Carlo Borer's Sculpture '422' Completed

Pictures (c) by Carlo Borer.

In previous posts I reported about the design and fabrication process of this piece of art.

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.

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.

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

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.

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.

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.

Exploring the PQ Strip

Helmut Pottmann and his colleagues form the Geometric Modeling and Industrial Geometry resarch unit at TU Vienna describe the concept of the PQ strip in this paper. The strip consists of planar quad faces. Fig. (c) by H. Pottmann et al.



By lofting the edges, a developable surface is generated. Developable surfaces can be flattened or unrolled without distortion.



Download a grasshopper definition of the PQ strip here.

Kusner Minimal Surface

Discovered by Rob Kusner, Director of the Center for Geometry, Analysis, Numerics and Graphics (GANG). The Kusner surface is an inversion of the Boy surface. The three discs are planes "reaching infinity".

Le Topologicon

Le Topologicon is a great comic book about topology. The author is Jean-Pierre Petit who discovered the first parameterization of the boy surface in 1981. He is also the co-founder of the non-profit organization Savoir-sans-frontiers. The book deals with moebius strips, klein bottles, boy surfaces, magnetic fields, surface stiching and a lot more interesting topics. In my opinion, the Topologicon is a must-read for everyone dealing with surface modelling, mesh modelling, subdivision surfaces, surface tesselation, vector fields, deformations, or topological space. Some example pages:

Homotopy of Boy and Roman Surface

In topology, a smooth deformation from one surface in an other is called homotopy. The parameterization of this homotopy can be found on wolfram's page.

The grasshopper definition, download here.

This figures from Werner Boy's paper (1903) show the topological construction of the boy surface.

A proposal for a pavillion by Christophe Delsart and Yvan Ngnodjom based on the boy surface for the ARPAM project.

Boy Surface

The Boy Surface has been found 1901 by Werner Boy. One obtains a topologicaly equivalent surface by attaching a Möbius strip to a disk along its boundary. Like the Möbius strip this surface is nonorientable.




This beautiful parameterization, based on complex numbers, was discovered in the late 1980s by Rob Kusner and Robert Bryant. In this post the boy surface has already been mentioned on eat-a-bug.



Download the definition here

Cairo Tiling

This pentagonal tesselationis called cairo tiling.

3D2REAL: A wall based on the cairo tesselation, designed and build by ILEK students.

Extension of the congress center Davos, currently under construction. The roof structure is based on the cairo tesselation. By Degelo Architekten, Engineer: Dr. Schwartz Consulting

The cairo pattern seems to be a blend of a diamond pattern and a chinese pattern.

Download the definition here.

Read Colors From File

The RGB chanels ploted as surface

The input image

This VB script component reads the RGB values from a jpeg file
Download the grasshopper definition ColorSurface.zip here

Formula Tower

At digital matters, Tobias Wallisser from LAVA presented his Superformula Tower concept. Inspired by this beautiful idea, I created a grasshopper definition for towers based on a simpler formula.

The floor plate shapes are based on this formula:

(in polar coordinates)

r(φ) = 1 - a(1 + Sin(φ b))

FormulaTower.zip