Written by Mike Sterling for Canadian Community News
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Sierpinski's Triangle stage one to five
Should we trust our own common sense?
As a skeptic, I have little regard for the words "Well, it's just common sense". I heard that over and over again during the Joint Review Panel hearings on the low and intermediate Deep Geologic Repository in Kincardine. Those same words were used in deputations before Saugeen Shores Town Council with the same intent.
But, not to dwell on the hearings in particular, just how common is this special sense possessed by anyone delving into complex areas? Can we rely on it?
Should we depend upon our version of common sense or should we listen carefully to people who have studied the complexity of problems? Let's look at some simple examples of how far off common sense can be..
Anything worth considering is not common and the truth is really uncommon. In fact truth is the rarest and most precious thing we have. If in a lifetime, we can see it once or twice, we are lucky. It is hard to find and hard to use properly. It is hidden and shy. It takes hard work to reveal it.
To get at the truth, we have to be careful of time forward and back and distance large and small. Time is particularly troublesome. Time fools us. Scale fools us.
Physics, engineering and mathematics in many ways are studies of the very large and very small. We also have the study of time that can warp in the very large. What is time anyway?
We as humans are 'stuck' in our own view of scale. We have difficulty seeing, feeling and understanding cosmic forces. We have difficulty dealing with quantum mysteries in the very small. We also don't do well with radioactive substances and the time it takes to decay them. Some are fast and some slow..
Philosophy is also filled with ideas that are tied closely to scale. Researchers in the past considered themselves natural philosophers. They studied the world and the universe from their position in time and space. Leonardo knew his skills and tried to use them all. His views on the world were broad and in depth.
Once we move deeper and much smaller into subatomic concepts, we change our sense of scale altogether. We have to leave our 'common sense' at the entrance to changes in scale. On the other end , where the very large is measured in cosmologic light years, we have to be very careful too. How do we understand the microwave signature of the Big Bang? Is it open to ouf common sense? Can we look in the past almost 13.4 billion years?
We get in trouble when we lose track of our scale in relation to the measures of the very large and very small that we are trying to understand. As soon as we go to the very small or very large, we have to be uncommonly wary.
As soon as we try to peer into the future in hundreds or thousands of years, we lose track of our relationship to our own time scale. We live in the short. Nuclear decay of certain types lives in the long. What will concrete poured today look like 100,000 years in the future? What will plastic look like?
Let's illustrate two well known paradoxes that involve scale and the fabric of our thought. The large and small will be examined. The first is easy to understand, the second a bit harder.
Mathematics and Science are full of truth that is uncommon. I wrote about half-life and the infinite series 1/2+1/4+1/8+ .... = 1 in the article below.
That series sums to one and just a few additions will show you how fast it tries to get near one.
It is the same series that the Greek Zeno studied. He was not studying the half-life of radiation, but the nature of the world he thought he understood. He was using his form of common sense to interpret the world around him.
But, near a sum or close enough is not what you might think, if we enter the world of the infinite.
Some series are well known for their rapid convergence to a number. These are taught in high school, but not in depth and not remembered well. Why?
Well, our head hurts when we think about the infinite. Our teachers are not fond of headaches either. Don't worry, the Greeks got a headache too. Infinite series are usually way in the back of the book and described on a page or two. Sometimes school is out before the student gets to infinite series. We could walk up and down our main street and ask people about infinite series. We'd find few answers.
How about the harmonic series made famous in music theory and known to Pythagoras? How common is the sense of this musical incarnation?
Try to sum it. Does it converge to a number? It must, many would say. It's just 'common sense.' Others would guess that it goes on forever, but what it sums to, they cannot say.
Some might do their sums as the British say with a spreadsheet like Excel or a calculator long dusty hiding in a drawer.
A pencil is a waste of that pesky thing called time. Anything is a waste of time in trying to find the sum of the harmonic series.
Maybe somebody who is persistent, will find out that the first million terms of the harmonic series sum to only 14.357. So the end is in sight or is it? Just a few more additions and we'll have it or will we?
Being persistent they ask for more time and say: Let me try 10,000,000 or more additions and I'll find an answer that is good enough.
