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Proceedings of the IASS Annual Symposium 2016
“Spatial Structures in the 21st Century”
26–30 September, 2016, Tokyo, Japan
K. Kawaguchi, M. Ohsaki, T. Takeuchi (eds.)
Copyright © 2016 by <Zhao Ma, Pierre Latteur, Caitlin Mueller>
Published by the International Association for Shell and Spatial Structures (IASS) with permission.
Grammar-based Rhombic Polyhedral Multi-Directional Joints
and Corresponding Lattices
Zhao MAa*, Pierre LATTEURb, Caitlin MUELLERa,
a* Massachusetts Institute of Technology, Department of Architecture
77 Massachusetts Ave, Cambridge, MA 02139, United States
x__x@mit.edu
b Université catholique de Louvain, Louvain School of Engineering EPL/IMMC,
Civil & Environmental Engineering,
Abstract
This paper presents a new type of joints derived from a historic wooden puzzles. A rule-based
computational grammar for generating the joints is developed based on the relationship between the
three-dimensional shape and its two-dimensional “net”. The rhombic polyhedra family, cube, rhombic
dodecahedron, and rhombic triacontahedron are fully explored in the paper as they are the central part
of the joints and is enclosed by all the elements around. This paper also presents lattices derived from
these joints, and provide structure analysis to compare the structural performance of these lattices. The
non-orthogonal lattices provide new way of looking at lattice structures and also provide new
challenges for designing assembly techniques for both human, robots, and drones.
Keywords: polyhedron, lattice, joint, puzzle, drones
1. Introduction
Conventional building structures rely on using large scale construction material such as concrete and
steel to gain enough strength for stability. They need either on-site casting or prefabrication, the latter
of which will also need large equipment to install. Besides, these construction techniques consume a
huge amount of energy and are often non-reversible construction methods. To respond to the need for
a sustainability building industry and energy-wise construction environment, old ways of construction
and assemble systems need to improved, and new ways need to be invented.
However, the increasing application of robotic fabrication & construction are making more impact to
this field. The change of tools often affects the change of related methodologies to suit the way these
tools are used. Robots like robotic arms, drones, etc. are able to achieve works with higher degree
complexity in a faster efficient way, but at the same time are limited to the payload they can work on,
which requires assembly-like structural systems. Each member of the system need to small and light
enough to handle and can be connected with strong enough joints to form a larger scale whole.
Meanwhile, realization of different algorithms through programming not only enable robots and
drones to achieve their work but also provide more freedom and possibility that we can design
structures with higher degree of complexity and that are difficult to assemble by human labour. These
algorithms also populate the number of design methods and allow us to design in ways we cannot
imagine. For instance, computational shape grammars are available to design and generate building
units for unconventional structures parametrically.
With the notion of structural efficiency, in other words occupying more space with less materials, the
most common and conventional way is to use lattice-based structure, such as trusses, tensegrity
structure, deployable & foldable structures. However, most structure of this type are orthogonal (some
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
2
with diagonal members as triangulation). From a historical perspective, these types have simpler logic
that are easier to understand and can be designed for variation easily and quickly. Since all members
are either vertical or horizontal, it is also easier to model, analysis and simulate dynamically in finite
element software. The design of the joints can also be simplified to fewer types as all members came
from the same direction.
Meanwhile, non-orthogonal lattice structures, though having been studied and developed in micro
fields (chemistry, electronic engineering, etc.), have not been developed much for reasons of
constructability. The difficulties may lie in the design of joints, assembly process, the tolerance, etc.
But new tools and methods we are engaging now provide an opportunity to flow back and re-examine
these decisions we made throughout the river of history, to learn from old techniques, and discover the
valuable pearls we thought was gravel before.
Figure 1: Kongming Lock [ http://de.wikipedia.org/wiki/Bild:SechsTeileHolzknoten.jpg]
This paper starts from one of the most ancient wooden puzzle in Asia, the “Kongming Lock” (Figure
1), a derivative of building joints for Chinese traditional wooden architecture, and mathematically and
computationally presents the logic and rules for the generation of the puzzle. The paper also discovers
the grammars that generate the only other two more complex derivatives of the same type. By
understanding the relationship of different members, new non-orthogonal lattices will be generated
and presented at the end of this paper.
2. Literature review
2.1 Polyhedron and lattice
The application of polyhedra in structural field has been widely in different scales, ranging from the
creation of the roundest soccer ball (Huybers [1]) to building scale polyhedral domes (Wester [2]).
Lattice structure formed by some of these polyhedra can also achieve free-form structure to suit for the
architectural need (Davis [3]). Variations of polyhedral structures such as geodesic domes have also
been built and patented (Buckminster [4]), the joints of which connect five members coming from a
morphed surface plane (Ohme [5]).
