Insane Imaginings, Random Reveries & Other Creative Cogitation

Numbers, there are many kinds of numbers:

• The Natural Numbers, 1, 2, 3… are counting numbers that a child can figure out with their fingers and toes etc.  Depending on who is doing the definitions the natural numbers may or may not include the number 0,. Natural numbers do NOT include numbers such as that have digits after the decimal place such as 2.25 , nor do the natural numbers include negative numbers.
• Whole Numbers are the Natural Numbers plus Zero plus the Negative Counterparts of the Natural Numbers,  …-3,-2,-1,0,1,2,3…; Whole Numbers are also called Integers.  Again Integers/Natural Numbers do NOT include numbers with digits after the decimal point.
• Rational Numbers are numbers that can be expressed as a ratio of two Integers (excepting division by zero), therefore are numbers like 1/1, 2/3, -5/8 etc.  Rational Numbers can include numbers whose decimal equivalent terminates such as -5/8 = -0.625, or numbers whose decimal equivalent does not terminate, but repeats instead such as 2/3=.66666…
• Irrational Numbers are numbers that cannot be expressed as a ratio of two Integers, for example, the square root of two.  Irrational numbers do not include numbers involving the Imaginary Unit.
• Complex Numbers are numbers numbers involving the square root of negative 1, also called the Imaginary Unit, i.  Numbers that are multiples of the imaginary unit are called Imaginary Numbers.
• Both Rational and Irrational numbers can be Algebraic.  An Algebraic Number is any number that can be found as a root of polynomial equation with Integer coefficients, such as the square root of 5, which would have the integer coefficient 2, or negative cube root of 10, which would have the integer coefficient -3.   Numbers like the square root of 2 are irrational numbers, but they are algebraic.  If n is the lowest possible degree of such a polynomial, the roots are algebraic of order n. The square root of two is algebraic in order 2, nevertheless the square root of two is an Irrational Number.
• Transcendental Numbers are NOT Algebraic.  The most famous transcendental numbers are Pi, the ratio of a circle’s circumference to its diameter, and Euler’s Number, which is the base of the natural logarithms among other things.
• Just to complete the picture, Real Numbers are numbers that are NOT Complex Numbers.  They can included Rational and Irrational Numbers; they may be Algebraic or Transcendental.  Real Numbers cannot include numbers involving the Imaginary Unit, but Complex numbers can have Real Numbers as components.

Although some Transcendental Numbers, such as Pi and Euler’s Number are cool, they are not mystical, nor are they particularly rare; technically almost all Real and Complex Numbers are Transcendental, since the Algebraic Numbers are countable, but the sets of Real and Complex Numbers are uncountable. All Transcendental Numbers are Irrational, since all Rational Numbers are Algebraic, but NOT all Irrational Numbers are Transcendental as some Irrational Numbers are Algebraic (such as the square root of 2).

Now that some of language of numbers has been defined let’s consider the Golden Ratio, Phi. The Golden Ratio can be found in the proportions of the human body, the proportions of many other animals, plants, DNA, the solar system, art and architecture, music, population growth, the stock market.  It can derived mathematically, geometrically, or via the Fibonacci Series.

The Golden Ratio: the ratio of the sum of the quantities to the larger one equals the ratio of the larger one to the smaller one. Consider the whole length of something, lets call this length, C, where C is composed of two smaller unequal sections, A and B, such that C = A + B. Say B is the larger piece and A is the smaller piece. If C/B = B /A then the division of C into parts A and B is proportioned to the Golden Ratio.  This dimensioning considered most pleasing to the human eye, probably because these dimensions are found so frequently in nature. If (A+B)/B = B /A, then both equal Phi, the Golden Ratio.  Phi is approximately 1.6180339887498948482045868343…

Since the Golden Ratio, Phi, is the solution to the equation, x² - x-1 =0, which is Algebraic in order 2, Phi is NOT a Transcendental Number.

The Fibonacci Series is a series of numbers with a recursive relationship; it simply expressed as the sum of the preceding two numbers in the series: starting at 0 and 1, then the next number is 1=1+0, then number after that is 2=1+1, then 3=2+1, then 5=3+2, then 8=5+3, etc.  The first 20 terms of the Fibonacci Series are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181…

The Fibonacci Series also frequently occurs in nature. We have 2 hands, 5 fingers, each finger has 2 knuckles and 3 segments-all Fibonacci numbers. The number of flat surfaces on a banana – usually 3 or 5 – again a Fibonacci Number. Flower seed heads often have a certain number of spirals to pack seeds in such a manner that the seeds have the same amount of space-the number of spirals is usually a Fibonacci Number. Plants also frequently arrange their leaves according to the Golden Section; If the entire circumference is proportional to 1.618 then the angle of rotation is usually 0.618, which is the lesser of the sections in the Golden Ratio 1.618 = 1/0.618. When the lesser section refers to an angle of rotation it is called the Golden Angle. The tendency of plants to wind themselves using the Golden Angle and otherwise arrange themselves with Fibonacci numbers is called Phyllotaxis; an estimated 90% of plants exhibit the tendency.    But not all plants follow phyllotaxis, sometimes sweet peppers have 4 chambers instead of 3, some flowers have 4 petals, such as a fuchsia,  or 6 petals, such as a crocus. Still it would seem that the Golden Ratio and Fibonacci numbers must have a connection.

