How to Cherish the Earth, Save the World, and Spread Peace and Brotherhood Throughout the Universe

.

Think big!!  After you’ve spent most of your adult life trying to be helpful one blade of grass at a time, you have to seriously start thinking big.  Just keeping my lawn mowed is not going to make much of a difference to the neighborhood, you’re telling yourself.  Taking home one little pup from the animal shelter doesn’t begin to make a dent in the hundreds of pets who are destroyed every day across the nation.  More aggressive action is needed.  (But you have to take the pup home.)

Seriously, though, someone must keep an eye on the big picture.  You’ve done your share for twenty years, but when you look around, things are actually getting worse.  You pick up a hamburger wrapper one evening, but there’s a million kids out there throwing out hamburger wrappers with great determination.  It’s as though they’ve told themselves: this place needs more hamburger wrappers on the street, to be just perfect.

It’s time to step back, and write down some basic principles; some Axioms for doing everything that needs to be done.

We can’t stop doing our part.

• There’s no point in retiring early, liquidating all your assets and giving it all to charity, if you give up steadily doing the right thing at the local level.  You’ve got to keep on with minimizing your resource footprint, leave alone your carbon footprint.   You’ve got to keep grinding on.
• We have to keep alert for what others are doing that’s good.
• Isolation is counterproductive.  Anything that anyone else is doing is worth commending, and worth encouraging.

We must keep alert for new dangers of all sorts.

• New fires must be spotted, and put out.  We have to keep a lookout for large movements by people, businesses or governments that need studying.  In the long run, will they be destructive?  Will they use up resources that will be to the detriment of everyone? 
• We have to keep educating ourselves, because new threats must be countered with new knowledge.  With every advancement in science and technology, the potential for destructive exploitation multiplies, and we must be in a position to act knowledgeably.  It is no longer a time for knee-jerk reactions, if there ever was a time for it.

We have to pass on our attitudes and information to others, especially the young.

• Education is a huge part of the total program.  One of the biggest frontiers is how to mobilize the most effective PR techniques to forward the program of conservation, preservation, protection, and education.  We spoke about the organization called TED, which features exciting and effective speakers to address a carefully selected and influential audience.  They have stumbled onto this idea of making creativity in problem-solving sexy.

We have to recognize that we are not in this alone. 

• The feeling of being in a partnership with everyone is probably a bigger force for peace, generally, than other initiatives for promoting peace for its own sake.  Certainly peace needs no reasons for being a goal: if ever there was a null hypothesis, peace was it.  But in a world that takes conflict as a given, partnership for conservation is good motivation for peace.
• No one is too mean or too young or too anything to be enlisted in the war against senseless consumption of resources.

Identify all resources.

• Some are focused on fiscal resources, especially those who are chained to the dominant economic paradigm in which everything that is done has to be paid for with money that comes from somewhere.  All that they do is designed around how to stimulate certain activities, perceived by them as desirable, with money; how to stifle other activities, perceived by them as undesirable, by denying funding.  These people have to be taken seriously for the moment, because their paradigm will continue to be in force for a long time.  But not forever.
• Some are focused on energy resources, especially economic planners.  This is an industrial mindset, where early in the last century automobiles were the dominant industry, and it took a fair degree of energy to keep the production lines working.  Later in that century, the lack of gasoline threatened the health of the auto industry, reinforcing the belief that development and energy were inseparable.  This mindset is becoming a hindrance, though clearly we cannot do without energy.
• Some are focused on time resources, especially those who believe that time is of the essence for various programs, such as addressing global warming, or developing alternative energy sources.  The elderly are always conscious of their limited resources of time, for obvious reasons.  Busy individuals who are constantly battling the tendency to be given additional responsibility are naturally preoccupied with how much they can do effectively and well in the time they have.
• Some are focused on environmental resources, which is the one resource that cannot be restored easily.  Once an environment is destroyed, not much can be done about it.  Businessmen and traditional politicians tend to ignore the Environment because it is a paralyzing constraint, and highly inconvenient.
• Food resources preoccupy those who are concerned with world hunger.  We have been aware of food resources versus population for so long that we are fatigued.  But it is a real need, and if a sufficient stream of high-quality natural food is not going to be available in the future, as a result of depletion of environmental resources particularly, and the annexation of land resources for more lucrative deployment as industrial real estate or business premises, then we’re going to be faced with food shortages.  As long as business interests are trusted with decisions about land use, land will be funneled towards creating money for businesses rather than growing food for those who cannot afford to pay a lot of money for it.  If land availability gradually evaporates, we will have to make do with low-quality artificial food, and the present uproar about genetically modified foods will be laughable in retrospect.  We will have to start eating foods that are good for us but taste terrible, or food that tastes wonderful and kills us off fast.  If land is used wisely today, we can have food that is both tasty and nutritious.  We had better be careful of where this is going.
• We must not forget human resources.  Businessmen think of human resources as low-wage employees who can be treated like dirt.  But the real issue is how to make use of the human talent that is available to solve the important problems facing society, rather than enlisting them to entertain the bored masses.
• Cultural resources.  This is probably the one that is least thought about by most people.  But we are at one end of a vast chain of humanity, the memory of whose challenges and achievements and joys have the potential to encourage and inspire us.  It is easy to sacrifice priceless historical artifacts for the purpose of locating an office building, or entertainment, or personal glory.  Most of us are accustomed to regarding our own culture and its achievement as the epitome of refinement, and to disregarding the triumphs of other societies and peoples.  But this mindset is self-defeating, and at the least will alienate potential allies, and at the worst will destroy something that we can learn to be inspired with ourselves.  The richness and the diversity on this single planet is the greatest marvel in a marvelous universe.

wikiHow: A review of an unbelievably ambitious undertaking.

.

I stumbled on wikiHow, which describes itself as a clearing-house for information on almost any sort of DIY activity.  Visit it: it is amazing in scope, and miraculous in how it manages to accomplish most of what it sets out to do.

Teenagers, as we all know, have embraced the Internet with open arms, together with all aspects of technology.  While we look on such things as smart phones, faceBook, Itunes and such things with a healthy dose of skepticism, kids dive into them headfirst.

