Sophia Vyzoviti Joanna Sotiriou

Department of ArchitectureUniversity of Thessaly

Greece

Descriptive and Generative Models of Crumpled Sheets

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From Material Behavior to Form GenerationWhen a sheet of paper is crumpled, it acquires permanent scars displaying the distri-bution of stress along its surface. These irregular creases that wrinkle the paper sur-face reveal that crumpling has focused the stress at some points, exceeding there the limit yield of the material and leading thus to irreversible plastic deformations (Amar & Pomeau 1997). Crumpled sheets have been investigated by physicists with respect to their buckling and elasticity and by mathematicians through the geometry of de-velopable surfaces.

Cambou and Menon produced a series of statistical descriptions of forced crumpling experiments, focusing on the distortional stress a flat aluminum sheet is subjected to while becoming a crumpled ball (Cambou & Menon 2011). What we find particularly interesting in this research is the predictability of such a transformation of matter, the possibilities of pattern recognition within the random folding and further simulation with precision. One of the most intriguing observations made during their X-ray mi-crotomography experiments reveal that the internal three-dimensional geometry of a crumpled ball is in many respects isotropic and homogeneous. A crumpled sheet consists of multiple flat layers of material thus, revealing that in the case of crumpled paper, randomly applied forces result to uniformly distorted geometries.

Diamant and Witten, observe a surprising condition of symmetry in the wrinkles and folds of thin coated surfaces on fluid substrates (Diamant & Witten 2013). What is quite interesting about their research is the identification of a condition of ambivalence in this material behavior. Given a certain compression, a sheet floating on water, displays alternative deformations between symmetric and anti-symmetric folds with no cost of energy.

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Narain, Pfaff and O’Brien investigate techniques methods for simulating the plastic de-formation of crumpled sheets. They employ dynamic remeshing in order to improve existing simulation techniques and particularly a framework of adaptive mesh refine-ment. According to the researchers “high in-plane stiffness causes deformation to be concentrated along narrow creases and at point singularities” (Narain, Pfaff & O’Brien 2013). What we find interesting in this research is the representation of a thin sheet of material as a finite element mesh composed of triangles. Dynamic remeshing allows accurate and efficient representations of plastic deformation and fold and crease for-mation by aligning mesh elements with the emerging detail in the material.

Focusing upon the morphology of crumpled sheets, we discuss methods of transcrib-ing their material behavior into a form generation process that can be integrated in the development of a design tool for architects. A fusion between the properties of the material and its digital representations is orchestrated in a sequence of form studies. Initiating with instinctive physical crumpling experiments, we proceed with rational-ized representations constructing descriptive and generative models of crumpled sur-faces. Digital outputs are fabricated as investigations of material efficient static and kinetic crumpled surface prototypes. We expand upon representations of crumpled surfaces that we consider significant in the discourse of folding in architecture.

Fig. 1

Crumpled sheet of photocopy paper.

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Folding as a design medium diverges from the already confirmed architectural trends and notions since it calls for an aggregate approach rather than a fragmentary archi-tectural reading. It is also important to consider the fold as a continuously differenti-ating entity. As such a folded geometry should be recognized as a malleable entirety which encapsulates the architectural information in advance, translated into program-matic relations and flows.

According to Lars Spuybroek “Continuity in architecture does not mean organically smoothed-out forms but architecture of singularities – sudden changes in a system that organizes the previous state by a matter of shifting scale, like column-line to vault-surface.” (Spuybroek 2008). Thus, such a quality allows folded surfaces to respond to spatial in-quiries by transforming not into aggregates of fragments but into catalytically inter-connected elements. This emergence property of folded forms revokes the dualism of the intentional and the unintentional, the essential and the accidental.

Folding in the context of architecture, has brought forward innovative spatial concepts: smoothness, the redefinition of boundaries, architectural landscapes with no defin-able beginning or end; evolving continuums - an analysis of which exceeds the scope of the current paper. However, if we espouse the notion that folding can surpass pure aesthetics and become a competent architectural medium and design tool, we need to approach the fundamentals of parametric and computational design, in respect to folding’s dynamic nature. Consequently, to construct an efficient randomly folding generative design tool, a careful collection, rationalization and finally organization of the knowledge hidden behind the actual act of physical crumpling is mandatory. If we follow Delanda’s logic that everything is material, “generative design processes must also contain traces of materiality which, while unconventional, or even accidental, must be understood in order to define the relationship of generative design to architecture” (Krunic 2009). Therefore, revisiting the microscopic articulation of a crumpled paper surface in physical space, we focus on the points of elastic deformation.

Architecturally Significant Descriptions of Crumpled SheetsAs a form generation process, crumpling introduces an avant-garde quality in the output: random and stochastic folded surface conditions. As an architectural concept, crumpling consolidates a recent tendency for a microscopic rearticulating of the surface’s texture that captures causal interactions among its component parts, enhancing its ‘response’ ability.

