Krauskopf, B

Osinga, HM

Storch, Benjamin

English

The steel sculpture Manifold consists of an 8 cm wide closed band of stainless steel that winds around in an intricate way, curving and coming very close to itself. It is based on a complicated mathematical surface, known as the Lorenz manifold, which has an important role in organising the chaotic dynamics of the well-known Lorenz equations. Namely, this surface consists of all points that, under the force field generated by the Lorenz equations, end up at the origin of the three-dimensional phase space. This is special because all other points go to the famous Lorenz butterfly attractor. The Lorenz manifold can be found and represented numerically by a set of smooth closed curves consisting of points that lie at the same geodesic distance (given by the length of the shortest path on the surface) from the origin. Any band between two such curves illustrates an aspect of the geometry of the surface. As is explained in this paper, the sculpture Manifold represents a choice of band that is motivated by aesthetic, practical and mathematical considerations. The goal was to create an element of dynamicism while only hinting at the underlying surface

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                          Krauskopf, B., Osinga, H. M., & Storch, B. (2008). The sculpture Manifold:a band from a surface, a surface from a band.Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.The sculpture Manifold:a band from a surface, a surface from a bandBernd Krauskopf and Hinke M. OsingaDepartment of Engineering MathematicsUniversity of Bristol, Queen’s BuildingBristol BS8 1TRUnited Kingdomb.krauskopf@bristol.ac.ukh.m.osinga@bristol.ac.ukBenjamin StorchBrynawelPlas Llwyn OwenLlanbrynmairPowys SY19 7BGUnited Kingdomben@benjaminstorch.co.ukDecember 2007AbstractThe steel sculpture Manifold consists of an 8 cm wide closed band of stainless steel that winds around in an intricateway, curving and coming very close to itself. It is based on a complicated mathematical surface, known as the Lorenzmanifold, which has an important role in organising the chaotic dynamics of the well-known Lorenz equations.Namely, this surface consists of all points that, under the force field generated by the Lorenz equations, end up atthe origin of the three-dimensional phase space. This is special because all other points go to the famous Lorenzbutterfly attractor. The Lorenz manifold can be found and represented numerically by a set of smooth closed curvesconsisting of points that lie at the same geodesic distance (given by the length of the shortest path on the surface)from the origin. Any band between two such curves illustrates an aspect of the geometry of the surface.As is explained in this paper, the sculpture Manifold represents a choice of band that is motivated by aesthetic,practical and mathematical considerations. The goal was to create an element of dynamicism while only hinting atthe underlying surface.1 IntroductionThe existence of the sculpture Manifold is a direct result of the 2006 Bridges Conference held in London,where the authors met for the first time. It was a meeting of minds between two mathematicians with acomplicated and aesthetically intruiging surface and an artist and metal craftsman who has been researchingthe creation of surfaces with negative curvature for several years.The development of numerical methods for the computation of complicated surfaces such as the Lorenzmanifold, introduced in more detail in Section 2, has been the research topic of Krauskopf and Osingafor quite a number of years. As it turns out, the result of their computational method translates directlyinto crochet instructions [9]. After mounting the crocheted work with some wires, a reasonably accuratethree-dimensional image of the Lorenz manifold can be obtained. This representation of the surface incrochet attracted considerable media attention and was presented at the 2006 Bridges Conference in London.Starting from a discussion of the curvature properties of the Lorenz manifold, the plan was born to producean artwork based on it in steel with the techniques perfected by Storch [12]. The idea was that this wouldallow for a better representation of the geometry of the surfaces, which is characterized by both positive andnegative curvature [11]. The challenge in such an endeavour is to balance aesthetic considerations with theconsiderable practical problems of staying close to the mathematically prescribed form of the surface.2 The Lorenz manifoldThe Lorenz manifold is a two-dimensional surface of initial conditions of the well-known Lorenz equa-tions [7], which are given as the differential equationsx˙ = σ(y− x),y˙ = ρx− y− xz,z˙ = xy−β z.