Topologic: tools to explore architectural topology

Abstract

Buildings enclose and partition space and are built from assemblies of
connected components. The many different forms of spatial and material
partitioning and connectedness found within buildings can be represented by
topology. This paper introduces the ‘Topologic’ software library which
integrates a number of architecturally relevant topological concepts into a
unified application toolkit.

The goal of the Topologic toolkit is to support the creation of the lightest, most
understandable conceptual models of architectural topology. The formal
language of topology is well-matched to the data input requirements for
applications such as energy simulation and structural analysis. In addition, the
ease with which these lightweight topological models can be modified
encourages design exploration and performance simulation at the conceptual
design phase.

A challenging and equally interesting question is how can the formal language
of topology be used to represent architectural concepts of space which have
previously been described in rather speculative and subjective terms?

Full text

316 AAG2018 317
Topologic: Tools to
explore architectural
topology
Robert Aish, Wassim Jabi, Simon Lannon, Nicholas
Mario Wardhana, Aikaterini Chatzivasileiadi
Robert Aish
robert.aish@ucl.ac.uk
Bartlett School of Architecture, University College London, United Kingdom
Wassim Jabi
jabiw@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Simon Lannon
lannon@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Nicholas Mario Wardhana
wardhanan@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Aikaterini Chatzivasileiadi
chatzivasileiadia@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Keywords:
Non-manifold topology, idealised model, material model
316 AAG2018
317
Abstract
Buildings enclose and partition space and are built from assemblies
of connected components. The many different forms of spatial and
material partitioning and connectedness found within buildings can
be represented by topology. This paper introduces the ‘‘Topologic’’
software library which integrates a number of architecturally relevant
topological concepts into a unified application toolkit.
The goal of the Topologic toolkit is to support the creation of the
lightest, most understandable conceptual models of architectural
topology. The formal language of topology is well-matched to the data
input requirements for applications such as energy simulation and
structural analysis. In addition, the ease with which these lightweight
topological models can be modified encourages design exploration and
performance simulation at the conceptual design phase.
A challenging and equally interesting question is how can the formal
language of topology be used to represent architectural concepts of
space which have previously been described in rather speculative and
subjective terms?
1. Introduction
This paper focusses on the conceptual issues surrounding the use of
topology in architecture. It builds on previous research and proof of
concept studies (Aish and Pratap 2013; Jabi 2014; Jabi et al. 2017). Other
concurrently published papers describe in greater detail the implemen-
tation of the Topologic toolkit and specific applications of Topologic in
building analysis and simulation (Jabi et al. 2018; Chatzivasileiadi, Lannon,
et al. 2018; Wardhana et al. 2018).
Topology and in particular non-manifold topology are vast subjects that
span algebra, geometry and set theory. It is beyond the scope of this paper
to delve into the mathematical constructs and proofs that precisely define
non-manifold topology. Topology has applications in biology, medicine,
computer science, physics and robotics among others. Since the motiva-
tion for this research is to address the needs of architects and engineers,
this research focusses on a specific application of non-manifold topology
in the representation of significant spatial relationships in the design of
buildings using computer-aided three-dimensional geometric processing.

316 AAG2018
317
Topologic: Tools to
explore architectural
topology
Robert Aish, Wassim Jabi, Simon Lannon, Nicholas
Mario Wardhana, Aikaterini Chatzivasileiadi
Robert Aish
robert.aish@ucl.ac.uk
Bartlett School of Architecture, University College London, United Kingdom
Wassim Jabi
jabiw@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Simon Lannon
lannon@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Nicholas Mario Wardhana
wardhanan@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Aikaterini Chatzivasileiadi
chatzivasileiadia@cardiff.ac.uk
Welsh School of Architecture, Cardiff University, United Kingdom
Keywords:
Non-manifold topology, idealised model, material model
316 AAG2018 317
Abstract
Buildings enclose and partition space and are built from assemblies
of connected components. The many different forms of spatial and
material partitioning and connectedness found within buildings can
be represented by topology. This paper introduces the ‘‘Topologic’’
software library which integrates a number of architecturally relevant
topological concepts into a unified application toolkit.
The goal of the Topologic toolkit is to support the creation of the
lightest, most understandable conceptual models of architectural
topology. The formal language of topology is well-matched to the data
input requirements for applications such as energy simulation and
structural analysis. In addition, the ease with which these lightweight
topological models can be modified encourages design exploration and
performance simulation at the conceptual design phase.
A challenging and equally interesting question is how can the formal
language of topology be used to represent architectural concepts of
space which have previously been described in rather speculative and
subjective terms?
1. Introduction
This paper focusses on the conceptual issues surrounding the use of
topology in architecture. It builds on previous research and proof of
concept studies (Aish and Pratap 2013; Jabi 2014; Jabi et al. 2017). Other
concurrently published papers describe in greater detail the implemen-
tation of the Topologic toolkit and specific applications of Topologic in
building analysis and simulation (Jabi et al. 2018; Chatzivasileiadi, Lannon,
et al. 2018; Wardhana et al. 2018).
Topology and in particular non-manifold topology are vast subjects that
span algebra, geometry and set theory. It is beyond the scope of this paper
to delve into the mathematical constructs and proofs that precisely define
non-manifold topology. Topology has applications in biology, medicine,
computer science, physics and robotics among others. Since the motiva-
tion for this research is to address the needs of architects and engineers,
this research focusses on a specific application of non-manifold topology
in the representation of significant spatial relationships in the design of
buildings using computer-aided three-dimensional geometric processing.