Ha! .... Not so fast. Ha, not so far. You will run out of digits on your computer and that's no proof. The harmonic series is not amenable to 'common sense'.
The harmonic series has no limit and it is infinite in sum. That's good for celestial music, I guess, but bad for those who rely on their 'common sense'.
It's easy to prove that it has no limit. It was first shown by a cleric Nicole Oresme (ca. 1320-1382). He was a French theologian, economist and mathematician. Notice the breadth of his interests.
The Greeks had a hard time with infinity. Zeno posed the paradox of a man running in a race. He has to get half way and once he is there, he has 1/2 left and he has to go half that way and that's 1/4 left. You get the picture.
Repeating 'forever', he never is quite there says Zeno because we must go on infinitely, halving as we go. What a treadmill that is! So, we again encounter the series that is related to half life. This is not the harmonic series at all.
It took centuries until Georg Cantor made the paradox not so alarming. He pointed out that 1 can be expressed as 1/2+1/4+1/8+1/16+ .... Why not? There is no difference Both are the same in the limit.
One is also equal to .99999999 ... It all has to do with those three little periods at the end that mean go on forever. Let's see what a little 7th grade algebra can do.
(1) Let x = .9999999999 .... (note the dots) What is x really? Can we find out?
Multiplying both sides of equation (1) by 10 we get equation (2):
(2) 10*x =9.999999999 ...
Subtract equation (1) from (2) and we toss out all but two nines.
(3) 9x = 9
Dividing both sides of equation (3) by 9
(4) x = 1
So you see it is quite clear that .9999999999 ... = 1.
This had been known for centuries, but Georg Cantor put everything on firm ground in the late 1800s. He met with resistance and was hospitalized for mental depression, but he persisted. Even the great scientists of his age were afraid of infinity. He endured the headache of the infinite.
He showed us the truth. The sum of a convergent series is a perfectly good number. He also proved that infinity has degrees. Some forms of infinity are very different than others. The counting numbers have a certain form of infinity shared by the rational numbers, but the number line from A to B no matter how we measure, has a different degree of infinity.
Zeno would not have been deterred, however. The Greeks thought that the world was either made of some sort of magical 'oil' (my term) or it must consist of tiny particles that we think of as the Greek idea of atoms.
The oil idea is one that conveys something that is uniform and dense and overcomes friction. It is something we can move through easily, but it leaves no trace. Later it took on a name aether or ether.
Even the earth must move through it, but if it does, how come all our mountains are not worn smooth by the 'stuff' of the cosmos? Maybe it's dark matter?
It must be this magical oil. There are no edges or discontinuities in the Greek continuum, but somehow these tiny Greek atoms allow us to go about our business.
If the universe is continuous and made of some 'glob' of stuff that we 'waltz' around in and not tiny particles, then we would have to traverse every term in an infinite series to go from A to B. That would take infinite time and the Greeks rejected that too. Anything to do with the infinite was anathema to some of them.
But if reality is made of tiny particles, we can go from A to B with impunity. We sometimes do it out of habit, if it happens to be the distance from our easy chair to the refrigerator. When we go to get a beer, we are using the half-life infinite series that sums to one. How about that?
There was not much progress on the atom until the late 1800s. Even as late as 1912 some scientists resisted the atomic theory. Notice it still caries the last name of theory instead of fact. It is fact none the less.
So Zeno's paradox is not subject to what people call 'common sense'.
Sierpinski's Triangle after a few iterations.
We now move to a famous geometric shape that also is not amenable to 'common sense', at least not mine. Nature is full of these things.
What we can say about it seems to defy what many call 'common sense'. It is called Sierpinski's Triangle after the man who popularized it. He was a fan of Cantor and taught the first university course on set theory and Cantor's ideas.
Here is a warning. Don't be looking for Sierpinski's Triangle to end up as a single triangle. It will be like an artistic framework, but oh well, you'll see.
Figure 1 shows an interesting design. The YouTube illustration of it, is impressive too. (see below, when you have time)
To understand Sierpinski's triangle start with an equilateral triangle filled in with black with an arbitrary area of 1 unit. Look first at figure 2 above on the far left and the filled in black triangle.