(1)The right-hand side of the Lorenz equations (1) can be thought of as defining an instantaneous force of agiven magnitude and direction that depends on the position in the three-dimensional (x,y,z)-space. Given apoint particle at some initial position in (x,y,z)-space, the solution curve or trajectory through this point isthe path that the particle travels under the influence of the force field (1). Trajectories can be computed bynumerical integration. The Lorenz equations were introduced by Edward Lorenz in the 1960s as an exampleof a deterministic system that exhibits unpredictible chaotic dynamics [13]; see [3] for a popular account.For the classic parameter values σ = 10, ρ = 28, and β = 2 23 one finds the famous Lorenz butterflyattractor by integration from practically any initial position. The Lorenz manifold, on the other hand, consistsof all initial positions in (x,y,z)-space for which the particle ends up at the origin (x,y,z) = (0,0,0) instead.Mathematically, the origin is a saddle point with one unstable and two stable directions, and the Lorenzmanifold is its stable manifold. As such, dynamical systems theory guarantees that the Lorenz manifold is atwo-dimensional smooth surface. This surface is an important geometric object that helps one to understandchaos; see [1, 2, 8] for details.While the Lorenz manifold is well defined as a two-dimensional surface, there is not explicit or implicitequation that describes all the points on it (as is the case, for example, for a sphere). It is defined only‘indirectly’ by the property that it consists of trajectories that end up at the special point (x,y,z)= (0,0,0). Asa result, the Lorenz manifold can only be solved via numerical approximation. There are several techniquesfor doing this; we refer to [6] for an overview where the Lorenz manifold is used throughout to illustratethe differences between the methods. Our approach is based on the idea that one can build up the surfacefrom near the origin as a set of concentric smooth closed curves. The first such curve or ring is chosen asa small circle in the plane that is spanned by the two attracting directions; this choice amounts to a linearapproximation of the surface that can be found directly via a straightforward eigenvector computation. Anew ring is added to the collection at each step. It is found by determining the points that lie at a givensmall distance from and, under the force field, pass through the ring that was added in the previous step.In this way, the surface is grown with equal speed in all radial directions. Each computed ring correspondsto a so-called geodesic level set, that is, the arclength of the shortest path to the origin over the surface is(approximately) the same for all points on a ring. We refer to [4, 5] for the mathematical details of thismethod.Figure 1 shows the Lorenz manifold computed with our method up to the geodesic level set at distance140.75. The surface is rendered transparent except for a 20-units wide band of geodesic level sets that coversthe last computed rings. Two different view points are shown, where each time the z-axis is the vertical axis inthe picture. A first observation is that the Lorenz manifold is symmetric, that is, invariant under rotation over180◦ about the z-axis. This is a direct consequence of the respective symmetry of the Lorenz equations (1).In particular, the z-axis itself is invariant under the dynamics and it is part of the Lorenz manifold.3 From surface to sculptureFrom an aesthetic point of view, the Lorenz manifold is a new complex form with a fascinating geometry.We find its geometry particularly appealing because the smooth complex form gives a sense of motion anddynamics to the static object. While it is not immediately clear that the surface is related to point particles(a) (b)Figure 1: The Lorenz manifold computed as a collection of geodesic level sets. A darker outer band isshown on the transparently rendered surface. The two panels show two different view points obtained byrotation about the z-axis, which is the vertical axis in the images.flowing to the origin, the shape does conjure up an image of gracious movement and, as such, somewhatcaptures the underlying mathematical meaning. The complexity of its geometry, however, presents a seriouschallenge when one wants to turn the Lorenz manifold into a sculpture.The steel sculpture uses the Lorenz