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72 AAG2018 73
Beyond the basket case:
A principled approach to
the modelling of kagome
weave patterns for the
fabrication of interlaced
lattice structures using
straight strips.
Phil Ayres, Alison Grace Martin, Mateusz Zwierzycki
Phil Ayres
phil.ayres@kadk.dk
CITA, Denmark
Alison Grace Martin
alisonmartin57@gmail.com
Independant 3D Weaver, Italy
Mateusz Zwierzycki
mateuszzwierzycki@gmail.com
The Object, Poland
Keywords:
Kagome, triaxial weaving, basket weaving, textiles,
mesh topology, mesh valence, mesh dual, fabrication,
constraint modelling, constraint based simulation,
design computation
72 AAG2018 73
Abstract
This paper explores how computational methods of representation
can support and extend kagome handcraft towards the fabrication of
interlaced lattice structures in an expanded set of domains, beyond
basket making. Through reference to the literature and state of the
art, we argue that the instrumentalisation of kagome principles into
computational design methods is both timely and relevant; it addresses
a growing interest in such structures across design and engineering
communities; it also fills a current gap in tools that facilitate design
and fabrication investigation across a spectrum of expertise, from the
novice to the expert.
The paper describes the underlying topological and geometrical
principles of kagome weave, and demonstrates the direct compatibility
of these principles to properties of computational triangular meshes
and their duals. We employ the known Medial Construction method
to generate the weave pattern, edge ‘walking’ methods to consolidate
geometry into individual strips, physics based relaxation to achieve a
materially informed final geometry and projection to generate fabri-
cation information. Our principle contribution is the combination of
these methods to produce a principled workflow that supports design
investigation of kagome weave patterns with the constraint of being
made using straight strips of material. We evaluate the computational
workflow through comparison to physical artefacts constructed ex-ante
and ex-post.
1. Introduction
The term “weaving” covers a broad range of textile production methods.
Common to all is the principle of material interlacing to generate local
systems of friction-based reciprocity. This imbues resulting artifacts with
robustness through structural redundancy, resilience through friction-ba-
sed junctions, efficient use of material and potent aesthetic qualities.
These attributes have long been exploited in a diverse range of use are-
nas, through craft-based tacit knowledge or engineering-based explicit
knowledge, to produce lightweight artifacts with emergent properties
that offer advantage beyond those of the constituent materials.
74 AAG2018 75
1.1 Kagome
Kagome represents a particular class of weave which, in many ways, is
conceptually closer to braid. Where conventional weave is defined as the
interlacing of two distinct sets of yarns (warp and weft) at right angles
to each other, braid is defined as the interlacing of three or more distinct
sets of yarns (or ‘‘weavers’’) at oblique angles to each other [1]. In kagome,
the geometrical archetype arranges these three sets as a regular trihex-
agonal tiling with a vertex configuration (3.6)2 and p6 symmetry.
Figure 1: A regular planar sparse kagome weave comprising three dis-
tinct sets of weavers. The underlying pattern is a trihexagonal tiling.
The physical properties of these lattices are determined by the interplay
between combinatorics (valences of vertices and faces, connectivity, and
topology), geometry (vertex positions) and material attributes (mechani-
cal and geometric). Tacit understanding of this interplay allows the crafts
person to fabricate close approximations of arbitrary design targets.
74 AAG2018 75
1.2 Motivation
Kagome represents a highly principled method for producing complex
curved geometries with a single mesh structure, without the necessity
of joinery or the fabrication of nodes. The self-bracing capacity, greater
shear resistance (compared to biaxial weave), ability to rigorously control
geometry, high redundancy, ability to locally repair and potent aesthetic
qualities, make triaxially woven structures an attractive target for investi-
gation across a diverse range of design and craft practices, including
architecture. However, without means for visualisation and interrogation,
complex design targets can remain challenging for experts to strategise
and realise (keeping account of the number of weavers, their crossings
and potential self-crossings, calculation of material requirements, asses-
sing discretisation due to material lengths, etc.), and remain out of reach
for those without a tacit craft understanding.
