Fibonacci Numbers: Math Meets Mysticism

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Image Of Sunflower via WikipediaFibonacci (not a real name), translates to "Son of Bonacci." This brilliant mathematician and philosopher was endowed with a natural (or perhaps supernatural) curiosity regarding numerical series and patterns. The Fibonacci Series or Sequence is perhaps the most amazing key ever discovered to the actual design of the world we live in. It is a numerical structure and relationship that somehow underlies the essence of virtually all living creatures and systems. A friend of mine ( a clergyman) once referred to this amazing string of numbers as "God's own creation  equation." My graduate school thesis advisor, himself a stunningly brilliant man, stated that [paraphrasing] "Fibonacci unearthed something every bit as important as the Watson-Crick Model Of Human DNA."

As for me, my feelings about this incomprehensibly enormous epiphany, and about the mind which was driven (or directed) to find it, can be summarized thus:


"Einstein developed a Theory, but Fibonacci discovered a Fact."

-- Douglas E Castle
  
TNNWC Management Consulting Services - "Catalysts For Organizational Growth And Profitability"
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Fibonacci Number Formula

The Fibonacci numbers are generated by setting F₀=0, F₁=1, and then using the recursive formula:F_n = F_n-1 + F_n-2in order to get the rest. Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...


This sequence of Fibonacci numbers appears all over mathematics and is pervasive throughout nature -- it is a recurring theme, a mysteriously consistent pattern in the structure of many living things, as if all were designed utilizing the same magical engineering program.


We see this Fibonacci design everywhere, if we know what we are looking for.They are intimately connected with the golden ratio, for example the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.


The elements of the Fibonacci design also appear in biological settings, as mentioned earlier, such as branching in trees, arrangement of leaves on a stem, the fruit spouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.


The Golden Ratio (computed based upon any number in the Fibonacci Sequence divided by the number in the Sequence immediately preceding it), or the Golden Rectangle (based upon consecutive Fibonacci numbers as the measurements of its sides) are seen in the artwork of Leonardo DaVinci, in the design of numerous architectural structures, and in all manner of painting, sculpture and design.


In nature, it is the most prevalent theme; in art, engineering, science and even economics, Fibonacci Sequence, the Golden Ratio, the Golden Rectangle and the Golden Mean, either due to their embedded existence in the Human subconscious or because they are consciously and deliberately incorporated by us, as a certain standard of balance or beauty, into a system, design or form. It is rather like the obelisk in Arthur C. Clarke's book (later, a movie directed by Stanley Kubrick), "2001: A Space Odyssey."


The actual Golden Ratio (signified by the Greek letter Phi) is approximated at

We can always find the next number in a Fibonacci Sequence, by simply adding the two consecutive numbers in the Sequence that immediately preceded it. But there is a problem...[isn't there always?]


If we wanted the 100th term of this Sequence, it would take lots of intermediate calculations with the recursive formula to get a result. Can there be an easier way? Of course! Would I ask a rhetorical question without knowing the answer? Well? Would I?


Yes, Virginia, there is an exact formula for the n-th term! Before I reveal this formula to you, please know that the letter Phi stands for 1.6180339887, and that "Sqrt" is an abbreviation for "square root," although it does sound like the name of a carbonated soft drink.


The formula follows:


a_n = [ Phiⁿ - (phi)ⁿ ]/Sqrt[5].
where Phi=(1+Sqrt[5])/2 is the so-called golden mean, and phi=(1-Sqrt[5])/2 is an associated golden number, also equal to (-1/Phi). This formula is attributed to Binet in 1843, though known by Euler before him. Mathematicians, unlike Braintenance Bloggers, tend to be egomaniacs.


Even medical research scientists like to have diseases (and even entire syndromes!) named after them...biologists like to have newly-found species named after them. Heck -- there's even a company ("Star Registry") or something like that will officially name a distant star after someone you love, and issue a certificate of authenticity (on fine paper) to you for you to present to the object of your affection.

Some of us are obviously brilliant (and we know who we are) -- others are merely megalomaniacs.