In a microscopic analysis of a crumpled sheet, one could distinguish the two main de-formation classes: the plastic deformation, revealing points where the applied forc-es have exceeded of the material limit yield and the elastic deformation points. The former materialize as permanent and irreversible wrinkles on the paper. The latter, oc-cur as the most intense folds of the surface. Since the elastic deformations of a crum-pled paper are reversible, the final product is dynamic and therefore, kinetic.

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Crease Patterns as Deployable TessellationsThe morphology of a crumpled membrane is a network of straight ridges or folds that meet at sharp vertices (Lobkovsky & Witten 1996). The first descriptive model is based on prior research of the author (Vyzoviti & Dimitriou 2012) and comprises a crumpled surface reconstruction on the basis of developed crease patterns that relies on the ra-tionalization of this complex network of ridges or folds into a deployable tessellation.

The process of rationalization focuses on the recognition of patterns and regularities in the complex network of ridges and folds. The analysis distinguishes between es-sential, form generative folds -which we further on refer to as high lines, and second-ary folds -that we refer to as texture. Crumpled surface analysis intends to construct a minimum necessary description, a crease pattern that is deployable. This representa-tion integrates the fundamental origami rule that edges are two colorable, that is, they are either concave creases (valleys) or convex creases (ridges). In this model Freeform Origami software developed by Tomohiro Tachi (2010) is employed to reconstruct a crumpled sheet, to simulate the paper’s buckling and also to extract three-dimension-al mesh geometry, suitable for design development and fabrication.

Crease Patterns as Non-Directed Graphs Microscopic analysis of the complex network of creases that constitute the crumpled surface informs the second descriptive model. The crease pattern of a crumpled tessel-lation is represented as a non-directed graph constituted by vertices and edges.

Fig. 2

A rationalized crumpled crease pattern (left) and its software augmented tessellation (right).

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Since the process indicates a vertex driven system, we proceed to the rationalization of the vertices movement. Such a type of unit testing helps us draw further conclusions on the direct relations between a single vertex and the connected edges. An ontol-ogy that captures local fold-ability around a single vertex is developed employing the graph theory principle of degree.

The ontology was constructed and later on employed for the formation of the primi-tive conditionals algorithm. Since the procedure is mainly vertex driven, the first step called for an assortment of the various types of the generated vertices. Thus, a first de-gree upwards moving vertex (convex) is described as Vm, a downwards moving vertex (concave) is described as Vva and a vertex belonging in a high line trace (second degree of convex) is described as Vh. It is quite important to note here that for purposes of

Fig. 3

Crumpled surface tessellation represented as non-directed graph. In this case: Graph size = 142 (number of edges) and Graph order = 43 (number of vertices).

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visual abbreviation on the graph itself, a first degree upwards vertex is symbolized as “+” and a high line trace vertex is symbolized as “++”, symbols inscribed in the vertices, graphically presented as circles. Driven by the two colorable origami rule which refers to the nature of the graph edges, an assortment of the various types of edges was mandatory. An upwards moving edge (mountain) is assigned to the abbreviation of Xm and the downwards moving edge (valley) is represented in the algorithm as Xva. However, we also had to assort the mountains and valleys that were connected to the graph perimeter for the purposes of the algorithmic conditionals. Thus, the abbrevia-tions Xmp and Xvavp were introduced, referring to the upwards and downwards moving edges which are also connected to the graph perimeter, respectably.

In order to gain total control over the dispersed vertices and describe their relative movement in accordance to the inter connected edges, the concept of “degree” was introduced. More specifically, each vertex was given a total degree “dvn” which was di-vided in the mountain degree of the vertex (mdvm), or the total of xm connected to it, and the valley degree of the vertex (vdvm), or the total of xva connected to it.

Finally, two terms, belonging to the graph theory, were used in order to define the overall characteristics of the final graph. More specifically, the Graph Size is presented here as |X| and refers to the total number of the graph edges, whereas |V| symbolizes the Graph Order, giving the total number of the vertices consisting the graph.

Towards a Generative Algorithm of Crumpled TessellationsCombining knowledge from the descriptive – explanatory models, a generative algo-rithm is further composed, serial executions of conditionals that describe iterative ver-tex consignation.

In the first step of characterizing the nature of the vertex, we observe the edges con-necting to it. If the totals of the mountains and the valleys are not equal, we proceed to tracing which one preponderates in number.

In case the number of the valleys is greater than the number of the mountains, the vertex is characterized as Vva and thus, moving downwards. In the opposite case, we check the ratio of mountains to valleys. If the ratio is greater (yet not equal) than two (2), the vertex belongs to a high line trace (Vh, ++). If not, the vertex is a Vm and there-fore moving upwards (+).

