Two Mathematics I Didn’t Learn in School Describe Everything

I remember thinking, as I was studying plane geometry in high school, this is all very nice, but nothing is shaped this way.  The world is not made up of circles, rectangles and triangles as pure forms.  But, I enjoyed geometry because for some reason that I couldn’t identify everything made intuitive sense.  How could that be?  It turned out that the real geometry was a little mor complicated, but related.

The Sound of PHI

Many of us claim to be miserably inept at mathematics and yet our world from the design of the atom to the design of galaxies and the universe are expressed in a mathematical ratio variously called Fibonacci’s number, the golden mean, the golden ratio and PHI.  We are at our core mathematically constructed beings, and we see PHI in everything and appreciate it without realizing it.

The reason we don’t realize it is summed up in a joke.  A fish runs into a scuba diver.  The diver asks, “How’s the water?”  The fish replies, “What’s water?”

Fibonacci’s sequence is 1,1,2,3, 5, 8, 13, 21 etc. where each number is the sum of the previous two numbers.  The ratio is 1 + √5)/2 or roughly 1.618 and it is seen in everything in nature. 

When I first learned of this ratio I was dubious.  I learned it from an artist, and we all know about artists. She claimed it described the shape of a nautilus shell, the shape of people’s faces we think of as beautiful, that it was everywhere and she couldn’t believe I got through college without knowing about it.  I told her that I got through college without knowing how to draw beer from a keg, too.

I constructed graph paper and drew out the spiral that is present in snail shells, studied the proportions of people’s faces, and other examples, and found as claimed that our concept of beauty is based on PHI.

The relationship between PHI and Chaos:

“In modern mathematics, the golden ratio occurs in the description of fractals, figures that exhibit self-similarity and play an important role in the study of chaos and dynamical systems.” ~ Encyclopedia Britannica


I am fascinated with economics, so forgive me if I spend a lot of time on the large and small movements of markets.

In a recent article on technical stock trading and the use of PHI, further examples of the innate nature of PHI were given.  The ratio of female bees to males in a hive is 1.618.  The seeds in the concentric rows of sunflowers are arranged according to PHI.

PHI is used to analyze the movement of a stock by technical traders in order to anticipate a sustained movement in one direction allowing them to make trades for profit.  Unfortunately, PHI does not predict the beginning or end of a trend; that depends on another mathematical approach.

It has been known for a long time that the movement of the entire stock market is based not on value, but on emotion, that is on “investor confidence”.  Paul Krugman, Washington Post editor, predicted the collapse of the stock market if Donald Trump was elected president.  That hasn’t happened yet.  The ups and downs of markets; that is the periods of expansion and contraction, don’t follow PHI at least in any way that I can see.  I looked at the average length of expansions – 38.7 months – and the average times of contractions – 17.5 months – the relationship to complete cycles, the lengths of the longest contraction (the Great Depression) the second longest contraction (the Great Recession) the longest expansion (the Clinton Years) and can’t find anything that looks like PHI.

Economic cycles follow another mathematic model; chaos.  Chaos does not mean randomness. Chaotic systems like the markets and weather systems are affected by a great many factors, any one of which, over time, may make predictably large differences in the outcome.  It is what has been termed the “butterfly effect”. The problem is that in order to predict an outcome all of the infinite number of variables must be known and measured at inception.

In simple harmonic systems like a vibrating guitar string amplitude is determined by how hard one plucks the string.  Frequency, which determines pitch, is determined by the length of the vibrating string.  Changing the pitch involves changing the wavelength by fretting the string at a different point.  Thus amplitude and frequency are unrelated.

However, in markets it appears that they are in some way;

“Economist Claudio Borio of the BIS has shown that boom-bust cycles are increasing in amplitude over time; thus we should expect an even bigger crisis by 2020 than 2008 was.


Technically the frequency is decreasing while the amplitude is increasing, similar to what you might see in a (sic) Inverse Sornette wave.


Financial imbalances have been exacerbated by overly accommodative monetary policies run by the big central banks like the Fed, PBOC, BOJ, and ECB.” ~ from

Didier Sornette’s book, “Why Stock Markets Crash: Critical Events in Complex Financial Systems” looks not at the “triggering event” in the hours or days before a crash, but at the years and months prior to the crash.  He attributes what he calls “collaborative speculation” as the culprit that causes prices to rise at an accelerated rate creating a bubble.  The results look like this:

Oversold Market, 1929 crash, and Years Following.

Markets depend on optimism and fear for their movements, not cold analysis.  In large systems like markets and weather the degree of predictability becomes less over time.  The “cone of uncertainty” meteorologists use in tracking the path of hurricanes expresses this characteristic very well.

As previously stated their behavior (chaotic systems) in nature is impossible to predict because these systems are extremely dependent on initial conditions, and there is no way to know all of the initial conditions.  The oddity of chaos is that on a large scale it appears to be very orderly.  By setting the initial conditions chaos can be demonstrated using fractal geometry in very beautiful images.

