Quasi-crystalline geometry for architectrual structures
QUASI-CRYSTALLINE GEOMETRY FOR ARCHITECTURAL
Barbara Weinzierl1) and Ture Wester2)
1) Stud. Arch. 2) Associate Professor
Royal Danish Academy of Fine Arts, School of Architecture, Copenhagen, Denmark.
The purpose of the paper is to investigate some possibilities for the application of Quasi-
Crystal (QC) geometry for structures in architecture. The basis for the investigations is
A: to use the Golden Cubes (the two different hexahedra consisting of rhombic facets where
the length of the diagonals has the Golden ratio) as basic elements for aperiodic 3D
geometries (Ref.1) and
B: to raise aperiodic Penrose tilings (Ref.1) and its binary substitutions from their 2D basis
into 3D QC geometries and describe the structural behaviour for these spatial configurations.
The structural qualities are based on stability considerations for 3D structures consisting of
nodes connected with hinged bars (lattice action), and hinges plates connected by shear
connections (plate action) (Ref.2, 3). The QC structure was considered basically as an ideal
lattice structure stabilised with the necessary number of ideal plates (Ref.4).
As the “Average Valency” (VA) of the nodes (i.e. the number of bars connected to the node) is
essential for the kinematic stability of large clusters of QC units and as there seems not to
exist any proof for this number, a simple counting showed that the VA - for increasing number
of nodes - converges to be identical to a square 2D periodic lattice (Valency equal to 4) and to
a cubic 3D periodic lattice (Valency equal to 6).
An interesting quality for building purposes is the fractal quality of the binary substitution as
described in the paper.
The QC type of crystallisation was discovered and published in 1984 by Dan Schechtman
(Ref.5). The diffraction images showed 3D icosahedral symmetry, which among crystallo-
graphers were believed to be impossible and “forbidden”. 2D aperiodic tiling with fivefold
symmetry using a fat and skinny rhombic tile was found by Roger Penrose (Ref.6) a decade
before. These so-called “penrose tiling” turned out to be a 2D version of the 3D QC.
QC steel alloys, which are harder but more brittle than normal steel, are recently
commercially produced by e.g. Sandvik AB in Sweden for high quality accessories as e.g.
razor blades for electric shavers, surgery instruments etc.
Haresh Lalvani (Ref.7) and Koji Miyazaki (Ref.8) have done remarkable research on QC
geometry including the quality that aperiodic QC can be regarded as a periodic 5D cubic
lattice projected down to 3D. Tony Robbin (Ref.9) has worked with QC architecture and
created a large-scale QC sculpture at the DTU in Lyngby, Denmark.
An example of a building in which penrose patterns and QC geometry have been used as a
design idea is RMIT’s Storey Hall in Melbourne by the group ARM (Ashton Ragatt
The new QC pattern of crystallisation broke totally with the traditional conception of the
morphology of close packing cells and symmetry orders of crystals introducing aperiodic
fivefold symmetry in 3D space.
The 3D QC geometry can form a close-packing configuration consisting of two different
“distorted” cubes with identical rhombic facets. The relation between the facet's diagonals is
the Golden Ratio. All cells have therefore identical faces and identical edges, and all nodes fit
to a regular dodecahedron all oriented translationally identical. The valency of a node may
vary from 4 to 12.
A 2D penrose pattern is the projection of a specific layer in an infinite 3D QC. In a penrose
tiling there are eight different combinations of rhombs to create so-called vertex-stars, hence
one way of building a QC is to stack such vertex-stars overlapping in 3 D (Ref.1).
Another possibility is to build a Rhombic Triacontahedron (see fig.3) from ten skinny and ten
fat rhombic cubes and using them as cores for a QC, and fill the gaps with rhombic cubes in
an aperiodic order.
Another interesting geometry is found by making a binary substitution of a penrose pattern.
Each fat rhombus is replaced by a vertex-star of three fat ones and one skinny rhombus, and
each skinny rhombus is replaced by a vertex-star of one fat and two skinny rhombi (Ref.1).
This means that two binary vertex-stars may be used as building blocks of the binary sub-
stitution pattern. A binary substitution tiling is still aperiodic but has no fivefold symmetry.
An additional possibility is to make this substitution only partly by making “switches”. These
are made by substituting only one rhombus of the binary vertex-stars with another binary
vertex-star in the same scale of the pattern. This method creates stronger undulating structures
than the ordinary binary substitution, although periodic parts are unavoidable (Fig.1).
