Printer Theory

It's a worthwhile question, you might want to stop here and think about it!

One possible answer is, just make images that have plenty of pixels and are shaped more or less like pieces of paper. The printer can scale to fit, it'll be fine. That isn't a bad answer, but it's also not much help when you come to the moment of actually having to pick an image size.

Another possible answer is, why don't we have both? That is, make two sets of images, one for 8½ × 11 and one for A4. That isn't a bad answer, but in general it just sweeps the problem under the rug. Unless the image content is going to be generated by some kind of system that understands page layout—TeX, a web browser, a word processor, et cetera—sooner or later you'll need to decide how you want to place the images onto the two different sizes of paper.

Eventually we'll need some actual numbers.

paper name	8½ × 11	A4
paper size	8.5 × 11 in	210 × 297 mm
standard margin	1 in	25 mm
usable area	6.5 × 9 in	160 × 247 mm
usable area (in)	6.5 × 9 in	6.3 × 9.7 in

We'll also need to think about the number of dots per inch that we want to provide. In my simple view of things, these are the two reference points:

• 300 dpi used to be the number for low-end laser printers, but now it's near the low end of the range for normal household printers. It's fine for photographs.
• 600 dpi used to be the number for high-end laser printers, but now it's the number for low-end laser printers. It's nice for text and diagrams.

Since urticator.net is entirely made of text and diagrams, 600 dpi is the natural choice here.

It's just a tiny detail, but the question of whether to write “dpi” or “DPI” is interesting to me. If I were writing a lot of prose containing the phrase “dots per inch”, I might shorten it to “DPI”, but when the phrase is attached to a number, the neuron for “unit of measurement” lights up more than the neuron for “acronym”, and that leads me to prefer “dpi”. Years ago the same kind of thinking led me to pick “mph” over “MPH” in all the essays about driving. Also, let me make the intermediate step explicit: I think of units of measurement as being mostly in lower case. Of course there are exceptions to the rule, like megawatts (MW), gigahertz (GHz), and so on.

But, let's not wander too far off topic. The next answer I considered is what I'll call the fair answer. When we scale an image to fit, generally some fraction of the usable area will be wasted. What if we arrange for that fraction to be the same for both sizes of paper? A little bit of math leads to the not-too-surprising result that the fraction is the same when the aspect ratio of the image is the geometric mean of the aspect ratios of the usable areas of the two sizes of paper. And, here are the aspect ratios.

paper size	8.5 × 11 in	210 × 297 mm
aspect ratio	1.2941176	1.4142857
usable area	6.5 × 9 in	160 × 247 mm
aspect ratio	1.3846154	1.54375

You might notice that the aspect ratio of A4 paper is close to root 2 (1.4142136). That's by design; see ISO 216 for many interesting details. Another interesting detail is that A4 is one of the few ISO 216 sizes with an aspect ratio that's not only close to root 2 but also a good rational approximation to root 2. In fact, it's a convergent (because 297/210 reduces to 99/70).

		1	2	2	2	2	2	2	2	2
0	1	1	3	7	17	41	99	239	577	1393
1	0	1	2	5	12	29	70	169	408	985

But, let's not wander too far off topic. To compute the geometric mean, first we multiply 9/6.5 by 247/160. The factors of 13 in 6.5 and 247 cancel, leaving 171/80 = 2.1375. Then we take the square root to get 1.4620192, and that's the fair answer.

Unfortunately, that's still just an aspect ratio, not an image size. But, given an aspect ratio and a paper size, we can obtain an image size in the following way.

1. Imagine a rectangle with the given aspect ratio.
2. Scale the rectangle to fit the usable area of the paper. The rectangle now has a definite physical size.
3. Multiply that physical size (in inches) by 600 dpi.

Let's see how that works in general. Suppose we have an aspect ratio r that's between the aspect ratios of the usable areas of 8½ × 11 and A4 paper. On 8½ × 11 paper, the rectangle will be thinner than the usable area, so it will scale to full height, 9 inches, and the width can be computed.

w	= 5400 / r
h	= 5400

On A4 paper, though, the rectangle will be thicker than the usable area, so it will scale to full width, 6.3 inches, and the height can be computed.

w	= 3780
h	= 3780 r

Unfortunately for fairness, the result is that we now have two image sizes, not just one.

However, when I was messing around with the equations, I noticed an amazing thing. If you plot the two widths as functions of the aspect ratio, there's one place where the curves intersect; and if you plot the two heights as functions of the aspect ratio, there's one place where the curves intersect; and it's the same place! That is, there's one aspect ratio where we get the same image size and physical size for both sizes of paper. The aspect ratio is 5400/3780, the image size is 3780 × 5400, and the physical size is … 6.3 × 9 inches. It was at about that point that I realized that the result was less like an amazing mathematical coincidence and more like the broad side of a barn. But, it's still a very nice result. Here are some additional features that make it irresistible.

• The aspect ratio 5400/3780 reduces to … 10/7 (1.4285714)! For some reason I'm fascinated by the number 7; see for example Fractions, or Multiplication in Base 7, or in fact the whole decimal expansions section.
• Did it bother you that I used the approximate value 6.3 inches instead of the exact value 160 mm? It should have! Amazingly, it turns out that the approximation is an unusually good one, so good that the number of pixels is the same either way. To a few more decimal places, the exact value is 6.2992126.
• Because the physical size is the same, if you construct some tesseract models in Europe and some in the United States, they should all fit together perfectly.

So, that's the final answer! I hereby declare 3780 × 5400 to be the official image size for printable images on urticator.net. If you're printing on 8½ × 11 or A4 paper, that should work well, and if you're doing something else, well, there are plenty of pixels.

I imagine you might also be wondering which essays contain printable images. Right now there's only one, Tesseract Model, but I hope that in the near future I might add one or two more. If I do, I'll be sure to make them link to this essay so that you can find them in the backlinks (at right).