Figure 2: Triaxially woven structures produced using straight maple
strips. Regular (left) and arbitrary (right) geometries are clearly gover-
ned by the interplay between introduced topological defects, material
stiffness and material geometry.
By intersecting the underlying principles governing the interplay of topo-
logy and geometry in triaxial systems with computational representation,
a platform for expanded exploration of these systems can be establis-
hed. This holds relevance to a wide variety of current and emerging
domains of application.
76 AAG2018 77
Figure 3: Two pre-relaxed kagome patterns approximating design geo-
metries. The weave topology is directly derived from inherent properties
of the design mesh (valence) and the weave pattern directly derived
from geometric attributes of the mesh dual (connected edges and their
lengths).
In this paper, we present a method for generating weave patterns with
the constraint that they be fabricated from straight strips of material.
Our motivation for working with straight strips relates to supporting the
future exploration of kagome applications at scales ‘‘beyond the basket’’,
where efficient use of material becomes a poignant issue. We address
key representational challenges including the generation of appropriate
topology, or mesh valence, to achieve a design target, together with the
relaxation of the mesh to simulate material performance – both of which
hold influence over final shape. In addition, we demonstrate the extrac-
tion of fabrication instruction and the physical making of computationally
developed design targets. We position this work in connection with
the literature to: 1) differentiate it from related approaches (specifically
related to the use of geodesics); 2) identify the open challenge that our
work addresses; 3) cite computational methods that we build upon.
Finally, we discuss our contribution, identify its limits and offer
trajectories for future work.
76 AAG2018 77
2. Topological principles governing
kagome geo
metry
The archetypal kagome lattice is a woven version of a tri-hexagonal
tiling; the weavers in one direction incline at an angle of 60° to those of
the other two directions, and the lattice, consisting of equilateral triang-
les and regular hexagons, will cover an infinite flat plane (Fig. 1).
2.1 Single curvature
Single curvature of the kagome lattice is easily achieved by bending
the plane, creating a developable surface. If the axis of curvature exists
across the centre points of opposite edges in the unit hexagon, one set
of weavers will act as arches perpendicular to this axis. If the axis of
curvature exists across opposing vertices of the unit hexagon, one set
of weavers will act as beams parallel to this axis. Limits on the radius of
curvature are dependent on the mechanical properties of the material.
Figure 4: Single curvature is easily achieved in the regular triaxial lattice
and can follow any line of hexagonal symmetry – across opposite edge
centres privileges arches, across opposite vertices privileges beams.
2.2 Double curvature
Breaking topological symmetry of a regular trihexagonal tiling by the
introduction of geometric singularities will induce double curvature [2].
These topological defects, or ‘‘lattice disclinations’’, are the mechanisms
that introduce in-plane strains and result in out-of-plane deformation [3].
Positive Gaussian curvature results from the introduction of <6 sided
cell. Figure 5 shows physically woven examples in which a single cell has
been substituted; firstly with a pentagon, then a quadrilateral and finally
a triangle. Of note is the way in which deformation out-of-plane becomes
more pronounced as edges are removed from the substituted polygon.
Figure 6 shows physically woven examples of negative Gaussian
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curvature resulting from cell substitution with a polygon of side >6; firstly
a heptagon, then octagon and finally a nonagon. Here, it is the increase
in sides of the substituted polygon that results in a more pronounced
curvature. Despite changes in topology through the introduction of
disclinations, the vertex valence of the materialised lattice is maintained
at v4 throughout, corresponding to the local crossing of two weavers.
Figure 5: Introducing disclination in the regular lattice by substituting
a <6 edge count polygon produces positive Gaussian curvature. From
top to bottom, each row decreases an edge – pentagon; quadrilateral;
triangle.
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Figure 6: Introducing disclination in the regular lattice by substituting a
>6 edge count polygon produces negative Gaussian curvature. From top
to bottom, each row increases an edge – heptagon; octagon; nonagon.
Weaving disclinations provides the means to locally distort the lattice,
causing a controlled deformation of the surface out of plane. Strategic
combinations of disclinations, informed through tacit knowledge, allow
the crafts person to realise specific and diverse design intent (Fig. 7).
