Tuesday, June 14, 2011

Moving On: Fourier Series

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Well, its about that time to start thinking about putting this blog to rest.  I do this for a couple of reasons:

1.  It takes a ton of time to research each topic.
2.  It takes a ton of time to figure out how to "translate" each topic is a way normal humans can understand.
3.  My appointment as a visiting professor is now up and I'm returning back to the world of engineering.
4.  There are a ton of science bloggers out there who have more free time on their hands than I do.

With that being said, I plan to hone my focus more.  I've decided to concentrate in the future on the best way I tend to learn...visually.  As we all know, psychiatrists tend to lump everyone into 4 types of learners:  Visual, Auditory, Read/Write and Tactile.  For some reason, science academia chose to focus on auditory and read/write learners with lectures, note taking and textbooks.  This is one of the reasons academia fails so spectacularly in science education: a large majority of people are visual/tactile learners.  I have insisted many times that EVERYONE can fully understand science and math, the problem is that it is presented in the wrong format which has, unfortunately, been institutionalized by academia.  If there is any group that innovates slower than molassas, its an academic faculty.
I've been toying with the idea of "Visual Math" leading up to "Visual Science" as a future blog endeavor.  Mostly because these types of learners are traditionally underserved, and also because that is the type of learner I am myself.

I will leave you with an clear example:  Fourier Series.


Here is the way a well known and respected engineer attemps to explain a fourier series in a well known engineering manual:

"Any periodic waveform can be written as the sum of an infinite number of sinusoidal terms, known as harmonic terms.  Such a sum of terms is known as a Fourier Series, and the process of finding the terms is Fourier Analysis.  Since most series converge rapidly, it is possible to obtain a good approximation to the original wave-form with a limited number of sinusoidal terms."

-Lindeburg, "Mechancial Engineering Reference Manual"

To a read/write or even auditory learner, this may work fine.  For the majority of us however, this explanation probability leaves your eyes watering, ears bleeding or wanting to through the text out your window.

Now let us contrast Lindeburg's written explanation with the following visual explanation:




Now I bet that makes MUCH more sense to most people.  If you now go back and read Lindeburg's explanation, it will also make more sense.  For most people, the why's of the science need to be illustrated visually before one can move into the details.  A fundamental failing of science education that I hope to address with a new blog.

Wednesday, February 2, 2011

Some Perspective...

Ever wonder what the other planets in our solar system would look like if they were as close to us as the moon?

 

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Tuesday, February 1, 2011

Massive Storm Moving Across the US

As seen from orbit.  Not a bad time to be in Florida...

 

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Friday, January 21, 2011

Wednesday, January 19, 2011

The Luckiest Friday the 13th You Will Ever Have

Apophis is the name given to asteroid 99942. Apophis is also the greek word for "the destroyer". I'm not sure who came up with the name, but its another case where either scientists or the press can be rather irresponsible in the nomenclature.
Here's Neil Tyson discussing this remarkable event from a few years ago:



The good news is that Apophis will not hit the Earth on April, Friday the 13th, 2029. In fact, it will be a once in a lifetime event for us all to witness. The following picture will help illustrate just how close apophis will come to hitting Earth:
apophis-2029.jpg
The concerning aspect of Apophis is what happens after 2029. The "keyhole" Dr. Tyson talks about is a worst case scenario in which case Earth WILL get hit in the year 2036, which would result in widespread destruction of the west coast of the US. The chances that Apophis hits this "keyhole" is 1 in 250,000. A rather remote possibility. However, keep in mind your chances of getting hit by lightning are 1 in 500,000. Therefore this is a concern for scientists. It could be an even bigger concern in further out years and its eventual contact with Earth may just be a matter of time.
No matter what happens, at the very least, humanity will have at least 7 years warning to engineer a solution.
But, on April, Friday the 13th, 2029, look up at the sky and count your lucky stars that Apophis' orbit isn't even slightly different.

Thursday, January 13, 2011

Learn Something Today: The Fear of Numbers

Numbers can seem scary to lots of folks, but they needn't be any more intimidating than letters.  Equations need not be any more confusing than sentences.  In the ancient world, less than 1% of society was educated, but within this 1%, nearly all were equally versed on both language and math.  Today, the country's literacy rate is at all time highs while its math literacy rate continues to drop.  There is no reason for this.