However, most of these applications of polyhedra focuses on using the polyhedra as domes, surface
structures that enclose certain spaces, or on producing space trusses composed of repetitive polyhedral
geometries. Few has focused on using polyhedra as joints to generate complex lattice structures whose
members are placed towards multiple directions other than orthogonal ones. Some special polyhedral
joints were designed to provide multi-directional connection, such as the triakonta system (Elliott [6]),
and the KK system (Chilton [7]), as well as related patents (Jeannin [8]). But most of these
applications either use the polyhedron as a multi-directional provider or invent special cases to suit the
need. Few has been focused on the lattice structure that different polyhedral joints can provide.
2.2 Traditional Asian joints and wooden puzzles
The puzzle of "burr" refer to a series of puzzle families composed of different members interlocked
with each other by carefully cut notches. The most common type of the puzzle is the six-piece burr,
also called “Puzzle Knot”, “Chinese Cross” or “Kongming Lock”. It is the most well-known and
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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presumably the oldest of the burr puzzles. It is actually a family of puzzles with the same finished
geometry and basic shape of the pieces, with different notches. This family of puzzles was first solved
mathematically and completely by mathematician Bill Cutler in 1994 with the help of computational
analysis, as in Cutler [9]. The notches on the pieces for each of these puzzles are often different from
each other, resulting in different assembly processes.
Other burr families, such as Altekruse, Chuck, Pagoda, are also developed throughout the time (Coffin
[10]). These puzzles in general can be divided into two categories: puzzles whose members share the
same basic shape (Figure 1) and puzzles whose members share different basic shapes (Figure 2).
Figure 2: a puzzle with different shaped members
[https://commons.wikimedia.org/wiki/File:Pagoda_Burr_Puzzle.jpg]
Most of the burr puzzles are made with square notches. There is a small category of puzzles are made
with diagonal notches, often called “stars” (Slocum [11]). However, if examined carefully of the
bounding box of the members of these puzzles, they are actually the same type of the puzzles above.
However, most of the puzzles were made through physical experiments by cutting/gluing (Coffin [12])
blocks. For complex puzzles, this process is exhausting and even impossible as it is hard to keep the
tolerance down after multiple cuts. Besides, there are few general tools to understanding the geometric
and assembly logic behind these puzzles. Since most of these puzzles are symmetric in space, it is
reasonable to infer certain rule based systems will help in computing these geometries through
iterative processes.
2.3 Grammatical computation
As a well-known type of rule-based design method in design and computation field, shape grammars
is a commonly used method to generate multiple design possibilities by defining a set of allowable
shape transformations. Once formulated, the rules can automatically generate new designs diversely.
Stiny has provided many two-dimensional examples in his book "Shape: Talking about Seeing and
Doing" (Stiny [13]). Esher’s published tile designs are also examples for 2D tessellation and are
examined in a computational perspective on analytical methods for design (Ozgan et al. [14]). 3D
grammar and algorithms have also been developed to analysis polyhedral object pattern (Wang [15]),
or to extend the frames in conceptual design representation (Albert et al. [16]). More advanced
implementation of using shape grammars on convex polyhedra adapts grammar based tools to wider
range of geometries (Thaller et al. [17]).
2.4 Research question
The topology design of almost all lattice structure is actually a 3D tessellation problem about how to
use basic polyhedral units to occupy the space. Research and application in reality are mostly focus on
cubic units which results in orthogonal-looking structures that are both easier to understand and
construct. However, thanks to the increasing use of computer and digital fabrication methods such as
robotic fabrication, drone construction, lattices with higher complexity and constructability are more
easily to understand, fabricate and build. It may even benefit the process and logic of these fabrication
and construction methods if designed carefully to suit the machines. This paper will discuss: 1.
grammatical algorithms for traversing faces of three rhombic polyhedrons: cube (regular hexahedron),
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
4
rhombic dodecahedron, rhombic triacontahedron; 2. space lattice structures derived from the joints of
these three rhombic polyhedrons; 3. structural assessments of these lattices.
3. Shape Grammar for Traversing Polyhedra
3.1 6-piece geometric puzzle and shape grammar
The puzzled in Figure 1 can be regarded as a cluster of six sticks intersecting each other. An
interlocking state can be achieved by removing parts from each sticks. Since the number of ways that
sticks can be arranged symmetrically and spatially is very limited, it is useful to examine the question
with unnotched members.
By examining this the puzzle, a center cube (Figure 3) can be found as the intersection part shared by
all the six members. The six members each corresponds to one of the six faces, and a grammar that
demonstrates the relationship (the rule) between every two members can be found. By computing the
same grammar repetitively, we can generate the overall geometry of the puzzle after 6 iterations
(Figure 4).