However, in an inexhaustible space of possible combinations, enlisting
computation becomes a relevant tool for exploring, searching and navi-
gating this space.
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Figure 7: A diverse variety of artefacts demonstrating results from
strategic combinations of disclinations.
3. State of the art
In this section, we highlight relevant literature restricted to computational
representation of weave patterns and related computational methods
with a particular focus towards architectural design. We briefly cover
methods for establishing and refining mesh topologies, approaches
to weave in general, approaches to kagome representation in particu-
lar and provide a summary that identifies the open challenge that we
address.
3.1 Mesh topology and refinement
With a focus on mesh representations that have relevance to archi-
tecture, Schiftner et al. provide a method for refining triangular design
meshes such that the incircles of mesh faces form a packing – a CP
mesh [4]. This class of mesh is directly related to the kagome pattern,
which can be produced by connecting the centres of tangent incircles.
As precise CP meshes are rare, an optimisation algorithm is used to re-
fine a mesh towards an approximation of the design target. The mesh is
generated by producing an isotropic centroidal Voronoi diagram which is
iteratively relaxed using Loyd’s algorithm. However, this leads to random
placement of singularities which is undesirable if aiming to achieve regular
geometries. Use of the mesh operators, edge collapse, edge flipping and
edge splitting is a common method for locally refining the topology of
mesh as described in Narain et al. [5] and allows approximate locating of
required valence in the required position.
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3.2 Approaches to weave pattern representation
in general
Computational representation of weave patterns in general have been
well studied, however, the majority of these relate to biaxial weaving or
braiding. In most cases, the representation task is approached using the
tiling method described by Mercat [6] in which a predefined tile dictio-
nary defining local weaver geometry and crossings can be applied to a
quad mesh. This has been applied in the context of arbitrary manifold
design meshes [7], and with specific focus on braided structures [8, 9]. In
these cases, the principled approach to representation, which considers
interlacing and constraints related to fabrication, provides workflows and
tools for realising complex morphologies that are directly producible.
However, these tools operate with quad meshes which are less suited
to the kagome representation task. In another approaches, modelling
proceeds through direct manipulation of explicit geometry [10]. This is
not deemed to be a viable approach for the task considered here, consi-
dering the opportunity for exploiting the close affinity between the data
structures of triangular meshes and kagome pattern principles, and the
culture of use surrounding meshes for design expression.
3.3 Approaches to kagome pattern representation
in particular
Within architectural design specifically, approaches for defining kagome
patterns tend to exhibit shortcomings by either: 1) only considering a to-
pologically regular trihexagonal tiling; 2) exploring geometrical outcomes
of fixed and predetermined topologies; 3) abstracting out the weaving
principle such that the mechanical properties gained by interlacing are
sacrificed, whilst maintaining the topology of the trihexagonal tiling.
In the first case (which is often coupled with the third case) complex
geometries are achieved by a distortion of the regular grid rather than
conforming to the principles for achieving curvature described in the se-
ction above [11, 12]. This can present significant challenges for fabrication
strategies, junctioning methods and material use. In the second case,
relaxation of pre-determined and fixed topologies can result in principled
patterns, but impedes fluid design investigation due to a lack of ‘‘on-the-
fly’’ topology editing methods.
Kagome patterns have also been explored as a derivative of a gene-
ral approach to free-form surface segmentation using geodesic pattern
82 AAG2018 83
generation [13, 14]. The cited literature describes two approaches –
N–patterns from level sets, and the use of a regular trihexagonal web of
geodesics – but also identify limits in both cases. Pottman et al. acknow-
ledge that the level set approach produces webs of curves that are as
geodesic as possible, but deviations are inevitable in conditions of strong
Gaussian variance [13]. Deng et al. point to the fact that true geodesic
webs do not exist in general and that adequate surface approximation is
not always possible [14].
