The Math Of Crumpled Paper.

As a teacher, I see lots of crumpled paper ending up in the trash can.  Occasionally, I've used crumpled paper in an activity to help students practice their math but I didn't realize that someone had explored the mathematics behind crumpled paper. 

Some mathematicians at Harvard discovered that the lines in crumpled paper follows certain predictable patterns.  When you crumple a piece of paper, it creates stresses in the paper.  To relieve the stress, the paper breaks into a series of flat facets and each face is separated by ridges that are raised.

If you open the crumpled piece of paper, facets and ridges appear disordered but the mathematicians discovered that the total length of the paper increases logarithmically each time your unfold and recrumple the paper again and again.  The mathematicians knew that this process could be repeated multipole times with the same results but they didn't know how it happened.

They used sheets of Mylar which is a type of shiny, polyester film used by NASA in their spacesuits.  They studied how the flat areas in the Mylar broke up into smaller and smaller pieces as they crumpled, uncrumpled, and crumpled the sheets again and again. Unfortunately, the networks of ridges are irregular making it difficult to to define the flat area or facets in the sheet.  So one of the mathematicians hand traced every facet out by hand using Adobe Illustrator and Photoshop.

The person hand traced the flat areas for 24 (4 inch by 4 inch) sheets of Mylar.  Each sheet could take between hours and days to finish. On average, the sheet had 880 facets after a few rounds of crumpling but one had over 3,800 facets. Many of the sheets when completed looked like abstract art hung in a museum or like the colored world maps you find in many classrooms.

As the paper is crumpled again and again, the larger regions break down into smaller regions much like a pebble on the beach breaks down. Physists have a theory called fragmentation theory that explains why rocks break down into smaller pieces and the mathematicians discovered their data on crumpled papers fit that theory.  This allowed them to predict the logarithmic scaling they'd observed in sheets of Mylar.

During their research, the took one sheet of Mylar and crumpled it up 70 times.  They noticed that after a few times of scrunching it, they were unable to discern new ridges but upon closer observation, they noted that the sheet never stops forming the ridges but the ridges increase at a logarithmic rate.  

I can just hear a couple of my students giving me the "So!"! in response to this.  This discovery has applications in other fields such as the macroscopic folding in the Earth's crust, the microscopic crimping of graphene membranes found in high end batteries and superconductors, and the future design of small thin smartwatches. 

I love reading about things like this since it reinforces the idea that math explains the world.  Let me know what you think, I'd love to hear.  Have a great day.