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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;

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;

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).

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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

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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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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

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