There are four (4) basic types of numbers:

1.  Whole Numbers (example: 42).  These are the oldest of numbers and date back over 2500 years to ancient Greece.  To the greeks at this time, the number "1" (one) had a different meaning than we use today.  They considered "1" as a whole unit.  Numbers to them would be things that "make up" or "compose" the complete unit.

At this time however, it became important for shepherd's to be able to understand a convey the size of their flocks for business purposes.  They needed a way to describe size by numbers, this is where whole numbers were born.  The limits of "counting" at this time were 1, 2, perhaps 3, and then "flock" or "a lot".  As time progressed, more and more "counting" numbers were added to the Greek numeral system.

These numbers were not meant to be divided.  There was no need for it.  A "divided" cow wasn't worth anything because it would be dead.  Therefor, most common ancient greeks had no understanding of things like fractions.

Some famous shepherds:
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Eventually, the ancient greeks realized a problem though:  there was no number, either Whole or Rational that would describe certain geometric shapes.  For example, Pythagorus could not find a number that would be the exact ratio between a circle's circumference and area, otherwise known as "pi".  The ancients also knew of another weird ratio call the golden mean that seemed to keep occurring everywhere in nature, yet no one could figure its exact "number."  For these reasons, greek mathematicians had to develop the idea of "irrational numbers".

(Much, much later, the decimal system (example: ".5" is the same as "1/2") was developed which conveys rational numbers in an even more simplified form.)

3.  Irrational Numbers ("Pi", certain "square roots"). Rational and Irrational numbers are a case study into how math and science really needs to learn to update its terms as times change.  If you are confused by the descriptions "rational" or "irrational", you are not alone.  Today, its hard to think of any number not being "rational".  Even more so, why even label a "fraction" as rational?  The reason we have this descriptive confusion is based on time:  2500 years ago, there was a very good reason the greeks found some numbers to be rational, and others irrational.

The Greeks knew the numbers of Pi and the Golden Mean had to exist.  They were real and could be "made".  Circles were common.  Shapes and architecture commonly used the golden mean, so how could this ratio not exist?  It didn't make sense to them, hence to the ancient greeks, these numbers were not rational, but irrational.  In fact, they were considered so irrational at the time that Pythagorus, the most famous mathematician of his age, was said to have thrown one of his students overboard after the student proposed the idea of irrational numbers while on a voyage.
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Pythagorus being a bore in math class 2500 years ago

The rule of identifying irrational numbers became much easier after the invention of the decimal system. A number is irrational if it has a non repeating, yet infinite decimal.  For example "Pi" is approximated at 3.14.  But this is not the exact value of Pi.  It can be further refined as 3.1415926535, yet even this is not correct.  In fact, Pi has been calculated to extend the decimal out to thousands of places.  The truth is, there is no exact number of Pi.  The decimal places stretch out to infinity.  If you ponder this a little, you too, may feel a little bit of the ancient greek's frustration and consider it "irrational" as well.

4.  Imaginary Numbers ("4i" or "12i").  These are the weirest of all numbers.  Here again, the term "imaginary" is not terribly accurate or helpful in today's context.  A better term would be "invented".  These numbers do not exist.

Imaginary numbers consist of a real number (ie, whole and rational numbers), and an "invented" number represented as "i".  "i" is simply the square root of -1, a number that does not exist.  Its kind of like trying to divide a number by zero, its impossible.  However, in some fields of science and math, calculations become much easier to deal with and more efficient if "i" really did exist.  Therefor, we separate the "real" part fromt he "invented" part.  In most cases, these numbers tend to meaningless unless you are dealing with vectors (to be covered later).

As a general summary from wikipedia:

"For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. For instance, fractions such as ⅔ and ⅛ would be meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. In the same way, negative numbers such as –3 and –5 would be meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits[2]. Similarly, imaginary numbers have essential concrete applications in a variety of scientific and related areas such as signal processingcontrol theoryelectromagnetismfluid dynamicsquantum mechanicscartography, and vibration analysis."

...as you can see...there is nothing to be afraid of here.  Once you know the context, numbers are pretty easy to understand.

Monday, January 10, 2011

Science Provides Real, Tangible Optimism

A tip of the hat to rgower over at reddit for this.  NASA could use a good PR campaign right now.