Mathematical interlude - The divine ratio, Phi

At the moment we are doing chemical calculations, an important part of the sylabus as these will lay the groundwork for much of the rest of your course, so we'll need to really understand what's going on here, hence why I am taking it so slowly to make sure that none of you get left behind.

In class the other day I mentioned the constant Phi, whose value is 1.618. This is a dimensionless constant as it is a ratio, and it's a ratio found so commonly in nature that it has it's own name: The Divine Ratio. More about the Divine Ratio in a minute.

When I say a "dimensionless ratio" I see a lot of you looking back at me blankly - let me clarify.

If we take a line and measure it we get back a measurement with units in, lets say for simplicity, metres. The unit of this measurement is metre. The dimension of this measurement is 1, as it's a straight line, which in theory takes up only one spatial dimension. It's one-dimensional or 1D.

If we then take the same line and move it "upwards" in space it traces out a new object. If we travel the same distance "up" as the length of the orignal line we get a new shape built, a square. The square now has units of metres-squared or metres to the power of 2. The shape exists now in a second dimension. This goes on and on by taking your new object and pulling it into the next dimension to create the next-dimensional object. *(see end notes for further comments on dimensional increase)

If we take a square and divide the area by the length of the line we arrive back at the original line length and the original unit, metre, and hence the original dimension.

If we however, take the square and divide it be another square, or take a square number and divide it by another square number, what do we get?

Something that has no units. A ratio. This tells us how many times our square can fit into the space occupied by the dividing square.

This ratio is dimensionless, as is the ratio of a line (1D) divided by a line (1D). In fact ALL ratios are dimensionless.

The point of the lesson on Wednesday was to point out that the RAM is nothing but a RATIO of masses, which compares the mass of the element you are looking at with another mass. The mass that was chosen was the mass of the carbon atom, or specifically part of a carbon atom. In order to give us a constant to compare to the mass of a standard Carbon-12 divided into 12 was chosen as the standard.

The name RELATIVE Atomic Mass tells us that it's not ACTUALLY an atomic mass, as that would have units of mass, namely the atomic mass unit or even grams if you wanted. But this tells us how many times more or less it's mass is compared to something else.

So Hydrogen has a mass equal to one twelth of a carbon-12, Lithium, 7 times etc. Note that the RAM is the Relative Atomic Mass of a number of atoms and hence it can be a non-integer number (not whole number) like Chlorine.

As this is only a ratio and has no units, in calculations it does not represent ACTUAL quantities, only it only tells us how much more one elements mass is than another. e.g. it's true to say "The mass of Oxygen is 16 times that of Hydrogen and 4 times that of Helium" - this statement would hold true if we were talking about any quantity of oxygen, hydrogen or helium. It's a COMPARISON. It's also true to say that "The same NUMBER OF ATOMS of oxygen would have a mass 16 times that of the SAME NUMBER of Hydrogen and 4 times that of the SAME NUMBER of helium"

Anyway back to the Divine Ratio. The divine ratio is dimensionless, as a ratio always is. It is the most beautiful example (in so many ways) of the dimensionlessness of ratios as it is seen in nature in the division of many different dimensions.

Plants, Animals and human beings all display proportions that follow the divine ratio. Try it next time you have a tape measure. Measure the distance between the top of your head and the floor. Then divide that length by the distance between your belly button to the floor. The number that comes out is 1.618. Phi.

Try this one - hip to floor divided by knee to floor. Or this one while you're sitting at your desk - finger joints. Toes. Spinal divisions. In fact the human being is such a testimony to Phi that Leonardo da Vinci himself drew his human figure, The Vitruvian Man in these proportions and this was long regarded as the most perfect human figure in art. The name of the painting derives from the Roman Architect Marcus Vitruvius who praised the Divine Proportion in his text De Architectura. You can read more about the Divine Ratio in Dan Browns "The Da Vainci Code"

Phi is everywhere - in shells, trees, mountains, coastlines, clouds and other fractals. Everywhere. The number of males to females in any bee colony is phi.

If you want to read about fractals but don't want to pick up a maths book I strongly recommend, Michael Crichtons Jurassic Park. If you do want to pick up a maths book then I would recommend popular science such as "From here to infinity" by Ian Stewart (A lecturer at my old university)

And of course there is a wealth of information to be found on the web too. Have fun!

Phoenix

*End notes - In Edwin A Abbot's classic mathematical novel "Flatland" he describes the process beautiully as "being pulled "up but not north". The story describes the encounter of A Square, an inhabitant of Flatland where all beings are 2-dimensional, who one day meets a divine being from the next dimension, A Sphere. Described with the fanaticism of a avid mathematician Abbott paints a witty and thought provoking picture of the narrowmindedness of society, through a discourse in geometry, and opens the minds eye to the possiblity, or should that be certainty of dimensions above the limits of our perception. Written in the 19th century, Abbotts novel predated Einsteins advances into the next dimension (relativity) by almost half a century.

- Extended -
At your present level of study, a thorough knowledge of dimensional analysis is not really important as you deal with so few variables in your equations. As your study increaseas it becomes much more important to check that the dimensions on either side of your equations balance, and this is why I feel it necessary to introduce you to the concept of dimensions, even though it's not on the sylabus as such. If you choose to take this knowledge on then that's great, if not then you'll learn it later anyway, it is your choice.

A point of note though - a dimensionless constant, though dimensionless, still can have a very profound effect on your final answer as these are ratios. Take Pi for example. This is but a ratio of the circumference over the diameter of a circle. Though dimensionless it appears in one of the first equations you will have ever learnt. The point here in chemistry is that the RAM is a dimensionless constant, a ratio of the mass of a given amount of an element against 1/12 of the mass of a carbon-12 atom.
Other dimensionless constants you may have met include Hubbles constant, e and now Phi. If it's a ratio it's dimensionless by it's very definition, because it is one quantity divided by another quantity of the same units. Hence, in actual fact, the big numbers in our balanced equations that we refer to as the number of moles are actually dimensionless ratios showing us proportions of reactants