User:Ceoil/Tall Trees
This paper describes the geometrical studies underlying the design and manufacturing of Anish Kapoor’s Tall Tree and the Eye sculpture.
The use of digital form finding techniques with gravity simulation, explicit history tools, together with the study of sphere packing and curved-mirror reflections, allowed the development of a geometrical model that could adapt and change accordingly to the design and structural progress from the initial stage to the construction phase.
SYNOPSIS
[edit]The sculpture is the result of a collaboration between Arup and the artist Anish Kapoor, exhibited at London’s Royal Academy of Arts in 2009 and at Bilbao’s Guggenheim in 2010. It comprises 73 mirror-polished stainless steel spheres, stacked to a height of 14 metres, which creates the appearance of weightless floating bubbles rising into the sky. Each sphere has an average diameter of 1 metre and a wall thickness of just 1–2 mm – a fragility that presented one of the major challenges of the design process. For this reason the sculpture has required an inner structure of three carbonated steel masts, linked together by curved bracing elements and connected to a steel base frame at ground level. The hollow spheres have been fabricated with a high-pressure water technique and then were mirror-polished. When approached by the artist with a concept of stacking several mirror spheres, the Arup AGU (Advanced Geometry Unit) team examined and developed multiple methods and design options.
SPHERICAL PACKING
[edit]A first approach was analysing spherical packing theory in order to achieve an ideal design that would: ∙ Avoid visibility of connections between the spheres. ∙ Avoid visibility of any structural element. ∙ Present the minimum necessary quantity of structure. ∙ Be based on a simple construction sequence.
CHIARA TUFFANELLI, MULTI-SPHERICAL MIRRORED SCULPTURE ARUP
[edit]As soon as the design process began, the focus was to find the most efficient way of packing spherical objects and whether there is a rule that facilitates constructing and controlling tangencies between numerous spheres. The regular packing system would provide modularity that facilitates the design process, constructability and the fabrication process. The two most efficient regular packing systems where the highest density arrangement (0.74) can be achieved are: ∙ The hexagonal close packing (HCP) = where the sphere’s centres lie on the tetrahedron vertices’. ∙ And the face-centred cubic packing (FCC) = where the spheres centres are located on half octahedron vertices. For this reason placing the spheres centres on either an octahedral or tetrahedral packing configuration would establish an effective way of controlling tangencies between all spheres. But, at the same time, the regular packing system would give a far too regular arrangement strongly perceivable to the spectator. The irregular packing system, on the other hand, would provide the visual effect of a casual layout but would also require extra structural elements. This packing arrangement has been explored through the Reactor toolset in Autodesk 3ds Max, which enables simulation of complex physical scenes. The Havok’s physics technology, used in Reactor, provides a dynamic environment for the objects in the scene (for example assigning gravity force or collision power between rigid bodies once created in the 3D space). The results of these investigations on both regular and irregular packing characterised the sculpture’s final geometrical outline. It therefore combines the two systems, both for structural and aesthetical reasons.
REFLECTION ANALYSIS
[edit]Studies on the reflection properties of mirror spheres were carried out to enhance the control of the visual impact that the sculpture would have on future 1: The Tall Tree and the Eye sculpture in the Royal Academy of Arts courtyard, London. 2–4: Sierpinski and Apollonian gasket, a tetrahedron of rendered spheres and a photograph of the sculpture. 3 2 observers. Once basic convex mirror properties were explored, several 3D computer models were then developed in order to visualise the effect of reflection on multiple mirror-tangent spheres. Photo-realistic renderings provided images that were then verified on site as very close to reality. Issues such as how reflection is affected by the observer, how angles and distances between spheres affect the reflection or how the context will be distorted in the reflection, were the starting point for further considerations. For example, different angles, relative positions and diameters between several spheres can affect the result of the reflection. Far more interesting reflections can also be achieved when the spheres centres lie on a plane non-orthogonal to the viewer. Furthermore, an opening at the lower level of the sculpture was designed in order to allow the visitor to enter the sculpture and gaze up at the endless reflections as if into infinity.