Wikipedia, so often maligned by academics who are frustrated by the potential for finding unsubstantiated information there, or information not endorsed by authorities, is in actual fact very useful for practical purposes.  The whole point of Wikipedia is that it does not wait on authorities to endorse anything.  Anyone can contribute information; it is simply flagged as unsubstantiated if there are no links to pages with established reliable information, or references to printed sources.  If you want to use tentative information, you can.  Kids love Wikipedia, simply because it is easy to use, and is relatively complete.  They can Google stuff, too, but with Google you get directed to the most popular websites, not necessarily the most informative ones.

wikiHow appears to have evolved into a home for teenagers who feel the urge to give advice.  The sheer volume of advice on matters exclusively of interest to teens is stunning.  You get instructions, for example, about how to design a pony in Ponyville.  (Ponyville, if you didn't know, is the center of the My Little Pony universe.)

Some of the advice is startling in the sheer nerve the author must have had to offer the advice in the first place; for instance an article about how to get back on your feet after serving a prison term.  When you think about it, though, where would an ex-con go, to get disinterested advice?  Out of the generosity of his heart, one of them has put together a surprisingly down-to-earth set of suggestions, and in fair good sequence, too.

I signed on as a volunteer, and every time I visit the site, there is a message for me begging me to patrol recent edits to posts.  (There is obviously an apprenticeship period, where senior volunteers look over your shoulder.  So far I have received Thumbs Up for good editing.  Evidently younger editors are not careful readers, and sometimes miss problems in contributions entirely, or misunderstand the intention of a contribution.  They're hardly to blame, because often the original intention is utterly opaque, which is the kindest thing I can say about some of the articles.

The articles range from being totally inane, to being utterly useful and inspired in their simplicity and effectiveness.  The simplicity must come from the original article; the editorial process, because of the nature of the software, is more conducive to band-aids and superficial correction than wholesale restructuring.  The routine editing that a volunteer is urged and invited to do is never likely to correct more than spelling, grammar (rarely) and punctuation.

There is also a big problem with voice.  When a sexagenarian like me (my browser's spell-checker flagged 'hexagenarian', which it probably associates with witchcraft) is editing an article by a thirteen-year-old, just trying out her compositional wings, especially one who writes just as she speaks, in African-American dialect, it is likely to end up reading very differently than it started out.  In fact, it could easily turn off other young readers because it sounds as if it originated with an adult.

Luckily they have a forum, and I've raised some issues there.

If any of my readers have the time for it, I urge you to sign up with wikiHow, and help things along.  My advice is to take it slow; getting too involved too fast is likely to burn you out.  It is far easier to keep your smile on if you don't get frustrated by tackling too many tasks on the site right away (even if they want you to).

Other places you ought to think of supporting with your participation are

• Amazon.com reviews.  If you've bought stuff on Amazon, consider reviewing them, despite the fact that your review helps the website.  Amazon hardly needs our help, but the customers do.
• The Distributed Reading Project of the Gutenberg Project.  This is in the same tradition as wikiHow, and has a similar apprenticeship process, even more stringently carried out.  But they have a more difficult task, and unless they have capable volunteers, they'll continue to process useless pabulum, rather than the priceless public domain books that are just a little harder to get into e-book or simplified pdf format.

Arch

The Paranoia about Food Additives Detracts from Legitimate Concerns

.
There are times at which I wish I knew more Chemistry (than the far more than average amount that I do), and I was recently reminded of this.

People have been concerned for nearly a century (ever since the Food and Agriculture Industries turned to Chemistry in a big way in the 1900's) about the consequences of ingesting large amounts of chemical additives in food.  Recently there was a conflagration on some social media, with a post titled "Eight Foods we Eat in the U.S. that are Banned In Other Countries," which caused me some alarm as I followed the link.

Whenever a concern is raised about something--take the Global Warming phenomenon, for instance--in the electronic society we live in, the concern spreads faster than wildfire, and, unfortunately, finds resonance in people who are nervous, but who don't have the scientific background to assess the relative significance of various aspects of the concern.  So, depending on circumstances, a person could get bombarded, quite by accident, by a number of worries about food additives, which kicks them off on a Wild Google Chase.  Depending on what pops up on their search on Google (and remember, that service, which is also the Mama of this blog you're reading now, is fine-tuned to give the searcher what is likely to grab their interest from among the literally thousands of links that they could give you in response to your query) it is likely to reinforce their worry, rather than set it to rest.  Luckily for me, a blogger on In The Pipeline (?I'm not exactly sure whether he was quoted there, or whether that is where his blog lives,) responded in detail, and mostly convincingly.  You should read this, just to absorb some biology and chemistry principles, which will generally provide a context for discussions of this sort.  My crude summary of some of the issues is as follows:

A.  We eat a large variety of things daily that are poisonous in large doses.

We know this already; almost every medicine we take is typically a poison in large doses.  The word poison, which we use in a colloquial sense, can be probably given an exact meaning, but remember the toxicity of anything depends on the circumstances: the person, what else you've eaten, how much of it you've eaten.  Being a diabetic, even sugar in large doses would kill me for certain (though if I don't eat sugar for long enough, that would probably kill me, too).  Even Vitamin C can be toxic, depending on how it is delivered.  I'm betting that a huge amount of it administered intravenously continuously for a long time is going to be pretty devastating, especially if the body can't get rid of it fast enough.  (The body does indeed excrete excess Vitamin C, but how fast is fast?)

B.  Some of the things we do eat, that could be poisonous in large doses, are eaten because of their beneficial effects. 

Iron, salt, Chromium, Magnesium, Potassium, you name it.  We gotta have them.  Water.  Other things we eat TCBPILD ("that could be poisonous in large doses)" is because they're not beneficial, but are unavoidable. 

C.  Sometimes it is because it is convenient for commercial food manufacturers to put it there, and they don't care about us enough to not put it in there, like trace amounts of pesticides, which we must be concerned about.

I was also recently made aware of the fact that when non-stick pans are allowed to get overheated ---e.g., you forgot to put the food in it one it was hot--- they release some rather nasty chemicals both into the air, and into the food.  It all depends on how much overheated the pan must get before this happens, and whether the pan is safe after it has been carefully cleaned.  Generally speaking, Teflon (and its successors) is an amazingly useful material, but it can easily be abused.  If you tend to be careless when cooking, either use some other sort of cooking receptacle, or give the cooking to someone else.  (In this century we have come to regard being careless with the cookware as a god-given liberty, justified by the Bill of Rights.  That's progress, I suppose!)

