The Math of a Crumpled Piece of Paper Is Insanely Important.

Your balled-up receipt contains multitudes. 

  • Mathematicians at Harvard University recently discovered paper crumples in some surprisingly predictable patterns. 
  • When you crumple the paper, creases appear to relieve stress in the sheet. 
  • It turns out the total length of these creases accrues logarithmically as you repeatedly compact and unfold the sheet of paper.

If you’ve ever uncrumpled a crumpled piece of paper, you may have noticed the creases form in geometric patterns. But you probably didn’t! It’s a crumpled piece of paper!

Well, next time, look closer. Because it turns out there’s some beautiful math in those chaotic creases.

As you scrunch up a sheet of paper, it creates stresses in the sheet. To relieve that stress, the paper breaks up into a series of flat facets, each separated by raised ridges. When you open the sheet of paper back up, the landscape of geometric creases appears disordered at first glance.

Mathematicians at Harvard University recently uncovered something wholly unexpected about these wads of balled-up paper. Their experiments in Nature Communications show the total length of each crease in the scrunched-up paper increases logarithmically as you compact and unfold the sheet over and over again—that is, the crumpling process follows a general, predictable rule.


From conducting previous research, the mathematicians knew “the logarithmic growth of crease length was a robust phenomenon that could be reproduced over and over again but we were missing the physical basis for why this happens,” Harvard’s Jovana Andrejevic, the lead author of the new paper, said in a prepared statement.

To uncover the physical explanation for why paper crumples in this unexpectedly predictable fashion, Andrejevic and her team focused on the flat sections (which they call “facets”) of a crumpled sheet. Specifically, they studied how the facets in crumpled sheets of Mylar—a type of shiny, polyester film used in NASA spacesuits to protect astronauts from radiation—break up into increasingly smaller fragments as the surfaces are scrunched up again and again.

That quickly became something of a Herculean task. The network of ridges in a crinkled piece of paper are irregular, making it difficult to define the areas of the facets that they define in the sheet. As a result, the crease data is muddied and confusing, making it hard for automated computer algorithms to accurately interpret.

Harvard’s Chris H. Rycroft, Ph.D., one of the paper’s coauthors, told The New York Times that Andrejevic, who is also an artist, solved this problem “in spectacular style, by tracing the facets all out by hand.”

Using Adobe Illustrator and Photoshop, Andrejevic hand-traced the patterns in 24 sheets. It took her hours, and sometimes days, to complete this process. On average, the 4×4-inch squares of Mylar each possessed 880 facets after a few rounds of crumpling, but one had a staggering 3,810 facets.

The resulting images are stunning, not dissimilar to the abstract art you might otherwise see hanging on the walls at the Met. The facets in the crumpled sheets of paper sort of resemble the countries on colorized world maps.

“If you look at how these regions evolve as the paper is crumpled again and again, you see that the larger regions break up into several smaller regions, just like how a pebble on the beach will break up into smaller pieces over time,” Rycroft said in the prepared statement. 

Physicists already have a theory that explains how rocks break down into smaller pieces to become sand, for instance. This is known as fragmentation theory. By applying the mathematics of this theory, Andrejevic and her team found it aligned with their own data and could predict the logarithmic scaling they noticed in sheets of Mylar that were crumpled over and over. 

The researchers crumpled one sheet up to 70 times, but noticed after a few scrunching sessions, it was hard to discern much difference between the creases. Upon analysis, they found although a sheet never actually stops forming new creases, it does so at a logarithmic rate. That means with more and more crumples, there are fewer and fewer new creases. 

Even though we’re talking about the dynamics of squished paper, this work isn’t trivial; there are practical reasons to study the way in which paper collapses under stress. You can observe something similar in the macroscopic folding of the Earth’s crust, and in the microscopic crimping of graphene membranes used in high-performance batteries and supercapacitors (the scrunched structures add more surface area for electrochemical reactions).

Uncovering the mechanics of crumpling is crucial for the future design of thin, wearable devices like smart watches. That’s because engineers need to use increasingly smaller, more flexible batteries to shrink down the overall devices. 

So the next time you crunch up some junk mail or a gum wrapper, just know it contains multitudes.

Source: Popular Mechanics