In contrast to these geodesic methods, which operate from proper-
ties of a surface (which in practice is generally approximated by a mesh),
our approach operates directly on properties of the mesh and form-finds
the final geometry through a relaxation procedure. This models the
actions of the local reciprocal systems, which, in practice, we find causes
material strips to deviate from true geodesics due to induced torsions
often arising in areas of pronounced double curvature. In short, the use
of geodesics to derive kagome patterns cannot cover all cases that can
be materialised in practice, whereas a principled kagome pattern can
always be derived from a manifold triangular mesh [15].
The strong affinity between kagome lattice patterns and computa-
tional triangular manifold meshes have been described by Mallos and
implemented in the context of a kagome design and fabrication tool
[ibid]. However, to our knowledge, this tool does not implement a step
that allows the consideration of kagome patterns resulting from straight
members – a case that requires relaxation of the kagome geometry with
specific relaxation constraints.
3.4 Identifying the open challenge
In summary, and in reference to the state-of-art presented here, we
can state that whilst there exist a number of methods and algorithms
related to the kagome representation task, to the best of the authors
knowledge, a holistic computational approach that aids designers by
coupling specific fabrication constraints with the principles for “real-time”
exploration of arbitrary kagome topologies and geometries, remains an
open challenge.
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4. Computational approach
Our approach to achieve a principled and generalised method for
kagome representation, of arbitrary geometries, makes use of various
algorithms and methods described in the literature; we declare these
below. The contribution of this paper is to draw these together to fulfill
the representation task with a focus on fabrication using straight strips
of material. The representation task has three stages:
1. topology generation
2. kagome pattern generation
3. relaxation to final geometry
4.1 Topology generation
Using the low-polygon modelling method [16], a coarse triangular mesh
approximation of the desired geometry is created. In the example shown,
the target geometry to model is a existing kagome “socket” condition
comprising a regular planar face intersected by a singularly curved tube.
The transition exhibits negative Gaussian curvature (Fig. 8).
The topology of the low-poly mesh is adjusted to establish the
required valence structure. Adjustment is done using conventional mesh
refinement operations; edge splitting, edge flipping and edge collapsing [5].
Figure 8: The target geometry to model is a detail of an existing
kagome weave with negative Gaussian curvature (left). This is
coarsely approximated with a low-polygon mesh (right).
84 AAG2018 85
Mesh valence of a regular planar tiling is 6, positive Gaussian curvature
requires <6 (but >2) valence and negative curvature requires >6 valence.
In this case, six valence 7 conditions around the rim of the transition and
regular valence 6 conditions to the stem have been introduced. Once the
refined valence structure is established, intermediary mesh operations
such as relaxation (as in the case shown in Fig. 9) or mesh subdivision
can be applied.
4.2 Kagome pattern generation
The mesh dual is obtained and decomposed into a data structure of
individual vertices and their three connecting edges. A new vertex is
then placed at the centre of each connecting edge and these three new
vertices connected with a closed polyline. This operation essentially
truncates the original vertex, creating a new facet that represents the tri-
angular element in the kagome lattice. The operation is equivalent to the
medial construction method described by Mallos [15]. At this point, the
weave pattern is purely visual and contains no information about weaver
continuity; all higher edge faces of the lattice are visually inferred from
their surrounding triangles.
The list of truncated face polylines is now converted into a data
structure that represents individual weavers. The polylines are exploded
into individual linear elements and then “walked” to find connected seg-
ments that meet a criteria of minimum angular deviation. Once weavers
have been identified, they are locally displaced in an alternating pattern
(up/down) along the surface normal vector at crossing points to model
interlacing. Once interlaced, each weaver is converted into a triangular
mesh approximating the material strip width using the method described
by Vestartas et al. [9]. At this stage, meshes may exhibit areas of interse-
ction as can be seen in Figure 11 (right).
4.3 Relaxation to final geometry
The weaver meshes are relaxed using the constraint-based solver
Kangaroo2 for Grasshopper. Additional constraints are added to ensure
weavers relax into developable geometries approximating straight strips,
and to prevent collisions and intersections between weavers – thus
preserving the structure of interlacing. Having found the final geometry
84 AAG2018 85
Figure 11: The edges of the kagome pattern are “walked” to construct
individual weavers (left). Weavers are then displaced normal to the
surface to model interlacing, and then meshed according to material
geometry (right).