But then, I digress. This article is dedicated to Fibonacci, who discovered something truly magnificent and inspiring; something so great that its implications are felt in every single discipline of course of study...from art, to the financial markets, to theology, and beyond.


Douglas E Castle
[http://aboutDouglasCastle.blogspot.com], and [http://www.LinkedIn.com/in/douglascastle].

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Strengthen Your Mind: Word Games!

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Every word has numerous meanings and may be used in numerous contexts. Knowing this, we create puns, spoonerisms, oxymorons, paraprosdokians, paradoxes, riddles and other recreational wordplay.

One of the greatest benefits of this wordplay is the accessing and  exercising of multiple parts of the brain. This serves to strengthen our creative and associative abilities, to ward off senile dementia and intellectual degradation and to sharpen our analytic and verbal skills.

As a side benefit, my beloved Braintenancers, we also entertain ourselves and eachother. 

Here are some twisted definitions which will be of interest to all, and in particular, to those of us privileged to be Baby Boomers. Thanks to Ms. Jane Portnoy for her contribution to this article.

Let's let 'em roll...

ADULT:
A person who has stopped growing at both ends
and is now growing in the middle.

BEAUTY PARLOR (if you remember this terminology, you’re showing your age):
A place where women curl up and dye. 

CHICKENS:
The only animals you eat before they are born and after they are dead.

COMMITTEE:
A body that keeps minutes and wastes hours.

DUST:
Mud with the juice squeezed out.

EGOTIST:
Someone who is usually me-deep in conversation.

HANDKERCHIEF:
Cold Storage.

INFLATION:
Cutting money in half without damaging the paper.

MOSQUITO:
An insect that makes you like flies better.

RAISIN:
A grape with a sunburn.

SECRET:
Something you tell to one person at a time.

SKELETON:
A bunch of bones with the person scraped off.

TOOTHACHE:
The pain that drives you to extraction.

TOMORROW:
One of the greatest labor saving devices of today.

YAWN:
An honest opinion openly expressed. 

WRINKLES:
Something other people have,
Similar to my character lines.

I hope you've enjoyed this journey into the the world of smart-aleckisms. Next time (sigh) we'll be dealing with a mathematical challenge.

Douglas E Castle [http://aboutDouglasCastle.blogspot.com]

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Figures Of Speech: Malaprops & Moronyms.

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We bloggers (and the occasional speechwriter) use dictionary.com for quick word definitions, etymological stories (there's one below) and for alternate word choices. For this last, we use the Thesaurus feature. It ("Dic," as we chronic abbreviators and vulgar insinuators often call it... for short), is enormously useful, despite the fact that it is so laden with ads, pop-ups, pavilions, flash graphics and other distractions that the quick bit of information which you seek is often hidden ("obscured" would have been a snazzier word choice, but I'm in a rush) on the page.

While some people (mostly 23-year old internet billionaires or Baby-Boomers taking a break from looking for an employment opportunity which they'll never find) think this "word search" or "improve your visual acuity"- type exercise is fun, I detest it, although I wouldn't put that in writing. Being the Baron Of Braintenance (my hobby) as well as the modest Chairman of TNNWC Management Consulting Services (an adult job), I must keep all of my options and resources open. Flexibility and adaptation are crucial business skills.

Today, I needed an alternative for the word malapropism, and stumbled upon Mondegreen, which is defined and discussed below, courtesy of Dic. When children learn things by being encouraged to parrot what they are hearing from adults, they become mondegreen-manufacturers. For example, I remember singing "My country tis a bee, sweet lamb of liver tea, of thee I see..." And I wasn't the only one. It was a pandemic.

This mondegreenation (not exactly a Lingovation, but a grammatical extrapolation) is a product of a) not listening too closely, and b) not understanding the meanings of the actual words. Acquired early, these survive late into adulthood and can be the cause of embarrassment.

As children, we were afraid to run out in the cold for fear of catching "ammonia" and going into a "comber."

