Modelling Seashells, Part 2: An Exercise in Abstraction.

This week I describe an artistic exploration I went on after modeling as realistic looking of a seashell as I could using Rhino and Grasshopper (described here). That model was a surface defined by two different curves that were replicated, at increasingly smaller scales, around a logarithmic spiral.

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Once I had done this, I decided to experiment. I wanted to create new forms by making this model more and more abstract. The first thing I did was to extract the curves that defined the model, and turn them into wires. Here’s the 3D printed result:

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One of the two curves that define this shell form is bumpier than the other. The next step in abstraction was to create a similar wireframe structure, without the bumps. The resulting model is quite mesmerizing when it is spun!

I liked this design very much, so I played with it a bit. Shrinking it down to a smaller size turned out to make a nice pendant. Fabricated even smaller still, and paired with its mirror image, made a lovely pair of earrings.

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Finally, I decided to try to triangulate it. This proved to be the most difficult task. Each triangle is flat, which is in conflict with the fact that the curvature of the model gets more and more extreme as you move toward its apex. The result is that somewhere along the line some compromises had to be made, and those compromises resulted in a form that is no longer clearly recognizable as a seashell. In some sense this was the ultimate goal: to create a truly new and unique form.

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In a week or two I’ll write about my experiences fabricating a version of this triangulated model that was over 5 feet tall!! In the meantime, you can order any of the above models from Shapeways here.

Modelling Seashells, Part 2: An Exercise in Abstraction.

Knitting and Printing the Figure-8 Klein Bottle

This week I’ll describe one strategy for representing 4-dimensional objects in 3-dimensions, using both modern and traditional media. While more than three spatial dimensions is a foreign concept to most people, Mathematicians have been studying theoretical objects in four dimensions and more for over a century. (Some contemporary Mathematicians even think about infinite-dimensional things, whatever that means!).

Each new dimension simply gives you more freedom to move. For example, if I draw a figure-8 curve on a piece of paper without lifting my pen, then it necessarily crosses itself. However, by introducing a break at the intersection, our brain tricks us into interpreting the 2-dimensional picture as a representation of a 3-dimensional loop with a twist.

In the same way one can imagine a four-dimensional object. Our familiar three dimensions become like the paper. By introducing breaks, we can interpret one part of the surface as passing right through another, without intersecting it. The surface is not actually broken: it has just been temporarily lifted into the fourth dimension and returned, like the breaks in the drawing above.

The Klein Bottle is a classic four-dimensional mathematical object. There are various models used to help us visualize it. The most common one looks a lot like a vase with a bent “nose” that seems to intersect itself. Here is an image of some beautiful glass Klein bottles that you can buy from Cliff Stoll at Acme Klein Bottle. I highly recommend checking out his website!

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The deficit of these models is that they intersect themselves (as they must in three dimensions), while the actual four-dimensional object has no self-intersections. These models are analogous to drawing the twisted loop pictured above as a figure-8 curve, without lifting your pen.

To create a representation of the Klein Bottle with no self-intersections, I used a different model. Another way to think about the Klein Bottle is to sweep a figure-8 (with a break at the crossing to avoid the self-intersection) around a circle, while rotating 180 degrees. It’s hard to visualize, because this can only be done in four dimensions. If we try to construct it in three dimensions, it will intersect itself. However, by representing a skeleton of the model, and making the wires inter-weave, one can avoid self-intersections. Here’s a Rhino render of my design (modeled with Grasshopper):

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Here’s the 3D printed version, available at Shapeways in small and large versions.

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In the close-up on the right you can see how the individual wires that define the model are interwoven, so that there is no place in the model where it actually intersects itself.

This model got me thinking about making things with more traditional fabrication techniques. At the same time I had an undergraduate thesis student, Joi Chu-Ketterer, who wanted to explore knitting and crocheting  mathematical surfaces. This kind of thing has been done a lot for hyperbolic surfaces. See, for example, the magnificent coral reef of the Institute for Figuring. I’d also seen many knitted tori and spheres. Cliff Stoll has some neat knitted Klein bottle hats on his website, but I had never seen anyone try to knit the figure-8 Klein bottle. The nice thing about yarn is that you can interweave the strands, to achieve the same kind of avoidance of self-intersections. It’s a painstaking process to thread the strands through each other while knitting them, but Joi did it!

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Knitting and Printing the Figure-8 Klein Bottle

Modelling Seashells

Anyone who has ever seen my work should have expected that sooner or later I would write a post (or two or three) about seashells….

There are many places in nature where there appear to be mathematical laws at play. For example, pinecones exhibit the mathematical phenomenon of phyllotaxis, tree branching can be modeled with various fractal-generating techniques (e.g. Iterated Function Systems),  and tiger stripes are apparently governed by reaction-diffusion equations. Personally, I find the most beautiful example of apparent mathematics in nature is the way seashells follow certain kinds of curves called “logarithmic spirals.”

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The key to understanding logarithmic spirals lies in the fact that after each turn of the spiral,  everything shrinks by a certain fixed factor. (I’ll assume this factor is 1/2 here). So, for example, if the radius of the first turn is 100, then the radius of the second turn is half that, or 50. The third turn has a radius that is half that again (i.e. 25), etc. The heights of each turn of the spiral obey the same law: if the height of the first is 100, then the second will be at height 50, the third at height 25, etc. Because the radius and height are always in constant proportion, the resulting curve lies on a cone.

Using some basic techniques from an area of mathematics called Differential Geometry, we can find three special vectors at each point of the spiral, called the Tangent (T), Normal (N), and Binormal (B).  (Conveniently, Grasshopper has pre-programmed algorithms to find these vectors for us.)

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Together, N and B define a plane that is perpendicular to the curve:

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To make a basic seashell shape, we simply place circles in these planes. The important part is that the radius of these circles must obey the same scaling law. So if the radius of the first circle is 16, then the radius of the circle that appears after one full turn will be 8, the radius will be 4 after two turns, etc.

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To make things look more realistic, one can use curves other than circles. In the following images (top view on the left, side view on the right), I used two different hand-drawn curves. They’re similar, but one of them has a slight bump. I then placed these curves (always respecting the same scaling rule) around the spiral, alternating so that there were two copies of the smooth curve for each copy of the bumpy one…

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…and made a surface out of them.

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Making this object 3D-printable was the next challenge. And it turned out to be a huge one! First of all, you can’t 3D-print an object with infinitely thin walls. The walls must have some thickness. However, the standard thickening tool (the “Offset” function) doesn’t work on this design! This is because the diameter of the smallest curve, at the top of the seashell, is less than the amount you have to thicken to make the object printable.

To deal with this issue, I created a second seashell, inside the first, using only the smooth curves. The trick to making it all work was to make the inner shell have fewer turns than the outer one, so that it could be offset inwards. That left the smallest part of the seashell model solid, which is fine because it’s deep inside where no one can see.

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Finally, it was time to add some color. This was done by finding an image of an actual seashell texture on the internet, and wrapping that image separately onto the inner and outer surfaces.

Here’s a photograph of the final design, 3D-printed in full-color “sandstone” at Shapeways. You can order your own here!

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And yes, it does “work.” You really can hear the ocean in it.

Modelling Seashells