Derek Lowe, who wrote the response to the Eight Foods article, did not allay my nervousness about food hormones that are routinely administered to cattle, and antibiotics that are administered to almost all farm animals not in response to diseases, but as preventatives.  Big Farms do things whose harmful effects will not be established for many years, by which time, the Big Farms are betting, they will have made enough profits to pay off any lawsuits, or just go bankrupt, with the money safely in the Cayman Islands, or wherever.  Do please check out the facts for yourself; I'm being lazy not to do it for you, but I don't think a sedentary lifestyle alone can account for some of the beefiness of some of the younger members of our society.  I'm not alone in suspecting that hormones administered to cattle have something to do with it.  Derek Lowe does address something close to this issue, but I'm not convinced.  At any rate, eating beef is not going to kill us because of an additive.  It can kill us more directly because of its essential beefiness.

Now, I agree with Derek Lowe about fears about Arsenic: it is found naturally in the environment, and we had better get used to a little more of it in the environment than we've had in the distant past, just because of how much we tend to use the stuff in various convenience materials (e.g. rat poison), and how all that tends to go in the landfill, and finally, how runoff from the Landfill will eventually end up in the water.  We just can't afford to be too upset about teensy-weensy bits of Arsenic all over the place.

But, if any of you finds yourself in a position to influence policy, here is the scary part: unlike vitamin C, as far as I know both Arsenic and Lead tend to be accumulated in the body.  Now this is not intended to send you into a panic or a coma, but prolonged exposure, even at moderately high levels, to environmental Arsenic and Lead is bad, because of the cumulative effect.  Being in a room containing lead paint for a week is probably not at all dangerous.  (Lead paint is pretty good at keeping the lead trapped in the paint matrix.)  But having lead paint in your bedroom ceiling is very worrisome indeed.  I might not be an expert on these issues, but I believe that if I have alerted you appropriately to concern about Lead in the environment, that's probably good, and if you take my remarks to mean that having enormous doses of lead paint in your environment is nothing to worry about, that was not my intention.  Just keep a sense of proportion about these dangers, and be aware of them, especially where young children are concerned.

Must we be concerned about food additives?  I think we must.  The occasional loose cannon who goes berserk on Fb only distracts from the real concerns.

Must we be concerned about Global Warming?  I think we must.  It is true that a statistical analysis of temperatures around the globe over time does not reveal a warming trend that rises above the variations that have been observed (or inferred indirectly) over geological time.  But science has indeed shown that human activity has resulted in warming, or aggravating the warming.  But that warming is smaller than warming that took place before people were on the planet!  Nevertheless, we're adding to the warming, and, to put it mildly, consequences of continued warming is going to be very uncomfortable for most of us.  It is like being on a leaky boat.  Suppose we're all bailing like mad, but the boat continues to sink, because of natural causes.  Just imagine learning about a number of folks who are actually poking little holes in the hull.  Well, they say, the boat is sinking anyway!  What harm will a few holes do?  What difference will a few extra holes make?

Some people believe that we were put on this planet to make holes in the hull, and they justify themselves by saying that the hull has been leaking for centuries.  I know it isn't a perfect analogy, but that's how it looks from where I'm sitting.

Arch.

Arch Explains the Miracle Formula.

.
A couple of days ago, I supplied readers of this (sadly mathematical-leaning) blog a formula for finding the square root of numbers between 3 and 10 approximately.  The formula was:

which, admittedly, is a formula that only its mother could love, and would make strong men weep.  How, you would have asked yourself, does Arch come up with these disgusting formulas?  (Or, formulae, if you're a stickler for rectitude, always supposing there is such a word as rectitude.)  They're enough to cause cancer in small mammals!

Actually, I can explain the idea fairly easily, starting at a very crude approximation formula indeed, and proceeding to a formula that is a little more tricky, from where it is just a small hop of faith to the one that I gave.

The first step is just to use a line.  The square root of 4 is 2, and the square root of 9 is 3.  The first fact is represented by (4, 2), where 4 is the number, and 2 is the square root.  The second fact is represented by (9, 3).  For all numbers between 4 and 9, and a little to the left of 4, and a little to the right of 9, the line joining the two points (4, 2) and (9, 3) serves as an approximation to the square-root curve.  If you want to see the line and the curve side-by-side, or rather, superimposed for comparison, here they are:

Observe the blue Square Root curve, with the two points (4, 2) and (9, 3) plotted in.  The line we can use as a low-budget way of finding square roots of numbers between about 3 and 10 is the green line.  To find a square root approximately, find the point along the number line across the bottom that corresponds to your number, say 6.  Now go up until you hit the green line; that's the rough square root!  If you want the exact square root, go up to the blue line, and that point gives you the exact square root!

The rough square root of 6 is, according to this line, 2.4.
The exact square root is, according to the little calculator I have on my PC, 2.449489742783178 (and even that is approximate, as most of you realize, only it's certainly a lot more accurate than 2.4).

Now, how does one go about getting this 2.4 approximate value?  Well, as you must have suspected, it is a matter of ratios and proportions, and your Dad could have probably figured it out, even if he had no mathematics training whatsoever, simply by using the sorts of skills they taught people back before calculators.  It is called interpolating.  You say, well, 6 is 2/5 the way from 4 to 9, so the square root of 6 must be roughly 2/5 the way from 2 to 3, and you get 2 + (2/5)1 = 2.4!!!

Now, in order to proceed, we need to do a tiny bit of algebra.  I mean, algebra was invented to further the progress of mathematics, and though I'm trying to minimize the algebra, I can't avoid it completely.  All I want to do, (and I'm trying not to sound obnoxiously apologetic) is to write the ratio-and-proportion method above using x's.  The formula is

What's going on here?  Well, we get the number "2", multiplied by a red fraction, and the number "3", multiplied by a purple fraction.  The numbers 2 are 3 are just the two square roots at the two ends of our line.  Now look at the two fractions, which contain the x's.  If you put in x = 4, the second (purple) fraction becomes zero.  Check it out: the top becomes zero, and with it the entire fraction.  On the other hand, the red fraction simplifies to 1.  Magic!  So the formula becomes just 2.