318 AAG2018 319
We can contrast this approach with more conventional representations
of buildings as a collection of physical building components, typically
modelled as manifold solids, as demonstrated by Building Information
Modelling (BIM) applications. While BIM can be used to model the physical
structure of the building, architecture is usually conceived in terms of an
overall form and a series of related spatial enclosures (Curtis 1996). This
spatial conceptualization is a key aspect of architectural design because
it directly anticipates how the resulting building will be experienced.
However, there are no practical design tools which support the creation
of this spatial representation of architecture. Non-manifold topology
is ideally suited to create a lightweight representation of a building as
an external envelope and the subdivision of the enclosed space into
separate spaces such as rooms, building storeys, cores, atria, etc. This
lightweight representation also matches the input data requirements for
important analysis and simulation applications, such as energy analysis,
(Ellis, Torcellini, and Crawley 2008).
Conventional BIM applications, in contrast, do not explicitly model the
enclosure of space. Although it might be possible to indirectly infer the
enclosed spaces from the position of the physical building components,
the fidelity of this representation depends on the precise connectivity of
the bounding physical components, which cannot be relied upon. Even
if this approach was viable, the level of detail of BIM models is often too
complex for this type of analysis (Maile et al. 2013). Detailed BIM models
are also cumbersome to change which may inhibit design exploration at
the conceptual design stage.
One option might be to explore spatial modelling with existing solid
modelling applications. However most of these applications are based
on conventional manifold modelling techniques and do not support
non-manifold topology. Indeed, many regular manifold modelling applica-
tions treat non-manifold topology as an error condition.
The objective of this research is to develop design tools based on
precise topological principles but presented in ways which are under-
standable by architectural users who may have little previous experience
of topology. The intention is that Topologic can be an effective interme-
diary between the abstract world of topology and the practical world of
architecture and building engineering.
318 AAG2018
319
2. Background
2.1 The distinction between manifold and non-manifold
Topology
In a previous paper (Aish and Pratap 2013) the following distinctions
were made between manifold and non-manifold topology:
“A 3D manifold body has a boundary that separates the enclosed
solid from the external void. The boundary is composed of faces,
which have (interior) solid material on one side and the (exterior) void
on the other. In practical terms, a manifold body without internal voids
can be machined out of a single block of material.”
“A non-manifold body also has a boundary [composed of faces]
that separates the enclosed solid from the external void. Faces are
either external [separating the interior (enclosed space) from the
exterior (void)] or internal [separating one enclosed space (or cell)
from another]. Furthermore, a non-manifold solid can have edges
where more than two faces meet.”
2.2 The distinction between an idealized and a
material model
One of the key themes which runs through this research is the distinc-
tion between an ‘‘idealised’’ model (of a building) and a ‘‘material’’ model
of the physical building components. An early demonstration of this
principle was made in 1997 (Aish 1997) and further developed (Hensen
and Lamberts 2012).
Typically, idealised models are far less detailed than material models,
therefore lighter and more easily edited. In addition, the different
topological components of the idealised model (faces, edges, vertices)
can be used as the ‘‘supports’’ for related building components in the
material model. The connectivity of the components in the material
model need not be directly modelled. Instead this connectivity can be
represented through the topology of the idealised model.
2.3 Previous research
The case for non-manifold topology as well as its data structures and
operators for geometric modelling were comprehensively set out by
(Weiler 1986). In his introduction, Weiler explains why non-manifold
topology is needed:

318 AAG2018
319
We can contrast this approach with more conventional representations
of buildings as a collection of physical building components, typically
modelled as manifold solids, as demonstrated by Building Information
Modelling (BIM) applications. While BIM can be used to model the physical
structure of the building, architecture is usually conceived in terms of an
overall form and a series of related spatial enclosures (Curtis 1996). This
spatial conceptualization is a key aspect of architectural design because
it directly anticipates how the resulting building will be experienced.
However, there are no practical design tools which support the creation
of this spatial representation of architecture. Non-manifold topology
is ideally suited to create a lightweight representation of a building as
an external envelope and the subdivision of the enclosed space into
separate spaces such as rooms, building storeys, cores, atria, etc. This
lightweight representation also matches the input data requirements for
important analysis and simulation applications, such as energy analysis,
(Ellis, Torcellini, and Crawley 2008).
Conventional BIM applications, in contrast, do not explicitly model the
enclosure of space. Although it might be possible to indirectly infer the
enclosed spaces from the position of the physical building components,
the fidelity of this representation depends on the precise connectivity of
the bounding physical components, which cannot be relied upon. Even
if this approach was viable, the level of detail of BIM models is often too
complex for this type of analysis (Maile et al. 2013). Detailed BIM models
are also cumbersome to change which may inhibit design exploration at
the conceptual design stage.
One option might be to explore spatial modelling with existing solid
modelling applications. However most of these applications are based
on conventional manifold modelling techniques and do not support
non-manifold topology. Indeed, many regular manifold modelling applica-
tions treat non-manifold topology as an error condition.
The objective of this research is to develop design tools based on
precise topological principles but presented in ways which are under-
standable by architectural users who may have little previous experience
of topology. The intention is that Topologic can be an effective interme-
diary between the abstract world of topology and the practical world of
architecture and building engineering.
318 AAG2018 319
2. Background
2.1 The distinction between manifold and non-manifold
Topology
In a previous paper (Aish and Pratap 2013) the following distinctions
were made between manifold and non-manifold topology:
“A 3D manifold body has a boundary that separates the enclosed
solid from the external void. The boundary is composed of faces,
which have (interior) solid material on one side and the (exterior) void
on the other. In practical terms, a manifold body without internal voids
can be machined out of a single block of material.”
“A non-manifold body also has a boundary [composed of faces]
that separates the enclosed solid from the external void. Faces are
either external [separating the interior (enclosed space) from the
exterior (void)] or internal [separating one enclosed space (or cell)
from another]. Furthermore, a non-manifold solid can have edges
where more than two faces meet.”
2.2 The distinction between an idealized and a
material model
One of the key themes which runs through this research is the distinc-
tion between an ‘‘idealised’’ model (of a building) and a ‘‘material’’ model
of the physical building components. An early demonstration of this
principle was made in 1997 (Aish 1997) and further developed (Hensen
and Lamberts 2012).
Typically, idealised models are far less detailed than material models,
therefore lighter and more easily edited. In addition, the different
topological components of the idealised model (faces, edges, vertices)
can be used as the ‘‘supports’’ for related building components in the
material model. The connectivity of the components in the material
model need not be directly modelled. Instead this connectivity can be
represented through the topology of the idealised model.
2.3 Previous research
The case for non-manifold topology as well as its data structures and
operators for geometric modelling were comprehensively set out by
(Weiler 1986). In his introduction, Weiler explains why non-manifold
topology is needed:

320 AAG2018 321
‘‘A unified representation for combined wireframe, surface, and solid
modelling by necessity requires a non-manifold representation, and is
desirable since it makes it easy to use the most appropriate modelling
form (or combination of forms) in a given application without requiring
representation conversion as more information is added to the model.’’
Non-manifold topology allows an expansion of the regular Boolean
operations of union, difference, and intersection. This expanded set
includes operators such as merge, impose, and imprint. For a full
description of non-manifold operators, please consult (Masuda 1993).
Representing space and its boundary was the focus of early research
into BIM (Björk 1992; Chang and Woodbury 1997) and into ‘‘product
modelling’’ (PDES/STEP) (Eastman and Siabiris 1995) and was proposed
as an approach to the representation of geometry definition for input to
Building Performance Simulation in the early design stages (Hui and
Floriani 2007; Jabi 2016). However, this is not emphasised in modern
BIM software where the building fabric is represented through manifold
geometry and energy models from are derived from the fabric models.
Separately, non-manifold topology has been successfully used in the
medical field to model complex organic structures with multiple internal
zones (Nguyen 2011; Bronson, Levine, and Whitaker 2014).
Our focus is to create a schema which separates abstract topological
concepts from domain specific and pragmatic concerns of architecture,
engineering and construction. We maintain this separation, but also explore
important connections: how buildings can be represented by topology and
how a topological representation can potentially assist architectural users
in the conceptualisation and analysis of new buildings. Therefore, our
focus is not to create new non-manifold data structures, but rather to har-
ness existing geometry and topology kernels in an innovative way; indeed,
it is completely feasible that the Topologic schema could be implemented
with different data structures or with different kernels.
A comprehensive and systematic survey of topological modelling
kernels, which support non-manifold topology, was carried out by the
authors and published elsewhere (Chatzivasileiadi, Wardhana, et al.
2018). Features and capabilities of kernels were compared in order to
make an informed decision regarding what underlying kernel to use.
Popular geometric kernels, such as CGAL, were discounted due to their
inability to represent higher dimensional entities such as CellComplexes
and for their more limited set of irregular Boolean operations.
320 AAG2018
321
3. The Topologic toolkit
The core Topologic software is developed in C++ using Open Cascade
(https://www.opencascade.com/) with specifi c C++/CLI variants deve-
loped for different visual data fl ow programming environments (Wardhana
et al. 2018). Topologic integrates a number of architecturally relevant
topological concepts into a unifi ed application toolkit. The features and
applications of Topologic are summarised in Figure 1 and Figure 2.
Figure 1: The Topologic application toolkit summarised in eight key
points.

