yellow wood...

Two vectors diverged in a yellow wood…

We’ll call the green vector A and the red vector B. The cross product of A and B is written as A x B (read as “A cross B”) and is calculated using the following formula:

A x B = |A| |B| sinθ n

In this formula, |A| is the magnitude (or length) of vector A. |B| is the length of vector B, θ (theta) is the measure of the angle between A and B, and n is a unit vector, which helps you figure out which way the cross product points in three-dimensional space. And what exactly is the cross product of A and B, I hear you asking? It’s a third vector (really, a pseudovector…don’t ask) that is perpendicular to both A and B. The blue arrow below represents the cross product of the green and red vectors.

Hot cross vectors

Hot cross vectors

You’re probably thinking “Great…so what?” Well, the cross product has a lot of practical uses, particularly in the field of engineering. Cross products are used in some calculations involving torque, a force that causes things to spin. Yes, despite what sounds like a lot of made-up mathematical nonsense, it does have real world value.

Now then, on to the heart of this joke: why can’t you cross a vector with a scalar? Because you specifically need two vectors – two quantities that have directions. Since a scalar has no direction, you cannot cross a vector with a scalar. Insert uproarious laughter.

Now that you thoroughly understand vectors and cross products, here’s a follow-up joke to send you on your way. Enjoy!

Q: What do you get when you cross an elephant with a banana?

A: |elephant| |banana| sinθ n

Mandelbrot Mirth

Q: What does the B stand for in Benoit B Mandelbrot?

A: Benoit B Mandelbrot

Benoit Mandelbrot was born in 1924 in Warsaw, Poland. He rose to prominence as a mathematician who developed fractal geometry, which is all about measuring the “roughness” of nature. It also produces some interesting shapes, called fractals, which are known for being self-similar.

A self-similar fractal looks the same on the small scale as it does on the large scale. If you could examine the fractal pattern from far away, then zoom in to one point in particular…you’d get the same image. What’s that you say? You want to experience zooming in on a fractal pattern while electronic dance music plays in the background? As you wish!

These patterns are generated by a formula that performs a series of calculations on each point in the pattern and assigns colors based on the outcome of the calculations. Although these calculations may be performed by hand, the process is slow (very slow, in fact) and laborious. The development of computers really helped mathematicians explore Mandelbrot patterns in arbitrarily high detail.

I know what you’re thinking: this is all very pretty but what practical value does it have? Well, as I said before, Mandelbrot originally developed fractal geometry as a means of describing the roughness of things found in nature, like, say, coastlines or mountain ranges. Fractal geometry has also led to the development of cell phone antenna that maximize reception while minimizing space.

Now, about that B: According to his obituary in The Guardian, Benoit Mandelbrot added the B himself in order to make his name more distinctive. It didn’t stand for anything in particular (much like the S in Harry S Truman). However, if you assume that the B stands for Benoit B Mandelbrot, then it works beautifully as an analogy of his life’s work. Zoom in on the B and you get…Benoit B Mandelbrot. Zoom in again and you get…Benoit B Mandelbrot. Zoom in again and…well, you get the point.

Not an actual fractal.

Not an actual fractal.

Whether Mandelbrot had a humorous motive in choosing the letter B as his middle initial, we may never know. I will say this much: it’s hard to imagine that a man as clever as Mandelbrot would have completely missed the connection.