Figure 3: central cube of the puzzle
Figure 4: computation process
During the computation process, the six corresponding faces of the center cube are also traversed with
no repetition. No matter how the geometric shape of the six members change, the relationship between
each member and its corresponding face as well as the grammar will maintain the same. Thus, the
grammar for computing the shapes is equivalent to a grammar that traverse all the faces of a cube
without overlapping.
3.2 “Net” and 2D computation
By unfolding the cube into a planar diagram, which is called a “planar net” (Buekenhout & Parker
[18]), the grammar can be transformed from 3D to 2D. For a cube, there are in total 11 distinct nets
(Figure 5) exist (Weisstein [19]) and more than one grammar can be defined for each net based on the
number of rules in each grammar. The single grammar mentioned above belongs to the 6th net, which
visually has a repetitive relationship between every two connected square.
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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Figure 5: eleven distinct nets for the cube
Following the shape grammar defined in Figure 6, we can easily compute the result as shown in
Figure 7. Notice here that though only 6 steps are shown here as we are computing the 6th net of the
cube, this iteration can continue infinitely if only considering the 2D layout.
a. initial shape b. shape rule (with label)
Figure 6: grammar for computing net traversal
Figure 7: grammar computation
The number of distinct nets increases extremely fast depending on the number of faces of the
polyhedron. For instance, the dodecahedron and icosahedron each have 43,380 distinct nets
(Buekenhout & Parker [18]).
a. cube b. rhombic dodecahedron c. rhombic triacontahedron
Figure 8: the only three polyhedra with rhombic faces
In the book Geometric Puzzle Design, Coffin [10] mentioned that faces of the enclosed center must be
rhombic (square is a special case of rhombic), and there are only three isometrically symmetrical
solids with such faces: the cube, the rhombic dodecahedron, and the rhombic triacontahedron (Figure
8). For the reason mentioned above, this paper will not examine all the distinct nets of the rhombic
dodecahedron and the rhombic triacontahedron, but will only show one of the nets that have a
grammar with less rules (Figure 9).
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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a. rhombic dodecahedron b. rhombic triacontahedron
Figure 9: nets and computation sequence for rhombic dodecahedron and rhombic triacontahedron
3.3 symmetrical geometry computed from 3D shape grammar
If the grammar is computed in 3D space, the computation will traverse all the faces on the three
polyhedra respectively. The corresponding linear members of each face of the polyhedron will
compose a complex spatial geometry (Figure 10) in which no member is intersected with any other
member. The space enclosed by all the members is the corresponding polyhedron.
Figure 10: geometries composed by linear elements corresponding to the faces of the central
polyhedra
4 Lattice with polyhedral joints
Most of the existing research or projects focus on developing polyhedra as the primitive shape of
dome structures or extending the edges of polyhedra to form space frames. In this paper, the author
will discuss the possibility of forming space lattices by using the three polyhedra mentioned above as
joints, and extending the linear members corresponding to the faces of these polyhedra. This will
produce space lattices some of which will have non-orthogonal connected space units.
4.1 Lattice of cubic joints
As the faces of the cube are oriented orthogonally, the lattice produced by the joints based on the 1st
geometry in [Figure 8] will still be an orthogonal style. The difference is that there are two parallel
linear elements coming from each joint in each direction, and in total 6 members oriented in 3
directions – as the result of 3 pairs of parallel faces in the cube. By orienting multiple cubes into a 3D
matrix, we can create lattices from these cubic joints by adding linear members along their faces
(Figure 11.a). This type of lattice can be extended infinitely in space and defines different geometric
topologies.
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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a. cubic b. rhombic dodecahedral
Figure 11: polyhedral tessellation, matrix, and corresponding lattice
4.2 Lattice of rhombic dodecahedral joints
Similar to the cubic case, we implement the same method to the rhombic dodecahedron joints (2nd
polyhedron in Figure 8). From Figure 10, it is clear that there are three parallel linear elements coming
from each joint in each direction and in total 12 members oriented in 4 directions. The 3D matrix for
orienting the polyhedral joints is also derived from the tessellation of the basic shape – the rhombic
dodecahedron.
However, unlike the case in section 4.1, linear members coming from neighboring do not share the
same axis thus will not coincide as their length are extended, and creating new jointing location in
space. The polyhedron enclosed in these new joints are also rhombic dodecahedron (Figure 11.b). If
extending continuously in space, we can also find the simplified lattice for this polyhedral joints. The
lattice unit is shown in Figure 12. It is composed of the wireframe of rhombic dodecahedron and the
four axes for the 2nd symmetry shown in Figure 13.
Figure 12: the unit of rhombic-dodecahedron-joint lattice
Figure 13: three types of symmetry of rhombic dodecahedron
4.3 Rhombic Triacontahedral Joints
For rhombic triacontahedron, the circumstance is different as it is not 3D tessellation shape – unlike
cube and rhombic dodecahedron, rhombic triacontahedron cannot fill three dimensional space by
orienting itself.
