Advances in Free-Form Masonry

This model of a free-form compression-only vault has been build using rapid prototyping. It stands just by gravity and friction. It is part of a research project of the BLOCK Research Group at the ETH Zurich. Two publications describe details about the structural models and the design process.


Form and force diagrams in plane, the force distribution in the network, and the form of the network in 3d.
The form-finding process is based on TNA, a method developed by Philippe Block.




Pictures and movie (c) by BLOCK Research Group

New Evolutions in ZipShape

Christoph Schindler, co-founder of schindlersalmerón, develops the principle of ZipShape since 2007, in order to build curved elements from cnc-manufactured plain material without molds.


An element consists of two differently slotted panels that only interlock if bent to the desired curvature.


A grasshopper definition worked out during a workshop at the CITA Copenhagen shows the dependency between slot geometry and global curvature.


The newest development allows the fabrication of developable twisted surfaces.


This principle has been applied to a sculpture produced and tested at the Bern University of Applied Sciences in Biel.

Twisted Beams on Surface

Twisted beams with custom cross-section curve aligned normal to a given surface.


Download the gh-definition here.

PQ Mesh Design: Morpharchitecture Workshop - Lyon 2010

The gh definition presented here has been used for the design of a shell structure build from sheet material at Les Grandes Ateliers, Lyon, by architecture students of Ecole nationale supérieure d'architecture de montpellier and the TU Delft.



The pq design tool ensured the planarity of the quadrilateral faces, that allowed the fabrication of the shell from flat sheet material.


Photo by Robert Vierlinger.

Developable Surface Optimization

The artist Carlo Borer asked me to implement tailored developable surface modelling tools for rhino.


Using Grasshopper together with the evolutionary algorithm solver Galapagos, an optimization routine for lofted surface strips towards developable surfaces has been implemented.


Gaussian curvature of original and optimized surface. For developable surfaces, the gaussian curvature is zero.


Carlo Borer: 401, 2009.

Benoît Mandelbrot has died

Benoît Mandelbrot, father of fractal geometry and chaos theory, has died last thursday. The idea of self-similarity in nature and mathematics is described in his great book The Fractal Geometry of Nature.


The picture above shows the Mandelbrot set, a set of points in the complex plane, the border of the set is a fractal.


Not only coastlines, fern, and the vegatable romanesco exhibit self-similarity, but also stock price charts. Mandelbrot describes a fascinating, unorthodox view on finacial theory in his book The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin and Reward.


Mandelbrot's TEDTalk from 2010.

Architecture and Governmental Violence

Stuttgart 21, the project for a new train station in Stuttgart, Germany, started with a winning competition entry by Christoph Ingenhoven and Partner, Frei Otto, Büro Happold, Leonhardt and Andrae, in the year 2000. The picture shows Frei Otto's model used for form-finding of the roof. The main functional idea behind the project is the transformation of the existing dead-end station into an underground through station.

A strong opposition against the expensive billion-euro project has been formed by the citzens of Stuttgart.

Frei Otto is now a strong opponent to the project, his position can be found in Stuttgarter Zeitung (in German). A strong supporter of the project is Werner Sobek, Professor at TU Stuttgart and successor to Frei Otto. His statment to the project can be found in Stuttgarter Nachrichten (in German).

On last Thursday, September 30, the police used water canons and tear gas against peaceful protesters, who tried to protect the trees in the park against cut down. BBC and The Guardian reported about this incident. Since chancellor Angela Merkel strongly supports the Stuttgart 21 project, it seems to become a hot election issue, even in national scale.

Interactive PQ Mesh by Daniel Piker

This great piece of work is based on Kangaroo, a physics engine for grasshopper.The original video and a lot of more great stuff can be found on Danie Pikerl's vimeo account. The concept of circular and conical meshes is described in a paper written by H. Pottmann and J. Wallner.

Planar Quad Mesh Design Pt. 2

Some examples:




The principles of the tool are:
- all nodes are located on a ray through the point in the direction of the normal
- the distance from input surface to pq mesh is adjustable
- two of the edges are controlled by the graph-mapper

Planar Quad Mesh Design Pt. 1

A parametric tool for the design of planar quadliteral meshes based on uv-grid and normal directions of a design-driving NURBS surface.

The glazed freeform grid shell of the hippo house in the berlin zoo, has been designed by Schlaich, Bergerman + Partner. It consists of planar quadliterals, the geometry is based on a translational surface.

In this paper, a comprehensive discussion about geometric aspects of surface tesselation, by H.Pottmann et. al., can be found. A direct parametric approach, developed by Gehry Partners and Schlaich, Bergemann & Partners is described in this paper.

Download the definition for the planar quad mesh tool here.

Advances in Architectural Geometry 2010

Since Friday, the proceedings volume of the Advances in Architectural Geometry Conference 2010 is available. The book is published by Springer Verlag.

The Bubble Room

An underwater dome, shaped by the buoyancy force of an oxygen bubble. This beautiful piece of subaqueous architecture has been build by Jordan Needham and his familiy at the bottom of a Nevada mountain lake. Found on the boingboing blog.


The pictures are screenshots from the movie below.

