Towards a methodology to assess adaptability in

SSS10 Proceedings of the 10th International Space Syntax Symposium

017

Towards a methodology to assess adaptability in educational spaces: An entropy approach to space syntax

Carolina Coelho Department of Architecture, Centre for Social Studies, University of Coimbra, Portugal [email protected]

Mário Krüger Department of Architecture, University of Coimbra, Portugal [email protected]

Abstract

This paper aims to explore the concept of adaptability applied to educational facilities and to propose a comprehensive methodology to identify and assess this concept within educational spaces, in which Space Syntax plays a significant contribution on the adaptability of the individual spaces analysed, as well as on the whole spatial system.

The research recognises the relevance of adaptable educational architecture towards evolving pedagogical, technical and social needs, for a more complete spatial answer and a better building performance over time. Adaptability follows Krüger’s (1981) definition as “the ability of the built form to maintain compatibility between activities and spaces, as those vary". The research question resides in “how” and “by what means” can a contemporary school building be considered on its degree of adaptability, recognising the significance of the formal classrooms as content providing places, as well as other informal learning spaces where peer communication bears a recognised relevance towards educational experience and student achievement.

This problematic addresses the approaches on building adaptability carried out since the 1970’s, with a critical review based on recent advancements. The originality of the presented methodology resides in a crossing of outcomes from distinct spatial assessment processes, aiming to describe how adaptability is portrayed and what are its key variables to assess it. It is applied to distinctive learner-centred environments, in order to conclude on potential adaptability discrepancies between them. As a combined process, the methodology is composed of three stages, in which the subsequent complements the previous, as follows:

1. Probabilistic and combinatorial models, following Fawcett’s studies (1976), focus on maximising entropy, as an “index of adaptability” (Baird, 1972), considering the most adaptable one, to be the most probable macrostate amongst all the microstates (Krüger, 1984). Thus, the use of a mathematical approach to determine the entropy of each space provides a quantitative measure of adaptability.

2. Subsequently, a Space Syntax analysis focused on the parameters of integration, depth and connectivity, based on both the axial lines that cross each convex space analysed in stage one, and on the convex spaces themselves, informs on “how” and “why” a space is being used and whether spaces with analogous entropy levels, allocate different uses. These findings display information on whether spatial morphology bears relevance for the school’s occupancy, namely to activity schedules and spatial density of natural movement.

C Coelho & M Krüger Towards a methodology to assess adaptability in educational spaces: An entropy approach to space syntax

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mailto:[email protected]

mailto:[email protected]


3. The final stage comprehends a systematization of the results, concluding on whether potential spatial feasibility and effective spatial occupancy coincide, providing an outlook on the relevance of spatial morphology, towards use and adaptability. It also informs on adaptability variations between formal and informal, productive and supportive spaces (Krüger, 1992). Its final outcome potentially provides a comprehensive outlook on spatial analysis and a methodological development on architectural research, to be applied to other design briefs.

Keywords

Adaptability, educational spaces, methodology, entropy, integration.

1. Adaptability in school buildings

Definition, relevance and contemporaneity

“Educational facilities need to accommodate both the known and identifiable needs of today, and the uncertain demands of the future.” (OECD, 2001, p.vii)

This paper recognises the relevance of adaptable educational architecture towards evolving pedagogical, technical and social needs, for a more complete spatial answer and a better building performance over time. The research question resides in “how” and “by what means” can a contemporary school be considered on its degree of adaptability, recognising formal classrooms as content providing places, as well as informal spaces where peer communication bears an acknowledged relevance towards educational experience and student achievement.

Much addressed in the 1970s, OECD defined adaptability as: “the quality of a building which facilitates adaptation; adaptation may require relocation, replacement, removal or addition in respect of either the constructional elements, services or the finishes of the building - essentially large magnitude/low frequency change” (1976, p.10). For this research, adaptability follows Krüger’s (1981) definition as “the ability of the built form to maintain compatibility between activities and spaces, as those vary", which implies the building’s potential to allocate a range of activities without any physical transformation.

In 1957 the British Ministry of Education had already suggested that changeability in educational pedagogies had to be accounted for in post-war school building construction. Later, Weeks (1963) mentioned the need for adaptability to avoid an early obsolescence. Fawcett (1978) pointed out that adaptable strategies were related to spatial “attributes”, which was later developed by Duffy (1990) and Brand (1994) with his seven “shearing layers of change”, which lead to diverse adaptable solutions.

Due to current pedagogical developments, educational spaces have undergone alterations in their organization and materialisation, recognising the school as a “catalyst for change” (Worthington, 2007, p.15). In fact, an adaptable school building, by coping with current and future unpredictable changes will enhance its performance and answer the community’s needs, as a more endurable and sustainable building, considering adaptability as an attribute for exemplary educational facilities today (OECD, 2011).