Indeed, the Golden Ratio, Phi, and the Fibonacci Series are intimately connected.  As the numbers in the Fibonacci Series increase, the ratio of successive terms tends toward a certain number and that number is the Golden Ratio; in mathematical terms the limit of the ratio of successive terms in a Fibonacci Series as the terms go to infinity is the Golden Ratio. Lets see this in action: Keeping 3 digits accuracy behind the decimal, the ratios of successive terms in the Fibonacci Series are  Undefined = 1/0 (division by zero is not permitted), 1.000 = 1/1, 2.000 = 2/1, 1.500 = 3/2, 1.667 = 5/3,  1.600 = 8/5,  1.625 = 13/8, 1.615 =21/13, 1.619 = 34/21, 1.618… 55/34, 1.618 = 89/55, 1.618 = 144/89 etc.  As you can see by the 9th iteration the ratios of successive terms of the Fibonacci series have converged to the Golden Ratio to 3 digits of accuracy behind the decimal.

From an artist standpoint using 1.618 as an approximation to the Golden Ratio should develop pleasing ratio in the scales most of us work in.  Therefore if you wish to section 36-inch-long canvas into the golden ratio then 36/1.618 = 22.25 inches will be the larger dimension and 36-22.25 =13.75 inches will be the smaller dimension. Also  22.25/13.75 = 1.618 as it should.

Yours in art – Jake

Artist, AKAJake.com Come Experience the Art!

So I admit it. I am a geek.  The beauty of the numbers has long fascinated me.   It is one of the reasons I wound up getting an advanced degree in physics.  I considered engineering,  mathematics, and even art as my college major, but physics won out on the fun factor.  Engineering was too pedestrian, Art too risky, and the mathematicians I saw really didn’t look like they were having a good time.  I wound up getting a masters degree in nuclear physics. but I am not doing anything with it these days.  If I can say one thing about getting a masters degree (which probably applies to any masters degree, not just physics) is that graduate school taught me to think, to use the tools my undergraduate degree gave me.  Physics taught me to look at the big picture as many solutions to Physics problems apply to a great many very different types of problems.

But I digress, I really still am a beauty of the numbers gal.

So one problem, which is kind of like the problem of all problems, is the P verses NP problem.  What is P verses NP? In a nutshell, the big question is whether P = NP or P ≠NP;  if P = NP then every problem has a efficient solution and we can find it efficiently. One might postulate all problems are solvable in an infinite amount of time, but what good is that?  Only God, perhaps, has an infinite amount of time to solve problems; in the real world problems need to be solved in a finite and reasonable amount of time. The amount of time is the crux of P verses NP. Are problems really as complex as they appear? Is the only solution brute force over an infinite amount of time?

By way of example, consider we have a large number of wedding guests we need to seat at tables at the reception.  We might want to pair them up into groups of two people who like each other, so that no one is lonely.  In 1965 a guy named Jack Edmonds developed an efficient algorithm to solve this particular problem, and having an efficient algorithm means, among other things, you can feed it to a computer and solve the problem in a reasonable amount of time. Jack Edmonds pioneered the lingo for P verses NP.  Problems with efficient solutions are the “P”, in P verses NP.  This particular problem is called the “matching problem.”

But there are many other ways to seat our large number of wedding guests that are by far more difficult to solve.  Suppose we want to make groups of three wedding guests who like each other, OR large groups of wedding guests who like each other, OR we want to seat guests around a round table such that guests, who can’t stand each other, are not seated next to each other, OR group  three couples at the wedding with other couples in a mutually agreeable triads, OR the evil converse social science experiment where only guests who can’t stand each other are grouped together in one way or another (hey, maybe the wedding planner has a thing for the groom).  All the problems that have efficient, verifiable solutions are “NP”.

So if P = NP then every problem has an efficiently verifiable solution AND we can find it efficiently.  On the technical side, even if this statement is true, we still may not have the algorithm to efficiently solve a given problem. The P verses NP problem is an abstraction, therefore just because we may do something doesn’t necessary mean that we can…

Naturally this question has ramifications well beyond computing algorithms, such as mathematical proofs, encryption, DNA sequencing etc.

Why think about it at all?  The P verses NP problem is one of the seven Clay Mathematics Institute Millennium Prize problems; solving it is worth $1,000,000.  Not a bad chunk of change, but also realize proving P = NP  might solve other Millennium Prize problems as well, each worth $1,000,000.  However you should know that many theorists believe P ≠NP. Don’t get your hopes up.