320 AAG2018
321
‘‘A unified representation for combined wireframe, surface, and solid
modelling by necessity requires a non-manifold representation, and is
desirable since it makes it easy to use the most appropriate modelling
form (or combination of forms) in a given application without requiring
representation conversion as more information is added to the model.’’
Non-manifold topology allows an expansion of the regular Boolean
operations of union, difference, and intersection. This expanded set
includes operators such as merge, impose, and imprint. For a full
description of non-manifold operators, please consult (Masuda 1993).
Representing space and its boundary was the focus of early research
into BIM (Björk 1992; Chang and Woodbury 1997) and into ‘‘product
modelling’’ (PDES/STEP) (Eastman and Siabiris 1995) and was proposed
as an approach to the representation of geometry definition for input to
Building Performance Simulation in the early design stages (Hui and
Floriani 2007; Jabi 2016). However, this is not emphasised in modern
BIM software where the building fabric is represented through manifold
geometry and energy models from are derived from the fabric models.
Separately, non-manifold topology has been successfully used in the
medical field to model complex organic structures with multiple internal
zones (Nguyen 2011; Bronson, Levine, and Whitaker 2014).
Our focus is to create a schema which separates abstract topological
concepts from domain specific and pragmatic concerns of architecture,
engineering and construction. We maintain this separation, but also explore
important connections: how buildings can be represented by topology and
how a topological representation can potentially assist architectural users
in the conceptualisation and analysis of new buildings. Therefore, our
focus is not to create new non-manifold data structures, but rather to har-
ness existing geometry and topology kernels in an innovative way; indeed,
it is completely feasible that the Topologic schema could be implemented
with different data structures or with different kernels.
A comprehensive and systematic survey of topological modelling
kernels, which support non-manifold topology, was carried out by the
authors and published elsewhere (Chatzivasileiadi, Wardhana, et al.
2018). Features and capabilities of kernels were compared in order to
make an informed decision regarding what underlying kernel to use.
Popular geometric kernels, such as CGAL, were discounted due to their
inability to represent higher dimensional entities such as CellComplexes
and for their more limited set of irregular Boolean operations.
320 AAG2018 321
3. The Topologic toolkit
The core Topologic software is developed in C++ using Open Cascade
(https://www.opencascade.com/) with specifi c C++/CLI variants deve-
loped for different visual data fl ow programming environments (Wardhana
et al. 2018). Topologic integrates a number of architecturally relevant
topological concepts into a unifi ed application toolkit. The features and
applications of Topologic are summarised in Figure 1 and Figure 2.
Figure 1: The Topologic application toolkit summarised in eight key
points.

322 AAG2018 323
Figure 2: Boolean Operations implemented in Topologic.
322 AAG2018
323
3.1 Class hierarchy
The Topologic class hierarchy is designed to provide the architectural
end-user with a conceptual understanding of topology. It also functions
as an ‘‘end-user programmers’ interface” (EDPI). This user-oriented
class hierarchy is distinct to the implementation-oriented class hierarchy
within the Topologic core.
The Topologic superclass (Fig. 1, section 1) is abstract and imple-
ments constructors, properties and methods including a set of Boolean
operators. These operators can be used with both manifold and non-
manifold topology (Fig. 2). Topologic implements the expected concepts
such as: Vertex, Edge, Wire, Face, Shell, and Cell. The interesting addi-
tional topological concepts are:
CellComplex which is a contiguous collection of Cells and is
non-manifold.
Cluster which is a universal construct and allows any combination
of topologies, including other ‘‘nested” Clusters, to be represented. A
Cluster may represent non-contiguous, unrelated topologies of different
dimensionalities.
3.2 Topological relationships
Topologic supports the building and querying of three different types of
topological relationships (Fig. 1, section 2)
Hierarchical relationships: between topological entities of different
dimensionality. These relationships are created when a higher dimensional
topology construct is composed from a collection of lower dimensional
topologies. Subsequently the compositional relationships may be queried:
cellComplexes =
vertex.Edges.Wires.Faces.Shells.Cells.
CellComplexes;
Conversely, the decompositional relationships may also be queried, for
example from higher dimensional topologies down to the constituent
collections of lower dimensional topologies:
vertices = cellComplex.Vertices;

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Figure 2: Boolean Operations implemented in Topologic.
322 AAG2018 323
3.1 Class hierarchy
The Topologic class hierarchy is designed to provide the architectural
end-user with a conceptual understanding of topology. It also functions
as an ‘‘end-user programmers’ interface” (EDPI). This user-oriented
class hierarchy is distinct to the implementation-oriented class hierarchy
within the Topologic core.
The Topologic superclass (Fig. 1, section 1) is abstract and imple-
ments constructors, properties and methods including a set of Boolean
operators. These operators can be used with both manifold and non-
manifold topology (Fig. 2). Topologic implements the expected concepts
such as: Vertex, Edge, Wire, Face, Shell, and Cell. The interesting addi-
tional topological concepts are:
CellComplex which is a contiguous collection of Cells and is
non-manifold.
Cluster which is a universal construct and allows any combination
of topologies, including other ‘‘nested” Clusters, to be represented. A
Cluster may represent non-contiguous, unrelated topologies of different
dimensionalities.
3.2 Topological relationships
Topologic supports the building and querying of three different types of
topological relationships (Fig. 1, section 2)
Hierarchical relationships: between topological entities of different
dimensionality. These relationships are created when a higher dimensional
topology construct is composed from a collection of lower dimensional
topologies. Subsequently the compositional relationships may be queried:
cellComplexes =
vertex.Edges.Wires.Faces.Shells.Cells.
CellComplexes;
Conversely, the decompositional relationships may also be queried, for
example from higher dimensional topologies down to the constituent
collections of lower dimensional topologies:
vertices = cellComplex.Vertices;