2. Methodology

Sequential description, originality and relevance

“Describing change from within can be a considerable challenge. Still, change is a fascinating topic and has brought forth a plethora of concepts and theories across different academic disciplines.” (Sailer, 2014, p.xx).


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This quote in the latest JOSS, Vol 5, No 2, entitled Changing building typologies, corroborates change as a contemporary and relevant issue in the design of buildings. Particularly, when applied to case studies, current research on this problematic may provide “actionable knowledge” (Elliot, 2001, p.555) towards future designs.

Accordingly, this paper proposes a comprehensive methodology to identify and assess adaptability within educational spaces, with a critical review based on advancements from the approaches carried out since the 1970s and the crossing of distinct spatial assessment processes, aiming to describe how adaptability is portrayed and what are its key variables to assess it. It is applied to distinctive learner-centred environments, in order to conclude on potential adaptability discrepancies between them. As a combined process, the methodology is composed of three stages, in which the subsequent complements the previous, as follows:

1. Probabilistic and combinatorial models, following Fawcett’s studies (1976a), focus on maximising entropy, as an “index of adaptability” (Baird, 1972), considering the most adaptable one, to be the most probable macrostate amongst all the microstates (Krüger, 1984). Thus, the use of a mathematical approach to determine the entropy of each space provides a quantitative measure of adaptability.

2. Subsequently, a Space Syntax analysis focused mainly on the attributes of integration, depth and connectivity, based on both the axial lines and convex spaces, informs on “how” and “why” a space is being used and whether spaces with analogous entropy levels, allocate different uses. These findings display information on whether spatial morphology bears relevance for the school’s occupancy, namely to activity schedules and spatial density of natural movement.

In fact, this study will focus on both convex spaces and axial lines, assuming that both moving and static activities bear potential on knowledge spread, whether on formal scheduled classes or on spontaneous activities held in “spatial units” (Hertzeberger, 2009, p.11), “articulated classrooms” (Hertzeberger, 2008, p.24) and “learning streets” (ibid, p.113). Consequently, the methodology will consider both situations by two methods of analysis; space syntax and entropy approach, in order to provide input on the potential correlation amongst the two approaches.

3. The final stage comprehends a systematization of the results, concluding on whether potential spatial feasibility and effective spatial occupancy overlap, providing an outlook on the relevance of spatial morphology, towards entropy and ultimately adaptability. It also informs on adaptability variations between formal and informal, productive and supportive spaces (Krüger, 1992).

Its relevance lies in the analysis of the correlations between both approaches – entropy and space syntax – for the different distributions analysed. Finally, this paper will conclude on the relevance of this potential correspondence towards adaptability, and ultimately the learning process. Its final outcome potentially provides a comprehensive outlook on spatial analysis and a methodological development on architectural research, to be applied to other design briefs.

After the sequential depiction of the three stages, this chapter will focus on the presentation of the selected case study, which is Quinta das Flores School in Coimbra, Portugal. This is a building from 1968 with a scattered pavilion layout, which has been modernised in 2008-2009 under an adaptive reuse process, in the context of a secondary school modernization programme implemented in Portugal since 2007.

This school holds a particular curriculum where regular and artistic teaching share common spaces. Therefore, this widespread school community enables higher levels of peer interaction, socialization and ultimately cross-curricular informal learning. It also acts as a urban and community hub, enhanced by its artistic nature, which leads to frequent public exhibitions, as well as a special openness to external organizations that use these spaces outside the school hours. This attests this school as a high complexity case study on adaptability in educational facilities.


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For this paper, the methodology will be focused on the new building from 2008, which holds the common spaces and embodies the main entrance through a new street façade, which underlines its representativeness towards the city and the community.

3. Entropy approach

Interpretation, experimental application and conclusion

“The pursuit of adaptability can be seen as maximising the looseness of the fit of the known activity schedule into the first-build physical schedule […] Thus the concept of entropy seems likely to be a very useful tool.” (Fawcett, 1976a, p.7)

Stage one of this methodology is based on an entropy approach to adaptability. It follows Fawcett’s (ibid.) assumption that adaptability can be measured in a “precise numerical value” (Fawcett, 1978, p.55) provided by an adaptability model (ibid.).

A sample of spaces will be analysed according to their features and respective potential for activity allocation. This can be measured as the entropy of that distribution as “a unique, unambiguous criterion for the amount of uncertainty in a probability distribution” (Fawcett, 1976b, p.17).

For this research, entropy will be calculated through Shannon’s formulation (1949), later developed by Jaynes (1957) and Tribus (1969):

S = -∑i pi ln pi

Hillier also pointed out the possibility of analysing entropy by means of a mathematical formulation: “In other words, entropy relates the notions of order and chaos into a single concept, but at the same time gives it a much more precise and limited reference to the world. […] It permits the concept to be captured in a formal mathematical expression as well as through words.” (Hillier, 1996, p.57).