The mathematics in chaotic systems is not that complicated, but they are strange.  They depend on the use of imaginary numbers and the concept of partial dimensions. We are used to thinking of a line as having one dimension, a square as two dimensions, and a cube as having three dimensions.  In the math of chaos a dimension may be something between a line and a square, or a square and a cube.


The first people to find what became known as fractal geometry were British cartographers trying to determine the length of the coast of Britain.  They found that the more detailed the map, the longer the coast line became.  However, the visual expression of chaos depended on the use of computers because the equations depend on repeatedly reentering the product of the last equation.  Benoit Mandelbrot, an IBM worker was able to display the first example of a fractal in what has become known as the Mandelbrot set.

Mandelbrot set

This fractal, which appears very ordered, is actually the product of chaos. The small details in fractals express the golden ratio.

Fractals and chaotic systems are everywhere in nature.  Mountain ranges and coastlines are the result of chaos.  The beauty of nature seems to lie in the mathematics of large systems.

Returning to Markets:

The increasing amplitude and frequency of economic boom / bust cycles may be spurred by two factors.  One factor is the continued effort by big banks to eliminate any management of economies and by protection of banks when they come up short during economic contractions.

Speculators, especially large banks and the extremely wealthy that can actually manipulate investor confidence and produce bubbles, stand to make large profits by selling toward the end of expansions, and buying at the bottom of contractions.  Being insured against the failure to get out of the market before a bust cycle begins effectively makes large investors immune to the risks of speculation.

Managed economies should flatten the curve and decrease the frequency of expansion and contraction.

An economic system that depends on growth may also be fueling the increasing amplitude and frequency of booms and busts. 

Predictions by some economists are that the economic system that has existed through recent history that depends on continual growth will come to an end by 2050. 

Two South African researchers have proposed that 1.618… describes not only the shape of the nautilus shell, the shape of Hurricane Katrina, the shape of an elephant’s tusk or the horn of a kudu, and the organization of the solar system, but is the “constraint of space time topology”.  In short, it is the constant that describes the universe.,

Fractals are based on chaotic equations, but the individual elements are based on PHI. 

And, just in case you are not yet convinced of the omnipresence of PHI, this is the DNA double helix:

Information on Fractal geometry can be found here.

Views: 335

Comment by Anna Herrington on January 24, 2018 at 10:31am

Once I learned about other layers of numbers and mathematics applications and meanings in high school thanks to a very progressive and inclusive thinking geometry teacher, that's when I grew to love the universe of mathematics. Enjoyed this post, even if you're missing the artist's flair in discussing  ; )

I admit, this rather shocked me: "When I first learned of this ratio I was dubious.  I learned it from an artist, and we all know about artists."

I hope you learned that one underestimates an artist at one's own risk/ignorance.

Comment by marilyn sands on January 24, 2018 at 1:05pm

Wow - what an amazing Post - but STILL can't figure out why Stocks are up & the majority of people don't favor Trump or what he has done.

Comment by Tom Cordle on January 24, 2018 at 1:31pm
So much meat in this and so much to respond to, I hardly know where to begin.

I had heard of the Golden Ratio, another thing I learned from my artist Uncle Mike. But I confess to having forgotten about it, since it had little practical value in my life. That is until I managed a Nautilus Fitness Center in downtown Orlando. I learned some fascinating things about the magical shape of the chambered Nautilus. One of those things was the fact that the shape of the cams on the exercise machines followed the shape of the Nautilus. The inventor of those machines was an eccentric genius named Arthur Jones, and no, not our Arthur Jones.

I have a little experience with guitars, as well, and as you may know, the guitar is only nominally in tune, that is to say, it may be in tune at the nut and at fret twelve, but not fully in tune on all the strings on all frets in between. There are several designs and fretting systems that attempt to compensate for this, including the most popular by Buzz Freiten. More here:


Frankly, most people wouldn't even know this is a problem, unless they have a pretty damned good set of ears – but a few I've met do. For most pickers, it doesn't matter since we spend a good deal of time bending strings, which obviously puts them in and out of tune. That is a common practice in popular music, and even in classical musical minute bends on stringed instruments contribute to an effect called vibrato.

Vibrato is a staple of vocalists; if you've ever listened to Dolly Parton sing, you've heard it in spades. Some singers, Lee Greenwood for one example, don't seem to have that ability, but achieve a similar effect through other vocal machinations.

You might also note that country singers have a habit of sliding into notes (see the late Johnny Cash). Some of this is by design, and some of it is a result of the genre's instrumentation, which historically included slide/dobro guitar and fiddle. Banjos are the butt of a thousand jokes about never being quite in tune.

My bottom line? All of these "shortcomings" are close enough for practical purposes, including making some pretty damn good music.

I'm sure this is more than you wanted to know about all this. And I'm sure Kosher knows more about all this than I do.
Comment by Tom Cordle on January 24, 2018 at 1:36pm
Almost forgot – I wonder if there's any research that shows the correlation between income inequity and depressions/recessions? I raise that question because I read recently that income disparity in the the worst it's been since – wait for it – 1929.
Comment by Rosigami on January 24, 2018 at 3:53pm

Wonderful post, Rodney! As an artist and musician I can appreciate the geometry behind them, bringing a lovely order to the beauty that is possible.