Take out the first white triangle. The three black triangles that are left have a combined area of 3*(1/4)= 3/4 and what was removed has area 1/4.
In the next step there are nine black triangles each with area 1/16 with combined area of 9/16. The white removed so far has a combined area of 7/16. So far, so good. 7/16 + 9/16 = 1 as it should.
Continuing in this manner the black triangles that are left at each step are 3/4 of what they were before removing area. We can represent this as (3/4)n as n goes to infinity.
But, as n goes to infinity something odd happens. Even though we are removing only 1/4 of the black at each step, we have in the limit nothing left. Let's say that again. What is left has zero area, but the shape is tremendously complex. It is very regular though. We will measure the perimeter of the complex shape now.
But before we do, we have to note that the sides of an equilateral triangle of unit area are 1.519671, not 1. One is the area, but the real value of the sides might complicate the arithmetic. We will deal with it, however.
So while we are getting rid of all the area, what is happening to the perimeter? The black triangles' combined perimeter follow the sequence 3, 9/2 27/4, 81/8 ... . Changing this into an infinite sum to measure the perimeters, we have:
3 + 9/2 + 27/4 + 81/8 + ....
3 + 32/2 + 33/22 + 34/23 + ....
Of course this number grows without bound. At this point we could put 3*1.519671 ... back in as k so that the perimeters are correct.
Perimeter = k + k2/2 + k3/22 +k4/23 + ....
As you can see, the perimeter grows very quickly because k is greater than one.
In the end or at the limit we have a shape with zero area and infinite perimeter. How about that for 'common sense'?
So how far did 'common sense' take us along our route of discovery? Not far, I would say.
Also, note that Sierpinski's Triangle ends up not as a triangle, but a very complicated shape made up of triangles that some call Sierpienski's Gasket.
Can you find our half-life sequence growing somewhere in Figure 2? Yes, it appears along any original side.
Seems appropriate, doesn't it? It's also another example of the series adding to one or in our case to 1.5719671 along any original edge. Why? Let's let k = 1.5719671 ... and look at just one side.
So k = k/2+ k/4 + k/8 + k/16 ... = 1.5719671 .. which is what we expect. The infinite series sums to the right value.
Many such seemingly paradoxical situations can be brought forward, especially in physics. The Greeks did not like them. They did what they could to suppress much of this and sadly a great deal had to be rediscovered much later.
Maybe it was lost because of the fire at the library at Alexandria, who knows? Maybe Archimedes knew it and never bothered to write it down?
How can light act both like a particle and a wave? When it acts as a wave, what is the medium in which it travels? Does some form of ether exist or as we called it the magical oil?
The particle nature of light would be easy to understand for us if we had different eyes. Low light would hit our retina's like a slow beat. The photons would strike our receptors in a steady beat. If we turned the light up to bright, the photons would strike faster and we could see that light does in fact act like a particle.
These are all questions of uncommon sense. They have been the subject of deep study by the best brains available.
In our everyday life, it is a good idea to have respect for subjects that at first, many claim, subject to their common sense. It takes a lot of pluck to say to experts: "But, isn't it just common sense....?"
Pick any document that was submitted seriously to the Joint Review Panel in the hearings and see how far common sense takes you.
This is especially true, if you have experts lined up on one side who have studied a topic for a long time.
We have to step back and try to learn a bit before we jump to our version of what sense is. The lesson for us is to hold off and see the depth of the problem first.
So, what is all this about? What triggered my dismay over all this? Dismay is too mild. Discouragement is more like it. I was annoyed by the 'common sensors'. They made no sense at all.
The idea is this and it is a simple one.
The truth is uncommon. If you are for or against something, make sure you respect the facts that underpin the arguments and don't be so foolish as to say that a complex subject can be turned upside down with the two words 'common sense'.
If you encounter those words in the future, challenge the 'common sense' person to present facts not some version of reality that they dreamed up or heard from the common sense grapevine. It is ok to say: "Well, I don't know, but it seems to me that thus and so are true, but I'm open to expert knowledge"
Don't let some strange and unverified version of somebody's common sense drown the facts.
You'll get a feel for how Sierpinski's Triangle deals with the infinite with this Youtube Video
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Saturday, April 05, 2014