Posted tagged ‘Golden Ratio’

Getting Mathematical on Weeds.

May 31, 2011

Dandelions amaze me.

We tend to take them for granted.

“Huh.  They’re just stupid weeds. ” , many of us might say.

But if you look closer, they’re actually quite beautiful.

And if you zoom in REALLY close, you’ll find something even more amazing.

Notice, there’s definitely a spiral pattern there.

If you connect the dots, you can definitely count 13 curves in the clockwise direction.

But if you connect the dots in a counter-clockwise direction, you get 21 curves.

Now, remember those numbers, (13 and 21), while I digress for a bit.

Consider this mathematical sequence of numbers.

0,  1,  1,  2,  3,  5,  8,  13,  21,  34,  55,  89….

For those of you who don’t recognize this,   these are Fibonacci numbers, where any given number is the sum of the previous two.

It’s quite simple:

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

2+ 3 = 5…and so on.

Now, if you draw a series of squares,  based on the Fibonacci sequence, and you get something like this:

And if you draw a continuous arc though each square,  it forms a spiral seashell pattern, like this:


This is called the Fibonacci Spiral

Now, let’s take my 13 clockwise red-curves:

And if I take them, one by one, and superimpose them on the Fibonacci Spiral, I get this:

Kinda fits, doesn’t it?

Same thing if I take the 21 counterclockwise curves…

Again, each curve also seems to fit, when superimposed  on the counter-clockwise Fibonacci spiral:

Now, let’s just recap:

I zoomed in on a photo or a dandelion, connected dots and generated some rough curves.

And the shape of these curves fit a spiral based on the Fibonacci sequence.

Not to mention, the number of clockwise and counter-clockwise spirals are 13 and 21.
Which are Fibonacci numbers themselves.

What’s going on here?   Is this magic?   Or  a fluke?


Actually, this is no accident.

You see, Nature tends to like Fibonacci numbers.   For example, you rarely see flowers with 4 or 6 petals.  But you see many with 3, 5 or 8.

Flowers seed pods are also arranged this way.  The number of spirals are always Fibonacci numbers…one clockwise, one counter-clockwise.

In this case, with my dandelion,  it was 13 and 21.  With larger flowers (like Sunflowers), you’ll find numbers 34 and 55.

But why Fibonacci numbers?

Basically, it has to do with Nature trying to optimize itself.  With flowers, if seeds are arranged in Fibonacci spirals, you can fit more of them onto the plant,  and you get more bang for your buck.    There’s a good interactive exercise that demonstrates this.

I won’t get into the whole mathematical explanation.   But you can find some good discussions here and here.

It’s not just dandelions.  You’ll also find Fibonacci sequences with pine cones, pineapples and asparagus and seashells.       Plant leaves are arranged in Fibonnacci spirals, to optimize the sunlight they recieve.

Fibonacci numbers are everywher in Nature   More examples are shown here.


It’s pretty amazing, when you think about it.

Take an abstract concept.   A sequence of pure, unadulterated numbers:

0, 1,  1,  2,   3,  5,  8,  13,  21,  34….

And it’s architecture upon which much of Creation is built.

It’s staring at us, in our face.

The miracle of Pure Math, combined with Mother Nature.

Even with a lowly dandelion.

..and THAT’s why they amaze me.




Perfesser Friar’s Favorite Science Facts.

October 20, 2008


If the Universe was infinitely  large and infinitely old, the entire sky would be ablaze with starlight.

The fact that we see blackness in the night sky proves that the universe is of a finite age, and that it is expanding.

If you don’t believe me, read more about Olbers’ Paradox.


We remember from High school physics that (neglecting friction), that all objects fall at the same speed.

But it’s really awesome when you can see this first-hand.

Take a large coin, and place a tiny piece of paper on top of it.

Drop the coin.   It acts as a wind break, protecting the paper from wind resistance.

They both fall together.

How cool is THAT? 😀


No engine, no matter how perfect, is 100% efficient.

Not all of the energy from the burning fuel will go towards making the wheels turn.

Some of the energy will ALWAYS be wasted as excess heat.

Yes, Friar, you might ask, but who are we to say we can’t make a perfect machine?

What if we could develop a Magical Wonderful Engine, with frictionless pistons and gears that were perfectly oiled, everything was perfectly insulated, and everything ran perfectly smoothly?

Surely, THEN, we’d be able convert all the fuel energy into increasing our gas mileage?

Well, such an Engine exists…in La-La land, in our imagination.

It’s called a Carnot Engine.

But even if we could build one, a Carnot Engine STILL wouldn’t be 100% efficient.

We’d still waste some of the energy as heat.

And this isn’t just an opinion or theory.   It’s a mathematical proof.

It’s in the Official Rule-Book of How the Universe Works.

It’s the 2nd Law of Thermodynamics.


Related to the above.

That nasty 2nd Law is a bitch.    It limits the maximum output that our power generating plants can achieve.

In theory,  if some of our power plants were perfect Carnot engines, we could get efficiencies approaching 60%.

But in the Real World, nothing is perfectly frictionless or perfectly insulated.    So our actual coal and gas plants might typically be only 30% efficient.

That means for every 100 watts of heat we get from from burning fuel, at the most, maybe 30 watts will go towards making electricity we can use.

The remaining 70 watts will get dissipated into hot air up the stack, or wasted in heating the river water.

Hardly seems fair, does it?

Yet we can’t do a damn thing about it.

Oh well.  There’s no such thing as a free lunch.

Ask the 2nd Law.


Atoms are mostly empty space,  with almost all the mass centered in the very small nucleus.

Our planet has a density of about 5500 kilograms per cubic meter.  But if you focus on just the proton and neutrons, the density of the atomic nucleus is quite high: about
1018 kilograms per cubic meter.

That’s about the same density of a neutron star, which for all intents and purposes, can be considered to be a giant atomic nucleus.


The theory of general relativity dictates that time passes more slowly under a strong gravitational field than it would under a weaker one.

You might think, oh, that really only applies for massive objects like black holes and neutron stars.

But we can actually measure this in every day life, on planet earth.

For example, our GPS units would not be accurate, if the techno-geeks didn’t take into account the time dilation factor for the satellites located far above the earth.

And in 1971, an experiment was done where they put atomic clocks on plane that flew around the globe, and compared them to an atomic clock that stayed on the ground.

Taking it account the planes’ speed and altitude, they predicted that the clocks on the planes should have run a few nanoseconds faster, because they where higher up, where the effect of Earths’ gravity was slightly less.

And the clocks did run faster, exactly as predicted.

(Way to go, Einstein!)


Take the Fibnoacci series of numbers:   1, 1, 2, 3, 5, 8, 13, 21, 35, etc.

Each number is the sum of the previous two.

So what, you might ask?

Well, take the ratio of any two consecutive numbers.

The further you go down the series, the closer this ratio approaches 1.6180339….

The is called Φ, or the Golden Ratio, and it’s been know since antiquity.  It has significance in mathematics and geometry.

For some reason, humans find this number pleasing.

We’ve built temples and pyramids and composed paintings, even our credit cards are based on this ratio.

But it’s not just something we happend to make up.

You’ll  find the Fibonacci sequence in nature, in flowers and sea shells.   Our bones and anatomy appear to be based on Φ.

That’s a pretty cool number, if you ask me.