Figure 10: The mesh dual is obtained (left) and each vertex “truncated”
to generate a visual kagome pattern (right). This pattern does not yet
describe individual weavers.
Figure 9: The mesh is refined by collapsing, splitting and flipping edges
to modify the valence according to the required curvature (left). A pre-
liminary relaxation has then been performed after adding an additional
layer of outer triangles in the plane to encapsulate the valence 7
conditions (right).
86 AAG2018 87
through relaxation, fabrication information can now be extracted (Fig. 12).
Weaver lengths are easily determined, and being developable, projected
as unrolled strips and marked with crossing points indexed with other
weavers or self-intersections. Physical limits on material length can
inform weaver discretisation, ensuring sufficient material cross-over for
splicing.
5. Two cases
In this section we briefly present two case studies that examine
relationships between a computational representation and a physical
artefact – one constructed ex-ante and the other ex-post modelling.
The first study demonstrates the use of our approach in the context of a
simple fabrication exercise. The second study demonstrates the use of
our approach in the context of computational design exploration.
Figure 12: The modelled weavers are relaxed to ensure they corres-
pond to straight elements and the final weave geometry is form-found.
Fabrication information is then extracted and includes length of strips,
strip ID’s and strip crossing ID’s. This information is applied to the weave
representation (left) and as material layout (right).
86 AAG2018 87
Figure 13: Extraction of fabrication information to produce a woven
stadium of revolution.
5.1 Case 1: Stadium of revolution
In this first case, we aim to construct a physical weave from computationally
generated fabrication information. A stadium of revolution, or ‘‘capsule’’ geo-
metry, is defined as the design target. This geometry comprises a cylinder
with single curvature and two hemispherical caps. Drawing upon the princip-
les governing double curvature in kagome lattices, we expect the hemisp-
herical portion to contain pentagonal ‘‘defects’’ to achieve local synclastic
curvature. Each pentagon included in the mesh increases the aggregate
angular deficiency by π/3, therefore a triaxial mesh with 6 pentagons will
make a hemisphere. The rest of the lattice can be achieved using a regular
88 AAG2018 89
hexagonal tiling. We follow the modelling steps described in section 4 to
determine how many weavers, their respective lengths, crossings with other
weavers and self-crossings. We see from this analysis that the woven figure
comprises 6 simple rings of length cca. the circumference of the cylinder,
and two longer weavers with multiple self-crossing points. This is verified
with the physically weaving shown in Figure 13 (bottom right).
5.2 Case 2: The distorted helix
In this second case, the kagome helix is woven prior to any computational
modelling. Rather than aiming towards verisimilitude of the model, we
demonstrate how the relaxation step can provide exploratory insights
through simulating the interplay of material behaviour and topology. The
helix is modelled and the mesh refined, but in this case disclinations are
randomly placed within the mesh. As the relaxation proceeds and weaver
geometries straighten according to our fabrication constraints, and local
sites of curvature emerge where hexagons have been substituted with
synclastic curvature inducing pentagons, or anticlastic curvature indu-
cing heptagons. In this case, we demonstrate how computation provides
an accessible and fast (compared to physical weaving) exploratory tool
to assist the designer in searching the inexhaustible space of possible
disclination combinations, and potentially discovering novel aesthetic
expressions.
Figure 14: A physically woven helix with mesh disclinations placed to
realise a regular geometry (left) compared to a simulation where discli-
nations have been randomly located (right). This shows the necessity for
the relaxation step, but also suggests interesting geometric articulations
and ‘‘organic’’ expressions of a corrupted ideal.
88 AAG2018 89
6. Towards architectural and structural
applications
the instrumentalisation of a principled computational approach to kagome
pattern generation and representation has broad applicability. Within
architecture, hexagonal tiling patterns have been exploited to stunning
spatial effect by Shigeru Ban in projects such as the Pompidou Metz
and Nine Bridges golf club. However, in these cases, double curvature is
achieved through a distortion of the regular hexagonal tiling. The resulting
geometry is realised through complex shaping of stiff curved laminated
members. In such a context, the application of kagome topology principles
for achieving complex geometry could offer a more rational approach to
geometry with the implication of greater efficiency in fabrication.
