Curved Curiosities

I came across the Oakes Twins through working at the Science Gallery in Dublin. They’re going to be exhibiting there very soon – and, as it is my job to talk to people about the various exhibits in the gallery, I did some research… and became completely absorbed and sidetracked by the mathematical background to what they are doing.
They’re Ryan and Trevor Oakes, identical twins in their early thirties who make extraordinary drawings, that are also kind of like scientific investigations into the very nature of sight and human vision.

I love that they paint on a curved canvas because according to them, a person’s field of vision is shaped like a sphere, wrapping around us, the radius of the sphere extending towards infinity (at least when you’re looking up at the sky.) When you think about this it is totally obvious, but it needed somebody to realise it. Kind of like the invention of perspective in the Renaissance.
This, they say, was probably their most important insight (I emailed them asking for information and got a beautifully-written, enthusiastic description of their work and their way of working. How often do you get that from people you just email out of the blue?)

Oakes Twins Maths thingIn the beginning their works were just line drawings, and they then progressed on to working with light and colour in REALLY interesting ways.
They say that the story of sight is the story of spherical shapes interacting – the light that emanates from any light source emanates out in a sphere, and every particle of light that hits a surface rebounds back in a “dome of radiating light” to borrow a phrase from Trevor, and then that massive collection of particles, air saturated with them, hitting our sphere-shaped eyes, gives us our sphere-shaped field of vision.

I love this – spheres are wonderful. They are everywhere in the biggest and most impressive parts of nature – the sun, the moon, the solar system – and in the most minute parts – (frogspawn, atoms).

There’s a strange thing that happens with their canvas, something I came across in one of their catalogues. The catalogue explained that they had been puzzling over the problem for a while.

The canvas is built out of strips of card which when laid out on a table look like the image above, and when strapped together form a spherical section. Mathematically speaking, it becomes, when strapped together, the area bounded by the something like this curve: z = x2 + y2 and the lines z = 0 and z = 1.
When the whole shape is turned upside down, you get the same thing only upside-down. (Obviously.)
But something else happens when you lay out the strips and then individually turn each one upside down, and then strap them together again.

Monkey_saddle_surface.svgIn fact, you get something quite different, that happens to have a wonderful name: a Hyperbolic Paraboloid. Or a “Saddle Curve.” It looks a bit like this image above – like a pringle, or a saddle. I like these shapes – in maths they are interesting because there is a point in the middle that is both a “local maximum point” and a “local minimum.” If you could stand there you’d be at the top of a hill and at the bottom of a valley simultaneously.
The equation for this is different, although similar to the equation for a spherical section – something like z = x2 – y2 bounded by z = 0 and z = 1 as before.

The weird thing is that this doesn’t happen when you use different kinds of shapes to construct, say, a hemisphere. Imagine a hemisphere cut into lemon-shaped slices which are then laid out on a table. Turn all of these upside down one by one, strap them together and you still get a hemisphere.

I thought this was quite mind-blowing.
I could understand why the curved canvas shape changes into a hyperbolic paraboloid – because essentially you are inverting one of the axes by turning each strip upside down but keeping the other axis the same, while when you turn the whole shape upside-down, you are rotating the whole plane – but I couldn’t understand why the same thing doesn’t happen with the hemisphere lemon shapes.

I knew you couldn’t represent a hemisphere by the same equation as above, z = x2 + y2. You need a different one, say z = √(x2 + y2 – 1). And there is a difference in that a hemisphere is a projection of a circle onto a sphere, so it sort of fits into it, while the curved canvas is the result of projecting a different kind of shape, a rectangle, onto the circular, curved surface of the sphere.

But still I couldn’t understand why they should behave so differently, being just different sections of the same shape.
I couldn’t figure it out until I had done the equivalent of cutting up a football and actually laying it out (I used, not a football, but an orange peel. ) When I had laid them out I realised what should have been obvious from the beginning: while the canvas strips vary in shape, the ones at the top and the bottom looking different, with more curvature, than those in the centre, each orange peel strip was identically shaped. And so when rotated through 180 degrees its shape didn’t change at all, and so when put all together again an identical sphere would be created. (Assuming I’d cut the orange peel slices perfectly.)Eva's Image 3

And when I thought about it this made perfect sense, because the hemisphere, unlike the curved canvas, has the same level of curvature all around. It isn’t an imposition of a new kind of shape upon a very different one, there’s no squaring of the circle involved, and so none of the stretching to accommodate the extra corners is involved.

I’d like to be able to put this more precisely, mathematically speaking – to explain it in terms of the equations of each of the shapes. I can see that one of the equations can easily have part of itself turned upside down by putting a negative sign in front of the y2 – while with the hemisphere equation this is not so simple. If anyone has any ideas they would be much appreciated.

While trying to draw a hyperbolic paraboloid using perspective, I also became completely sidetracked by the issue of how to use perspective perfectly…. and it turns out that Perspective has a strange and wonderful mathematical history. But that’s another story.

Image Credits:
  • View from the roof of the Union Square Cinema: from catalogue of Compounding Visions: The Art of Ryan and Trevor Oakes, copyright Trevor Oakes, Ryan Oakes, Lawrence Weschler.
  • Saddle curve  By Nicoguaro – Own work, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=48854230
  • Longitudinal Sections Collage – Nutmeg Tree Stories original.
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