324 AAG2018 325
or
vertices =
cellComplex.Cells[n].Shells[n].Faces[n].Wires[n].
Edges[n].Vertices;
Lateral relationships: these occur within a topological construct when
the constituents share common topologies of a lower dimensionality.
adjacentCells = cellComplex.Cells[n].AdjacentCells;
adjacentFaces = shell.Faces[n].AdjacentFaces;
Connectivity: The path between two topologies can be queried.
path = topology.PathTo(otherTopology);
3.3 Idealised representations
Three different idealized models are considered (Fig. 1, section 4)
Energy Analysis: a CellComplex can represent the partitioning and
adjacency of spaces and thermal zones.
Structural Analysis: a Cluster can be used to represent a mixed-dimen-
sional model, with Faces representing structural slabs, blade columns
and shear walls, Edges representing structural columns and Cells
representing building cores.
Digital Fabrication Analysis: a CellComplex can represent the design
envelope where topology can inform the shape and interface between
deposited material (Jabi et al. 2017).
Circulation Analysis: a dual graph of a CellComplex can represent the
connectedness of spaces.
324 AAG2018
325
3.4 Cell as a space or as a solid
A Cell is defined as a closed collection of faces, bounding a 3D region.
However, this same topology can represent two distinctly different
application concepts: a Solid and a Space (Fig. 1, section 5). A Solid is
interpreted as a single homogeneous region of material and its boundary
defines where the material ends and the void begins. This is the inter-
pretation of the Cell as used in ‘‘Solid Modelling’’ and BIM applications.
A Space is a more abstract concept and may include an implied
conceptual distinction between the material which is ‘‘contained’’
(represented by the enclosed 3D region of the Cell) and the ‘‘container’’
(represented by the Faces of the Cell). A Face may represent a
boundary which is intended to be materialized with a defined thickness
or may represent a ‘‘virtual’’ (e.g. adiabatic) barrier which is not intended
to be materialized.
Solids and Spaces have exactly the same Cell topology, but the
domain specific semantics and expected behaviour of this topology may
be different. Consider a boolean ‘‘difference’’ operation representing
a hole drilled into a Cell (as a solid). A new part of the Cell boundary
would be created, but the result would still be a Cell.
What result would the user expect if the same Cell represented a
Space? Would the boolean ‘‘difference’’ only apply to a specific Face
(as part of the Space’s boundary)? Would the user expect the boolean
operation to create an internal boundary within the selected Face?
Would the user expect this operation to destroy the integrity of the
enclosure, changing the Cell into an open Shell?
This example helps to explain the difference between a material
model (the Cell as a Solid) and an idealised model (the Cell as a
Space). More generally this example demonstrates the need for the
architectural users to customise the application of abstract topological
concepts with the domain semantics which suits their purpose.
This relationship between application semantics and abstract con-
cepts works both ways. Sometimes more generally applicable concepts
emerge by abstracting ideas from other specialist domains. For example,
the concept of a topological Cell may have originated as an abstracted
analogy of a biological cell, with similarities in terms of the homogeneity
and continuity of the contained 3D region and the role of the cell wall as
a closed container with selective permeability (Fig. 3).

324 AAG2018
325
or
vertices =
cellComplex.Cells[n].Shells[n].Faces[n].Wires[n].
Edges[n].Vertices;
Lateral relationships: these occur within a topological construct when
the constituents share common topologies of a lower dimensionality.
adjacentCells = cellComplex.Cells[n].AdjacentCells;
adjacentFaces = shell.Faces[n].AdjacentFaces;
Connectivity: The path between two topologies can be queried.
path = topology.PathTo(otherTopology);
3.3 Idealised representations
Three different idealized models are considered (Fig. 1, section 4)
Energy Analysis: a CellComplex can represent the partitioning and
adjacency of spaces and thermal zones.
Structural Analysis: a Cluster can be used to represent a mixed-dimen-
sional model, with Faces representing structural slabs, blade columns
and shear walls, Edges representing structural columns and Cells
representing building cores.
Digital Fabrication Analysis: a CellComplex can represent the design
envelope where topology can inform the shape and interface between
deposited material (Jabi et al. 2017).
Circulation Analysis: a dual graph of a CellComplex can represent the
connectedness of spaces.
324 AAG2018 325
3.4 Cell as a space or as a solid
A Cell is defined as a closed collection of faces, bounding a 3D region.
However, this same topology can represent two distinctly different
application concepts: a Solid and a Space (Fig. 1, section 5). A Solid is
interpreted as a single homogeneous region of material and its boundary
defines where the material ends and the void begins. This is the inter-
pretation of the Cell as used in ‘‘Solid Modelling’’ and BIM applications.
A Space is a more abstract concept and may include an implied
conceptual distinction between the material which is ‘‘contained’’
(represented by the enclosed 3D region of the Cell) and the ‘‘container’’
(represented by the Faces of the Cell). A Face may represent a
boundary which is intended to be materialized with a defined thickness
or may represent a ‘‘virtual’’ (e.g. adiabatic) barrier which is not intended
to be materialized.
Solids and Spaces have exactly the same Cell topology, but the
domain specific semantics and expected behaviour of this topology may
be different. Consider a boolean ‘‘difference’’ operation representing
a hole drilled into a Cell (as a solid). A new part of the Cell boundary
would be created, but the result would still be a Cell.
What result would the user expect if the same Cell represented a
Space? Would the boolean ‘‘difference’’ only apply to a specific Face
(as part of the Space’s boundary)? Would the user expect the boolean
operation to create an internal boundary within the selected Face?
Would the user expect this operation to destroy the integrity of the
enclosure, changing the Cell into an open Shell?
This example helps to explain the difference between a material
model (the Cell as a Solid) and an idealised model (the Cell as a
Space). More generally this example demonstrates the need for the
architectural users to customise the application of abstract topological
concepts with the domain semantics which suits their purpose.
This relationship between application semantics and abstract con-
cepts works both ways. Sometimes more generally applicable concepts
emerge by abstracting ideas from other specialist domains. For example,
the concept of a topological Cell may have originated as an abstracted
analogy of a biological cell, with similarities in terms of the homogeneity
and continuity of the contained 3D region and the role of the cell wall as
a closed container with selective permeability (Fig. 3).

326 AAG2018 327
Figure 3: The cell wall as a separator and as a connector, in biology and
in architecture (with acknowledgement to Wix, 1994).
3.5 Apertures and Contexts
A Face may have internal boundaries which may represent an aperture.
The location of an aperture within the host Face is defined by a Context.
Apertures can represent windows or doors. (Fig. 1, section 6) (The
representation of Apertures is discussed in more detail in section 4.4
‘‘Regional Topology’’)
3.6 Material representations
While all Cells have a common topology (a closed 3D region bounded
by Faces) different configurations of Cells may be generated from
different types of foundational topologies using different geometric
operations (Fig. 1, section 7), for example:
Point location connector components: may be based on Vertices.
Linear components such as columns or beams: may be based on Edges
(or Wires) using operations where a cross section Wire is extruded
along a path.
Area based components such as slabs, floors, walls may be based on
Faces: using offset operations with a specified thickness and direction.
326 AAG2018
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Volume based components such as a containment vessel may be based
on Cells using thin-shell operations and a specified wall thickness.
Conformal cellular structures, used in 3D printing, may be based on
CellComplexes.
Complex sub-assemblies of material components can be modelled as
Clusters.
3.7 Integration of idealized and material models
The integrated BIM model uses the idealized non-manifold spatial model
to define the location and connectivity of the material model. (Fig.1,
section 8). The defining centre lines or centre faces of walls and floors
of the material model may be offset from the edges and faces of the
idealized model. We can now appreciate the difficulty of attempting to
reverse the direction of the arrow to recover an idealized spatial model
from a material model.
In traditional BIM, the 3D material representation is the defining
model while the drawings are the derived models. With architectural
topology the idealized non-manifold topological representation becomes
the defining model and the 3D material representation is now a derived
model.
The idealised non-manifold spatial model acts as a useful conceptual
and practical intermediary between the user and the material model (Fig 4).
In this workflow the user is not manually placing specific material com-
ponents on specific Faces or Edges of the idealised model. If such a
workflow had been adopted, then any change in the idealised topology
might have removed these specific Face and Edge and orphaned (or
potentially deleted) the material components. Also such a change to the
idealised topology might have created new Faces and Edges which the
user would be required to populate with material components.
Instead, the populating of the idealised topology is rule-based using
the Visual Data Flow programming tools available in the host applica-
tion. The rule-based generation of the material model allows alternative
building configurations to be easily explored via the manipulation of the
idealised spatial model as previously suggested (Aish and Pratap 2013).

