Old Pappus’ Book of Mathematical Calligraphy Andrew Stacey∗ 17th November 2011 Abstract It is a truth universally acknowledged, that a mathematician in possession of a good theorem must be in want of notation. 1 Introduction It is almost as universally acknowledged that an alphabet of merely 26 letters is insufficient to express the vast richness of mathematical ideas. Whilst there may not be many articles that truly use more than 26 symbols, other factors weigh in to ensure that it is desirable to have a wide range to choose from. In particular, it is good style to use notation consistent with that of other authors in ones field, and to avoid notation that clashes. Notation should be used to aid comprehension, not to hinder it. Whilst there are times when it is appropriate for an author to say ∀♥ > 0, ∃♣ ∈ N : ∀♦ > ♣, |♠♦ − ♠| < ♥ I am sure that most readers would rather that he didn’t. A common practice in mathematical writing is to use similar letters for similar things. I find that my default assumption for the English alphabet is that of Table 1. (“Locally constant” means that when used then they should be considered as constant, but the constant is arbitrary.) Your assumptions may be different; nonetheless, I would expect to see the same sort of grouping in that similar letters represent similar things. This grouping is something that we have come to expect and is something that a good notation system can exploit. Using similar letters for things is a way of saying, “These are related.”. But herein lies the problem for there are many types of relationship. Let us consider two groups, G and H. Now for pairs of elements, g1 , g2 ∈ G and h1 , h2 ∈ H, we can ask whether or not there exists a homomorphism γ : G → H which takes g1 to h1 and g2 to h2 . ∗ The author is a professional mathematician and amateur calligrapher. His mathematical statements, of which there are none, should be viewed as reliable whereas his calligraphic comments, of which there are many, should not. 1 Letters a, b, c, d e f, g, h i, j, k, l, m, n o p, q, r s, t u, v w x, y z Meaning locally constant numbers, 2.718281828904590 . . ., functions, locally constant integers, too easily confused with 0, functions, variable real numbers, either vectors or the components of w, either a vector or a complex number or function, variable real numbers, variable complex number Table 1: The author’s standard interpretation of letters Here, we have several types of relationship. The two things designated G and H are similar things (they are both groups). The elements g1 and g2 are elements of G, whereas h1 and h2 are elements of H. The homomorphism is a third thing and so we have brought in a third alphabet and used the thing that most looks like a g from that alphabet. This illustrates a common strategy: when we have more than two types of relationship, we need to use more alphabets. The difficulty with this is twofold. Firstly, there may not be a clear correspondence between the new alphabet and the old; it is standard that α represents a and γ represents g, but what represents c? Secondly, by using symbols from an unfamiliar alphabet we lose the quickness of association that familiar symbols bring. An alternative is to use the same letters but from different fonts. The choice of g for an element of G is actually an example of this. A more extreme example is the common notation for the Lie algebra of a Lie group: g of G. In days of yore, this was self-limiting. Articles with fancy fonts required careful handling, often with the symbols written by hand. In modern times, such archaic methods seem . . . archaic. With the plethora of fonts available, it is possible to have a whole panoply of symbols all related and all in different fonts. This freedom does have a negative side. It is now possible to use obscure fonts in articles and presentations. However, from time to time it is desirable to actually write a symbol by hand, either on a chalkboard or, dare I say it, a piece of paper. The temptation, when faced with a complicated symbol in a complicated font, is to replace them all by just “squiggle” (unicode character U+21DC). This is less than perfect as it breaks the relationship that the choice of notation was designed to preserve. One can easily imagine an extreme case. A mathematician writes an article involving some concepts that he chooses to express using letters from a variety of fonts; say g, G, g, and G. This mathematician is then asked to speak on that 2 Figure 1: Thick and thin calligraphic pens article and decides, either willingly or unwillingly, to present the material on a chalkboard. He then has to write these symbols on the board. One of the listeners to this talk likes to take notes. She then has to copy these symbols from the board to her pad of paper. Unfortunately, the speaker is a little indistinct, both in speech and writing, and the symbol g is never clearly explained. So the listener writes down some approximation of it, but it is such a poor approximation that when she looks back at these notes six months later1 then it isn’t clear if this symbol was g, ℘, or something else entirely. In conclusion, to enjoy the freedom of the wide variety of fonts available, mathematicians need to learn how to write these symbols both on the chalkboard and paper. 