Metamagical Themas, page 37
Interpolating Between an Arbitrary Pair of Typefaces
The worst is yet to come, however. Presumably Knuth did not wish us to take his rhetorical question in such a limited way as to imply that the numbers 6 1/7 and 1/4 were important. Pretty obviously, they were just examples of arbitrary parameter settings. Presumably, if METAFONT could easily give you a 6 1/7-point font that is 1/4 of the way between Baskerville and Helvetica, it could as easily give you an 11 2/3-point font that is 5/17 of the way between Baskerville and Helvetica-and so on. And why need it be restricted to Baskerville and Helvetica? Surely those numbers weren't the only "soft" parts of the rhetorical question! Common sense tells us that Helvetica and Baskerville were also merely arbitrary choices of typeface. Thus the hidden implication is that, as easily as one can twiddle a dial to change point size, so one can twiddle another dial (or set of dials) and arrive at any desired typeface, be it Helvetica, Baskerville, or whatever. Knuth might just as easily have put it this way:
The ability to manipulate lots of parameters may be interesting and fun, but does anybody really need an n-point font that is x percent of the way between typeface TI and typeface T2 ?
For instance, we might have set the four knobs to the following settings:
n: 36
x: 50 percent
TI: Magnificat
T2: Stop
Each of these two typefaces (see Figure 13-2) is ingenious, idiosyncratic, and visually intriguing. I challenge any reader to even imagine a blend halfway between them, let alone draw it! And to emphasize the flexibility implied by the question, how about trying to imagine a typeface that is (say) one third of the way between Cirkulus and Block Up? Or one that is somewhere between Explosion and Shatter? (For these typefaces, see Figure 13-2.)
A Posteriori Knobs and the Frame Problem of Al
Shatter, incidentally, provides an excellent example of the trouble with viewing everything as coming from parameter settings. If you look carefully, . you will see that Shatter is indeed a "variation on a theme", the theme being Helvetica Medium Italic (see Figure 13-2). But does that imply that any meticulous parametrization of Helvetica would automatically yield Shatter as one of its knob-settings? Of course not. That is absurd. No one in their right mind would anticipate such a variation while parametrizing Helvetica, just as no one in their right mind when delivering their Nobel Lecture would say, "Thank you for awarding me my first Nobel Prize." When someone wins a Nobel Prize, they do not immediately begin counting how many they have won. Of course, if they win two, then a knob will spontaneously appear in most people's minds, and friends will very likely make jokes about the next few Nobel Prizes. Before the second prize, however, the "just-one" quality would have been an unperceived fact.
This is closely related to a famous problem in cognitive science (the study of formal models of mental processes, especially computer models) called the frame problem. This knotty problem can be epitomized as follows: How do I know, when telling you I'll meet you at 7 at the train station, that it makes no sense to tack on the proviso, "as long as no volcano erupts along the way, burying me and my car on the way to the station", but that it does make reasonable sense to tack on the proviso, "as long as no traffic jam holds me up"? And of course, there are many intermediate cases between these two. The frame problem is about the question: What variables (knobs) is it within the bounds of normalcy to perceive? Clearly, no one can conceivably anticipate all the factors that might somehow be relevant to a given situation; one simply blindly hopes that the species' evolution and the individual's life experiences have added up to a suitably rich combination to make for satisfactory behavior most of the time. There are too many contingencies, however, to try to anticipate them all, even given the most powerful computer. One reason for the extreme difficulty in trying to make machines able to learn is that we find it very hard to articulate a set of rules defining when it makes sense and when it makes no sense to perceive a knob. It is a fascinating task to work on making a machine capable of coaxing shy knobs out of the woodwork.
FIGURE 13-2. A series of diverse typefaces: (a) Magn f cat; (b) Stop; (c) Cirkulus; (d) Block
This brings us back to Shatter, seen as a variation on Helvetica. Obviously, once you've seen such a variation, you can add a knob (or a few) to your METAFONT "Helvetica machine", enabling Shatter to come out. (Indeed, you could add similar "Shatterizing" knobs to your "Baskerville machine", for that matter!) But this would all be a posteriori: after the fact. The most telling proof of the artificiality of such a scheme is, of course, that no matter how many variations have been made on (say) Helvetica, people can still come up with many new and unanticipated varieties, such as: Helvetica Rounded, Helvetica Rounded Deco, Helvetican Flair, and so on (see Figure 13-3).
No matter how many new knobs-or even new families of knobs-you add to your Helvetica machine, you will have left out some possibilities. People will forever be able to invent novel variations on Helvetica that haven't been foreseen by a finite parametrization, just as musicians will forever be able to devise novel ways of playing "Begin the Beguine" that the electronic organ builders haven't yet built into their elaborate repertoire of canned rhythms, harmonies, and so forth. To be sure, the organ builders can always build in extra possibilities after they have been revealed, but by then a creative musician will have long since moved on to other styles. One can imagine Helvetica modified in many novel ways inspired by various extant typefaces. I leave it to readers to try to imagine such variants.