If you put in x = 9, the first (red) fraction becomes zero.  The second fraction becomes 1, so that the formula reduces to 3, magic again!  So this is a formula that gives the correct y-values at the two points we want.

Next, I have to convince you that this formula represents a line.  Algebra specialists can tell that by just inspection: there are no squares, no cubes, no nothing; in fact, just a tiny bit more algebra simplifies the formula to .4(9–x) + .6(x–4), or .2x + 1.2.  This one is obviously the equation of a line, and it is the correct line, if you put in x = 4 (which gives you 2), and x=9, which gives you 3.  So, if you want a very rough approximation indeed, use this one, for any numbers between 3 and 10.

I know I am very lazy when I'm checking the Web, so I should give you a table of square roots from this formula, and very accurate square roots, so that you can do a number-by number comparison of the exact versus approximate values.  (Don't forget that the 'exact' values are only exact up to however many decimal places I'm going to give you, and I think I will give you, er, 5.  Just to save space.  Here you go:

x	√x -approximate	√x -"exact"	Difference
3.0	1.8	1.73205	0.068
3.5	1.9	1.87083	0.029
4.0	2	2.00000	0.000
4.5	2.1	2.12132	-0.021
5.0	2.2	2.23607	-0.036
5.5	2.3	2.34521	-0.045
6.0	2.4	2.44949	-0.049
6.5	2.5	2.54951	-0.050
7.0	2.6	2.64575	-0.046
7.5	2.7	2.73861	-0.039
8.0	2.8	2.82843	-0.028
8.5	2.9	2.91548	-0.015
9.0	3	3.00000	0.000
9.5	3.1	3.08221	0.018
10.0	3.2	3.16228	0.038

Whoa!  Can you dig it?

The approximate square roots are clearly increasing in uniform steps, and just as clearly, can't possibly be the exact square roots.  But they're certainly correct at two points: 4 and 9, no surprise!  But wait:

The exact square roots are horrible numbers (well, not particularly horrible...) and are sometimes slightly larger than the approximate value, and sometimes slightly smaller.  Actually, between 4 and 9 they're larger.  The exact square-root curve lies above the line, remember?

So, as you can see, we're going to do a lot better with a curve!

Now, let's be serious.  Non-math people know very few curves outside circles and parabolas.  A circle is just the wrong type of equation, unfortunately, and anyway, the formula of a circle has even worse square roots than a plain ol' square root, let me assure you.  A parabola, on the other hand, has the easiest sort of equation next to a line.  So next we're going to try a parabolic approximation.

The easiest way to do this is to borrow the exact square root of one of the numbers, ideally the one plumb spang in the middle between 3 and 10: er ... 6.5.  Yuck.  It's 2.54951, approximately.  Wait; let's try 6.25.  I'm pretty sure its square root is exactly 2.5 ... Yes, it is.  OK, let's use that one.

Here's how we use it.  It's little variation on the equation of the line, with red and purple fractions.  Each fraction is going to have multiple pairs of parentheses, and there are going to be three fractions.  Oh joy, you're probably exclaiming!  So here goes.

The formula, please.   (Drum roll.)
It is:

You're probably wondering what this formula looks like once it is "simplified".  That's what I was afraid of.  Arch is getting old, I sweat and strain, body all aching and racked with pain... expand that quadratic, lift that bale, tow that barge, ...

I got lazy.  I got Wolfram Alpha to do it.  The result was

-0.00808081 x2 +0.30505051 x + 0.909091

How good is this one?  Aha!  Now that we have three different approximation formulas, it does get interesting to see which ones are better!  The first way is to expand our table, to incorporate the quadratic, or parabolic approximation.  This is easy.  (For you.  For me, there's more towing of barges.  But I do it gladly. ... )

x	x -"exact"	x -linear	Diff	x-parabolic	Diff
3.0	1.73205	1.8	0.068	1.75152	0.019
3.5	1.87083	1.9	0.029	1.87778	0.007
4.0	2.00000	2	0.000	2	0.000
4.5	2.12132	2.1	-0.021	2.11818	-0.003
5.0	2.23607	2.2	-0.036	2.23232	-0.004
5.5	2.34521	2.3	-0.045	2.34242	-0.003
6.0	2.44949	2.4	-0.049	2.44848	-0.001
6.25	2.50000	2.45	-0.050	2.5	0.000
6.5	2.54951	2.5	-0.050	2.55051	0.001
7.0	2.64575	2.6	-0.046	2.64848	0.003
7.5	2.73861	2.7	-0.039	2.74242	0.004
8.0	2.82843	2.8	-0.028	2.83232	0.004
8.5	2.91548	2.9	-0.015	2.91818	0.003
9.0	3.00000	3	0.000	3	0.000
9.5	3.08221	3.1	0.018	3.07778	-0.004
10.0	3.16228	3.2	0.038	3.15247	-0.010

Our old linear approximations look a little too simple, in comparison to the parabolic approximations, which go into 5 decimal places.  For the most part, you can see that they have at least two decimal places of accuracy, so we could have rounded up the last decimal place.  (Giving 5 decimal places when you're only accurate to about 3 is considered misleading, and in terrible taste.)

Look at the new column of differences!  They're very small.  The worst errors are at the two extremes, 3 and 10.  As expected, the error is zero at 4,  at 9, and at  6.25!!!  How about that!

It only remains to show you all three curves superimposed, so you can see that the parabolic approximation lies closer to the square root than the line.

The parabola is in orange, which is difficult to see.  I mean, the orange curve and the blue square root curve are very, very close.  However, outside the range between 4 and 9, the two curves diverge dramatically.  Here's a larger-scaled picture:

(If you have mathematical friends, they might point out, if they're cruel, that actually both the blue curve and the orange curve are parabolas, but just ignore them.)

So anyway, the quadratic approximation is very good indeed, but the very first approximation I gave you is better still.  After a while, though, the additional accuracy is hardly worth the computational complexity, and using a curve of degree five will probably only gain you a couple more decimal places of accuracy, and give you a lot more work to calculate, when you actually want to get an approximate square root quickly.

