100 Essential Things You Didn't Know You Didn't Know (30 page)

Suppose A is the collection of all brown animals and B is the collection of all cats. Then the hatched overlap region contains all brown cats; the region of A not intersected by B contains all the brown animals other than cats; the region of B not intersected by A represents all the cats that are not brown; and finally, the black region outside both A and B represents everything that is neither a brown animal nor a cat.

These diagrams are widely used to display all the different sets of possibilities that can exist. Yet one must be very careful when using them. They are constrained by the ‘logic’ of the two-dimensional page they are drawn on. Suppose we represent four different sets by the circles A, B, C and D. They are going to represent the collections of friendships between three people that can exist among four people Alex, Bob, Chris and Dave. The region A represents mutual friendships between Alex, Bob and Chris; region B friendships between Alex, Bob and Dave; region C friendships
between
Bob, Chris and Dave; and region D friendships between Chris, Dave and Alex. The way the Venn-like diagram has been drawn displays a sub-region where A, B, C and D all intersect. This would mean that the overlap region contains someone who is a member of A, B, C and D. But there is no such person who is common to all those four sets.

96

Some Benefits of Irrationality

Mysticism might be characterised as the study of those propositions which are equivalent to their own negations. The Western point of view is that the class of all such propositions is empty. The Eastern point of view is that this class is empty if and only if it isn’t.

Raymond Smullyan

There is more to photocopying than meets the eye. If you do it here in Europe, then you will soon appreciate a feature so nice that you will have taken it for granted. Put two sheets of A4 paper side by side face-down on the copier and you will be able to reduce them so that they print out, side by side, on a single piece of A4 paper. The fit of the reduced copy to the paper is exact and there are no awkward extra margins on the page in the final copy. Try to do that in the United States with two pieces of standard US letter-sized paper and you will get a very different outcome. So what is going on here, and what has it got to do with mathematics and irrationality?

The International Standard (I.S.O.) paper sizes, of which A4 is one, derive from a simple observation first made by the German physicist, Georg Lichtenberg, in 1786. Each paper size in the so called A-series has half the area of the next biggest sheet because
it
is half as wide but just as long. So putting two sheets side by side creates a sheet of the next size up: for example, two A4 sheets make one A3 sheet. If the length is L and the width is W, this means that they must be chosen so that L/W = 2W/L. This requires L
²
= 2W
²
, and so the lengths of the sides are in proportion to the square root of 2, an irrational number that is approximately equal to 1.41: L/W = √2.

This irrational ratio between the length and the width of every paper size, called the ‘aspect ratio’ of the paper, is the defining feature of the A paper series. The largest sheet, called A0, is defined to have an area of 1 square metre, so its dimensions are L(A0) = 2
^¼
m and W(A0) = 2
^-¼
m, respectively. The aspect ratio means that a sheet of A1 has length 2
^-¼
and width 2
^-¾
, so its area is just ½ square metre. Continuing this pattern, you might like to check that the dimensions of a piece of AN paper, where N = 0, 1, 2, 3, 4, 5, . . . etc., will be

L(AN) = 2
^¼–N/2
and W(AN) = 2
^-¼–N/2

The area of a single sheet of AN paper will therefore be equal to the width times the length, which is 2
^-N
square metres.

All sorts of aspect ratios other than √2 could have been chosen. If you were that way inclined you might have gone for the Golden Ratio, so beloved by artists and architects in ancient times. This choice would correspond to picking paper sizes with L/W = (L+W)/L, so L/W = (1+√5)/2, but it would not be a wise choice in practice.

The beauty of the √2 aspect ratio becomes most obvious if we return to the photocopier. It means that you can reduce one side of A3, or two sheets of A4 side by side, down to a single sheet of A4 without leaving a gap on the final printed page. You will notice that the control panel on your copier offers you a 70% (or 71% if it is a more pedantic make) reduction of A3 to A4. The reason is that 0.71 is approximately equal to 1/√2 and is just right for reducing
one
A3 or two A4 sheets to one of A4. Two dimensions, L and W, are reduced to L√2 and W√2, so that the area LW is reduced to LW√2, as required if we want to reduce a sheet of any AN size down to the size below. Likewise, for enlargements, the number that appears on the control panel is 140% (or 141% on some photocopiers) because √2 = 1.41 approximately. Another consequence of this constant aspect ratio for all reductions and magnifications is that diagrams retain the same relative shapes: squares do not become rectangles and the circles do not become ellipses when their sizes are changed between A series papers.