FIGURE 13-3. Three "simple" offshoots of Helvetica: (a) Helvetica Rounded; (b) Helvetica Rounded Deco; (c) Helvetican Flair.
A Total Unification of All Typefaces?
The worst is still yet to come! Knuth's throwaway sentence unspokenly implies that we should be able to interpolate any fraction of the way between any two arbitrary typefaces. For this to be possible, any pair of typefaces would have to share the exact same set of knobs (otherwise, how could you set each knob to an intermediate setting?). And since all pairs of typefaces have the same set of knobs, transitivity implies that all typefaces would have to share a single, grand, universal, all-inclusive, ultimate set of knobs. (The argument is parallel to the following one: If any two people have the same number of legs as each other, then leg-number is a universal constant for all people.)
Thus we realize that Knuth's sentence casually implies the existence of a "universal 'A'-machine"-a single METAFONT program with a finite set of parameters, such that any combination of settings of them will yield a valid `A', and conversely, such that any valid `A' will be yielded by some combination of settings of them. Now how can you possibly incorporate all of the previously shown typefaces into one universal schema?
Or look again at the 56 capital 'A's of Figure 12-3. Can you find in them a set of specific, quantifiable features? (For a comparable collection for each letter of the alphabet, see the marvelous collection of alphabetical logos compiled by Kuwayama.) Imagine trying to pinpoint a few dozen discrete features of the Magnificat 'A' (A7) and simultaneously finding their "counterparts" in the Univers `A' (D3). Suppose you have found enough to characterize both completely. Now remember that every intermediate setting also must yield an `A'. This means we will have every shade of "cross" between the two typefaces.
This intuitive sense of a "cross" between two typefaces is common and natural, and occurs often to typeface lovers when they encounter an unfamiliar typeface. They may characterize the new face as a cross between two familiar typefaces ("Vivaldi is a cross between Magnificat and Palatino Italic Swash") or else they may see it as an exaggerated rendition of a familiar typeface ("Magnificat is Vivaldi squared") (see Figure 13-4). What degree of truth is there to such a statement? All one can really say is that each Magnificat letter looks "sort of like" its Vivaldi counterpart, only about "twice as fancy" or "twice as curly" or something vague along those lines. But how could a single “curliness knob account for the mysteriously beautiful meanderings, organic and capricious in each Magnificat Letter?
FIGURE 13-4. A transition from curved to whirly to superswirly: (a) Palatino Italic Swash caps; (b) Vivaldi caps; (c) Magnzcat caps. It is provocative to compare this figure with Figure 16-7.
Can you imagine twisting one knob and watching thin, slithery tentacles begin to grow out of the Palatino Italic `A', snaking outwards eventually to form the Vivaldi `A', then continuing to twist and undulate into ever more sinuous forms, yielding the Magnificat `A' in the end? And-who says that that is the ultimate destination? If Magnificat is Vivaldi squared, then what is Magnificat squared?
Specialists in computer animation have had to deal with the problem of interpolation of different forms. For example, in a television series about evolution, there was a sequence showing the outline of one animal form slowly transforming into another one. But one cannot simply tell the computer, "Interpolate between this shape and that one!" To each point in one there must be explicitly specified a corresponding point in the other. Then one lets the computer draw some intermediate positions on one's screen, to see if the choice works. A lot of careful "tuning" of the correspondences between figures must be done before the interpolation looks good. There is no recipe that works in general for interpolation. The task is deeply semantic, not cheaply syntactic.
For a wonderful demonstration of the truth of this, look at the little book Double Takes, in which artist Tom Hachtman has a lot of fun taking unlikely pairs of people and combining their caricatures. His only prerequisite is that their names should splice together amusingly. Thus he did "Bing Cosby (Bing Crosby and Bill Cosby), "Farafat" (Farrah Fawcett-Majors and Yasir Arafat), "Marlon Monroe" (Marlon Brando and Marilyn Monroe), and many others. The trick is to discern which features of each person are the most characteristic and modular, and to be able to construct a new person having a subtle blend of those features, clearly enough that both contributoi1 be recognized. For a viewer, it's almost like trying to recognize t~ parents in a baby's face.
The Essence of `A'-ness Is Not Geometrical
Despite all the difficulties described above, some people, eve scrutinizing the wide diversity of realizations of the abstract 'A'-concept maintain that they all do share a common geometric quality. They sometimes verbalize it by saying that all `A's have "the same shape' or are "produced from one template". Some mathematicians are inclined to search for a topological or group-theoretical invariant. A typical suggestion might be: "All instances of ´A' are open at the bottom and closed at the top” Well, in Figure 12-3, sample A8 (Stop) seems to violate both of these criteria. And many others of the sample letters violate at least one of them. In several examples, such concepts as "open" or "closed" or "top" or "bottom" apply only with difficulty. For instance, is G7 (Sinaloa) open at the bottom? Is F4 (Calypso) closed at the top? What about A4 (Astra)?