It might be possible to combine rhombic triacontahedron and its stellation, rhombic hexecontahedron,
or other polyhedra to become 3D tessellation polyhedras. This paper will not discuss these cases as
they are not relevant to the topic.
5. Structural assessment of lattices
5.1 Lattice Design
To compare the structural performance of the different lattices composed of joints proposed above, we
defined a testing geometry, with dimension of 10m x 10m x 15m, filled with these lattices in different
densities (Figure 14). The density is controlled by the number of subdivision of the 3D space, and is
calculated as the ratio of the total volume of the targeted lattice to the total volume of the bounding
box. Notice that the section area of the linear elements in each lattice may vary, due to the fixed total
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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volume under each density. The assessment platform is created under Rhinoceros v5.0, Grasshopper
v1.0, and Karamba v1.1.
Figure 14: testing volume filled with lattice of different density (base mesh 2x2, 3x3, 4,4)
By examining carefully about the lattices in section 4, we find there are two parallel linear elements
and three parallel linear elements in each direction coming from each joint. These pairs are not
connected besides the two ends. It is reasonable to simplify these pairs into single line beam element
and modify structural properties through Karamba, i.e. moment of inertia, loading type, etc. to
simulate the scenario. The simplified lattice for cube is a normal orthogonal space truss (Figure 14).
The simplified lattice of rhombic dodecahedron joints has 3 topological forms due to different
stacking methods, since rhombic dodecahedron has 3 symmetric axes [Figure 14]. However, only two
of them can have a vertical staking topology. The third will result in an overall inclination thus will
not be discussed here due to structural stability reasons (Figure 15).
a. version 1 b. version 2
Figure 15: rhombic-dodecahedron-joint lattices with different stacking methods
5.2 FEA Analysis
We defined two load cases for the testing geometry: 1. vertical loads to simulate the behaviour of a
column; 2. shear loads to simulate structural behaviour of a cantilever (Figure 16).
a. 100 kN surface load b. 50 kN surface load
Figure 16: two load cases for the testing geometry.
We vary the base mesh from 4x4 to 16x16 (only even division) similar to the one in Figure 14 and fill
the testing geometry with three type of lattices: the orthogonal lattice (F6), and two versions of the
rhombic-dodecahedron-joint lattice (F12_v1, F12_v2). By adding the two load cases on these lattices
respectively, we can visualize the results in Figure 17:
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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Figure 17: results of two load cases
For both cases, the F6 lattice performs better than the two F12 lattices. Under column load, strain
energy of all three lattices will converge as the density increases. While under cantilever load, the
strain energy of each lattice will fluctuate around its certain value and there is no sign of convergence
shown on the chart. The result shows that though designed as new type of lattices, the F12 lattices are
not structurally efficient, especially when building large scale structures where the limits of the
materials are more likely to reach. However, the complexity and aesthetic property of this type of
lattice provides more architectural possibilities in small scale structures (Figure 18), and opportunities
and challenges for the design of the joints.
Figure 18: lattice variations
6. Conclusion
By looking back to the historical wooden puzzle and the relationship between these puzzles and their
corresponding polyhedra, this paper provides a different perspective in bridging the old wisdoms with
the computational tools. It offers a computational grammar based generation logic to generating a
certain type of symmetric polyhedral complex geometries by bridging the 2D “net” with the 3D
geometry. Unlike many polyhedral research that focus on using the geometry as domes, this paper
provides several new kind of lattices derived from the polyhedral joints using a bottom-up logic, and
verified the structural performance of these lattices.
As a new perspective of looking at symmetric polyhedral, Future works of this research may include:
the possible lattice in the rhombic triacontahedron category; the variation and application of rhombic
dodecahedral lattices; the improvement of structural performance of these lattices; the design of
joints/notches for connecting these complex lattices in reality.
0
200
400
600
800
1000
1200
1400
1600
0 0.1 0.2 0.3 0.4 0.5
Strain Energy (kNm)
Density
Column Load
F6 F12_v1 F12_v2
0
1000
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3000
4000
5000
6000
7000
8000
0 0.1 0.2 0.3 0.4 0.5
Strain Energy (kNm)
Density
Cantilever Load
F6 F12_v1 F12_v2
0.01
0.1
1
10
100
1000
10000
0 0.1 0.2 0.3 0.4 0.5
Log of Strain Energy (kNm)
Density
Column Load
F6 F12_v1 F12_v2
500
5000
0 0.1 0.2 0.3 0.4 0.5
Log of Strain Energy (kNm)
Density
Cantilever Load
F6 F12_v1 F12_v2
Proceedings of the IASS Annual Symposium 2016
Spatial Structures in the 21st Century
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