I once dated a woman (in her 40s at the time) who asked me if I had "...ever seen the Three Stoogers as a child." I knew that she had intended to say "stooges," but the damage was already done. I responded (smart aleck that I was) by saying "Yes. That was the best way to see them." She looked perplexed, but grinned sweetly. I knew that my overnight bag was going to stay in the trunk of my car.

Collectively speaking, I like to refer to all of these errors in spoken communications as "Moronyms." Whether they are spoonerisms, malaprops, grammatical extrapolations [I invented that] or just dumb things like "Notary Republic," the term moronyms (a legitimate lingovation, and cleverly engineered to look like such real words as homonyms, and synonyms) covers the field.

Enjoy your mondegreens, my cognition-craving, brain-straining colleagues. And be certain to visit these other pleasure stations within the TNNWC InfoSphere for stimulation: http://aboutDouglasCastle.blogspot.com, http://TakingCommand.blogspot.com, http://SendingSignals.blogspot.com, and http://Links4LifeAlerts.com.

Douglas E Castle

mon·de·green

/ˈmɒndɪˌgrin/ Show Spelled[mon-di-green] Show IPA

noun

a word or phrase resulting from a misinterpretation of a word or phrase that has been heard.

Compare eggcorn.

See also malapropism.

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Origin:
1954; coined by Sylvia Wright, U.S. writer, from the line laid him on the green,  interpreted as Lady Mondegreen,  in a Scottish ballad.

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Pi: A magical number (and easier to say than "Phi")

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The number Pi is a mathematical constant. It represents the ratio of the circumference of a circle to its diameter. If you divide the circumference of any circle (or perfectly-formed, hand-tossed pizza) by its diameter (the measure of a straight line cutting the pizza in half), you will get Pi, which is equal to 22/7, or approximately 3.14. Now let's review Fabian's quandary, from a couple of days ago...

Background:

Fabian Focaccia (not his real name, which is Sal Monella)  is a struggling 'artist' who is working at a neighborhood pizzeria in Brooklyn, New York (not his real location, as he is in the Federal Witness Protection Program, but which is still the best geographical location to make a pizza purchase if you/ youse should ever get around to it) to pay his bills until he can sell one of his paintings. He is faced with a decision and needs your help. He has cardboard boxes for 'take out' pizza (this pizza parlor does a big take-out business -- even for Arizona...oops!) which are each three inches deep (irrelevant for solving this problem) and measure exactly 20 inches by 20 inches square.

The Pizza Box Puzzle (In Two Parts):

1) What is the circumference of the largest pizza (assume that it is hand-tossed and perfectly round) which can be placed in the box neatly, i.e., placing it flat without stuffing it in and distorting its perfect shape?

Answer: This is really just a circle inscribed in a square. Here's a picture:

As you can see, the diameter of the pizza must  be exactly 20" for it to fit snugly in the box. If that's the case, then the pizza would have a circumference equal to Pi times the diameter, or approximately 62.8".

and,

2) What is the circumference of the largest individual pie (out of two) which can be placed in the box if two pies of equal size are placed in a cardboard container of the same dimensions as in number 1, above?

Answer: This is very similar to the first question. But this time you've got to put two equally-sized pies in the box without mutilating them. Here's a picture:

The two pies, side by side, can still not have combined diameters greater than the 20" length of the box (some of you were going to try doing this using the hypotenuse obtained by cutting the box into two right triangles, but that doesn't work - ha!). Each of the two pies will have a diameter of 10", and each will have a corresponding circumference of 31.4".

BCNU soon.

Douglas E Castle
(http://aboutDouglasCastle.blogspot.com)

Don't forget to maintain that brain! Braintenance! (http://Braintenance.blogspot.com)

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Pizza, Pi and Packaging.

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Your mind is a muscle. You must use it to strengthen it. If you don't, it will atrophy and hasten your descent toward dementia (which is not the name of a town in the Midwestern USA). Following are a pair of practical problems involving Pizza, Pi and Packaging. Try 'em.