326 AAG2018
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Figure 3: The cell wall as a separator and as a connector, in biology and
in architecture (with acknowledgement to Wix, 1994).
3.5 Apertures and Contexts
A Face may have internal boundaries which may represent an aperture.
The location of an aperture within the host Face is defined by a Context.
Apertures can represent windows or doors. (Fig. 1, section 6) (The
representation of Apertures is discussed in more detail in section 4.4
‘‘Regional Topology’’)
3.6 Material representations
While all Cells have a common topology (a closed 3D region bounded
by Faces) different configurations of Cells may be generated from
different types of foundational topologies using different geometric
operations (Fig. 1, section 7), for example:
Point location connector components: may be based on Vertices.
Linear components such as columns or beams: may be based on Edges
(or Wires) using operations where a cross section Wire is extruded
along a path.
Area based components such as slabs, floors, walls may be based on
Faces: using offset operations with a specified thickness and direction.
326 AAG2018 327
Volume based components such as a containment vessel may be based
on Cells using thin-shell operations and a specified wall thickness.
Conformal cellular structures, used in 3D printing, may be based on
CellComplexes.
Complex sub-assemblies of material components can be modelled as
Clusters.
3.7 Integration of idealized and material models
The integrated BIM model uses the idealized non-manifold spatial model
to define the location and connectivity of the material model. (Fig.1,
section 8). The defining centre lines or centre faces of walls and floors
of the material model may be offset from the edges and faces of the
idealized model. We can now appreciate the difficulty of attempting to
reverse the direction of the arrow to recover an idealized spatial model
from a material model.
In traditional BIM, the 3D material representation is the defining
model while the drawings are the derived models. With architectural
topology the idealized non-manifold topological representation becomes
the defining model and the 3D material representation is now a derived
model.
The idealised non-manifold spatial model acts as a useful conceptual
and practical intermediary between the user and the material model (Fig 4).
In this workflow the user is not manually placing specific material com-
ponents on specific Faces or Edges of the idealised model. If such a
workflow had been adopted, then any change in the idealised topology
might have removed these specific Face and Edge and orphaned (or
potentially deleted) the material components. Also such a change to the
idealised topology might have created new Faces and Edges which the
user would be required to populate with material components.
Instead, the populating of the idealised topology is rule-based using
the Visual Data Flow programming tools available in the host applica-
tion. The rule-based generation of the material model allows alternative
building configurations to be easily explored via the manipulation of the
idealised spatial model as previously suggested (Aish and Pratap 2013).

328 AAG2018 329
Figure 4: An idealised spatial model built with non-manifold topology
can be used as a convenient intermediate representation to manipulate
a material model, involving:
a. creating a cell from a lofted solid.
b. dividing the cell using several faces, resulting is a
CellComplex.
c. the individual cells can be derived from the CellComplex.
d. introduce a cylinder outside the CellComplex.
e. move the cylinder into and imposed on the CellComplex:
new cells are created.
f. move the cylinder further into the centre: the cells update
accordingly.
g. h. i. corresponding material models are derived from the
NMT models in d, e, f.
The workflow includes detecting vertical and non-vertical edges,
sweeping a circle along vertical edges to create cylindrical columns
328 AAG2018
329
and a rectangle along non-vertical edges to create rectangular beams.
The depth of the beams are parametrically computed according to their
length. For visualisation purposes, the surfaces are thickened slightly
into solids and made translucent.
4. Using non-manifold topology to
represent relevant architectural concepts
Non-manifold topology embraces fi ve concepts with architectural
relevance:
4.1 Non-manifold Cell
A non-manifold Cell may contain internal Faces which are not part of
the external Cell boundary. Both sides of such internal Faces point to
the same enclosed region. The concept of a non-manifold Cell is
required to model internal ‘‘semi-partitions’’ of architectural spaces which
do not fully divide the cell. (Fig. 5)
Figure 5: Different confi gurations of non-manifold Cells.
4.2 Cellular Topology
Cellular Topology is implemented as a CellComplex, where some Faces
of the Cell are also the external boundary, while other Faces form the
boundary between adjacent Cells. Cellular Topology can be used to
model a building which is partitioned into different architectural spaces
(Fig. 6).

328 AAG2018
329
Figure 4: An idealised spatial model built with non-manifold topology
can be used as a convenient intermediate representation to manipulate
a material model, involving:
a. creating a cell from a lofted solid.
b. dividing the cell using several faces, resulting is a
CellComplex.
c. the individual cells can be derived from the CellComplex.
d. introduce a cylinder outside the CellComplex.
e. move the cylinder into and imposed on the CellComplex:
new cells are created.
f. move the cylinder further into the centre: the cells update
accordingly.
g. h. i. corresponding material models are derived from the
NMT models in d, e, f.
The workflow includes detecting vertical and non-vertical edges,
sweeping a circle along vertical edges to create cylindrical columns
328 AAG2018 329
and a rectangle along non-vertical edges to create rectangular beams.
The depth of the beams are parametrically computed according to their
length. For visualisation purposes, the surfaces are thickened slightly
into solids and made translucent.
4. Using non-manifold topology to
represent relevant architectural concepts
Non-manifold topology embraces fi ve concepts with architectural
relevance:
4.1 Non-manifold Cell
A non-manifold Cell may contain internal Faces which are not part of
the external Cell boundary. Both sides of such internal Faces point to
the same enclosed region. The concept of a non-manifold Cell is
required to model internal ‘‘semi-partitions’’ of architectural spaces which
do not fully divide the cell. (Fig. 5)
Figure 5: Different confi gurations of non-manifold Cells.
4.2 Cellular Topology
Cellular Topology is implemented as a CellComplex, where some Faces
of the Cell are also the external boundary, while other Faces form the
boundary between adjacent Cells. Cellular Topology can be used to
model a building which is partitioned into different architectural spaces
(Fig. 6).