Finally, all this mathematics is going to be of theoretical interest only, for the simple reason that when you need a square root, what are the chances it's going to be between 3 and 10, really?  But this is the sort of thing math people do, to begin with, to develop tools for studying the business.  Similar methods can be used to get any square root approximately, I assure you, but it is too difficult to show you here.

Arch.

Functions and Formulas: Mathematics for the Masses!

.
I have finally flipped my proverbial gourd!  I have decided to explain a little mathematics to my readers, who have probably gone through life saying to themselves: "I've had a wonderful run, in all my 45 years (or fill in some appropriate number; you are all very likely to be different ages, but probably very nice ages, each in their own way) without being confused by mathematics higher than multiplication.  I'm not about to start now."  Well, I'm writing this for my own satisfaction.  There's nothing a mathematician likes to do more than satisfy himself!  (I have sprinkled exclamation marks liberally throughout this post, in a vain attempt to draw in the gullible reader.)

Let's start with square roots.  You probably know that the square root of 9 is 3, the square root of 16 is 4, and, best of all, the square root of 25 is 5.  Or, should I say, 5!  (That's actually not advisable, because in mathematics, writing an exclamation right after a number means something entirely different.  Ok, so back to plain old 5.)

What does this mean?  It just means that 4x4 is 16, 3x3 is 9, and of course, without exclamations, now, 5x5 is 25.  You're probably wondering why anyone would go to the trouble of multiplying a number by itself, in the first place.  (Well, at least in the case of finding the area of a square, you have to agree that it comes in handy.  The number of square inches in a square twelve inches on a side is 144.  Conversely, if you happened to have a square of area 144, you work backwards and observe that it must be 12 inches on one side.

This brings us to the idea of a function.  The square root function is just a formula --a very simple formula, actually-- using a variable, x.  The formula tells you the side of a square, when you happen to know the area of it.  The formula in this case is

 

Now, this is where things get interesting.  It starts off by making a chart of a variety of numbers, and their square roots, which is an obvious enough thing for anyone to do.  I made this chart in Excel:

Frankly, I'm disappointed, and you must be laughing.  It certainly shows us the square roots, but fails to report whose square roots they are.  For example, I'd like to be told that the highest bar is the square root of 25, for heaven's sake.  What use is just the number 5?
So I'm going to make a different sort of chart, namely a scatter-plot.  I'll first make it, and then explain it.

Now, this is nice.  Each little diamond gives a number, for instance 9, and its square root, which is 3.  Perfect.

Notice that they have also joined the diamonds up into a sort of curve.

That curve is what I want to talk about.  It suggests that numbers between 9 and 16, for instance, might have square roots too.  I'm sure you know that every positive number has a square root, but the interesting thing is that the square root of a number between a and b will have a square root between the square root of a and the square root of b.  You might think that's obvious, but ... well, let's just say: other functions do not behave the same way.  (Informally, a function is some formula, like a square root.)

I'm obviously going to pick nice numbers that have square roots that are easy to find.  That's called stacking the deck.

Let's consider, er, I don't know ... how about 10.24?  (Heh heh!)  You'll find that its square root is 3.2.   If you had the patience and the motivation, you would find that starting from the point between 10 and 15 on the chart above that corresponds to 10.24 (I know; it's practically impossible to find exactly), if you go straight upwards to cross the pretty blue graph, you would cross it at a number that corresponds to 3.2 approximately.

In fact, 3.2 will actually be just a little above  the pretty blue line.  Why?  Because the pretty blue lines are actually crude approximations to a pretty blue curve.  Unfortunately I'm at home, and I don't have access to software that can generate the curve ... wait, I actually do.  Here you are:

(If you're really interested, I could superimpose the graph consisting of little lines and diamonds on top of this one, but you probably get the idea.)

So there it is.  The curve, or graph, consists of millions of dots that stand for square roots of numbers.  The x-position of the dot --for instance the dot named B in the graph above-- is some number, in this case 10.24, and the y-position, how high up the dot is, is the square root of the first number, in this case 3.2.

Though of course you can find square roots directly from the diagram, using some sort of measuring method, it is actually more a conceptual thing.  You can see, going from left to right, that square roots of numbers increase a lot slower than the numbers themselves.  For instance, the square root of 16 is 4, but the square root of 25 is just 5.  (This is the opposite of the way taxes increase, for instance.  You earn a little more, and your taxes are a lot more!)

Now, girls and boys, I'm going to do something a little strange.  I'm going to talk about approximating square roots.

Here's the deal.  I would like to give you a formula to find square roots fairly exactly, using a formula that only involves addition, subtraction, and multiplication.  These sorts of formulas are called polynomials.  All you need is a really cheap $2.99 calculator that does multiplication, addition and subtraction.  (You're probably thinking, well, those calculators can do square roots, too!  Ok, it's the principle of the thing.  I can even find you a formula to do more difficult things, but ... enough for the moment.)

I'm not actually going to tell you the process; I'm just going to give you the formula.  This one will only find for you square roots of numbers between 3 and 10, and very roughly, too.

where, as usual, x³ stands for x . x . x, and so forth.

Pretty cool, huh?  You're probably thinking that using the square root key on your calculator is much easier, but still, isn't it interesting that you can approximate one kind of function, a square root, with an entirely different kind of function, namely a polynomial?

The approximation isn't very good between 3 and 10, and much worse outside that range.  Just to show you how bad it is, here is a graph showing both of them, and you can compare:

The approximation is in green, as is its formula, and the square root is in blue.  The two graphs look almost exactly the same between the (x-) values of 3 and 10, but you sort of know that they can't be.  I could blow up the graphs, and you would see a slight difference, but I imagine only die-hard math folks would care.  It's enough for you to know that it is possible to approximate lots of functions --especially continuous functions-- with polynomials, which only involve addition, subtraction, and multiplication.

In case you were wondering why I'm making all this fuss about polynomials, it's simple.  Computers ultimately only do multiplication, addition and subtraction.  All this crazy stuff you see on the screen is based on only those operation, and simple logic.  To be sure, a computer has to do a lot of addition and multiplication to show you a website on a screen, but we all know that computers can do stuff very fast.  (Many programmers don't quite realize that the software they build is ultimately based on just addition and multiplication, because the computer itself translates program code into arithmetic at a level below the level at which most programmers work.  (The arithmetic aspects are built into the design of so-called chips, which is done once and for all when someone invents a chip, or when someone writes a compiler for a language; the rest of the time we generally ignore the fact that computers are essentially just adding machines in disguise.)