Background:

Fabian Focaccia (not his real name, which is Sal Monella)  is a struggling 'artist' who is working at a neighborhood pizzeria in Brooklyn, New York (not his real location, as he is in the Federal Witness Protection Program, but which is still the best geographical location to make a pizza purchase if you/ youse should ever get around to it) to pay his bills until he can sell one of his paintings. He is faced with a decision and needs your help. He has cardboard boxes for 'take out' pizza (this pizza parlor does a big take-out business -- even for Arizona...oops!) which are each three inches deep (irrelevant for solving this problem) and measure exactly 20 inches by 20 inches square.

The Pizza Box Puzzle (In Two Parts):

1) What is the circumference of the largest pizza (assume that it is hand-tossed and perfectly round) which can be placed in the box neatly, i.e., placing it flat without stuffing it in and distorting its perfect shape? and,

2) What is the circumference of the largest individual pie (out of two) which can be placed in the box if two pies of equal size are placed in a cardboard container of the same dimensions as in number 1, above?

Knowing the geometric properties of circles, triangles and squares certainly does come in handy when it comes to practical everyday matters. This is high-utility knowledge, and sometimes a great deal of dough may be at stake. [cue groaning].

Douglas E Castle

p.s. Answers within the next 2 days.

Other Blogs:


http://aboutDouglasCastle.blogspot.com
http://Links4LifeAlerts.com
http://TheGlobalFuturist.blogspot.com
http://TheInternationalistPage.blogspot.com

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That Mike Moran fellow is quite a bright blogger. If only I could learn to follow his basic rules, I probably wouldn't have to work so doggone hard. -DC

Oxymorons Abound. IQs: Running On Empty*.

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Oxymorons have engulfed civilization like a moth-bitten shroud. I had dinner with a friend several nights ago, and he asked "What the h*ll (he actually used "f**k!" but I don't have the audacity to put that in writing) is a 'Jobless Recovery' ?." As a trained economist (paper-trained at very least), I was being asked a real question -- not a rhetorical one. I replied, "It's political 'nonspeak' (an invented word or Lingovation) for a 'continued recession."

The question which we are always asking ourselves: "Is that an oxymoron, or is it just me?"

The following appeared on my Blogger screen this morning: 


Free Online Advertising

See What $75 of Free Google Ads Can Do For Your Business. Try It Now!

www.Google.com/AdWords 

Need I say more?

Douglas E Castle

"Braintenance - Now More Than Ever!" -DC

Blog Links:

http://Braintenence.blogspot.com

http://Links4LifeAlerts.com

http://TheGlobalFuturist.blogspot.com

http://BusinessAndProjectPlanning.blogspot.com

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*Technically speaking, you cannot actually "run on empty." The person who can run on empty can easily exceed his maximum, as well. Could he also be [from a Nationwide Ad on TV] "The World's Greatest Salesperson In The World?"

Riddles Build Imagination, Visualization And Creativity

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A riddle is a word puzzle which is designed to challenge (i.e., exercise) your mind. You use memory, associative intelligence, logic, vocabulary, semantics and a variety of other skills, all in motion at the same time in order to "solve" a riddle. I'll provide you with some mind-expanding (and highly-entertaining) riddle links at the end of this post. You sharpen your intellectual and verbal skill sets (together! yes!) every time you work through one of these cunning conundrums [hey...isn't that alliteration he's using?].


Douglas E Castle (my third-person, holier-than-thou-sounding eponymous alter-ego) prescribes at least one to five good challenging repetitions ("brain crunches," if you're Jared DiCarmine, or some other rising star in the fitness guru firmament) daily.


Following are two horrific examples of genuinely groan-worthy mind-poppers {you may wish to have children and persons with cardiac conditions or a propensity toward seizures leave the room -- also, it may be advisable to keep a bucket handy}:


Q) What is the difference between a pregnant pause and a constipated silence?


A) The outcome.


Q) What has four wheels and flies?

A) A garbage truck.



I'll quit while I'm ahead...


Douglas E Castle

Other blogs by this fellow:

http://Braintenance.blogspot.com
http://Links4LifeAlerts.com
http://MadMarketingTactics.blogspot.com
http://www.TNNWC.com


Some Links To Riddles:
 

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