Go home Numerology…you’re drunk.

Pay no attention to the man behind the curtain...  Image taken from soundofheart.org
Pay no attention to the man behind the curtain… Image taken from soundofheart.org

 

So, in unusual form, I’m gonna write another blog post about a week after my last one!  I like this trend…perhaps I’ll keep it up.

I have been “Facebooked” the following video quite a bit over the last month or so.  It’s about the fantastic numerology of the circle.

While I do appreciate the sentiment, I felt compelled to share why this video is straight-up bullshit.  This will happen in three parts: the first being addressing why there actually are 360° in a circle (spoiler: it’s not anything mystical), the second being why the number 9 is not any cooler than any other number, and the third being wrap-up.

Part the First : Why are there 360º in a Circle

First off, let me say that there is no absolutely concrete answer to this…that has been lost to antiquity.  There are, however, some very compelling comments to be made.  In order to understand them, I need to tell you about the sexagesimal number system.

We are quite familiar with the decimal system, also known as base-10.  The idea behind it is pretty simple.  When you write a number with multiple digits, say 231, it is agreed that each digit place, from right to left, represents a power of the number 10.  231, for instance, can be represented as 2(100) + 3(10) + 1(1).  That’s it.  Question is, why 10?  Well, our hands…one digit for each finger (hell, we even call our fingers digits)…when we run out of fingers, we move on to the next place.

Now, the idea that we have base-10 implies that we could have base-N, where N is whatever number we want, right?  Binary, or base-2, is the most obvious example.  In this numbering system, each digit place represents a power of 2 instead of a power of 10.  We only have 2 digits (0 and 1) that we can represent a binary number with.  So, if I write 10110, that means 1(16) + 0(8) + 1(4) + 1(2) + 0(1) …consequently, the number 10110 in base-2 is equivalent to 22 in base-10.  This kind of counting is useful when the thing counting only has 2 digits; switches can only be on or off.

There are all sorts of bases that can be used for counting…the fact that we use base 10 is not amazing, it’s just anatomically convenient.  In fact, the ancient people of the Americas, such as the Incas, Mayans, and Aztecs, used something called vigesimal, which is base-20.  They did this because the counted on their fingers and their toes.

If you go way back to the Babylonians, they invented a system referred to as sexagesimal, which is base-60!  What!?  Why!?  Well, try this…hold out your hands in front of you, thumb out on your left hand and all five digits (LOL) out on your right hand.  Now, using your right thumb, point to each segment of the 4 fingers on your right hand.  Of course, assuming no wood shop accidents, you will count 12 “segments”.  When you count them all, extend the next digit on your left hand.  Do this until you extend all your left digits and you get…60!  Not so crazy after all.

Besides that, 60 is a prolific number.   It is evenly divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.  It’s quite useful for trade; if all the neighboring city-states have different currency exchange rates, use a numbering system that can accommodate all of the them.  It was used for thousands of years by the Babylonians and other cultures of ancient Mesopotamia as well as the Egyptians, who really laid the foundations for things to come.  We still use this system today, though only for measuring time and angles.

So, what about the circle?  Turns out, many of the ancient cultures used a 360-day calendar.  Why?  It comes from ancient Mesopotamian view that the sun, the moon, and the stars moved along 360-degree circuits through the sky.  It was a stellar calendar that seems to be a nice compromise between the solar year (which is about 365 days long) and the lunar year (which is about 355 days long).  So, express your cyclical calendar as a circle and you get 360 days, of 360°.

It’s easy to see how this got appropriated by mysticism.  But, it is simply an issue of practicality.

Part the Second : Revolution No. 9

Side note : the linguistic idea that the division of a thing after the primary is called “second”, such as the section title I just typed, comes from this numbering system (hour, minute, second…degree, minute, second…).

So, in this video, why is the number 9 so fantastic and prolific?  Why does every division and inscribed polygon in a circle collapse down to 9?  Turns out, it’s not remarkable at all.  Well, I suppose it is, but certainly not for any mystic reason.

You may recall a neat trick to figuring out if a number is divisible by 9: add up all the digits of a number and if the result is divisible by 9, then so is the number.  What is that all about?  Well, it’s a simple mathematical property that I will illustrate here.

Take any number at all, decimals and everything.  To illustrate this in the most general way possible, I’ll write that number like this:

IMG_9281

where the “…” means there are as many digits as you want in either direction and the constants a, b, c, b, and e are whatever digits you want.  Now, remember what the decimal system means.  It means that we can rewrite the number like this:

IMG_6125

 

Now, I can write each one of those numbers in the parenthesis as (1 + something), like this:

IMG_5717

 

I can now distribute the number outside the parenthesis and then rearrange the terms to get something like this:

IMG_7214

 

Notice the thing that I underlined…it’s the sum of all of the digits of the number I chose.  But, look at the first part.  If I factor a 9 out, I have the following:

IMG_0947

No matter what, the first part is always divisible by 9…there’s a 9 in front of it!  It doesn’t matter what I pick for the digits.  The only thing that matters is if the second part (second LOL) is also divisible by 9. That’s the secret to the trick!  That’s why you can just add up all the digits and see if they’re divisible by 9.