330 AAG2018 331
Figure 6: Cellular Topology modelled as a CellComplex.
4.3 Mixed dimensionality Topological models
In non-manifold topology it is possible to construct a single topological
model composed of entities of different types and dimensionality. The
concept of a mixed dimensionality topology is implemented as a Cluster
and can be used to create an idealized model of the structure of a
building (Fig. 7).
Figure 7: A mixed dimensional model with Edges representing the
column centre lines and Faces representing floor slabs, blade columns
and shear walls. Cells are used to represent the building cores.
330 AAG2018
331
4.4 Regional Topology
In conventional topological modelling, higher dimensional topological
entities are constructed from lower dimensional ones. Higher
dimensional topological entities are connected because they share
common lower dimensional entities. For example, adjacent Cells within a
CellComplex may share a common Face.
However, in the domain of architecture there are other forms of con-
nectedness which cannot be directly expressed in this way. For example,
a column can be idealised as an Edge. A fl oor or ceiling can be idealised
as a Face. We intuitively understand that a column (Edge) may connect
a fl oor (Face) to a ceiling (Face), but how can this be described if the
column is in the middle of the fl oor and when there is no topology within
the defi nition of the fl oor and ceiling Faces which is shared with the
Vertices defi ning the column’s Edge? (Fig. 8).
Figure 8: Defi ning the ‘‘Context’’ to describe the connectedness of two
topologies where one entity exists within the region of the other entity and
when the two entities do not share any common constituent topology.
Similar issues arise when we consider an internal boundary within a Face.
For example the Face may represent a wall and the internal boundary may
defi ne an Aperture such as a window or a door. We intuitively under-
stand that the Aperture (as a single 2D region) is contained within the
2D region of the Face, with no shared topology.
To address these issues, Topologic introduces the concept of a
context to represent the connectivity between two topological entities
which do not otherwise share common topology. In this example, the
Aperture is the subject (representing a window) and is defi ned within
the region (or context) of the host Face (representing the wall). The

330 AAG2018
331
Figure 6: Cellular Topology modelled as a CellComplex.
4.3 Mixed dimensionality Topological models
In non-manifold topology it is possible to construct a single topological
model composed of entities of different types and dimensionality. The
concept of a mixed dimensionality topology is implemented as a Cluster
and can be used to create an idealized model of the structure of a
building (Fig. 7).
Figure 7: A mixed dimensional model with Edges representing the
column centre lines and Faces representing floor slabs, blade columns
and shear walls. Cells are used to represent the building cores.
330 AAG2018 331
4.4 Regional Topology
In conventional topological modelling, higher dimensional topological
entities are constructed from lower dimensional ones. Higher
dimensional topological entities are connected because they share
common lower dimensional entities. For example, adjacent Cells within a
CellComplex may share a common Face.
However, in the domain of architecture there are other forms of con-
nectedness which cannot be directly expressed in this way. For example,
a column can be idealised as an Edge. A fl oor or ceiling can be idealised
as a Face. We intuitively understand that a column (Edge) may connect
a fl oor (Face) to a ceiling (Face), but how can this be described if the
column is in the middle of the fl oor and when there is no topology within
the defi nition of the fl oor and ceiling Faces which is shared with the
Vertices defi ning the column’s Edge? (Fig. 8).
Figure 8: Defi ning the ‘‘Context’’ to describe the connectedness of two
topologies where one entity exists within the region of the other entity and
when the two entities do not share any common constituent topology.
Similar issues arise when we consider an internal boundary within a Face.
For example the Face may represent a wall and the internal boundary may
defi ne an Aperture such as a window or a door. We intuitively under-
stand that the Aperture (as a single 2D region) is contained within the
2D region of the Face, with no shared topology.
To address these issues, Topologic introduces the concept of a
context to represent the connectivity between two topological entities
which do not otherwise share common topology. In this example, the
Aperture is the subject (representing a window) and is defi ned within
the region (or context) of the host Face (representing the wall). The

332 AAG2018 333
user may optionally specify that the context defines a locational ‘‘link’’
between the subject and the host. Here the vertices of the subject are
defined in the parameter space of the host and are now dependent on
any changes which are applied to the host. (Fig. 9).
The context with parametric coordinates is only used when there is
no shared topology connecting the two entities (Fig. 10).
4.5 Variable topology
In architecture, spatial divisions may be ‘‘hardcoded’’ as distinct rooms
separated by physical walls. While buildings appear to be solid, one of the
central tenets of architecture is that the use of space within a building is
or should be flexible. We think of multi-use or reconfigurable spaces.
There appears to be no established architectural methodology which
prescribes how the topology of a building emerges. In fact, the archi-
tectural design process is quite imprecise. It may start with an occupancy
model and a description of the anticipated activities of the occupants.
Activities may vary in time and space. Activities may overlap. Alexander
(1965) noted that neither activities nor space could be adequately
described by a simple hierarchical decomposition. The process by which
activities get translated into specific spatial enclosures and the choice
as to which boundaries of these enclosures are actually materialised as
walls or are left as purely virtual, is often a matter of contention (Fig. 11).
Virtual partitions may also be used in the topological representation of
Figure 9: The option to ‘‘link’’ the subject topology to the host topology.
332 AAG2018
333
Figure 10: Given the intersection of an Edge (red) and a Face (grey) in
different configurations, then the concept of the context (with parametric
coordinates) is used when the resulting Vertex occurs within a region of
the intersecting topologies.
other building sub-systems. For example, an atrium may be considered
as a single continuous space, or it may be considered to be subdivided
into different air conditioning zones without physical partitions. Depending
on the simulation parameters, virtual Faces could be inserted and can be
represented in the analytical model either as adiabatic or diathermic.
More generally, architecture is often characterized by degrees of
spatial partitioning and connectedness. How can these different and
sometimes ambiguous architectural concepts of space be represented with
topology? Topology provides a formal way to represent connectedness, but

332 AAG2018
333
user may optionally specify that the context defines a locational ‘‘link’’
between the subject and the host. Here the vertices of the subject are
defined in the parameter space of the host and are now dependent on
any changes which are applied to the host. (Fig. 9).
The context with parametric coordinates is only used when there is
no shared topology connecting the two entities (Fig. 10).
4.5 Variable topology
In architecture, spatial divisions may be ‘‘hardcoded’’ as distinct rooms
separated by physical walls. While buildings appear to be solid, one of the
central tenets of architecture is that the use of space within a building is
or should be flexible. We think of multi-use or reconfigurable spaces.
There appears to be no established architectural methodology which
prescribes how the topology of a building emerges. In fact, the archi-
tectural design process is quite imprecise. It may start with an occupancy
model and a description of the anticipated activities of the occupants.
Activities may vary in time and space. Activities may overlap. Alexander
(1965) noted that neither activities nor space could be adequately
described by a simple hierarchical decomposition. The process by which
activities get translated into specific spatial enclosures and the choice
as to which boundaries of these enclosures are actually materialised as
walls or are left as purely virtual, is often a matter of contention (Fig. 11).
Virtual partitions may also be used in the topological representation of
Figure 9: The option to ‘‘link’’ the subject topology to the host topology.
332 AAG2018 333
Figure 10: Given the intersection of an Edge (red) and a Face (grey) in
different configurations, then the concept of the context (with parametric
coordinates) is used when the resulting Vertex occurs within a region of
the intersecting topologies.
other building sub-systems. For example, an atrium may be considered
as a single continuous space, or it may be considered to be subdivided
into different air conditioning zones without physical partitions. Depending
on the simulation parameters, virtual Faces could be inserted and can be
represented in the analytical model either as adiabatic or diathermic.
More generally, architecture is often characterized by degrees of
spatial partitioning and connectedness. How can these different and
sometimes ambiguous architectural concepts of space be represented with
topology? Topology provides a formal way to represent connectedness, but

