First, a quick overview of some technical bits. The user can select a range of dates from the map interface, and a new set of markers is rendered after an AJAX query to the backend PHP code, which requests the data from our database and aggregates it at the zip code and state level. The decision to do this two level aggregation was based on the performance hit that would be taken in trying to render on the order of 7000 icons for a year’s worth of data on a nationwide map. The MarkerManager class (which used to be part of the core Google Maps API) is used to display one set of icons when zoomed inside of what might be called “state level” and another set when zoomed out further. This adds a bit of extra complexity to our colormap choice.

There are a number of ways that this data could have been mapped into visual symbols. The size or shape of the glyphs could be related to the number of admissions, for example. The method that we are using maps the number of admissions to color (when this mapping is stored, it is called a colormap). The particular mapping that we use is important, because some mappings are better than others for certain tasks. For the markers on the map, we use a colormap that smoothly transitions from a blue at the low end to a red at the high end. The particular choice of blue to red leverages our cultural understanding of blue as “cold” and red as “hot” (note that sometimes colormap choice depends on cultural interpretation), a metaphor which works well as a representation of low vs. high admission rates. The combination of these two choices allows us to understand the general trends in the data without needing to refer back to the legend frequently. The drawback is that it is a bit more difficult to compare the value of two individual markers than it would be if we had picked a colormap of more distinct hues such as {red, orange, yellow, green, blue, purple}. We tried to make it easier for the viewer to make comparisons by limiting the number of colors in our legend.

The next important choice is how exactly to map the number of admissions to a particular color. To do this well, an understanding of the distribution of the data is necessary. Intuitively, we would expect that there will be high admissions numbers for the zip codes closest to the museum, and the trend will be toward increasingly lower admissions from more distant zip codes, and likely there will be many zip codes with very low, but non-zero admissions numbers. The best way to look at this objectively is with a histogram. I will take a number of approaches with the data from the year 2008, for which the highest raw admission count for a zip code was 12,375 (during the aggregation, 9 digit zip codes are merged into 5 digit zip codes, resulting in slightly higher numbers in some instances), out of 6806 zip codes.

This first histogram segments the zip codes into buckets in a linear fashion, with each bucket representing 10% of 12,375. As you can see, the vast majority of the zip codes end up in the first bucket. If we used 10 colors with this linear mapping, almost all of the glyphs would be blue. It would be great for identifying and comparing the few zip codes that have very high admission, but a lot of subtlety in the big picture would be lost.

This histogram also uses a linear segmentation, but with an upper limit (or clamp). Any zip code with 10 admissions or more are put into the red bucket. This would be fine for comparing admissions from distant zip codes, but we wouldn’t be able to distinguish visually between admissions from zip codes around Indianapolis.

This is where the beauty of logarithms comes to the rescue. Let’s use our example to walk through the process. First we determine the logarithm of our maximum number, 12375, in base 2 (working in base 2 is very efficient for computers). The result is 13.59. The highest bucket represents 12375 in the real world, and 13.59 in exponent-land. Each bucket represents a tenth of 13.57 in exponent-land, just as in our first histogram each bucket represented a tenth of 12375. We then simply derive the numbers that each bucket represents in the real world; 2 to the 1.359th power equals 3 (when represented as a whole number), for example. These whole numbers in the real world are then used to put the zip codes in the proper bins.

The result is that we have teased out more meaning to bestow upon our colors. There are still thousands of zip codes in those blue buckets, but they are the ones that are scattered across the nation on our map, so here we are aided by spatial distribution. In the red zone, there are dozens of zip codes in each bucket, and this is just what we want to be able to analyze the subtleties around Indianapolis where the icons are more spatially dense. It also turns out that this trend is pretty much independent of the date range, and the colormap still works out well at the state level.

On the actual map we use eight icons rather than ten to reduce clutter in the legend, but the algorithm is the same.

You might notice that we use a different type of colormap in the demographic windows. We actually tried using a smooth colormap for this data for the more linear demographics (Income, Age, etc.). The problem was that it was then very difficult to visually relate a particular shade of blue or green to a particular age or income level, which would almost certainly lead to eye strain and headaches. So here we used a more rainbow-like colormap to make that analysis easier.