Here is one drawing of many I did (several years ago) that were all based on the Fibonacci sequence:

Comment by koshersalaami on January 24, 2018 at 6:23pm

This is really cool. I had no idea the sequence was so ubiquitous. And I still don't get fractals. I also don't get how the ratios show up in the music. 

Concerning strings, there's another variable that determines pitch: tension. That's now tuning works. The ratios aren't exact in music mainly because of equal temperment. The only way to play equally in tune in all keys is to throw the entire system very slightly out of tune. I could go into more detail and I have in previous posts, it has to do with the cycle of fifths not coming back to its exact starting point. 

Comment by Ron Powell on January 24, 2018 at 9:47pm

Given the gist of this post, my sense of it is that there is a relationship between PHI and evolution...

It would be interesting to know whether everything is evolving at the same rate even if not evolving at the same time...

There's a heavy duty piece of jazz in "The Sound of PHI".

I wish I had sufficient gifts to bring it out....

This piece is insightful and provocative. Nicely done....

Comment by Rodney Roe on January 25, 2018 at 4:56am

Thanks for reading.

@Anna, that comment was tongue-in--cheek.  The woman who told me that is a basket maker who was part of the same artist co-op that i belonged to.  I don't really call myself an artist.  I've done pottery for about 20 years, and it's one of the things I lay awake at night thinking about, but i have little formal training or education. I have the greatest respect for artists.  

@Marilyn, I don't know about Trump.  Nothing about him has followed the rules.  As far as what affects investor confidence, it's complicated.  I don't think Trump is responsible for the soaring Dow anymore than I think Obama was responsible for the crash in 2008 or the recovery.  Having said that, I hear that Trump is planning to replace the highly respected director of the Fed with someone with little experience and no advanced degree in economics. I'm making no predictions about what that will do, but I can't see how it can be good.

@kosh and Tom, I know a little about guitar and I appreciate your comments.  Strings stretch more on many guitars as you move toward the bridge because the bridge is higher than the nut.  When checking out a new guitar I check the harmonics at 12 against the depressed string tone to see how severe the discrepancy in tone is.  Jazz guitars actually have a neck that is parallel to the string despite the high bridge (probably to compensate for the height.) It makes playing easier and eliminates the angle problem.  

Tom your remarks about Dolly Parton and her vibrato made me think of Aaron Neville.  I don't know what he does.  Maybe something with his diaphragm that changes the force of air passing over his cords.  I can turn vibrato on and off, but i can't imitate Aaron.

Comment by Rodney Roe on January 25, 2018 at 5:07am

Tom, your question about income disparity was something I looked for.  I think that extreme disparity does lead to economic extremes, but i idn't find that specifically.  It would make sense in the respect that having wealthy individuals and organizations that with so much wealth that they can manipulate markets creates extreme bubbles in the market.  When those bubbles burst it is the small investor and the worker with a 401K that suffer most in terms of loss of equity, and the actual effect on the economy in which jobs are lost and wages reset at a lower level seem more extreme.

Ron, I haven't thought about your question.  I'll have to think about that. I think that the speed of evolution changes depending on extremity of conditions, but how that relates I don't know.

Comment by Rodney Roe on January 25, 2018 at 5:37am

@Ron, one of the problems that one runs into in researching this subject is push back from some groups who don't want to hear the news.  Climate deniers, creationists, and others attempt to say that either PHI is a myth, or that it is evidence of creative design.  In this video it is touted as "God's fingerprint".

I found it hard to research.

Here is a serious bit of research that set up an algorithm that would simulate chaotic evolution.  it is basically a thought experiment which is really all anyone can do since evolution of complex systems takes more time than we have.

An interesting experiment I read of involved placing an antibiotic on one side of a block of bacteria sensitive to the antibiotic.  In the same amount of time, in the same number of generations of bacteria (which divide about every 30 seconds) an organism emerged that was resistant to the antibiotic, which seemed to say that evolution occurs at a constant rate and may usually  produce solutions that have no problem.  When a problem does exist multiple solutions are often found.


You need to be a member of Our Salon to add comments!


Revelations and the Question

Posted by Ron Powell on February 21, 2019 at 5:30am 0 Comments

Clam Crimes Charged

Posted by Robert B. James on February 19, 2019 at 4:41pm 2 Comments

The True Definition of Treason

Posted by Doc Vega on February 19, 2019 at 2:01pm 10 Comments

The Vanity Ring - 4 - Sun on the Rocks

Posted by Workstudio on February 19, 2019 at 10:00am 0 Comments

Enemy of the Good

Posted by Tom Cordle on February 18, 2019 at 3:06pm 31 Comments

© 2019   Created by lorianne.   Powered by

Badges  |  Report an Issue  |  Privacy Policy  |  Terms of Service