One of these days I'm going to blog on how to compare formulas.  It is a fascinating idea that anyone can understand, but one that is usually only considered by specialists, which is a shame.  Of course, you can't use the information, but ... well, how much of the stuff I tell you can you really use, when you come right down to it?  Not a lot!

[P.S.  I suppose I ought to have tried to calculate the (approximate) square root of 10.24 using my polynomial, to see how close it is to the exact value of 3.2.  After all, this was the point of the whole exercise!

Well, I used Excel, and ... lo and behold, it gives: 3.203381 amazingly enough!  Just a little too high, but still, not bad, huh?  This is obviously a rounded value, and I didn't do it by hand, but, for what it's worth, there it is.]

Arch

This Post is an Experiment

.
I'm trying to put a music file in this post without going to all the trouble to make a video of it.  Most of the examples I want to give do not have relevant video to accompany it.  Some of my readers probably appreciate the music score I supply, but it is far more data to put on the Web than my humble posts deserve, let's face it.

OK, here goes:

Woo, woo!  That was a total success!!!!!

A few words about the tune:

I had featured this piece earlier; it is an aria from a CD which I wonder whether I should name, for fear of getting into trouble, or whether I should give the full reference to it, in the name of truth in advertizing, and hoping that some of you will decide to buy it:

OK, the latter, then: it is Vergnügte Ruh, from Cantata no 170, BWV 170.  [All the Cantatas have the same BWV number as their Cantata number, because Wolfgang Schmeider, who catalogued all of Bach's works, catalogued the Cantatas first, in their numerical order.  I do not know who assigned the number to the Cantatas; possibly the Bach Society (The Bach Gesellschaft, whose catalogue is called the Bach-Gesellschaft-Ausgabe).]  This performance is by Guillemette Laurens.

Warning: I have also modified the piece somewhat, for obvious reasons.

[Monday, June 10]
Here are a few audio files you might like.

 This first one is a recently generated file of the Serenade which I wrote a couple of years ago, and is almost my only composition (apart from a chorale-prelude written when I was nineteen).  The more I listen to it, the less satisfied I am with the musical ideas in it, though I did the best I could with what I began with.  Inexcusably, I have forgotten the exact scoring of this version; I think it is for solo strings and a clarinet.

[Added still later: no, that does not work, even for me.  There's something wrong with the protections on the file server.  The one below did not work either, I know; so much for being able to put a whole lot of files on for you folks...
Here is a (minimal) YouTube version of the piece.]

This next is the amazing opening chorus from J. S. Bach's Passion according to St Matthew, or Matthäuspassion, as it is called in German.  In case you weren't paying attention back in 2009 when I wrote about it at length, it is for two complete choirs and orchestras, probably located in two areas separated by at least several feet, and the piece is antiphonal, that is the sound goes back and forth between one chorus and the other.  (Gabrielli, in Rome, used the same principle in some of his best known pieces written to be performed exclusively in the Vatican, but, as the story goes, Mozart reconstructed the music from memory soon after he had heard it.)  The opening chorus uses the two choirs, as well as a children's chorus, probably situated between the two choirs and orchestras, in the middle.  The melody of the children's chorus is doubled softly in the organ.  To appreciate this wonderful piece one needs the full power of stereophonic sound, and most modern recordings try to do exactly that.  When creating a sound file in a computer, the matter is relatively easy.  You will probably find the stereo separation a little too extreme, but here it is, for what it's worth:
[As you know, the method I used at first did not work, so here is a link the old-fashioned way, via YouTube.  The video is absolutely minimal, as a protest against an unfriendly universe; but there is an interesting feature you will notice.

I had two files with this piece: an mp3 generated with my software, with the antiphonal choirs and orchestra as I described above.  However, I had used up all 24 lines of music that my software allowed, and when it came to adding the double bases, it wouldn't let me!  Can you believe it?  Bach does not specify double basses, but with such enormous musical forces, two entire choirs and orchestras (which, actually, could have been quite modest, separately), it was the practice to use proportionately heavy bass.

Well, in a spirit of adventure, I downloaded a trial version of the full software, Finale 2012, (the big brother of the PrintMusic I own,) which allows unlimited numbers of staves, so it was a mere bagatelle to add just two more: one line each of double basses for each choir and orchestra.  Of course, I had to take the opportunity to make it generate an mp3 for me, with these fabulous double basses.  Remember, this is slightly unfamiliar software, and it does things in slightly different ways than old buddy PrintMusic; it's like comparing a 1040EZ and 1040 for doing your taxes, you see?  By the time I had itemized my double-bass deductions, I had wreaked all sorts of havoc up and down the score.  Here is the tiniest possible video I could make easily.  (It is possible to make an even smaller video using Adobe Flash, but the software is completely screwing me up.)

This work, the St Matthew Passion has almost legendary stature within the Christian musical community, and old-time Bach fanatics find themselves sucked into the veneration of the work, and confuse musical veneration with religious veneration.  I'm reluctant to make disparaging remarks about the phenomenon, which you have to experience for yourself.  YouTube has a clip of Karl Richter, a sort of law unto himself, performing this same chorus.  It takes some 11 minutes, in contrast to the 5 that my (mechanically produced) version took.  (It was not taken at an insane speed, either; you probably thought it was a little too deliberate.)  But here is the Richter version.  The performers seem to feel that they are participating in a religious experience.  It is very difficult to sing at such a slow pace; the breathing is almost impossible.


Well!

Before you go away thinking that Karl Richter is a total loss as a musician, here is something that might convince you that his musicianship is more than salvageable.  He was an amazing performer on the organ and the harpsichord.  This is one of Bach's most celebrated works: the Brandenburg Concerto No. 5 in D major for Flute, Violin and Harpsichord.



Sock!Fight And now for something completely different. A song by my daughter's band

James Era

[To be continued.]

Why American Business nurtures a culture of Waste

.[
I think I have bitten off a little more than I can chew, here, but someone has to ask the hard questions, so that we can all think about this issue in easy stages.