Returning to the first hypothesis, if the total number of the mountains and valleys is the same, the study proceeds to the edges. More specifically, if the mountain edges connecting to the vertex in focus are greater in number than the corresponding valley edges, the vertex is a Vm (+). In case the mountain edges are less comparing to the total number of the valleys, the vertex is a Vva (-).

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In the last case where the number of mountain edges is the same with the total of the valley edges, we check the number of the edges that connect to the vertex and the perimeter of the graph. Characteristically, if the number of such an edge, which is also a mountain, is greater than the number of the valley edges, the vertex is a Vva (-) whereas if the specific hypothesis is false, the vertex is a Vm (+). In the last case where the total of the mountain edges connecting to the perimeter is the same with the corresponding valley edges, a graph bug is indicated and therefore, the plane is not deployable.

The algorithm begins with user-defined input which is either a sequence of same-color creases (Highline) or a total of randomly dispersed vertices (First Vertices Generation -VG1).

In case the designer prefers to initially inform the graph with edges and not dispersed vertices, the VG1 is consisted of the connections of the multiple edges describing the highline.

The next step calls for the mandatory formation of a set of valleys and mountains, pur-suant to the origami rule of interior and exterior angles, resulting to the Second Verti-ces Generation - VG2. In this step, the edges direction is random and not definite for the development of the graph.

Fig. 4

The primitive conditionals algorithm for vertex consignation.

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This step is followed by a theoretically infinite sequential formation of n VGs resulting to an arborescent graph consisting of neutral creases (algorithm loop). The graph or-der conditional provides the designer with control of tessellation complexity.

When the designer decides to end the loop, the total of VGn vertices are finally inter-connected creating the perimeter of the morph, thus closing the graph.

Once the formation of the graph is terminated, the procedure calls for the triangular tesselation of every shape consisting the total graph through the connection of the already-created vertices.

Using the conditionals algorithm and according to the desirable height of each vertex, the designer starts assigning certain edges to the two colors that, in this step, their nature is straightforward manifested.

The designer then has to define the color of the creases according to a set of rules, presented in the form of the aforementioned algorithmic conditionals. This set of re-strictions refer to qualities such as the preferable sign constraining the height of each vertex and moreover, the realization of deployment.

The procedure described leads to a functional and deployable pseudorandom tessel-lation.

Fig. 5

Procedural development of a pseudo-random tessellation.

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In this model Freeform Origami software developed by Tomohiro Tachi (2010) is em-ployed at the last phase to test the deployment capacity of the generated tessellation.

Conclusion and Extension: Prototyping with Architectural RelevanceBackward and forward chaining between physical and digital models is instrumental to this particular form generation process. As discussed above, in the case of crumpled surfaces, physical form studies may well provide starting points for the digital process – such as the quasi-physical ‘material computing’ studies described by Patrick Schu-macher (Schumacher 2007). Furthermore multiple digital outputs may be fabricated in media res providing catalytic information concerning the potential structure and performance of the generated crumpled surfaces. Therefore, extension of the research towards architecturally relevant prototyping of crumpled surfaces would require rigor-ous methods for a materiality of intermediate fabrication which is able to inform the generative models.

The most common problem concerns the leap in scale. The material behavior of thin sheets may be architecturally relevant when it is geometrically re-calibrated to define larger objects with the capacity to contain. The in between fabrication – a prototype scaled to fit available material resources is a key instrument in the development. In ad-dition to the sculptural qualities of random surface tessellations that are evident both in the initial instinctive physical crumpling experiments and the digitally generated deployable models, the prototypes ought to be enriched and refined within the digital narrative towards material efficient static and kinetic surface prototypes.

Fig. 6

Verification of deployment ability of generated pseudo-random tessellation in Freeform Origami software developed by Tomohiro Tachi (2010).

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In the series of experiments conducted in the academic studio of the University of Thessaly the geometric abstraction of crumpled surface tessellations has acquired a few significant material responses. The range of potential materiality, which includes lattice structures, articulated tessellations and multilayered surface laminates, is en-couraging. A distinction between soft and hard surfaces ought to be outlined: the soft ones are prone to be kinetic while the hard ones manifest surprising rigidity. Perhaps a general lack of purpose undermines the creativity invested in the morphogenetic process at its current state. Nevertheless the potential of intermediate materiality to inform generative models towards architectural prototype is extremely valuable. Con-sidering the current overabundance of digital fabrication installations and pavilions, the critical question is how innovative digital driven production may become embed-ded in the synergetic complexities of architectural practice.

AcknowledgementThe prototypes of Image 7 were fabricated by Maria Christu (left) and Serafim Papas (right) un-der the supervision of Sophia Vyzoviti at the Department of Architecture, University of Thessaly, Volos, Greece in the context of Folding Architecture Course http://www.arch.uth.gr/en/studies/course/779

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Fig. 7

Kinetic surface prototypes deriving from descriptive models of crumpled sheets.

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