In fact, depthampX (Varoudis, 2012) provides point depth entropy, by using Shannon’s formulation (1949) of uncertainty, hence displaying “how orderly a spatial system is structured from a certain location” (El-Khouly and Penn, 2012, p.5). Entropy is here understood according to Hillier’s definition (1996) and “corresponds to how easy it is to traverse to a certain depth within the system (low disorder is easy, high disorder is hard) […]” (Turner, 2001, p.7-8).

Moreover, according to Hillier “Entropy is maximal in a system when the system is in one of the macrostates for which there are the largest number of microstates” (1996, p.57). This definition can be paired with Tribus’s interpretation of entropy as “a measure of the uncertainty of the knowledge about the answer to a well defined question.” (1969, p.119).

For the current research, this can be translated into spaces’ “looseness of fit” (Fawcett, 1976a). If a space can potentially allocate several activities, the uncertainty about which activity is currently happening is higher than in a space with a single function. Consequently, a high entropy space will be more adaptable, in condition ceteris paribus (meaning that we assume that all other variables besides entropy are constant), because its allocation potential is also higher.


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Figure 1. Quinta das Flores School - spatial identification


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The first step towards the entropy analysis was the identification of 43 different spaces in the sample (Figure 1).

An original list of activities was then produced according to the specific curriculum of this school. In this activities’ sample, 5 domains were considered, from the most general to the most detailed scale, projecting a tree-shaped scheme, as follows:

Domain I) Corresponds to spaces’ educational domain: active learning environments or supportive environments.

Domain II) Corresponds to types of uses: pedagogical or non-pedagogical.

Domain III) Corresponds to types of activities: programmed or non-programmed. According to Krüger (1992) programmed activities are related to a functional schedule and a work routine (usually a weekly schedule), whereas non-programmed activities are related to spontaneous or non-repetitive activities. It is relevant to underline that non-programmed activities today may be pedagogical, due to the relevance of informality for the current learning model.

Domain IV) Corresponds to teaching regimes: regular, artistic and external, particularly meaningful for this school.

Domain V) Corresponds to activities in each teaching regime, detailed according to the subdivisions and previous domains.

Some of these activities are mutually exclusive, which is the case of learning activities in supportive environments, or circulation as programmed activities, which were removed. Hence, the overall activity sample lists a total of 81 possible activities (Figure 2).


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Figure 2. Activities sample.


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Figure 3. Feasibility matrix.


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These activities were then matched with the spaces, providing a feasibility matrix of the potential allocations, identifying each allocation activity to a space with a dark square (Figure 3). The sum of all the possible activity schedules leads to 908 potential allocations, assuming that only one activity takes place in a space for each time period considered.

At this stage, it was considered relevant to proceed with the calculations for two different distributions – a less and a more disaggregated one – in which the first focused on the types of activities (corresponding to domain III) and the latter on the teaching regimes (domain IV), which displays higher levels of activity allocations and hence a deeper complexity. Allocation matrices were constructed by identifying the distribution of the possible activities for each space according to the sections considered, which are clearly more extensive for domain IV. Subsequently, a distribution matrix was produced for each domain (Figure 4).

The entropy calculation for each of the spaces corresponds to the values encountered from the previous allocation matrices and their respective introduction in the entropy formulation.

As a practical demonstration this will be applied to space 1 (main hall) in the distribution for domain III:

1) In the feasibility matrix there were 48 potential allocations to this space.

2) The allocation matrix for domain III for space 1 indicated: 12 active learning, non-pedagogical, programmed activities; 15 active learning, non-pedagogical, non-programmed activities; 9 supportive, non-pedagogical, programmed activities; and 12 supportive, non-pedagogical, non-programmed activities. This adds to the total of the 48 activities, now subdivided in sections.

3) A distribution matrix divided the activities by the total allocations for this space: 12/48 (=1/4), 15/48 (=5/16), 9/48 (=3/16), 12/48 (=1/4). The simplified fractions are presented in both distribution matrices (Figure 4).

4) Finally, the entropy for space 1 is calculated:

S1= - (1/4 ln1/4 + 5/16 ln5/16 + 3/16 ln3/16 + 1/4 ln1/4) = 1,371


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Figure 4. Allocation and distribution matrices per domain. (Spaces 1 to 43 are identified in Figure 1 and Domains I to IV are identified in Figure 2)


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In a compared analysis of both distributions, and despite each one’s level of complexity, the entropy values are ranked similarly, meaning the highest and lowest entropy are consistent for both analysis and also with the results from the feasibility matrix. The following spaces were identified with the highest entropy in both distributions: library, orchestra room, music studio, auditorium, spare space, generic classroom and science lab. These results may reflect the school’s artistic and regular curriculum, which requires spaces with several features to allocate diverse activities of different nature.