HYPERBOLIC GEOMETRY
[edit]The research for the underlying principles behind mirrored reflection on multiple spheres led to the study of transformations in 2D space (such as inversions) and in the hyperbolic plane (such as Möbius transformation). They explain, at best, the effect of reflection on the spherical mirror surfaces. What happens in 2D inversion with respect to a circle, occurs in the 3D space with a set of tangent spheres under inversion to each sphere. For this reason the multiple-tangent spheres of equal diameter will produce a fractal reflection as shown in the computer-rendered image as well in the photograph of the sculpture in figures 2–4. This fractal pattern would then become increasingly complex and rich when people and surrounding buildings are introduced into the space and when packing is carried out in an irregular form.
THE DIALOGUE BETWEEN STRUCTURE & GEOMETRY
[edit]The form-finding process of the conceptual design had to integrate aesthetic requirements together with geometric and structural ones. If a 3D layout succeeded aesthetically it would then be exported into a structural analysis software (Oasys GSA Analysis) to verify the local and global stability of the sculpture. These results would then be used as feedback for further changes to the 3D model that, in a back-and-forth process, would generate further considerations. SPHERE TYPES To provide an insight into the geometrical attributes of the Tall Tree and the Eye sculpture a classification of 5–6: Although every single sphere weighs approximately 45kg, the result of the design process has been a stable and yet light structure appearing to be almost weightless. This effect has been achieved by means of an inner structure of three carbonated steel masts linked together by curved bracing elements and connected to a steel base frame at ground. 7: The Tall Tree and the Eye arrives on site. The protective layer is peeled away, the first reflections appear.
6 5 the spheres is necessary. They are all characterised by a different number of tangencies with each other and by the type of structure contained within: ∙ The mast spheres are the spheres that conceal the three carbonated steel masts and have at least two points of tangency with other spheres. ∙ The inner-bracing spheres are the spheres that enclose the curved-bracing structure (linked to the three masts) and have at least three points of tangency. ∙ The cantilever spheres are the spheres that will be supported by a curved cantilever structure connected to just one mast. ∙ The top-mast spheres are three extra mast spheres that have been inserted on site on the summit of the masts. ∙ The ground spheres are particular spheres that have been added at ground level for improved stability and enhanced reflections. Several geometrical constraints had to be established in order to fulfill visual and structural requirements. The first priority was to avoid visibility of the inner structure throughout the sculpture. This explains why all spheres listed above had to be tangent to either one, two, three or four others. What’s more, the diameter of the three masts had to be as small as possible. The three masts had to be located at an equal distance from the global point of origin, not exceeding 1 metre, so as to enable tangencies between the mast and the inner bracing spheres. Multiple alternatives for inserting and connecting the spheres on the structure were provided, always bearing in mind the geometrical constraints. It has then been a crucial task selecting which mast spheres would be connected to the inner-bracing spheres, because their location would determine the position and shape of the curved structural bracing and therefore the structural performance of the entire sculpture.
PARAMETERS
[edit]It was necessary to assign parameters that could completely define the sculpture’s geometry and would enable control of any changes required in an efficient way. Once defined, a 3D model that could adapt and change quickly, according to the design and structural progress, could be achieved either through implicit or explicit history tools that are tightly integrated with Rhino’s 3D modelling tool. The best solution for this project has been provided by the explicit history tool Grasshopper (a graphical algorithm editor) where parameters could be assigned and linked through several components. The best characteristic of this tool has been the fact that, in contrast to the implicit history tools, it could provide an immediate visual feedback and full control of each single component and stage of the process created by the user. Whenever one of the parameters changes, the whole model consequently adjusts to suit the initial requirements. In this way any geometrical variation required either for aesthetic or structural reasons could be rapidly exported into analysis models for structural tests. Finally, once the geometrical model was built and finalised, the polar coordinates of each single sphere was exported into a spreadsheet in order to enable the fabricator to control and rebuild the 3D model with any software. The process taken to create this soaring installation combining art, geometry, architecture and engineering, demonstrates a novel computational approach to exploring, modifying and exchanging design information 7 between artist, architect, engineer and fabricator.