Lined Paper from Another Dimension

What’s the coolest lined paper you’ve ever seen?

Trick question: you haven’t. I might as well ask you to name your raddest experience of flossing. It’s gonna be, at best, functional. “Cool” is beyond reach.

At least, it was until math teacher Matt Enlow came along.

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When I first saw his lined paper, in the art exhibit at the Joint Math Meetings in Baltimore, I was floored. I was flabbergasted. I was flummoxed and fascinated and many other participles that I do not usually apply to lined paper.

Matt’s paper is all of the greatest promises of math, rolled into one. (Flattened into one?) You want algebraic methods yielding geometric beauty? Check. You want computational tools building our wildest fancies into rigorous realities? Check. You want abstract ideas somehow transforming a concrete everyday experience? Check.

I asked Matt to explain the inspiration behind his paper, and he kindly obliged.

How did you get the idea for these parallel-universe papers?

Appropriately enough, I was inspired by Marc Thomasset’s Inspiration Pad.

Once I realized that I could make my own variations on that theme, my head started spinning with ideas. It became a minor obsession.

How did you make them?

I created the images in Mathematica, and printed them on standard 11″ x 17″ printer paper. I then hand-cut the corners, and hand-punched the holes.

I surprised even myself with how important verisimilitude was to me. For example, I had to go out and purchase a hole puncher that punched 5/16″ holes, because the only ones available to me in my house and classroom were 1/4″. And that simply would not do.

(I’m still a little dejected that I couldn’t figure out a simple, reliable way to print them two-sided.)

Let’s talk about the first one. How did you generate those wavy lines?

Not easily!

I thought of the paper as a carpet, under which a vertical sine wave was creeping upwards, as it also swept across the page from left to right.

I had to think about what a sine wave would look like “from above” with equally-spaced points on the curve. Those equally-spaced points would appear closer together at the steepest parts of the curve, and farther apart at the least-steep parts.

It involved a fair amount of inverse arc length computation — as in determining what intervals of my parameter produce a curve having a desired length. I didn’t even know what the finished product would look like, but I was pleased with the effect.

Onto the second. I love the 3D effect, like a bulb is emerging from the page. How did you create it?

It’s actually a similar approach to the previous one, where I pictured what a “curved” piece of lined paper would look like from above. In this case, if you imagine that this bump is a hemisphere, and imagine equally-spaced points on the half-circumference running vertically up the center, those points can be positioned precisely using trigonometry. From there I decided to use half-ellipses for the lines, and it was just a matter of determining the correct dimensions and position.

What’s the mathematics behind the third one? What’s the relationship between this image and the usual arrangement of lines on paper?

This one started as most of them do: an image in my head. I knew I wanted to preserve the orthogonality of the blue “lines” to the red one, but I didn’t know much else.

I decided that I wanted the other constraint to be that if you were to connect the “poles” at top and bottom with a line segment, the blue circles would cut it at equally-spaced points. So “locally,” toward the center of the page, it resembles a normal sheet of paper more closely.

And, finally, the last one. I find the white space very eye-catching, and a little disorienting. How did you create this 3D effect?

This was probably the simplest of all to create. It was just a matter of choosing two vanishing points (both off the sides of the paper), and connecting them to the points on the red line where the blue lines would normally intersect it.

Any plans to sell whole pads like this? I would pay top dollar for one.

You are far from the first person to ask me this! I am certainly interested in looking into the idea. One obstacle is that such a product would kind of defy all categories. In any case, I don’t suppose you have any publishing contacts?

Publishers reading this: Matt is open to offers! I will limit my fee to a humble 15% (of the paper, that is – much more valuable to me than the money).

NOTE: Matt has since added four new designs! Here they are: