Programmable matter, origami and the schoolgirl who solved the paper folding problem

Image Credit: d-Vision

Programmable matter is a material whose properties can be programmed to achieve specific shapes or stiffnesses upon command. This concept requires constituent elements to interact and rearrange intelligently in order to meet the goal. This paper considers achieving programmable sheets that can form themselves in different shapes autonomously by folding. Past approaches to creating transforming machines have been limited by the small feature sizes, the large number of components, and the associated complexity of communication among the units. We seek to mitigate these difficulties through the unique concept of self-folding origami with universal crease patterns. This approach exploits a single sheet composed of interconnected triangular sections. The sheet is able to fold into a set of predetermined shapes using embedded actuation. To implement this self-folding origami concept, we have developed a scalable end-to-end planning and fabrication process. Given a set of desired objects, the system computes an optimized design for a single sheet and multiple controllers to achieve each of the desired objects. The material, called programmable matter by folding, is an example of a system capable of achieving multiple shapes for multiple functions

Hawkes et al (2010), Abstract of paper entitled “”Programmable matter by folding”, PNAS.

Where would we be without paper? Sometimes we remember a tree and we recycle it. Sometimes we create a work of art so great that we want to preserve it. Usually we just scrawl and scrunch without thinking. But, some eco-friendly families use fridge magnets to leave messages. Others have a mini-whiteboard next to the phone. What about you? And then there’s Etch-a-Sketch and Magic Doodle that let you scribble away again and again and again. A few years back I read a copy of futurist and theoretical physicist Michio Kaku‘s 1999 book Visions: How Science Will Revolutionize the 21st Century and Beyond. Unlike many such books, what stood out was Michio’s approach. He researched and systematically-visited leading researchers around the world working on technological prototypes and patents. Smart technology then, was to be a product of the near-future. To be followed by a paperless society in the run-up to 2020. Now that it’s here and more and more homes feature digital photo-frames, we seem to be accelerating along the timeline of the future.

So, imagine my surprise this week as I read a new article in Scientific American entitled, Shifty Science: Programmable Matter Takes Shape with Self-Folding Origami Sheets. It reports on the latest success by researchers at Harvard University and the Massachusetts Institute of Technology (M.I.T.) who have invented the first real-life transformer, a device that can fold itself into two shapes on command. Wow. You program in the set of origami paper-folding instructions and flap, flap, flap… the object takes form in front of your eyes. If you think this is spooky, imagine that the same programmable matter is capable of folding itself into the shape of a 3D rendered image it is provided with! It may not be up to beating Origami experts in the optimum fold sequences just yet, but with advances in genetic algorithms, I doubt this will take long.

Image Credit: Scientific American

Already there have been significant advances in the mathematics of paper folding thanks to the efforts of origami theorists like NASA’s Dr Robert Lang and Professor Tom Hull. For an introduction, you could try reading this or you can make your own amazing shapes by following Tom’s instructions in the videos at his YouTube channel. Maybe you have seen some origami animals like frogs or cranes before. But what about origami rhinoceroses, dragons or tigers? Check these out! But, before you start having nightmares about your kids being woken up by a programmable matter dinosaur, perhaps we should ask what are the limitations? After all, we all remember the school-yard trick where you challenge sometime to fold a piece of paper in half 8 times and they cant! Or can they?

It used to be common knowledge that if you could fold a piece of without restriction, the height of a piece of folded paper would double in thickness each time it was folded. According to Wolfram’s Mathworld, “since one sheet of typical 20-pound paper has a thickness of about 0.1 millimeter, folding 50 times (if this were physically possible, which of course it is not) would produce a wad of height 1.13×10^11 meters, and folding one more time would make the stack higher than the distance between the Earth and Sun”. So it was until December, 2001.

Then along came a schoolgirl called Britney Gallivan of Pomona, California. She was told she would get an extra maths credit, if she took up the option of solving the problem of folding a sheet in half 12 times. After extensive experimentation, she folded a sheet of gold foil 12 times, breaking all existing folding records! And she did this using alternate directions of folding. But, the challenge was then redefined to fold a piece of paper 12 times. Impossible? She studied the problem and was the first person to realize the basic cause for the limits to the folding process. She derived folding limit equations for  folding in alternate directions and for the case of folding in a single direction using a long strip of paper. Her formula told her that to successfully fold paper 12 times, she would need about 1.2 km of paper.

After some searching she found a jumbo roll of toilet paper that would suit her needs. It was loooooooooooooog and cost a whopping $85. In January 2002, she went to the local shopping mall in Pomona with her parents, rolled out the jumbo toilet roll, marked the halfway point 600m along and folded. Then she marked and folded the paper a second time, and then again and again. After seven hours, she folded her paper for the 11th time into a skinny slab, about 80 cm wide and 40 cm high, and posed for photos. Making her grade, she then folded it one last time and wrote up her assignment in a 40 page pamphlet for the Historical Society of Pomona: “How to Fold Paper in Half Twelve Times: An Impossible Challenge  Solved and Explained“. Well done Britney. And, thanks for showing us that if we truly think then only the sky is the limit. You are an inspiration to all mathematicians.

Image Credit: Wolfram Research

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