manifold as inspiration. Rather than creating its entire image, weexplored the idea of selecting only part of the surface as a representation of the entire geometric object. Theselected band on the surface shown in Figure 1 has the elemental aspect of a surface, while at the same timeit gives the impression of a one-dimensional curve with its own dynamics. The mathematical meaning ofthe band and its association with geodesic level sets provides added insight into the geometry of the Lorenzmanifold when the band is visualised together with the surface. However, without the underlying surface theband becomes an interesting object in its own right. It has such a complex geometry that it is not immediatelyobvious what the underlying surface actually looks like.Figure 2 shows the actual steel sculpture (left column) together with a computer rendering of the re-spective band on the Lorenz manifold (right column). We found it quite amazing how well the steel bandcaptures the shape of the entire two-dimensional surface, at least for someone who knows what the Lorenzmanifold looks like. While the small band communicates the dynamics of a one-dimensional curve, it alsoreveals important elements of the two-dimensional geometry. Overall, the sculpture Manifold captures thecomplexity of the Lorenz manifold in a subtle way by how it curves in the three-dimensional space. Notethat the band is closed, but not knotted or twisted: it is orientable, that is, topologically equivalent to a simpleannulus. In other words, the steel band has two sides, which are distinguished by their surface finish: oneside is highly polished and the other is wire brushed.(a1) (a2)(b1) (b2)Figure 2: The sculpture Manifold (left column) and its computer-generated equivalent (right column); thetwo view points are the same as in Figure 1.4 Creation of the steel sculptureThe sculpture Manifold was created by using the data of the Lorenz manifold as computed with the algorithmin [4, 5]. We selected a band derived from computed rings that correspond to a steel band of 8 cm in an overallsculpture of 70 cm in diameter.The main challenge of creating the band from steel is that it is made by hammering flat segments intothe required curved shape. The data of the respective rings was loaded into CAD software and processed torender templates for the sculpture. To this end, we utilized the symmetry of the Lorenz manifold, and thus,of the band. There are precisely two intersection lines with the z-axis in the computer-rendered object, whichare located vertically at the top and bottom in Figure 2. A cut through those two intersection lines gives twohalves that are each other’s image under the symmetry. Each half was then divided into three sections: onelarge relatively flat section that roughly forms the bottom half of the band, and two smaller sections that formthe upper half with the highest curvature. The selection of the three (pairs of) sections was further based onthe necessity that they needed to be welded together along relatively straight sections in accessible locations.The three individual sections were created in the CAD software so that the steel pieces could be compareddirectly with the computer model. Furthermore, they were then manipulated in the CAD software to ‘flattenthem out’ into pieces that could be laser-cut from flat sheet steel.The two pairs of flattened pieces for the top half were cut from 1.2 mm stainless steel. The two flattenedpieces corresponding to the symmetric bottom sections were cut from 3 mm stainless steel, which ensuresthat the bottom half is strong enough to support the more winding and, hence, heavier top half of the sculp-ture. The flat pieces were hammered into their desired shape by hand, which took place under constantcomparison with three-dimensional computer-rendered images of the CAD model. The top part of the bandcontinuously changes in direction and curvature. Therefore, the right supporting tools (stakes) needed tobe applied to stretch and compress the sheet locally to the desired degree. While the flat sections alreadyshowed the most appropriate curvature in the plane, the relation between the inner and outer edge length isfurther altered in the transformation into the three-dimensional shape, where stretching of the inner or outeredge can be emphasised through the forming process. Furthermore, pieces were worked into their shape inpairs to ensure the symmetry of the overall sculpture. As can be seen in Figure 2, near the centre of thesculpture the steel band comes very close to itself. This made it