Creative Data Symposium

On 24.8.2010, I will give a lecture titled The Geometry of Forces at the creative data symposium at CITA, Royal Danish Academy of Fine Arts.

Parametric Paperstrip

This parametric model represents the geometry of a twisted straight paperstrip. Length, width, and twist are adjustable.


Some example strips. By definition these surfaces are developable.


The paperstrip as generative element. (c) by Michael Hensel and Achim Menges, Arch+ 188 p.20.


Download the definition here. Another interesting approach has been done by Daniel Piker.

Finds from the Seaside

On a trip to the seaside near Istanbul, Turkey I found some beautiful biological examples.

A nice example of an algae, ulva lactuca.


An eroded rabbit's skull from behind.


A wing of an insect, probably a cricket.

Geometry of Structural Form

I am glad to announce that the paper "Geometry of Structural Form" for the conference Advances in Architectural Geometry 2010 has been accepted. My recent research, done at the the ETH Chair of Structural Design, will be presented. Techniques for parametric structural design, based on graphic statics, are described in this paper.




This grasshopper definition has been inspired by Nicholas Grimshaw's Waterloo Station, London.


(c) Grimshaw Architects

Parametric Vault

Based on TNA, I implemented together with Philippe Block a grasshopper definition for the generation of structural freeform vaults. The image shows a structural surface with an non-uniform force distribution. This work is part of a resarch project done at the ETH Zurich.


This model of a compression-only vault has been build using rapid prototyping. It stands just by gravity and friction.


Implementation-wise the matrix computation routines from rhino SDK have been used inside grasshopper vb.net nodes. Download the grasshopper definition here. A tutorial video shows the use of the definition.

Thrust Network Analysis

Thrust Network Analysis, a new method for generating compression-only vaulted surfaces and networks was developed by Philippe Block as part of his PhD. Using projective geometry, duality theory and linear optimization, it provides a graphical and intuitive method, adopting the same advantages of techniques such as graphic statics, but offering a viable extension to fully three-dimensional problems.


Form and force diagram, and the resulting compression only network.


Parametric explorations of the indeterminacy of compression-only networks by the modification of the force diagram.



TNA has been used for the structural design of the Mapungubwe National Park Interpretive Centre by Architect Peter Rich.

Interactive Graphic Statics

Some nice examples of processing-based interactive graphic static applets have been developed at the MIT. Interactive Thrust is a project started by Philippe Block under supervision of John Ochsendorf.



A truss geometry and its force diagram by Luigi Cremona.


Another great interactiv webpage is Active Statics, developed by Simon Greenwold, Ed Allen and Waclaw Zalewski.

Graphic Statics

The french mathematician Pierre Varignon (1654-1722) introduced the funicular polygon and the polygon of forces in his work Nouvelle Mecanique ou Statique. He describes a way to construct the form of a hanging rope with attached weights graphicaly. Based on this principle, a technique called graphic statics has been developed in the 19th century. The basic principle of graphic statics is the reciprocal relation between force polygon and funicular polygon. Recomended books: Luigi Cremona: Graphical Calculus (1890) and Edward Allen,Waclaw Zalewski: Form and Forces (2009)

This grasshopper definition generates a funicular polygon for uniform loading. Download here.

The Honeycomb Vase

Honeycomb Vase, "Made by Bees", by Studio Libertiny.
Seen in the MoMA.

Rapid manufacturing is a process by which computer-controlled lasers solidify liquid or powdered resin to create a three-dimensional rendering of a digital design. By contrast, the Honeycomb Vase was produced through what the artist has called "slow manufacturing": Libertíny constructed a vase-shaped beehive scaffold (to be removed at the end of the process) and then let nature take its course. Forty thousand bees build the vase in one week.

Honeycomb like structure build by humans and machines has been studied by Andrew Kudless and by ILEK students.

Parametric Structural Design Lecture at MIT

In the 19th century, graphic statics, a geometric method for the analysis of the force distribution in structures, has been developed. This scientific, precise method has an intuitive and descriptive character based on reciprocal force diagrams. The recent emergence of parametric design techniques in architecture enables the construction of associative geometry in CAD systems. In this lecture, a novel approach will be presented, combining parametric tools with historic techniques of graphic statics. This highly interactive three-dimensional procedure provides direct control of form and forces at the same time. The aim of this approach is the integration of the flow of forces into the computer aided design process.

John Ochsendorf kindly invited Lorenz Lachauer to present at the MIT Masonry Research Group on Monday April 12. Recent research from the BLOCK Research Group and the Chair of Structural Design will be presented.

Developable Surface Modelling

A grasshopper definition for developable surface strips. Download here. Mårten Nettelbladt gives a nice description of developable surface types.

Developable Surfaces in Gehry's Architecture

Gehry's east river guggenheim project in NYC. The use of flexible sheet material like paper in physical models generates developable geometries.

Dennis Shelden, CTO at gt, describes in his phd thesis several ways to compute developable surfaces. Fig. (c) by Dennis Shelden.


Museum MARTa in Herford, Germany. Most of the roof surfaces are developable. Geometric modelling of the roof construction has been done by Jess Maertterer. Further discription on his page.