Category Archives: General

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:

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

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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:

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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.

Perspective

So, it’s this kind of day today…

IMG_3701…which always makes me introspective.  Consider the following…

About 13.8 billion years ago, the Universe came into existence, for whatever reason you choose to believe.  During that event, all of the energy that was, is, and ever will be spewed forth.  Some of it, after some time, coalesced into matter, mostly hydrogen mixed with helium in a 3-to-1 ratio, with a pinch of all the heavier elements for flavor.

In addition to creating matter, the primordial energy of creation also imparted all of these particles with kinetic energy, the energy of motion, transferred to them in the form of thermal energy radiated from space itself.

Nothing particularly interesting happened for about half a billion years, particles just kind of flew around the ever increasing expanse of space.  However, due to the fact that they had mass, the universe actually gave another type of energy to all matter…gravitational potential energy.  The particles attracted each other into great clumps of inert matter and, as they did so, converted this potential energy into kinetic energy, causing them to fall towards one another, increasing in speed.  As they got closer and closer, collisions became more and more common and this kinetic energy was rapidly transferred between them, something that we perceive as temperature.

At some point, the temperature became so great that the kinetic energy was enough to overcome the electrostatic repulsion between the like-charged particles, allowing in the strong nuclear force to “stick” them together and the process of nuclear fusion began.

Through potential energy, these nuclear furnaces converted the small amount of mass lost in the fusion process back into energy, which was radiated out into space.  After a while, the star would die, projecting it’s remaining matter back into the universe to seed the creation of the next generation of stars.

After (probably) three rounds of this, the Sun came into existence about 5 billion years ago.  What matter wasn’t consumed by its creation gravitationally collapsed to create the multitude of rocky bodies that orbited it, propelled by that same potential and kinetic energy.

On Earth, like on the other bodies, larger leftover chunks of rock and ice pummeled the surface.  Unlike the others, however, the distance was such that the energy emitted from the Sun was enough to keep the kinetic energy of the water molecules high enough to avoid forming crystals and freezing.  It was low enough, however, to keep them from flying apart completely and vaporizing.

Like the early Universe, nothing particularly interesting happened for another billion years.  However, the energy pouring onto Earth from the Sun allowed order to arise, fending off the ever-present influence of entropy.  Self-replication, powered by the Sun, gave rise to more and more complex structures.

About 350 million years ago, plants really began to take over, covering the Earth, powered by the Sun via photosynthesis.  They used it’s energy to create carbohydrates, made up of atoms of carbon and hydrogen provided by the destruction of the Sun’s progenitor.  This chemical potential energy could then be broken apart at a later time to fuel the process of life, fending off chaos…at least until the energy ran out and death occurred.

As the plants and their unused energy were buried, pressure and heat, yet again manifestations of gravitational potential energy, converted them into coal and natural gas.  Deep in the Earth they remained…until the surface order empowered by the Sun gave rise to us.

About 200 years ago, we figured out how to extract these materials. By burning them, we can release that chemical potential energy and do work.  Coal is burned to heat water, inciting a phase change to steam, imparting kinetic energy to a turbine blade, which rotates a magnet in a coil of wire, transferring that kinetic energy to the electrons, generating electricity.

These electrons are propelled to my home by electric potential energy, created by a difference of charge which acts like a form a pressure.  They travel through thousands of miles of copper, agitating the atoms along the way, generating heat and being radiated into space.

What energy survives this process is sent to its destination, guided by the electric potential created by the piezoelectric switch at the base of my water heater.  This potential is so great that the electrons are forced out of the conductor and into the air, jumping across the gap of the ignitor.  The kinetic energy of the electrons is high enough to raise the temperature above the flash point of the natural gas being passed through the same gap.  The energy is enough to break apart the 350 million year old methane molecules, which then mix with the oxygen, only present because life started releasing it into the atmosphere all those years ago.  The same life, amusingly enough, that created the methane in death.

This oxidization releases heat, which is transferred to the water through vigorous atomic collisions.  This hot water is then pumped through the house, driven by a pressure potential, to my shower, where it falls onto my skin.

The kinetic energy of the molecules is transferred to thermoreceptors in my skin, made possible by a 3-billion year battle between the Sun and entropy.  The energy is once again converted, this time used to create an electric potential on the thermoreceptor cell’s surface.  This induces an electrical potential cascade in my peripheral nerves, sending charge up my spinal cord, and into my thalamus, the switchboard of my brain.

My thalamus uses the energy conveyed from the electron flow to free proteins which bind with receptors in specific neurons, using the chemical potential energy to release acetylcholine into my bloodstream.  This activates the neurons of my cerebral cortex, again through a conversion from chemical potential energy to electrical potential energy.  This increase boosts my attention and that, combined with the increased blood flow caused by the vasodilatation induced by increased body temperature, gives me the idea to write this blog.

Cant_c985b7_1305487

And then I had a cup of coffee…

 

Life of π

Seems that three months have passed since I last wrote anything about anything here.  Turns out that actually participating in science is a time consuming endeavor.  Actually, it’s been a little over 3 months…3.19 months, actually…which is pretty close to…

pi-300x300 copy

Today is also Pi Day in the United states, as well as Albert Einstein’s b-day, so the power of math compelled me to post something.

The question is, why do we even care about π?  Why are people across America eating a slice of banana cream pie right now to celebrate a number?  Do Americans really need ANOTHER reason to eat pie?  Consequently, Pi Day will fall on Wednesday in 2018, the day on which Village Inn offers free pie…imagine the anarchy that will ensue!

What is it about π that has interested people for millennia and given it almost mythical powers?