Things are usually different in America and Canada. The American National Standards Institute (ANSI) paper sizes in use there, in inches because that is how they were defined, are A or Letter (8.5 in × 11.0 in), B or Legal (11 in × 17 in), C or Executive (17 in × 22 in), D Ledger (22 in × 34 in), and then E Ledger (34 in × 44 in). They have two different aspect ratios: alternately 17/11 and 22/17. So, if you want to keep the same aspect ratio when merging paper sizes you need to jump two paper sizes rather than one. As a result, you cannot reduce or magnify two sheets of one size down to one sheet of the size below or above without leaving some empty margin on the copy. When you want to make reduced or enlarged copies on a US photocopier you have to change the paper trays around in order to accommodate papers with two aspect ratios rather than using the one √2 factor that we do in the rest of the world. Sometimes a little bit of irrationality helps.

97

Strange Formulae

Decision plus action times planning equals productivity minus delay squared.

Armando Iannucci

Mathematics has become such a status symbol in some quarters that there is a rush to use it without thought as to its appropriateness. Just because you can use symbols to re-express some words does not necessarily add to our knowledge. Saying ‘Three Little Pigs’ is more helpful than defining the set of all pigs, the set of all triplets and the set of all little animals and taking the intersection common to all three overlapping sets. An interesting venture in this direction was first made by the Scottish philosopher Francis Hutcheson in 1725, and he became a successful professor of philosophy at Glasgow University on the strength of it. He wanted to compute the moral goodness of individual actions. We see here something of the impact of Newton’s success in describing the physical world using mathematics: his methodology was something to copy and admire in all sorts of other domains. Hutcheson proposed a universal formula to evaluate the virtue, or degree of benevolence, of our actions:

Hutcheson’s formula for the moral arithmetic has a number of pleasing features. If two people have the same natural ability to do good, then the greatest one that produces the largest public good is the more virtuous. Similarly, if two people produce the same level of public good then the one of lesser natural ability is the more virtuous.

The other ingredient in Hutcheson’s formula, Private Interest, can contribute positively or negatively (±). If a person’s action benefits the public but harms themselves (for example, they do charitable work for no pay instead of taking paid employment), the Virtue is boosted by Public Good + Private Interest. But if their actions help the public and also themselves (for example, campaigning to stop an unsightly property development that blights their own property as well as their neighbours’) then the Virtue of that action is diminished by the factor Public Good – Private Interest.

Hutcheson didn’t attribute numerical values to the quantities in his formula but was prepared to adopt them if needed. The moral formula doesn’t really help you because it reveals nothing new. All the information it contains has been plugged in to create it in the first place. Any attempt to calibrate the units of Virtue, Self-Interest and Natural Ability would be entirely subjective and no measurable prediction could ever be made. None the less, the formula is a handy shorthand for a lot of words.

Something strangely reminiscent of Hutcheson flight of rationalistic fantasy appeared 200 years later in a fascinating project embarked upon by the famous American mathematician George Birkhoff, who was intrigued by the problem of quantifying aesthetic appreciation. He devoted a long period of his career to the search for a way of quantifying what appeals to us in music, art and design. His studies gathered examples from many cultures and his book
Aesthetic Measure
makes fascinating reading still. Remarkably, he boils it all down to a single formula that reminds me of Hutcheson’s. He believed that aesthetic quality is determined by a measure that is determined by the ratio of order to complexity:

Aesthetic Measure = Order/Complexity

He sets about devising ways to calculate the Order and Complexity of particular patterns and shapes in an objective way and applies them to all sorts of vase shapes, tiling patterns, friezes and designs. Of course, as in any aesthetic evaluation, it does not make sense to compare vases with paintings: you have to stay within a particular medium and form for this to make any sense. In the case of polygonal shapes Birkhoff’s measure of Order adds up scores for the presence or absence of four different symmetries that can be present and subtracts a penalty (of 1 or 2) for certain unsatisfactory ingredients (for example, if the distances between vertices are too small, or the interior angles too close to 0 or 180 degrees, or there is a lack of symmetry). The result is a number that can never exceed 7. The Complexity is defined to be the number of straight lines that contain at least one side of the polygon. So, for a square it is 4, but for a Roman Cross (like the one shown here) it is 8 (4 horizontals plus 4 verticals):