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The mathematical magic of bending grids

Date:
August 24, 2020
Source:
Vienna University of Technology
Summary:
A mathematical discovery opens up new possibilities for architecture and design: For any desired curved surface a flat grid of straight bars can be calculated that can be folded out to the desired curved structure. The result is a stable form that can even carry loads.
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How can you turn something flat into something three-dimensional? In architecture and design this question often plays an important role. A team of mathematicians from TU Wien (Vienna) has now presented a technique that solves this problem in an amazingly simple way: You choose any curved surface and from its shape you can calculate a flat grid of straight bars that can be folded out to the desired curved structure with a single movement. The result is a stable form that can even carry loads due to its mechanical tension.

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.


Story Source:

Materials provided by Vienna University of Technology. Note: Content may be edited for style and length.


Journal Reference:

  1. Stefan Pillwein, Kurt Leimer, Michael Birsak, Przemyslaw Musialski. On elastic geodesic grids and their planar to spatial deployment. ACM Transactions on Graphics, 2020; 39 (4) DOI: 10.1145/3386569.3392490

Cite This Page:

Vienna University of Technology. "The mathematical magic of bending grids." ScienceDaily. ScienceDaily, 24 August 2020. <www.sciencedaily.com/releases/2020/08/200824110127.htm>.
Vienna University of Technology. (2020, August 24). The mathematical magic of bending grids. ScienceDaily. Retrieved November 14, 2024 from www.sciencedaily.com/releases/2020/08/200824110127.htm
Vienna University of Technology. "The mathematical magic of bending grids." ScienceDaily. www.sciencedaily.com/releases/2020/08/200824110127.htm (accessed November 14, 2024).

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