# What math do architects use

## Math and architecture : Free forms in the discrete network

The idea of the avant-garde in architecture changes from time to time. While in the 1920s she was concerned with social aspects and in the 1960s with technical utopias, today architects develop forms in their imagination that were never seen before and that were previously thought to be impossible to build. Here and there such ambitious structures were built, mostly with considerable effort. One thinks of Gehry's spectacular museum in Bilbao, Ben van Berkel's Mercedes Museum in Stuttgart or the strange bubble-shaped Yas Island Hotel in Abu Dhabi above the Asymptote Architecture Formula 1 racetrack, which shimmers in colorful LED light at night.

Designing and producing such apparently freely curved “free forms” is only possible with the latest computer technology. The architects call the creation process “parametric design”: the round shapes are geometrically broken down into small, flat parts, into triangles, squares or polygons. The smaller the parts, the closer you get to the ideal curved shape, but the challenge is to use large parts. In this way, curved surfaces can also be produced more cost-effectively. Parametric means that you use changeable parameters in the spatial model to be calculated - such as the length of the individual rods, the size of the partial areas or the radius of curvature. If you change one of these parameters, the system immediately calculates all effects on the overall design.

### Material consumption and production costs can be optimized

This is where the mathematicians at the Technical University of Berlin (TU) come into play with their research project "Freeform Architecture and Mathematics". Alexander I. Bobenko and Thilo Rörig solve these extremely complex problems with their discrete differential geometry. Your goal is to develop the ideal, "mathematically solid" model possible, in which, for example, quadrilaterals fit together at correct angles to form a curved overall shape.

The striving is not only mathematically and scientifically motivated, because such a geometrically perfected construct "looks beautiful", as Professor Alexander Bobenko shows with the design of a hall roof for a shopping center, which consists of a flat arched lattice structure. But architects, engineers and building owners also pursue certain goals: namely to optimize the construction with regard to material consumption and production costs.

It is true that freely curved shapes or "blobs" - as the bubble-like structures that emerged two decades ago are called - can be produced with a certain amount of effort, also because computers took over the control of the production machines and could, for example, produce hundreds of different facade panels for a building. But only the most recent mathematical models, as developed in the Collaborative Research Center DGD - Discretization in Geometry and Dynamics and at Matheon, enable more economical constructions: self-supporting, with serial elements, flat surfaces, torsion-free nodes and minimized profile thicknesses.

### The layman simply finds the structure beautiful

If you link the mathematical models with the programs of the structural engineer, the ideal result is an integrated draft that is designed and influenced by architects and engineers at the same time and ultimately even controls the production machines.

One application aspect is the building construction of free forms. In the art-in-building project “PBSA Tricolumn” at the Düsseldorf University of Applied Sciences, for example, a sculptural concrete column, Thilo Rörig calculated polyhedral parts of constant thickness and their complex scraps for the computer model, which - cut as formwork panels and put together to form the overall shape - the concrete cast made it possible in the first place.

However, the preparatory work for specific building projects is not the focus of the mathematicians at TU Berlin. Mostly they are looking for their own problems for which they research possible solutions. There is, however, a close collaboration with Helmut Pottmann from the Vienna University of Technology, who deals with geometric models for architectural applications and has worked on some of the buildings that have already been built. The result was the model of a hall with semicircular openings and a round roof opening, which consists of flat rectangles and whose construction thickness remains constant throughout. The layman simply finds the structure beautiful. Only the expert knows that this beauty is only possible through mathematics.

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