Mathematics in Nature

If you have an interest in mathematics, you will no doubt be aware of the number set known as the Fibonacci series.1

This ordered number set manifests itself throughout the universe, from plants (daisies, pine cones, sunflowers) to animals (the fins of dolphins, the spiral growth of sea shells), to enormous spiral galaxies.

The Fibonacci series is closely related to the mathematical value of π (Pi). Mathemiticians use π in probability theories, but it also exists almost everywhere in nature. For example, draw a series of parallel lines onto a sheet of paper each separated by a distance just a little greater than the length of a sewing needle. Now drop the needle haphazardly onto the paper as many times as you like. The needle will sometimes cross a line, and sometimes fall between the lines. But how many times will it miss the lines? The result of the test will always be at or near 2 over π, or 64%. This is a similar calculation to the length of an extensive, wavy, winding river when divided by the distance between the beginning and end of the river (distance “as the crow flies,” that is). The value always averages out to about π. Amazingly, π tells scientists which colour should appear in the rainbow (colour frequencies), and how middle C should sound on a piano (sound frequencies)! The value of π shows up everywhere around us; from the spherical shape of fruits, to the brilliant light of a supernova.

The Fibonacci series is also closely related to the Golden Ratio, also known by the Greek letter φ (Phi). The value of Phi cannot be fully represented by decimal fractions, as there is no pattern to the numbers, just as in π2. To 15 decimal places, the value of Phi is: 1·618033988749894. It has some amazing mathematical properties that we don’t have space for right here. We will mention a couple of interesting points though.

If you draw a line on a piece of paper, there are only two places on that line that you can mark out the Phi ratio:

Cutting the line using one of those points as above, then A = B ÷ Phi, and B = A x Phi. This same relationship continues no matter how many times we enlarge C by a factor of Phi. Hence, Phi is also called the Golden Ratio. This ratio occurs in nature over and over again. For example, the lengths of the bones in the human hand manifest the Golden Ratio from the largest bone nearest the wrist to the smallest bone at the end of the fingers. Note that these numbers are also in the Fibonacci series.

Other sections of the human body manifest the same length ratios: the legs, arms, head, etc. The bodies and wings of many insects and fish also follow the same pattern.

To obtain the value of Phi (note that this does not work accurately on any computer, as the full fractional value of Phi has infinite decimal places, as remarked above): Take a regular pentagon and draw a line connecting two of the vertices. Then divide the length of this line by the length of any one of the pentagon’s sides (they’re all identical anyway) and voilathere you have the value of Phi.

Notice how the value of Phi, the Golden Ratio, has a very special relationship with the Fibonacci numbers. With each successive number in the Fibonacci series, the ratio of that number to the previous one in the series gets closer and closer to the value of Phi.

Notice how, in the above table’s right-most ‘Ratio’ column, the successive values alternate between a little above to a little below the value of Phi with each progression; all the time getting nearer and nearer Phi, without ever actually reaching it!

Yet the Fibonacci series, φ (the Golden Ratio), and π are only three of many mathematical formulas that are built into the objects and creatures in nature.

Picture credit: Catawba Valley Community College

Cosmologist Max Tegmark of MIT3 has tried to explain this phenomenon by comparing our world to a computer simulation, a game. Professor Tegmark explains that if a player could be somehow transported into the game, all the properties of the game would seem to him to be ordered, structured, with many mathematical properties. However, the computer game was programmed by an intelligent and well-trained programmer. This includes conception, layout, aesthetic formation for playability, design for coding, graphics, lighting, and sound-effects, plus essential testing and bug-fixing, etc. If the computer simulation represents the world around us, then who or what does the programmer of the game represent? As the universe is evidently full of mathematical properties that contribute to the order, symmetry, and general arrangements of matter and forces manifest around and within us, then where did these properties come from? Who or what decided on the parameters of these properties? How have they come to be so superbly and impressively ordered?