[Why is it so important to focus on this issue?]
It seems obvious to me that waste is a problem.  The first word that springs to the lips is Landfill.  But waste is of many kinds: waste of time, waste of energy, waste of resources, and I suppose there are others: waste of energy, waste of talent, and so on.  We're finally getting to see that, though nobody likes waste in government, it has been assumed that in business, because profit is the principal motive in private enterprise, that waste control is built-in.  But it is not so; the culture of waste is actually driven by business.

A family may be able to afford to buy a new car every five years or so.  (The wealthier sector of the population can probably recycle cars more frequently.)  The same goes with a lot of things: computers, televisions, furniture, homes, buildings.  Everything is being built more cheaply, and unlike a half century ago, a used car simply does not have the quality in it to provide a used-car buyer with a reliable vehicle for a couple more years.  It used to be that the Japanese made cars that would last longer, but they've learned the lessons of planned obsolescence very well.  With advanced technology, it is actually possible to ration out the advancements slowly, so that a few months after you make a purchase, there's a new release of technological advancements all round.  (Manufacturers will not sell a new line until a competitor is about to release something comparable.)

Recycling buildings is sadder still.  All around us we see badly built buildings being torn down increasingly frequently (and sometimes we're glad, because the thing was such an eyesore).  But can the community afford to have buildings put up and torn down so frequently?  It provides jobs.  But can we afford to have waste being the engine of employment?
 
[What evidence is there to support this claim?]
There is little evidence I can produce to the reader except that I have seen waste in every place of business that I have visited, and have heard evidence in thousands of conversations with friends and acquaintances, especially those who have grown up in cultures that abhor waste, and who have come up against ridicule of the "Hey, this is America!" sort, and some of them have gone on to see embracing waste as a part of their assimilation into US culture, which is the saddest thing that has happened to them, among many other sad things.

Coping with News, Contributing your Mite

.
Well, this should be interesting; for the first time, I'm typing up a post on a tablet.  I usually put on a good number of images and details so that I almost  have to use a keyboard.  Anyway, it should be instructive.

This morning KT told me that it had been recently revealed that, contrary to the popular misconception, there had been considerable sexual misconduct by American GIs soon after the ceasefire of WW2, while the troops were waiting to be transported home.  Apparently a large number of troops had returned to Le Havre, and proceeded to have, in some instances, sex in public [See here for the original NPR story].  Numerous French women, we're told, having lost their men in the war, flooded to LeHavre to engage in sex (or prostitution) with the American soldiers returning from the front.  And of course it is those awful French girls who are to blame.  Er.

(That was too hard.  Back to the keyboard.)  This is typical behavior of soldiers all over the world, and we need not be particularly surprised.  It is just that that particular generation of US soldiery have traditionally been especially venerated, and their foreign escapades during the war, and the scrapes they got into shortly afterwards were kept out of the media for the most part, and, perhaps misguidedly, the US propaganda machine was able to portray them as being a cut above typical soldiers: the Great Generation.  All this public misconduct perhaps detracts from their greatness only a tiny bit, but it is just one more instance of the less-than-fabulousness of the Land of the Free and the Home of the Brave (and extremely red-blooded, ahem).

What can one do, when one is besieged by information that piles horror upon horror, misfortune upon misfortune, calamity upon calamity?  The environment, the economy, society, the bees, the genetically modified foods, Monsanto, Sara Palin, Cayman Islands, Social Security, Home Mortgage Crises, Congressional Lobbyists, Syria, Somali Pirates, Polygamy, and Fracking?  It's almost as if let there be one more hurricane, and you're ready to shoot yourself.

Well, all our friends --well, most of them-- are just as concerned as we are, but we're surrounded, out here, with determined conservatives.  (However, we're told, that the conservatives in our part of the world are pretty nuanced in their positions.  They're pro gun, but some of them are pro choice; they're conservative Christians, but some of them are not Creationists; they're pro fracking, but they're unhappy about Big Oil and pollution.)  We're careful about broaching Liberal Politics with people we don't know.  Mostly, we keep in touch, rather desperately, with like-minded people we know across the country, who post liberal stuff on Facebook.  I cautiously "Like" these posts, knowing that almost all my "friends" on Facebook are conservatives!  But they're my friends; it is weird.

People deal with this onslaught of unpleasant information in many ways.  (I'm talking about people who think like we do.)

For instance, one of KT's friends posts pictures of animals that were rescued.  There are horrible photos of the condition of the animal before the rescue, and after a period of recuperation, the animal looks perfectly happy and normal.  I'm not sure that the lady in question has anything really to do with animal rescue operations, but she appreciates what they do, and I am confident that if the opportunity arises for helping the operations financially, she would probably contribute.

Another of our friends fights tooth and nail to prevent drilling for gas in wild areas in Pennsylvania.  (The gas-friendly GOP administration has moved to open portions of Pennsylvania State Forests for drilling.  Drilling requires, of course, a lot more of a footprint than just the drilling pad.)  She attends every public hearing about opening up new areas for drilling, and speaks up.  She supports the only state representative from our area who has stood against drilling in nature preserves generally, and who has kept up the pressure to make gas companies pay more taxes specifically for the purpose of repairing roads and addressing spills and pollution caused by the drilling operations.  She runs for public office, and we're hoping she gets an opportunity to influence the process.  (Fiscal conservatives tend to equate energy with lower unemployment and an improved economy, generally.  This is probably true in the short run, but it will be a very short run indeed.)

Another of our friends is active in addressing the rising xenophobia across the country.  Even those who have generally regarded foreigners with some suspicion are beginning to see the danger of the rapidly spreading racism, and hostility towards Muslims.  A large proportion of African Americans are Muslims, and have absolutely nothing to do with religious extremists.  As one banner puts it, Islamic fundamentalists are to the typical Muslim as the Westboro Baptist Church is to Christians.

My few Liberal friends on Facebook keep up an onslaught of liberal propaganda, which, on the positive side, make me feel each morning that the world isn't completely coming to an end.  If all I could see was conservative opinions and disinformation (and there certainly is a lot of it) I could not deal with each new day!  This seems to be the hallmark of rational folks that I know, that they tend to be determined to keep up the pressure, hoping that the ignorance will be gradually eroded.

A lot of them are teachers.  A teacher is in a particularly vulnerable position, because if they incur the hostility of the parents of the kids in their classes, all sort of problems will arise.  So a teacher, for sheer survival, has to adopt a policy of teaching by example, rather than preaching.