There is also a fractal possibility in doing a binary substitution of a binary substitution as you
can quot;zoomquot; in and out of the pattern, and also change the scale without changing the pattern.
The borderline between the substitutions seems to create unavoidable periodic parts in the
Fig.1 Binary substitution of a Penrose Fig.2 Structure with fractal properties
structure with switches
The average valency (VA) in an infinite equal to the internal nodes of a huge 2D Penrose
pattern has been estimated by counting the VA of an increasing number of adjacent nodes. The
VA appears to converge towards 4. This is not surprising as the valency of a periodic tiling of
either the fat or the skinny rhombus is 4, because it is topologically equal to the square grid.
The similar procedure was
performed on a QC, and the
VA seems to converge towards
6 6, which is not surprising as
3D Quasi Crystal
the valency of the periodic
5 2D Penrose
packing of the fat or the
skinny rhombic cube is 6 - just
like the cubic grid. This
procedure is of course not a
solid proof, but a reasonable
1 7 15 20 21 29 40 42 48 51 52 62 68 73 76
reliable conclusion. VA is ab-
Number of Nodes
solutely essential for the
description of the structural
Graph 1. Average Valency (VA) of 2D Penrose converge
behaviour of Penrose and QC
to 4 and a 3D Quasi Crystal converge to 6 with
increasing number of nodes
For an infinite QC where V are Vertices, E are Edges, F are Faces and C are Cells the follow-
ing can be deduced: All cells have 6 facets and each facet share 2 cells then
F=3xC and as VA is six, then E=3xV.
The extended Euler Theorem for an infinite cell-configuration in 3D space is R0-R1+R2-R3=0
where R is the number of elements and the index is the dimension of these elements, which
means that V-E+F-C=0.
When these equations are combined we will get the following geometric equations for infinite
Quasi Crystal configurations consisting of the two golden rhombic cubes:
F=3xC E=3xV E=3xC F=3xV V=C E=F
3. STRUCTURAL BEHAVIOUR
For a large QC pure lattice structure E=3xV means zero internal geometrical and structural
redundancy but global instability because of the quot;missingquot; bars along the surface. Therefore
the configuration must be supplied with extra stabilising elements to reach kinematic stability.
There are of course many possibilities for placing these extra elements in order to stabilise the
structure. For a given QC pure lattice structure the Neutral Method (ref.4) the number of extra
stabilising plates or bars can easily be found.
The Neutral Method calculates the redundancy directly in a simple way, and is based on
Moebius' equation for kinematic stability for pure 3D lattice structures B+S ≥ 3xN; where B,
S and N is the number of Bars (edges), Supports and Nodes (vertices).
An unstable QC pure lattice configuration can be stabilised by extra bars - or maybe more
obviously - by inserting (identical) plates in the rhombic meshes. If the stabilisers are applied
to the QC-rhombs, a plate will have the same stabilising effect as a diagonal bar. If plates are
used they may be arranged in such a way that they articulate, emphasise and stress the
architectural idea of the structure.
According to the above-mentioned Neutral Method for the Golden Rhombic Dodecahedron as
QC, there are minimum 10 extra elements needed, for the Golden Rhombic Icosahedron as
QC minimum16 elements are needed. The Rhombic Triacontahedron as QC needs minimum
24 extra elements. As the surface of the Rhombic Triacontahedron, which is a semi-regular
polyhedron of second order, has 30 facets means that all 24 extra plates can be put on its
surface to stabilise also the inside of it and still keep 6 meshes open. There are many other
solutions for arranging the stabilising plates e.g. to emphasise horizontal or other parallel
planes (fig.3) – or flows of spirals etc.
For an quot;emptyquot; Rhombic Triacontahedron envelope with no internal nodes, all rhombs must
be supplied with a stabilising plate, and therefore closed.
For a space-truss consisting of one layer of QC cells the stabilising plates can be arranged in
rows along the so-called quot;snakesquot; (Ref.1 p.163), creating intersecting tubes. Except for the
ends, these tubes have no stabilising elements inside of them, as this would not add stability to
the system. In the more open parts of the spaceframe it is possible to either arrange the rest of
the needed plates or bars in a vertically standing position to the pattern, for the use of creating
holes or windows, or in the layer of the pattern to almost close the surface of the frame as a
roof or wall. (Fig.4)
Fig.3 A QC lattice Rhombic Triaconta- Fig.4 Stable single layer QC space-truss
hedron with stabilising plates as plate stabilised lattice.