334 AAG2018 335
Figure 11: The choice of spatial confi guration often starts with identi-
fying underlying activities of the occupants (1). These activities and their
spatial requirements may overlap. It may be inappropriate to describe
these as a simple hierarchical decomposition (with acknowledgement
to Alexander, 1965). The process by which activities are translated into
defi ned conceptual spaces (2) and are further translated into recognisable
enclosures (3) or into specifi c rooms (4) often refl ects architectural
intuition rather than a defi ned methodology.
334 AAG2018
335
when applied to architecture, it requires the user to choose what is being
connected.
If two adjacent regions have exactly the same contents with the
same behaviour and are so intimately connected that there is no effec-
tive barrier between them, then perhaps they should be considered as a
single region. So, the ultimate form of connectedness is the unification
of two adjacent regions into a single region or Cell. Therefore, a Cell
is more than just a continuous 3D region. It also implies that what is
contained represents a level of homogeneity, which has appropriate
meaning within the application domain.
If Cells represent spaces and Faces represent walls (or partitions)
then operations which add or remove the Faces of Cells within a Cell-
Complex can radically change the topology. The result of a modelling
operation to an existing topological construct may change the ‘‘type’’ of
that construct. The advantage of Topology is that it tells the architectural
users exactly what has been modelled in terms of partitioning and
connectedness and the type of the result (Fig. 12).
The general conclusion is that, where possible, the user should
define a single canonical non-manifold topology model describing the
maximal partitioning of space. Different subdivisions may be combined
to represent the spaces required for different activities. Different dual
graphs can be constructed as required by different analysis and simula-
tion applications (Fig. 13).
Figure 12: Editing operations to add or remove topological components
can have a radical affect, including changing the type of topological
construct.

334 AAG2018
335
Figure 11: The choice of spatial confi guration often starts with identi-
fying underlying activities of the occupants (1). These activities and their
spatial requirements may overlap. It may be inappropriate to describe
these as a simple hierarchical decomposition (with acknowledgement
to Alexander, 1965). The process by which activities are translated into
defi ned conceptual spaces (2) and are further translated into recognisable
enclosures (3) or into specifi c rooms (4) often refl ects architectural
intuition rather than a defi ned methodology.
334 AAG2018 335
when applied to architecture, it requires the user to choose what is being
connected.
If two adjacent regions have exactly the same contents with the
same behaviour and are so intimately connected that there is no effec-
tive barrier between them, then perhaps they should be considered as a
single region. So, the ultimate form of connectedness is the unification
of two adjacent regions into a single region or Cell. Therefore, a Cell
is more than just a continuous 3D region. It also implies that what is
contained represents a level of homogeneity, which has appropriate
meaning within the application domain.
If Cells represent spaces and Faces represent walls (or partitions)
then operations which add or remove the Faces of Cells within a Cell-
Complex can radically change the topology. The result of a modelling
operation to an existing topological construct may change the ‘‘type’’ of
that construct. The advantage of Topology is that it tells the architectural
users exactly what has been modelled in terms of partitioning and
connectedness and the type of the result (Fig. 12).
The general conclusion is that, where possible, the user should
define a single canonical non-manifold topology model describing the
maximal partitioning of space. Different subdivisions may be combined
to represent the spaces required for different activities. Different dual
graphs can be constructed as required by different analysis and simula-
tion applications (Fig. 13).
Figure 12: Editing operations to add or remove topological components
can have a radical affect, including changing the type of topological
construct.

336 AAG2018 337
5. Applying topology in analysis,
simulation and fabrication
Vitruvius distinguished between the practical aspects of the architecture
(fabrica) and its rational and theoretical foundation (ratiocination) (Pont
2005). Establishing topological relationships was found to be an essential
component of the setting out of the conceptual principles of a design
project (Jabi et al. 2017). Non-manifold topology was also found to be a
consistent representation of entities that can be thought of as loci, axes,
spaces, voids, or containers of other material.
This concept was previously explored by the authors in the context of
energy analysis, façade design, and additive manufacturing of conformal
cellular structures (Jabi 2016; Fagerström, Verboon, and Aish 2014; Jabi
et al. 2017).
5.1 Energy analysis
A proof of concept implementation of non-manifold topology for energy
analysis allowed the user to create simple regular manifold polyhedral
geometries and then segment them with planes and other geometries
to create a non-manifold CellComplex (Chatzivasileiadi, Lannon, et al.
2018; Wardhana et al. 2018). The tool can create complex geometry that
produces outputs that are highly compatible with the input requirements
for energy analysis software. Cells within the CellComplex are conver-
ted to spaces with surfaces, and bespoke glazing sub-surfaces, and set
to their own thermal zones.
5.2 Digital fabrication
A proof of concept implementation of non-manifold topology for digital
fabrication allowed a CellComplex to be conformed to a NURBS-based
design envelope (Jabi et al. 2017).The resulting model used topological and
geometric queries amongst adjacent Cells to create rules for depositing
material. These query results were used to identify boundary conditions
and to deposit material only where needed. This improved the material
efficiency and resulted in a higher mechanical and structural profile for
the 3D printed model.
336 AAG2018
337
Figure 13: Dual graphs can be constructed which describe alternative
connectivity of the Cells representing architectural spaces and used as
different analytical models.

336 AAG2018
337
5. Applying topology in analysis,
simulation and fabrication
Vitruvius distinguished between the practical aspects of the architecture
(fabrica) and its rational and theoretical foundation (ratiocination) (Pont
2005). Establishing topological relationships was found to be an essential
component of the setting out of the conceptual principles of a design
project (Jabi et al. 2017). Non-manifold topology was also found to be a
consistent representation of entities that can be thought of as loci, axes,
spaces, voids, or containers of other material.
This concept was previously explored by the authors in the context of
energy analysis, façade design, and additive manufacturing of conformal
cellular structures (Jabi 2016; Fagerström, Verboon, and Aish 2014; Jabi
et al. 2017).
5.1 Energy analysis
A proof of concept implementation of non-manifold topology for energy
analysis allowed the user to create simple regular manifold polyhedral
geometries and then segment them with planes and other geometries
to create a non-manifold CellComplex (Chatzivasileiadi, Lannon, et al.
2018; Wardhana et al. 2018). The tool can create complex geometry that
produces outputs that are highly compatible with the input requirements
for energy analysis software. Cells within the CellComplex are conver-
ted to spaces with surfaces, and bespoke glazing sub-surfaces, and set
to their own thermal zones.
5.2 Digital fabrication
A proof of concept implementation of non-manifold topology for digital
fabrication allowed a CellComplex to be conformed to a NURBS-based
design envelope (Jabi et al. 2017).The resulting model used topological and
geometric queries amongst adjacent Cells to create rules for depositing
material. These query results were used to identify boundary conditions
and to deposit material only where needed. This improved the material
efficiency and resulted in a higher mechanical and structural profile for
the 3D printed model.
336 AAG2018 337
Figure 13: Dual graphs can be constructed which describe alternative
connectivity of the Cells representing architectural spaces and used as
different analytical models.