Why doesn’t this work with all the numbers?  Well, it does work with 3 for the same reason; the first term is always divisible by 9, which means it’s also always divisible by 3.  So, if the second term is as well, the whole number is divisible by 3.  As for the other numbers, the key step in the process is the point where it write the numbers in parentheses as (1 + something).  It works because we use a base-10 system.  If I take any place value (1, 10, 100, 1000, so on) and subtract 1 from it, the result is always divisible by 9.  Because the number I pulled out was 1, I get a nice second term that is just the sum of the digits.  If I were to write the number in parentheses as, say, (2 + something), then that second term would have extra things in it and it wouldn’t just be as simple as adding up the digits; that’s why this only works with 3 and 9.

So, there’s no magic in it, it’s just the result of basing our counting system on the number of digits we have on our hands.

Protip : Bart and Lisa had to learn how to count in base-8 LOL
Protip : Bart and Lisa had to learn how to count in base-8 LOL

Part the Third : So what’s with the video?

The video seems magical because no matter what you do, all these digits add up to 9.  Well, there are 360° in a circle and 360 is divisible by 9.  So, if you multiply 360 by anything, the result is always divisible by 9.  No matter how you subdivide the circle, either by diameter chords (like the first part of the video) or by inscribing regular polygons (like the second part), you’ll always get a number that is divisible by 9.  Nothing to it.

As for the metaphysical meaning, well, what can I say.  Humans like to look at Nature, see patterns, and assign meaning to those patterns.  Sometimes they’re right, just as Newton’s laws.  Sometimes, they’re not, like astrology and numerology.  We are designed to pattern recognize and it’s so easy to assign meaning to nonexistent things.  The video presents a mystical link between geometry and arithmetic that seems magical, but it’s not…they’re just numbers.

The fact that we can abstract “things” in such a way as to even be able to discuss the “nine-ness” of a group of things or the fact that one can use mathematics to describe nature, that’s magic enough.  Seriously, think about the fact that I can say 9, and that it has meaning.  Primitive languages didn’t have that capacity.  They had, 1, 2, more than 2, and that was it.  The fact that we could eventually abstract number into its own concept is what made humanity the thing it is today.  None of the modern science we have would exist if it weren’t for that critical step.  That’s the mystical thing.

Let’s not make things more complicated than they have to be.

Thermostat’s broke, yo…

Image from http://sneerkat.com
Image from http://sneerkat.com

Man, it’s been a looooong time since I put up a new post.  That was not intentional.  It was part lack of motivation (or inspiration, as the case may be), and part being too busy doing science to sit down and write about it.  But, I’ve given myself a New Year’s resolution of sorts to give this blog the love that it deserves, so here it goes…

It’s cold outside.  Super cold, actually, thanks to the ridiculous wind that is accompanying our cold front.  Walking across campus to my office made me feel like David Attenborough traversing Greenland.  Twas good times.  (Edit: at least it was when I started writing this LOL)

In the grand scheme of things, however, it’s not even that cold.  I mean, it’s not even the coldest it’s been here, let alone truly cold.  While pumping liquid helium into the cryostat in our lab, I have experienced true cold; it was basically the exact opposite of the experience of opening your oven and inadvertently burning your face off.  You think -10 °F is cold, try -451 °F…

Temperature is an interesting thing, when you really think about it.  What exactly is it that we’re talking about when we refer to an object being “hot” or “cold”?  It all seems so arbitrary; hot or cold with respect to what?

Further, think about the fact that, unlike most other physical properties, things like mass or length or time, we don’t directly measure temperature.  Instead, we are always measuring some other physical quantities response to temperature.  In a mercury thermometer, for example, we are measuring the change in the volume of the mercury in response to a change in temperature.  In a fancier electric thermometer, like the one you put in your mouth when you’re sick, you’re measuring the change in electrical resistance of some component in response to a change in temperature.

Sometimes, the concept of temperature isn’t even defined.  Think about a piece of metal that you are heating with a torch.  What temperature is the metal?

Point is, temperature is wacky and we really take the concept for granted.  So, let’s think about it for a little bit!