2 Calligraphy The art of writing beautifully is called calligraphy. By using a little of this art of letter-forming, a mathematician can ensure that these strange symbols that she uses are clear both to others and, six months later on, to herself. There are many, many books that explain how to do calligraphy and which explain in detail how to form the letters. However, calligraphy for mathematicians is slightly different to calligraphy in general. The major difference is in the materials used. Either this will be chalk on a board, or a normal pen on paper. Whilst an experienced calligrapher can produce beautiful works of art using driftwood for a pen, it is more usual to start with a special calligraphic pen. These come in two sorts: thick pens and thin pens. Thick pens are usually held at a constant angle and drawn across the page, never being pushed or turned. A thin pen (or a copperplate pen) is usually used in a more fluid manner and it is common to vary the pressure to change the thickness of the line. Examples of the type of stroke produced with each are in Figure 1. The peculiarity of mathematical calligraphy is that whilst the fonts are often designed to be written with a thick pen, the tools are closer in output to a thin pen as they produce a stroke whose width does not depend on the angle at which the implement is held. There is a further difference between chalk and a normal writing pen in that chalk is less amenable to being pushed: drawing a circle with chalk will usually require two strokes whilst with a pen only the 1 She’s a very keen mathematician and actually reads again the notes that she makes. 3 gg Figure 2: The letter “g” in fraktur and ordinary mathematics font one. A third aspect of chalkboard mathematical calligraphy is that it should, by design, be clear at a distance. Thus intricacies are to be severely frowned upon. The problems are most acute for the chalk-wielder. She not only has to write an unusual symbol, it has to be clear from a distance (clear enough that the inexperienced note-taker can make a reasonable stab at copying it), and it has to look like a symbol from a font designed for a wholly different type of pen. No wonder many resort to squiggles! 3 The Fraktur Letter The fraktur style, of which g is an example, is one of the more common of the “fancy alphabets” in use in mathematics. It is therefore a reasonable choice for an example of showing how to go from a proper calligraphic letter to something suitable for a chalkboard. Let us begin with the letter itself, as displayed in Figure 2 next to a standard one for comparison. What is clear from this is the angularity of the letter. Although individual lines may curve (the lowest stroke being the most obvious), major direction changes are sharp. When converting to a chalkboard style, these are the characteristics that we wish to preserve (even to exaggerate). Let us begin by replicating the character in a calligraphic manner as in Figure 3. The left-hand rendering is the original character from the font, the middle is as if drawn with a calligraphic pen, and the right-hand version follows the same strokes but with an “ordinary” pen. Now, while this is reasonably faithful, it is not suitable for a blackboard. 4 g Figure 3: The fraktur “g”: original font, as if with a calligraphic pen, and the “skeleton” Figure 4: A more angular letter The thin line will be lost, and the angles could easily be smoothed out. The fact that the top left corner is continuous is a coincidence brought on by the thickness of the lines (this can be seen in the “skeleton” rendering). So we need to modify it to exaggerate the angularity, to make it feasible to draw on a blackboard, and ensure that it remains distinct from the ordinary g. The top of the letter has a nice dip rising to a point at the right. We can emphasise that. This is matched by the thin line at the bottom of the main part of the letter, but the thinness will be lost when rendered by chalk so we replicate (and mirror) the dip at the top. The side lines may as well be vertical. The top of the letter is now sufficiently angular, but the descender has no distinguishing features. A simple remedy for this is to make it sweep up instead of down. This also means that there is plenty of space for an exaggerated swash if so desired. The result is in Figure 4. Whilst this can no longer be called fraktur, it has the same angularity and thus will be clear on a chalkboard. Seeing and writing are two different things. Fortunately, the construction of the letter allows us to see how it should be written. This is a letter which is clearly distinguishable from an ordinary “g” and 5 1 2 3 Figure 5: How to write a “g” which is simple to write. It is therefore suitable for using on a chalkboard. Similar techniques can produce many more letters that are close enough to their “true” forms that their meaning is apparent, and which are simple enough that this meaning is conveyed even when they are drawn in chalk on a chalkboard. 6