In many foreign countries, of course, enlightened policies are usually adopted well before US authorities consider them seriously.  For instance, it appears that in Australia, where citizens were initially just as alarmed about gun control as they were here, the legislation has recently been passed, and gun-related crime is significantly down.  US citizens have an unreasonable faith in the superiority of our gun-toting criminals, and refuse to consider similar laws for gun control.

Few foreign countries permit religious lobbies to have a say in teaching of science, especially geology and evolution biology.  Americans alone are more religious than rational.  Even in Italy, the home of Catholicism, evolution is taught.  But across the US, there are science teachers who stubbornly refuse to toe the line in those school districts in which Creation Science is required.  Hooray for them.

The Boy Scouts of American finally adopted a policy that openly homosexual boys would be permitted to participate in Scouting, thus putting to rest a regrettable chapter in their  history, and in the history of the US.

To summarize, I can only say that despite the gallons of bad news we get every day, I make it a point to focus on the few ounces of good news that do make it past the censors, and I cheer.  We are helped by the fact that we do not watch TV.  (On those occasions when we're forced to watch it at a doctor's office or some such place, we're more amused than appalled at the degree to which local newscasters will go to dig up some dreadful little nugget of sensationalism with which to disgust their loyal viewers.)  I can only assume that people have been doing stupid things for millennia, but it's only recently that news organizations have become convinced that it's bad news that pushes up the ratings.  Is this true?  At any rate, I'd rather know now, however belatedly, that there were a good number of rapists and public adulterers among the Great Generation than to have been kept under the misconception that they were a more saintly lot than either their predecessors or their successors!  We've been scum for a long time, and it's good to know that we were consistent during WW2!  It doesn't bother rational people.

[Added later]
I was racking my brain to think what we do ourselves, KT and I, to put our spiritual finger in the figurative dyke, and I think it is recycling.  We put as much as we possibly can in our recycling container, and KT religiously takes it out to the recycling center whenever she has a spare minute.  Sometimes she takes the dog (but we're not ready to recycle the dog yet).  It turned out that Best Buy does not take in used batteries for recycling, which is the best thing Best Buy could possibly do for community service, quite honestly.  So we have an enormous pile of batteries, which we're going to regretfully throw in the trash, unless something better comes along pretty soon.

We recycle so aggressively that we often later discover that we've recycled things we happened to need.  Woe is we.  (Or woe is us, whichever is correct.)

On a slightly related note: a recent post by a Libertarian friend on Facebook asserted that many of the most popular charities, such as the Red Cross, and Goodwill, reportedly had CEO's who made enormous amounts of money.  They claimed that, generally speaking, very few cents of each dollar they received in contributions were actually used in charity.

Well, we have to put this in perspective.  A quick look at Snopes.com revealed an article on their website addressing this very problem.  (If you follow the link, make sure to go down to their response, which is quite far down the page, at the heading "Origins".  Everything that comes before only summarizes the accusations.)  A tool for measuring the effectiveness of charities has been developed (by, I believe, Charity Navigator) called their efficiency ratio, by separating the total budget of a charity (which has to be public, except where it is a religious organization, another good reason not to depend on religious charities, in my opinion) into (salaries + benefits), and (everything else), the latter presumably going towards their mission.  Today, it is not unusual to see the CEO of a charity earning an annual salary of around $400,000.  The U.S. President, for comparison, earns somewhere in the region of $350,000, as do members of Congress and Senators.  So yes, these people do earn a lot of money, but they're drawn from among the educated elite of the nation, and that sort of money is not out of line for a person who has to live or work in the vicinity of the bi-coastal megalopolises.  Most charities, it turns out, have an efficiency ratio of around 75%, the Salvation Army having a higher efficiency, and the Red Cross maybe a little on the low side.  So you decide: is the financial efficiency of your favorite International charity in line with your expectations?  If it is, then you should continue to support it.  If your standards are higher, you have to find more satisfying ways of spending your charity dollar.

If you'd like to contribute your mite where it really counts, here are some ideas.

* Hire someone to do your recycling for you regularly!  Some enterprising high-school kid might haul your stuff to the recycling center and sort it for you for $20 a week.  Okay, so you can't claim it on your income tax return, but if you persuade a neighbor who might be between jobs to do it, (or his wife, who is probably a little more likely to actually spend the money usefully!) you've done something really useful.  Two things, actually: you've spent money in your neighborhood, and kept some junk from the landfill!  (For those who live outside the US: The Landfill is the US euphemism for the enormous pile of trash that sits outside most towns.)  Three things, if it inspires the neighbor concerned to recycle his or her own trash.  Don't forget to recycle printer cartridges, cardboard cartons, and junk mail.  Note: if a company sends you regular junk mail, put it in a large envelope and mail it back to them.  Junk mail companies usually have a budget for recycling their own unused stuff, and will have to deal with this bounced-back junk mail in an acceptable manner.  In some communities, businesses such as mailing / sorting agencies are required by local laws to recycle.  Our local college, for instance, is required to recycle flat white paper; not nearly enough, in my opinion.  They should be recycling magazines and brochures too, because faculty get a lot of junk mail.

* This hiring someone to take care of things that you might be too unmotivated to do just struck me as a great idea.  I might hire a kid to organize my record collection.  I have spent close to $100 on duplicates of CDs that I already owned, because I had misplaced them, or put one in the wrong case, and then lost it.

* Cleaning up a section of highway is another great idea, especially if you can arrange to do it regularly with some friends.  This sort of thing is boring by yourself, but with a crazy buddy along, it might not be a chore at all, even if the buddy doesn't do much work.  I just had an idea: I wonder if our little honor society might not be persuaded to take on a section of highway ...

* One thing I must do, is to volunteer at the local library.  KT and I have found that our local library and its staff are always interesting people to be around, not to mention that the Library itself is just amazing.  They have a wonderful multi-purpose collection, and have all sorts of creative people putting together a storytelling event for kids, or a summer reading program, or an adult literacy initiative.  I would be content to shelve books, but I suspect that they have enough volunteers for whom shelving books is about as sophisticated as it gets...

So, listen to Lake Wobegon, and keep the good stuff in your sights, though you know that there's a lot of crap out there that does need addressing.

Arch