338 AAG2018 339
6. Conclusions
New design technologies often emerge in response to the limitations of
existing technologies and have the potential to benefit the architectural
design process. Understandably, the founding concepts and terminology
may be unfamiliar to architectural practitioners which may inhibit
adoption of these technologies.
The challenge in developing Topologic has been to maintain the
theoretically consistent use of topological concepts and terminology,
yet relate these to the more ambiguous concepts of space and
‘‘connectedness’’ found in architecture. The application of topology as
a direct link between architectural conceptual modelling and relevant
analysis applications is becoming established. A more challenging
task is to explore how topology can contribute to the way in which
architecture as the ‘‘enclosure of space’’ can be conceptualised.
Acknowledgments
The Topologic project is funded by a Leverhulme Trust Research Project
Grant (Grant No. RPG-2016-016).
338 AAG2018
339
References
Ale xA nde r, ch r is To ph e r. 1965.
“A City is Not a Tree.” Architectural Forum, 122 (1): 58–62.
Aish, rob erT. 1 997.
“MicroStation/J.” In Next Generation CAD/ CAM/CAE Systems, NASA Con-
ference Publication 3357, edited by Ahmed K. Noor and John B. Malone, 167–80. Hampton, Virginia:
National Aeronautics and Space Administration. Accessed June 25 2018 https://www.researchgate.
net/publication/320507941_MicroStationJ
Aish, rob erT, An d ApA rA Ji T pr ATAp. 2013.
“Spatial Information Modeling of Buildings Using
Non-Manifold Topology with ASM and DesignScript.” In Advances in Architectural Geometry 2012,
edited by Lars Hesselgren, Shrikant Sharma, Johannes Wallner, Niccolo Baldassini, Philippe Bompas,
and Jacques Raynaud, 25–36. Vienna: Springer Vienna. doi:10.1007/978-3-7091-1251-9.
bJö r K, bo- ch r isTe r . 1992.
“A Conceptual Model of Spaces, Space Boundaries and Enclosing
Structures.” Automation in Construction 1 (3): 193-214. doi:10.1016/0926-5805(92)90013-A.
bronson, Jon AThA n, Jo sh uA A. levi ne , An d ros s Wh iTAKe r. 2014.
“Lattice Cleaving: A
Multimaterial Tetrahedral Meshing Algorithm with Guarantees.” IEEE Transactions on Visualization and
Computer Graphics 20 (2): 223–37. doi:10.1109/TVCG.2013.115.
chA nG , T.W., An d ro be r T Wo od b ur y. 1997.
“Efficient Design Spaces of Non-Manifold Solids.” In
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CAADRIA.
chATz i vAs il e iA di , AiK ATe r in i, sim on lAn n on , WAs si m JAb i, nicholAs mAr io WArd hA nA , An d
rob e rT Aish. 2018.
“Addressing Pathways to Energy Modelling through Non- Manifold Topology.”
In SIMAUD 2018: Symposium for Architecture and Urban Design, edited by Daniel Macumber,
Forrest Meggers, and Siobhan Rockcastle. Delft, the Netherlands: Society for Modeling & Simulation
International (SCS).
chATz i vAs il e iA di , AiK ATe r in i, nicholAs mAri o WA rd hAnA , WAs sim JAb i, ro b er T Aish, A nd si mo n
lAn no n . 2018.
“A Review of 3D Solid Modeling Software Libraries for Non-Manifold Modeling.” In
CAD’18 - the 15th Annual International CAD Conference. Paris, France: CAD Solutions, LLC.
cur Ti s, Wi l li Am J. r. 1996.
Modern Architecture Since 1900. London: Phaidon Press.

338 AAG2018
339
6. Conclusions
New design technologies often emerge in response to the limitations of
existing technologies and have the potential to benefit the architectural
design process. Understandably, the founding concepts and terminology
may be unfamiliar to architectural practitioners which may inhibit
adoption of these technologies.
The challenge in developing Topologic has been to maintain the
theoretically consistent use of topological concepts and terminology,
yet relate these to the more ambiguous concepts of space and
‘‘connectedness’’ found in architecture. The application of topology as
a direct link between architectural conceptual modelling and relevant
analysis applications is becoming established. A more challenging
task is to explore how topology can contribute to the way in which
architecture as the ‘‘enclosure of space’’ can be conceptualised.
Acknowledgments
The Topologic project is funded by a Leverhulme Trust Research Project
Grant (Grant No. RPG-2016-016).
338 AAG2018 339
References
Ale xA nde r, ch r is To ph e r. 1965. “A City is Not a Tree.” Architectural Forum, 122 (1): 58–62.
Aish, rob erT. 1 997. “MicroStation/J.” In Next Generation CAD/ CAM/CAE Systems, NASA Con-
ference Publication 3357, edited by Ahmed K. Noor and John B. Malone, 167–80. Hampton, Virginia:
National Aeronautics and Space Administration. Accessed June 25 2018 https://www.researchgate.
net/publication/320507941_MicroStationJ
Aish, rob erT, An d ApA rA Ji T pr ATAp. 2013. “Spatial Information Modeling of Buildings Using
Non-Manifold Topology with ASM and DesignScript.” In Advances in Architectural Geometry 2012,
edited by Lars Hesselgren, Shrikant Sharma, Johannes Wallner, Niccolo Baldassini, Philippe Bompas,
and Jacques Raynaud, 25–36. Vienna: Springer Vienna. doi:10.1007/978-3-7091-1251-9.
bJö r K, bo- ch r isTe r . 1992. “A Conceptual Model of Spaces, Space Boundaries and Enclosing
Structures.” Automation in Construction 1 (3): 193-214. doi:10.1016/0926-5805(92)90013-A.
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“A Two-Level Topological Decomposition for Non-Manifold
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Modeling, 355–60. New York, NY, USA. doi:10.1145/1236246.1236297.
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340 AAG2018 341
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Geometric Models.” Computer-Aided Design 25 (2): 119–29. doi:10.1016/0010-4485(93)90097-8.
nGu ye n, Tru nc du c. 2011. “Simplifying The Non-Manifold Topology Of Multi-Partitioning Surface
Networks.” MA Thesis, Washington University. http://openscholarship.wustl.edu/etd/510/.
pon T, GrAhAm. 2005. “The Education of the Classical Architect from Plato to Vitruvius.” Nexus
Network Journal 7 (1): 76–85. doi:10.1007/s00004-005-0008-0.
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lAn no n . 2018. “Bespoke Geometric Glazing Design for Building Energy Performance Analysis.” In
MonGeometrija 2018, edited by Vesna Stojakovi . Novi Sad, Serbia.
Wei ler , Kev in J. 1986. “Topological Structures for Geometric Modeling.” PhD Thesis, Rensselaer
Polytechnic Institute.
Wix , Jef f re y. 1994 “Interoperability: a future context”. Paper presented at the CAMP Autodesk
Developer Group Europe Conference 1994, Autodesk.