In summary, colormap selection is a complex process involving aspects of mathematics, design, and the nature of the dataset. It’s one of those areas where art and science come together and the best practices of each can lead to successful communication of concepts and beautiful representations of data.

When I was very young, I had a special sense about written numbers. It’s hard for me to access that now, through all the years of education devoted to making sure I understood numbers in a standard way. But I still have a feeling about that early relationship, and sometimes I wonder how it might have developed if I hadn’t learned to be ashamed of it and to ignore it.

Here’s what I can recall: I knew the shapes of the numerals as indicators of the distinct characters of each. Though my sense for some of them has slipped out of reach, in the way dreams do, I can still feel the stronger personalities. The numeral five was intimidating in appearance, but in actuality quite sweet. Seven was both stern and judgmental. Eight had complexity and depth – and eight led to a painful collision with my first grade teacher, Miss Logan. She taught us to write eight with one continuous figure-eight line. Soon after, she exhorted us never to write it as one circle on top of the other – an idea that had, frankly, not occurred to me.

Once I heard about this forbidden way of making the image, I badly wanted to try it, to find out why it was seductive and wrong.  I hunched over my practice sheet to try what sounded to me like an ingenious alternative. The hurtful rap of a ruler on the back of my head shocked and scared me. I could hardly believe she caught me in the act so quickly and easily. Miss Logan’s efficient suppression of dissent gave me, early-on, the impression that privacy and experimentation had no place in the classroom.

Back to the personalities of numbers: you might think it’s just as well that this idiosyncratic notion of numerals having distinct natures signified by their visual forms was scared out of me. Even in first grade it was beginning to raise some dauntingly complex dimensions of arithmetic. What kind of psychodrama might be the sum of 8+7? If 5 were subtracted from 9, what interpersonal consequences would that equal? Left unmolested, I wonder if I might have been able to craft an alternative way of working with numbers that allowed me to derive answers that approximated my classmates’. I’ll never know. As it is, I developed a serious case of math phobia and went on to do poorly in math classes throughout my schooling – with only the slight exception of geometry, to which I was timidly attracted. It is only in middle age that I’ve come to terms with the fact that I am actually fascinated by mathematics as logic, and by the more philosophical implications of mathematics, rather than the computing tasks. I’ve also noticed that the concept of numbers having “natures’ isn’t entirely far-fetched when one considers mathematics as a system for describing relationships and processes.

My early sense about numbers may be one indication of something it’s taken me years to notice about myself: I believe I am a primarily a non-verbal thinker. Until I reached this hypothesis, I thought everyone thought approximately the same way.

Several years ago, I began asking my colleagues in the art museum education department how they thought – not what they thought, but how. Were they conscious of thinking in words, for example? I started this line of questioning because I realized that I was completely unable to describe or explain my experience of thinking. Of course I could mentally use words. If I needed to craft a statement of some kind and make decisions about the most effective wording, I could certainly rehearse the possibilities in my mind and make a choice. However, that would be a particular situation, very different from my ordinary, day-to-day thought/language processes. Truth be told, I had to admit that in my on-going mental life, words don’t play a part. In ordinary conversation, I do not plan or even know what words will come out of my mouth. I would even go so far as to say that the times I have jotted notes for a talk or to teach a class have led to my most lack-luster presentations. The notes always flummox me.  It’s taken me a while to trust myself, but I now feel that I am better off speaking “spontaneously.”

But back to the question about thinking that I posed to my co-workers: Most people seemed taken aback by the question and several mentioned that they had never considered how they thought. Upon reflection, quite a few said that they were conscious of words and sentences going through their minds. Several said they “heard” their thoughts as an on-going voice inside their heads. One person described being vaguely aware of punctuation in his thoughts! Another described dreams in which she read the narrative and conversations in a way that reminded her of the bubbles over the heads of comic book characters.

It was difficult to cover my own surprise at these revelations. Even now, as I type this anecdote into my laptop, I wish I could form the ideas on this screen with my hands. I wish that you could take them in with a probing – or a playful – gaze, rather than following various linear sentences to various open or dead ends. I don’t think with words.

How do you think? Can you describe your experience of thinking? Please let me know. I think others will be interested as well.