First off, where did the symbol π come from in the first place?  Previous to the year 1706, one had to be content to use the Latin phrase quantitas, in quam cum multiplicetur diameter, provenient circumferentia”, meaning “the quantity which, when the diameter is multiplied by it, gives the circumference”.  Clearly the essence of convenience.  Also, guess where all of our mathematical terms come from.

In 1706, my bro William Jones busted out the symbol for the first time in print, thereby relieving the hand cramping of his colleagues, choosing the symbol π (presumably) because it was the first letter of the Greek word περιφερεια, meaning “periphery”.

William Jones, Welch mathematician: Thanks Bro!
William Jones.  Thanks bro!

But what’s with the fascination over this particular ratio, the ratio of a circle’s circumference to its diameter?  Humans have long been enamored by numbers, giving symbolism to the digits 1 through 10 and various combinations thereof.  7 is lucky, 13 is unlucky, 1 represents the self, 2 represents unity with another, and so on.  Combine that mysticism with that which already existed with geometry for thousands of years and that gives some sense of the power π held over the ancients.

The Pythagoreans, pretty much the progenitors of the movement that lead to the Golden Age of Greek mathematics and cause of most middle-school student’s nightmares, believed that everything in the universe could be explained with number, specifically ratios of whole numbers.  They created a musical tuning such that all notes were related to each other by whole ratios, 2:1, 3:2, 4:3, and so on.  Everything divine in the universe could be expressed in this way.

The circle is, by most accounts, the simplest geometric figure you can create: a set of points all equidistant from some center point.  Pretty much the only things going on in a circle are the diameter and the circumference.  Needless to say, the ancients really wanted to find the magical ratio between the two.  They appear everywhere in Nature: all of the celestial bodies, bubbles, droplets of liquid.  What secrets could be unlocked finding this divine ratio?

The Greeks were like, well, ok, we can just  find the correct geometric construction and find it.  My bro Eudoxus came up with a pretty cool plan around 360 B.C. or so called the method of exhaustion.

Eudoxus.  Thanks bro!
Eudoxus. Thanks bro!

His plan was pretty simple, but ingenious.  Time to bust out my tools:

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Now, back in the day, the Greeks didn’t have a ruler per se, merely an unmarked straight edge.  Deal with it.  First up, draw a circle:

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Then, go about bisecting, drawing arcs and such…

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…until you can inscribe a polygon inside your circle.  In my case, I inscribed a regular pentagon.

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Then, using your inscribed polygon, bisect some more…

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…and then create a circumscribed polygon around the circle.

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Now, the perimeter of the inscribed shape is clearly less than that of the circle, while the perimeter of the outside shape is clearly greater than that of the circle.  If you find the perimeters of the two pentagons (30cm and 37.5cm, respectively) and then average them (yielding 33.75cm), you have a pretty good approximation of the perimeter of the circle.  Then you take the ratio of this average perimeter and the diameter of the circle (which is 10.6cm) and you get that π is about 3.1839 or so, not bad.  Give the polygons more sides and you get a better approximation.  Here, I’ve inscribed a hexagon and used it to construct a 24-sided polygon.

IMG_9442

Turns out that the ratio of its perimeter to its radius is 3.1385.  Better…

One hundred years later, my bro Archimedes came up with a pretty crazy idea in his work The Method, was was lost until recently.

Archimedes.  Thanks bro!
Archimedes. Thanks bro!

He made the bold observation that if you give the polygon an infinite number of sides, it becomes the circle!  He drew some 96-gons (we run out of cool names pretty quick, not unlike with elements) , came up with some formulae to find their perimeter and nailed down π  to between 3 10/71 and 3 10/70, or between 3.1408 and 3.1428.  Nice work, Archimedes.  If you continued this infinitely, you would get the true value of π.

And then Greece declined, Rome showed up, and Europe in general just stared out the window into the rain catatonically for a millennium and a half.

Then, in 1761, my bro Johann Lambert showed up and proved that π was irrational, screwing up the party for everyone…sorry.

Johann Heinrich Lambert...thanks, bro...
Johann Heinrich Lambert…thanks, bro…

The fact that π is an irrational number means that it cannot be expressed as the ratio in any form.  It is a never-ending, never-repeating decimal number: 3.1415926…  The issue was that the ancients didn’t know this (well, a Pythagorean “discovered” the irrationality of the square root of 2, but he was promptly killed for heresy) and the search went on for centuries to find the pattern.

By the 16th century, algebraic geometry had taken over and people were looking for an equation that could yield the value of π.  That didn’t last long, because my bro Charles Hermite came up with a method, later used by Ferdinand von Lindemann, to prove that π is not just irrational, it’s transcendental, that is, there is no polynomial of which π is a root.

Charles Hermite.  He has clearly had enough of your π bullshit, Europe.
Charles Hermite.  He has clearly had enough of your π bullshit, Europe.

At this point, all we can do is approximate π with infinite series and such, which we do to further and further numbers of decimal places, secretly hoping that math is wrong and we’ll find a pattern.  Actually, it’s mainly to prove processing power, but whatever.  At this point, the world record is held by Shigeru Kondo, who calculated π to 5 trillion digits.

 

Shigeru Kondo.  Infinite digits?  Come at me, bro.
Shigeru Kondo. Infinite digits? Come at me, bro.

So, π is basically the white whale of mathematics, tying together a linage of mathematicians over almost three millennia trying to answer a simple question:  what do I get if I divide this by this?  Indeed, things in the world are not a simple as they seem…except for the carbohydrates in this yummy pie.