The Concept of Temperature

Imagine two objects, say two cubes of metal, one “warm” and one “cool”.  You then bring them together so that they touch and then you observe what happens.  There’s all sorts of physical properties that we can observe, as stated above, things like volume and electrical resistance.  What you will observe is that after some disruption when they were brought together, eventually, everything stops changing.  The volumes, for instance, will change as soon as you touch the two cubes together, but eventually, the volume of each cube become constant.  When this happens, we say that the objects have reached equilibrium.  More specifically, they have reached what is known as thermal equilibrium.  Being in equilibrium doesn’t mean that the values of everything are the same; the cubes won’t necessarily have the same volume.  The values of volume will simply not be changing anymore.

The idea of temperature is that physical thing that tells you if you are in thermal equilibrium or not.  All systems have some sort of internal energy; consider, for instance, the kinetic energy that each molecule of the air around you has.  If you are in thermal equilibrium with your surroundings, life is great.  If, however, your surroundings have more internal energy than you do, Nature does what it needs to do to balance everything…by increasing your internal energy and decreasing the surrounds internal energy until you are in thermal equilibrium.  Of course, you perceive this as “getting hot”.  Realistically, when you heat up, your surroundings cool down, but your surroundings are so vast that you don’t really make a difference; they are like an infinite reservoir of energy.  Likewise if your surroundings have less internal energy than you do.  In this case, rather than your surroundings giving you energy, they take it away and you fell “cold”.

Back in the day, distinguished gentlemen, such as William Thompson,…

That's Lord Kelvin to you, son.
That’s Lord Kelvin to you, son.

…thought that this transfer was due to the flow of some unseen fluid they called “caloric”.  It was later established, much to Kelvin’s (pardon me…Lord Kelvin) credit that this idea was false, which ultimately led to the kinetic theory of gases and the idea that heat was transferred by atomic collisions.  He also gave us the important concept of an absolute temperature scale.  Of course, the unit of temperature, the Kelvin, one of seven fundamental units in Nature, was named after him.  Well played.

Victorian-era science was big on the weird fluids.  There was the caloric that transferred heat, the frigoric (seriously) that transferred cold, the phlogiston that was released during combustions, and let’s not forget the fantastic lumiferous æther through which light propagated.  But, I digress…

Chaos

In both cases, you are perceiving a disruption of your body’s “normal state of affairs”, a regulatory process called homeostasis.  Needless to say, we are fantastically ordered machines.  It takes a lot of energy to create and maintain that order…and Nature doesn’t like it.

There are a lot of laws that are thrown out that Nature must follow.  One that truly must be followed, however, is known as the Second Law of Thermodynamics.  This law can be stated in many ways, but the one most applicable to this discussion is that “the entropy of the Universe (a closed system) must always increase”.  Well, cool…what’s entropy.

Entropy is a deep subject.  The popular way to describe it involves the idea of chaos.  Think of water.  When frozen, the water molecules take on a rigid crystal structure and have a great deal of order to them.  As you add energy, these molecules become “agitated” until they finally break their crystalline bonds, thus becoming less ordered.  They still weakly interact, however, so they maintain a specific volume.  Heat the liquid even more and the molecules overcome this interaction as well and loose volume, becoming even less ordered, as the turn into a gas.  In physics, we would say that water vapor has more entropy than liquid water which, in turn, has more entropy than ice.  The second law says that this quantity, overall, must increase.  Nature does what it needs to do to ensure this happens.

What about a refrigerator?  Doesn’t it lower the entropy of the stuff inside?  Well, yeah…but it does this using a heat pump, that grill on the back of the device, which exudes heat into the surrounding environment, increasing its entropy.  The net effect is that overall entropy is increased.

Back to us and our homeostasis.  Based on this idea of order, we have pretty low entropy and, like the refrigerator, we radiate are ridiculous amount of heat back into the environment (some of us more than others…).  Our surroundings are constantly adding or subtracting energy from us, and we use the energy we extract from food (calories…at least we kept the word) to maintain order.  But the Universe is relentless.

I find your lack of entropy disturbing...
I find your lack of entropy disturbing…

It keeps chipping away until something finally breaks, homeostasis fails, and death occurs.  Without that inflow of energy, all of our complexity disintegrates and the second law is maintained.  Death is losing the battle against entropy.

Overall, however, we’re sticking it to the Universe.  Even though it destroys us, we serve to (minutely) raise it’s entropy over our lifetimes.  It turns out that entropy is tied to the amount of energy that can effectively do work.  When the entire Universe eventually comes to thermal equilibrium, work will not longer be able to occur.  You need “hot” and “cold” for energy to transfer.  If everything’s the same temperature, energy can’t flow and nothing interesting can happen.  All chemical reactions will cease, everything will become a near-absolute-zero soup and entropy will finally be maximized.  The inevitable heat death of the Universe…LOL the joke’s on you, Nature.

A depressing outcome in 100 trillion years…I guess being cold in the winter is the least of my concerns.  But at least my face won’t hurt when I go outside.