Mathematical formulas also occur in music; that is, in musical frequencies. For example, an Octave has a ratio of 2:1, a Fifth has a ratio of 3:2, a Fourth, 4:3, a Major Third, 5:4, a Major Tone [step], 9:8, and a Half Tone [half step], 17:16. The more precisely tuned to the given ratio, the sweeter the note that’s played! Pythagoras, Greek Mathematician of the 6th century BC, believed that these musical formulas gave evidence of a hidden order in the natural world.

Water molecule

Other ratios manifest in nature include: the 2:1 ratio of hydrogen to oxygen atoms in a water molecule; the number of times the moon orbits the earth compared to its own rotation, 1:1; the number of times mercury rotates compared to two of its revolutions around the sun, an exact 3:2 relationship.

That there is a preponderance of the occurrence of the number 3 in nature has often fascinated philosophers and mathematicians:

  • The earth, moon, sun system
  • Three basic dimensions: length, height, breadth
  • Three aspects of time: past, present, future
  • Three main components of atoms: neutrons, protons, electrons
  • Three atoms of the water molecule: 2 x hydrogen, 1 x oxygen
  • Chemical reactions have three results: heat, entropy, the final product
  • Plants manufacture three nutritional types that all animals and plants need: sugar, starch, protein
  • Three forms of matter (at normal temperature): solid, liquid, gas
  • The three main groups of the periodic table: non-metallic, metalloid, metallic
  • Elemental bonding is neatly divided into three: ionic bonds, atomic bonds, metallic bonds
  • Strict three-fold division in organic chemistry: simple bonds, double bonds, triple bonds
  • The above bonds come packaged with another threesome in that there are only three elements in existence capable of forming them: carbon, nitrogen, oxygen
Electron cloud

In the quantum realm, with particles of matter smaller than an atom, advanced mathematics governs their behaviour. For example, physicists no longer speak in terms of the rotation of an electron around the nucleus. It doesn’t simply “revolve” as does the moon around the earth. Its behaviour is referred to as a “cloud,” and the electron’s location at any given moment around the nucleus can be determined using mathematics. Physicists calculate the probable location of the electron relative to the nucleus, and when they experiment to find its actual (previous) location, it always matches the calculated probability!

Gru breaks the laws of physics (courtesy Universal Pictures)

A characteristic of the formation of galaxies and galactic clusters in the universe, that has been discovered in recent decades, has raised a considerable number of questions in the minds of cosmologists and physicists. Using mathematical calculations, together with the known laws of physics, they produce a model of how the universe expanded outwards from the Big Bang. And there is one very marked difference between what they calculate the arrangement of galaxies should be, and what the actual arrangement of galaxies is. According to the simulations, galaxies and stellar gases should be clumped in random areas throughout the universe, much like a lumpy porridge that hasn’t been stirred very well. There should be many areas without any matter at all, and many other areas clumped with matter. But what they find is a smooth uniformity. The laws of physics that govern the universe today are clearly different from the laws of physics that governed the early expanding universe!

This is also why there is an apparent contradiction between the laws inherent in the theory of relativity and the laws evident in quantum mechanics. Hence the statement that is repeated in The Fundamentals section:


Footnotes
1 If you are unfamiliar with these numbers, here is a brief description: This sequence is determined by adding up the last two values of the series to produce the next number in the series. The Fibonacci sequence starts with the number one, then the value of the previous number is added to form the next one. Thus, 1 + 0 is 1.

Sequence: 1 1

Adding the two numbers gives us the third number, 2. Thus: 1 + 2 = 3

Sequence:  1 1 2 3

Add the last two numbers again, 2 + 3:

1 1 2 3 5

And so on:

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 …

2 Except that π is known as a transcendental number, i.e. a real number (it’s also an irrational number) that is not the solution of any single-variable polynomial equation whose coefficients are all integers.

3 Author of the book Our Mathematical Universe and co-founder of the Future of Life Institute.


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