The problem with the METAFONT "knobs" approach to the `A' category is that each knob stands for the presence or absence (or size or angle, etc.) of some specifically geometric feature of a letter: the width of its serifs, the height of its crossbar, the lowest point on its left arm, the highest point along some extravagant curlicue, the amount of broadening of a pen, the average slope of the ascenders, and so forth and so on. But in many `A's, such notions are not even applicable. There may be no crossbar, or there may be two or three or more. There may be no curlicue, or there may be a few curlicues.
A METAFONT joint parametrization of two 'A's presumes that they share the same features, or what might be called "loci of variability". It is a bold (and, I maintain, absurd) assumption that one could get any `A' by filling out an eternal and fixed questionnaire: "How wide is its crossbar? What angle do the two arms make with the vertical? How wide are its serifs?" (and so forth). There may be no identifiable part that plays the crossbar role, or the left-arm role; or some role may be split among two or more parts. You can easily find examples of these phenomena among the 56 `A's in Figure 12-3. Some other examples of what I call role splitting, role combining, role transferral, role redundancy, role addition, and role elimination are shown in Figure 13-5. These terms describe the ways that conceptual roles are apportioned among various geometric entities, which are readily recognized by their connectedness and gentle curvatures.
For a remarkable demonstration of ways to exploit these various role-manipulations, see Scott Kim's book Inversions, in which a single written specimen, or "gram", has more than one reading, depending on the observer's point of view. Often the "grams" are symmetric and read the same both ways, but this is not essential: some have two totally different
readings. The essence is imbuing a single written form with ambiguity. Both Scott and I have for years done such drawings-dubbed "ambigrams" by a friend of mine-and a few of my own are presented in Figure 13-6, as well as the one on the half-title page. The strange fluidity of letterforms is brought out in a most vivid way by ambigrammatic art.
FIGURE 13-5. Examples off (a) role splitting; (b) role merging; (c) role transferal; (d) role redundancy; (e) role addition; and (f) role elimination. The idea in all these examples is that one smooth sweep of the pen need not fill exactly one coherent conceptual role. It may fill two or more roles (or parts of two or more); it may fill less than one, in which case several strokes combine to make one role; and so on. Sometimes roles can be added or deleted without serious harm to the recognizability of the letter. Angles, cusps, intersections, endpoints, extrema, blank areas, and separations often play roles no less vital than those played by strokes. '
Incidentally, it is most important that I make it clear that although I find it easier to make my points with somewhat extreme or exotic versions of letters (as in ambigrams or unusual typefaces), these points hold just as strongly for more conservative letters. One simply has to look at a finer grain size, and all the same kinds of issues reappear.
Chauvinism versus Open-Mindedness:
Fixed Questionnaires versus Fluid Roles
When I was twelve, my family was about to leave for Geneva, Switzerland for a year, so I tried to anticipate what my school would be like. The furthest my imagination could stretch was to envision a school that looked exactly like my one-story Californian stucco junior high school, only with classes in French (twiddling the "language" knob) and with the schoolbus that would pick me up each morning perhaps pink instead of yellow (twiddling the "schoolbus color" knob). I was utterly incapable of anticipating the vast difference that there actually turned out to be between the Geneva school and my California school.
Figure 13-6. Several ambigrams by the author. Decyphered, they say: "ambigram"; "ambigrams"; "winter"; "spring"; "summer"; "fall"; "Lee Sallows"; "Josh Bell"; "Alejandra" and "Magdalena" (reflections of each other); "Carol"; "Chopin"; and "Johann Sebastian Bach". All three composers' names utilize 90 degree rotation. Notice the extensive use of all the devices shown in Figure 13-5, namely role splitting, merging, transferral, redundancy,addition and elimination. See the half title page for a further ambigram by the author.
Likewise, there are many "exobiologists" who have tried to anticipate the features of extraterrestrial life, if it is ever detected. Many of them have made assumptions that to others appear strikingly naive. Such assumptions have been aptly dubbed chauvinisms by Carl Sagan. There is, for instance, "liquid chauvinism", which refers to the phase of the medium in which the chemistry of life is presumed to take place. There is "temperature chauvinism", which assumes that life is restricted to a temperature range not too different from that here on the planet earth. In fact, there is planetary chauvinism-the idea that all life must exist on the surface of a planet orbiting a certain type of star. There is carbon chauvinism, assuming that carbon must form the keystone of the chemistry of any sort of life. There is even speed chauvinism, assuming that there is only one "reasonable" rate for life to proceed at. And so it goes.
If a Londoner arrived in New York, we might find it quaint (or perhaps pathetic) if he or she asked "Where is your Big Ben? Where are your Houses of Parliament? Where does your Queen live? When is your teatime?" The idea that the biggest city in the land need not be the capital, need not have a famous bell tower in it, and so on, seem totally obvious after the fact, but to the naive tourist it can come as a surprise. (See Chapter 24 for more on strange mappings between Great Britain and the United States.)
