The mathematical magic of bending grids

The step into the third dimension

Suppose you screw ordinary straight bars together at right angles to form a grid, so that a completely regular pattern of small squares is created. Such a grid can be distorted: all angles of the grid change simultaneously, parallel bars remain parallel, and the squares become parallelograms. But this does not change the fact that all bars are in the same plane. The structure is still flat.

The crucial question now is: What happens if the bars are not parallel at the beginning, but are joined together at different angles? "Such a grid can no longer be distorted within the plane," explains Przemyslaw Musialski. "When you open it up, the bars have to bend. They move out of the plane into the third dimension and form a curved shape."

At the Center for Geometry and Computational Design (GCD) (Institute for Discrete Mathematics and Geometry) at TU Wien, Musialski and his team developed a method that can be used to calculate what the flat, two-dimensional grid must look like in order to produce exactly the desired three-dimensional shape when it is unfolded. "Our method is based on findings in differential geometry, it is relatively simple and does not require computationally intensive simulations," says Stefan Pillwein, first author of the current publication, which was presented at the SIGGRAPH conference and published in the journal ACM Transactions on Graphics.

Experiments with the laser scanner

The team then tried out the mathematical methods in practice: The calculated grids were made of wood, screwed together and unfolded. The resulting 3D shapes were then measured with a laser scanner. This proved that the resulting 3D structures did indeed correspond excellently to the calculated shapes.

Now even a mini pavilion roof was produced; measuring 3.1 x 2.1 x 0.9 metres. "We wanted to know whether this technology would also work on a large scale -- and it worked out perfectly," says Stefan Pillwein.

"Transforming a simple 2D grid into a 3D form with a single opening movement not only looks amazing, it has many technical advantages," says Przemyslaw Musialski. "Such grids are simple and inexpensive to manufacture, they are easy to transport and set up. Our method makes it possible to create even sophisticated shapes, not just simple domes."

The structures also have very good static properties: "The curved elements are under tension and have a natural structural stability -- in architecture this is called active bending," explains Musialski. Very large distances can be spanned with very thin rods. This is ideal for architectural applications.