Fig.5 Single-layer dome with torus-ring as Fig.6 Single-layer quot;Bubblequot; as combin-
combination of open plate and lattice. ation of open plate and lattice.
A single-layer envelope can be stabilised with a QC torus ring at its base if both are
constructed as a structurally closed combined lattice and plate structure (Fig.5), here shown
with plates as open frames. For a closed single-layer envelope (or quot;bubblequot;) all facets must
constructed as a structurally closed combined lattice and plate structure (Fig.6).
The aperiodicity of a QC configuration facilitates a huge number of different arrangements of
the rhombohedra. Therefore it may adapt to many different architectural requirements as
quality of space, scale, function, form, adaptability to landscape etc. not only regarding the
rhombohedra sizes, the overall shape and framework, but also the configuration and the
structural material of the added stabilising elements.
1) However, there are only eight different vertex stars and they repeat themselves within a
certain distance to each other, so there is a limited but very large set of space combinations
2) The fractal possibilities of QC’s are especially important for being able to use QC
structures in all scales and for all uses, from a skyscraper to a trash-bin, and also for the
detailing within a building (Ref.8).
3) The two unit rhombohedra can - as indicated earlier - also be separately arranged in a
strictly periodic way forming a distorted cubic grid.
As all three above-mentioned options can be combined in one configuration, there are an
almost countless number of possible combinations for architectural uses. It is up to the
Architect or Urban planner to investigate and select between all these possibilities.
Architecture using QC geometry is really not a stiff one-way modular kit - which would
rather limit - but is able to give more freedom to create structural and morphological
4.1 Architecture theory
The theory of post-modern deconstruction after Jacques Derrida (Ref.11) fits very well with
QC Architecture, as it can be seen as a displacement of existing cubic architecture with
twofold and fourfold symmetries with a representation of higher dimensional cubes which in
3D space physically visualises the gap between the binary terms of our western occidental
4.2 Urban scale
There are mainly three types of geometry in historical and modern city plans:
- the linear type, along a communication and transportation line e.g. a road or a river
- the concentric/radial type, around a fortification e.g. on a hill; or an important crossing of
roads or rivers
- the grid type, which already needs supervised and ordered urban planning from the
foundation of the city on e.g. Chinese cities of one planning concept.
The first two types are a logic reaction to the environment. The third type is a rational
decision of human planning and the need of ordering nature for different reasons.
It is indeed very interesting, that the diffraction pattern of a QC with its tenfold symmetry is
very similar to radial city plans. One idea is to superimpose a penrose pattern on a radial city
plan, so to say to show the virtual QC lying above the city, creating the city-plan with its
The penrose pattern would in general be a way of planning new city-grids, still ordering
spaces to a certain extent, but also regaining the liveliness of a more flexible and less
prohibiting grid like twofold or fourfold symmetry. Its “snakes” could be the streets; a vertex
star consisting of five fat rhombs in fivefold symmetry could be a street-circus or a plaza.
Chaos equals with Nature, and Order with Culture and Civilisation in our society. Cities grow
from village to megalopolis, become huge urban areas which need to be restructured to give a
more “natural” feeling of environment to the people that live in them, as real nature gets
pushed back to the worlds most hostile regions as deserts and beyond the polar circle.
The fractal possibilities by binary substitution can be used for constituting spaces in different
scales like the city centre, or the centre of a suburb and still the overall grid would not be
broken. Today’s cities are a mix of individuals from different cultures, countries, religions,
colours and continents. Therefore it has become necessary not to symbolise the equality of
individuals in a democratic system by a simple order, equal for all its users, but a complex
and diverse pattern with possibilities for all the different individuals coexisting in one place.
Architecture has always been aiming to control spaces and therefore also actions and users.
We need to give freedom and possibilities to develop now, so QC buildings on a penrose grid
would combine this freedom without totally loosing control of the city grid, and slipping into
anarchy - the essence of quasi crystallisation.
4.3 Wide Span and High-rise Buildings
QC buildings in a large scale can be used to room all kinds of uses and functions. The tilting
of the walls of QC buildings gives tension to the perception of the street space, as they seem
unstable and to come towards the pedestrian, not like the usual psychological stability of
vertical lines in buildings reacting to earth’s gravity. An example for this type of tension,
although not a QC, is the Royal Library Extension in Copenhagen by the Danish architects
quot;Smidt, Hammer & Lassen, a distorted cube also called the Black Diamond. QC buildings are
by this a real statement of human design and its interaction, and a starting point to overcome
The angles of the tilting floors can be used for large auditoriums, e.g. for an opera, concert-
halls or universities, horizontal split-levels can be mounted in a QC space-grid with connected
air-spaces, as horizontal plates hovering through space.