In the context of elastically bent structures, the attributes of mecha-
nical performance arising from interlaced material and efficient spanning
of space with straight strips of material have been demonstrated in
the CODA Jukbuin Pavilion. In this case, the weave principle of mate-
rial interlacing is maintained but double curvature is achieved through
material bending behaviour rather than steered by topology – the design
topology is a regular hexagonal tiling. This results in global curvature
effects but denies the possibility of highly localised double curvature.
Nevertheless, this work is of particular interest as it demonstrates the
transfer of interlacing principles at architectural scale.
In framing a direction for future work, our emerging hypothesis is
that architectural scale structures can be realised with full adherence to
kagome weaving principles, including material interlacing. This hypothe-
sis is supported by a comparative analysis of two hypothetical gridshells
which shows that a kagome gridshell outperforms a quadrilateral grids-
hell for a very similar construction cost [17].
Our outlook is towards the use of elastically bent members rather
than stiff curved laminated members. However, as we discuss above,
we see kagome principles being applicable in both contexts - in the
former, towards bending-active structures that adhere more closely to
their basket antecedents; in the latter, towards rationalised approaches
to geometry and fabrication. In the context of elastically bent structures,
principle challenges revolve around structural capacity. Yet despite this
challenge, the opportunities for material efficiency, a rationalised app-
roach to free-form geometry and efficient fabrication minimising the use
of connectors make this a compelling territory for further study.
90 AAG2018 91
6.1 Limits and future work
Where the work presented in this paper has limited itself to exploring
the task of kagome representation and simulation with consideration
to fabrication constraints, analysis of structural performance marks a
necessary next step – especially if seeking to explore architectural
applications. Preliminary investigations of model transfer to the structu-
ral analysis platform Autodesk Robot indicate that representational
outcomes generated by the approach described are poised to be taken
forward into this domain of analysis. In addition, the ability to computa-
tionally represent arbitrary kagome geometries and interrogate these
from a fabrication perspective, presents the compelling opportunity of
investigating robotic production.
7. Conclusion
This paper has presented a principled computational approach to the
task of kagome representation in arbitrary triangular meshes. Following
the literature, we have demonstrated the strong affinity between the
principles governing kagome patterns and intrinsic topological featu-
res of computational meshes and geometric features of their duals.
We have shown how design meshes can be manipulated to adjust the
baseline valence 6 structure that governs planar kagome tiling, upwards
and downwards to create sites of local double curvature. We have also
shown how the kagome pattern itself is derived from the mesh dual by
vertex truncation to the mid-points of connected edges – following the
medial construction method.
We have extended the state-of-the-art by intersecting this method
with physics based relaxation to allow simulation of the interplay between
topology and notional mechanical properties of weaver material, thereby
constraining results within the bounds of fabrication criteria – specifically
that patterns can be made from straight strips of material. This constraint
is seen to be a benefit for enticing transferability and use within domains
where material saving can be a key issue, such as architecture.
FInally, the approach presented here contributes a method that can
be computationally leveraged to explore and search the inexhaustible
domain of possible kagome patterns, and opening the possibility of this
search to be conducted by both the novice and the expert.
90 AAG2018 91
Acknowledgements
This work was partially supported by the European Union’s Horizon 2020
research and innovation program under the project “flora robotica”, FET
grant agreement no. 640959. The authors gratefully acknowledge the
participation of the 2017/18 master students enrolled at the Master of
Architecture programme CITAstudio: Computation in Architecture, KADK
in the “Triaxial Weaving & Bio-receptivity” workshop conducted over one
week in September 2017. Five large scale woven structures were produ-
ced, two of which are shown in Figure 2. The authors also wish to thank
the anonymous peer reviewers for their comments and suggestions to
improve the quality of the paper.
92 AAG2018 93
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