necessary to polish and brush the respectivesections before they were welded together.The welding itself is also quite a challenge because the tension in the steel sections is difficult to balanceduring the welding process. The key is to keep stresses to a minimum and balance them in such a way that theoverall required shape is attained. To achieve this, the four sections forming the top and central sections ofthe band are welded together first; here care must be taken to ensure that the resulting symmetric piece comesclose to itself in the central region, but does not actually touch. Similarly, the two pieces that form the bottomhalf (up to about the centre of the outer spiral in Figure 2) are first welded together along the symmetry axis.Finally, the entire sculpture is welded together along the two remaining seams. The finished sculpture ismounted onto a wooden base via a 30 mm pin that holds the band at the bottom along the intersection linewith the z-axis. In this way, the steel band can be rotated about the vertical axis of symmetry, so that it canbe enjoyed from any angle.The quality and precision of the sculpture is so good that it provides an almost-perfect rendering insteel of the computer-generated band. To illustrate the level of accuracy, we tried to match the view pointschosen for the computer images with those of the photographs of the sculpture in Figure 2. There aresome differences due to slightly different angles of projection, but overall the agreement is very convincing.However, nothing beats the experience of the real sculpture — and being able to look at it in detail and fromall angles!5 ConclusionsThe steel sculpture Manifold was born out of the wish to create a flowing and dynamic form on the basisof a special mathematical surface. It could only ever have taken shape by combining mathematical ideasfrom dynamical systems theory with the specialised hammering technique that allows for the introductionof negative curvature into sheets of metal. In short, the sculpture is an example of a successful interactionbetween arts and mathematics. While its intriguing shape can be enjoyed without any knowlegde of theunderlying mathematics, we hope that it may draw some viewers into the world of geometry.References[1] R. H. Abraham and C. D. Shaw. Dynamics — The Geometry of Behavior, Part Three: Global Behavior,Aerial Press, Santa Cruz California, 1982-1985.[2] E.J. Doedel, B. Krauskopf and H. M. Osinga. Global bifurcations of the Lorenz manifold. Nonlinearity,19(12):2947–2972, 2006.[3] J. Gleick. Chaos, the Making of a New Science, William Heinemann, London, 1988.[4] B. Krauskopf and H. M. Osinga. Two-dimensional global manifolds of vector fields. CHAOS, 9(3):768–774, 1999.[5] B. Krauskopf and H. M. Osinga. Computing geodesic level sets on global (un)stable manifolds ofvector fields. SIAM Journal on Applied Dynamical Systems, 2(4):546–569, 2003.[6] B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky,M. Dellnitz and O. Junge. A survey of methods for computing (un)stable manifolds of vector fields.International Journal of Bifurcation and Chaos 15(3): 763-791, 2005.[7] E. N. Lorenz. Deterministic nonperiodic flows. Journal of the Atmospheric Sciences, 20: 130–141,1963.[8] H. M. Osinga and B. Krauskopf. Visualizing the structure of chaos in the Lorenz system. Computersand Graphics, 26(5): 815–823, 2002.[9] H. M. Osinga and B. Krauskopf. Crocheting the Lorenz manifold. The Mathematical Intelligencer26(4): 25–37, 2004.[10] H. M. Osinga and B. Krauskopf. The Lorenz manifold: crochet and curvature. In R. Sarhangi andJ. Sharp (Eds.) Proceedings of 2006 Bridges London, Tarquin Publishing, 2006, pp. 255–260.[11] H. M. Osinga and B. Krauskopf. Visualizing curvature on the Lorenz manifold. Journal of Mathematicsand the Arts, 1(2): 113–123, 2007.[12] B. Storch. Anticlastic form — manifestating fields of tension. In R. Sarhangi and J. Barrello (Eds.)Proceedings of 2007 Bridges Donostia, Tarquin Publishing, 2007, pp. 189–194.[13] S. H. Strogatz. Nonlinear Dynamics and Chaos. Addison Wesley, 1994.

A survey of methods for computing (un)stable manifolds of vector ﬁelds.

Anticlastic form — manifestating ﬁelds of tension. In

Chaos, the Making of a New Science,

Computing geodesic level sets on global (un)stable manifolds of vector ﬁelds.

Crocheting the Lorenz manifold.

Deterministic nonperiodic ﬂows.

Dynamics — The Geometry of Behavior, Part Three: Global Behavior, Aerial Press,

The sculpture Manifold: a band from a surface, a surface from a band

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