A large Rhombic Triacontahedron with its five horizontal planes spiralling around the axis
would be one possibility for a communication organisation within such a building. The stairs
lead from one horizontal floor to the next, the elevator-shafts along the axis through the
middle, and the auditoriums leading down and away from the floors opening to the outside
views into all directions. (Fig.7)
QC domes and space-frames can be mounted over street spaces or plazas to create “Galleries”
looking like reptile skins growing over the city’s open spaces. Single layer QC structure can
also react and move with or against the landscape, creating amorphous convex and concave
spaces between the structure and the ground and in the quot;foldingsquot; of the undulating QC
surface. This can be used to cover large areas with temporary structures e.g. for fairs (Fig.8).
Fig.7 Spaces within a Rhombic Triaconta- Fig.8 Single-layer structure, stabilised
hedron with combined lattice and plate action.
The advantage is, that as ordinary twofold and fourfold building structures usually need flat
ground, and do not react at all to their environment with their unnatural shape, QC structures
are possible literally everywhere, not violating the Nature to fit them.
The fractal possibilities can be used to shift the scale within a building from big halls to
normal room size areas to an open space with small-scale space-frames above it. For example
this could structure an airport from the check-in hall to offices, waiting rooms, shops, and
communication and transportation areas.
It can also be used to create large buildings with about brick-size QC rhombohedra, as if the
rooms of the building where just carved out of a gigantic QC. The walls and floors of such
rooms would then be undulating in a small-scaled pattern, which would create the texture of
an organic grown structure.
QC structures for dwellings can be used in all the existing building types, as detached houses,
single houses, for rather low suburban housing areas, mixed with garden spaces, or middle-
scale housing blocks, with public park areas. If one rhombohedron is one room, there are e.g.
the eight vertex-stars that could be a way of organisation. In the penrose pattern the quot;snakesquot;
could be a possibility for the detached houses. The quot;snakesquot; could also be the streets, as the
borderlines for the housing clusters in various shapes with inner courtyards and gardens.
Another way is to cluster the single rhombohedra to a 3D QC into all directions of three-
Fig.10 Irregular stacking of rhombic
Fig.9 Regular stacking of rhombic
Fig.11 QC structure with growth Defect Fig.12 Interior Spaces of a QC
A Rhombic Triacontahedron could be the minimal living unit in a 3D cluster, with its
rhombohedra as the different rooms, with some split-levels or stepped podiums to create some
necessary straight floors. As there is regular and irregular ways of stacking these units and
attaching them to each other, houses can be adapted to different family sizes. The regular
stacking (Fig.9) creates holes and is not a complete QC aperiodic structure. However these
gaps can be used for courtyards and gardens, or shafts to bring daylight also to the lower
levels of higher clusters. The irregular stacking (Fig.10) creates a 3 D QC grid, in which gaps
for the above mentioned uses can be made by leaving out some rhombohedra.
There is also a mathematical algorithm for producing penrose patterns with incorporated local
growth defects, which yet retain the semblance of fivefold symmetry (ref.12, pp.86-96). These
non-tileable parts could grow through a real 3D QC like wormholes (Fig.11). QC-colonies
have almost no limits in how high or how far they can go, as each Rhombic Triacontahedron
can be constructed as a stable unit and therefore able to be a rigid building block in a larger
configuration. The QC grid in the cluster is supporting itself and all its parts (Fig.12).
It seems that the special QC geometry can be used in a new and advantageous way as a grid at
all levels for architectural structures, town planning, interior design etc. not least because of
the fascinating relation to both chaos and order.
Edwin A. Abbott's quot;Flatland - a romance of many dimensionsquot;, written in 1884 (ref.13)
inspires to consider if there might be a 5D world with 5D beings somewhere in our universe -
or rather beyond it - and how it might look like. As the 2D creatures in Flatland only perceive
two dimensions as living in a surface, we do not in our world have any sense which can
perceive a higher dimension than our usual three. As QC’s are regular and periodic in 5D
space a QC is just another quot;boringquot; cubic building for a 5D inhabitant It appears extremely
inspiring that we can build interesting architecture of their 3D representations. Higher
dimensional spaces and structures might be used as perfect tools for the architecture of the
third millennium. Only time will show if this potential is going to be utilised in our societies,
and not only as futuristic ideas for space stations and moon settlements. It is about time to
start something new.
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