The spaces with the lowest entropy are supportive and conceived for a particular activity, such as: storage, vault, archive, kitchen, kitchen storage, accesses, reception desk, ticket office, restroom, etc.

Nevertheless, some differences remain amongst both distributions (Figure 5). Although the same 7 spaces have been identified with the highest results in all three analyses, they have slight differences in the entropy rank, due to the accuracy of the values that go up to 3 decimal places. The fact that these calculations were based on two different distributions also provided higher values for the most disaggregated distribution from domain IV, which is the most complex and holds the widest activity allocations.

Subsequently, a correlation between the activities in the sample and the entropy of each space produced a scattergram that overlapped the two distributions (Figure 5). This graph confirms the previous hypothesis on the entropy values correspondence of both domains and that the features of the growth trend tend to be similar. But it also provides another conclusion: whatever the level of aggregation in the correlation between activities and entropy, the entropy tends to stabilise when the activities schedules increase. In short, from this analysis what is more significant in terms of adaptability, are the first 20 activities the system can allocate to spaces, and from that point the system is practically invariant.

Another consideration that adds further information to the entropy analysis is the introduction of two concepts: “axial line entropy” and “average axial line entropy”.

The “axial line entropy” is calculated as the sum of the entropy of all convex spaces intersected by this axial line, as follows:

AXIAL LINE ENTROPY: Si = Σk Sik

Si entropy of axial line i

Sik entropy of convex space k intersected by axial line i

The “average axial line entropy” is calculated, by dividing the axial line entropy by the number of convex spaces intersected, as follows:

AVERAGE AXIAL LINE ENTROPY: ASi = Σk Sik/NK

NK number of convex spaces intersected by axial line i

ASi average entropy of axial line i

Considering the axial line on the ground floor, paralleling the street façade with the longest length, it crosses the following spaces: library (18), main hall (1), corridor a and corridor b (2) and entrance (5). By summing up all their entropy, the axial line entropy corresponds to: S18+S1+S2a+S2b+S5 that equals 3,153 for the distribution for domain III and 8,647 for the distribution for domain IV. Its average entropy results from dividing these values by this number of spaces, which is 3,153/5 that equals 0,631, and 8,647/5 that equals 1,729, respectively for domains III and IV (Figure 10).

Crossing the axial line entropy with the entropy level of convex spaces, we conclude that the values can be very diverse. Clearly, both axial line entropy and average axial line entropy vary according to the entropy of the convex spaces that each axial line intersects.


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Figure 5. Correlation between activity allocations and entropy per domain. (Table and scattergram) (Spaces’ locations are identified in the school plans illustrated in Figure 1).


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4. Space syntax analysis

Interpretation, experimental application and conclusion

“[…] we know that although the parts of a building do not move, through their configurational differences they do affect the pattern of movement, in that other things being equal, the degree to which spaces are used for movement is a function of their configurational position.” (Hillier, 1996, p.394)

Having identified the spaces’ and axial lines’ entropy, stage two undertakes a space syntax analysis, aiming to assess the same attributes for both convex spaces and axial lines. This acknowledges, for the school brief, the relevance of studying individual spaces as well as their sequential layout and connectivity, which can trigger knowledge development:

“The space structure is admittedly highly broken up into ‘convex’ spaces — but there are always lines which link the convex spaces together, usually several at a time. […] But because people move in lines, and need to understand lines in order to know where they can go, this means that the space structure is easily intelligible from the point of view of movement.” (Hillier, 1996, p.116)

For the convex space analysis, the data gathering followed a sequential procedure. First, the final plans provided by the architect, considered to be the closest to the actual built space, constituted the basis in which the convex spaces were drawn. Then, in depthampX the spaces were linked and a convex map was produced. Then, a graph analysis was run in order to generate the visual images of the map according to the chosen attributes. Following depthampX’s colour scheme the most integrated space (HH radius n) is the main hall on the ground floor, followed by the corridor that accesses the auditorium’s dressing rooms, the corridor that accesses the administrative services and the library. Spaces such as the secretariat, the auditorium, main hall’s adjacent rooms and the cafeteria, are also highly integrated. These rooms correspond to common living spaces, shared by all, which corroborates once more the school’s cross-curricular teaching. On the first floor it is also the main corridor that is most integrated, followed by the adjacent spaces and the main corridors in the perpendicular sections of the building (Figure 6). According to the attributes summary the first floor is more integrated than the lower.

Generic classrooms and science labs, because of their location in one building section predominantly for regular teaching, present lower values of integration. Also the dance studio, the music room and the orchestra room, located in the extremities, are poorly integrated, despite having high levels of entropy. Consequently, from the rooms with high entropy only the library and the auditorium on the ground floor correspond to high levels of integration, whereas on the first floor this correspondence is not noteworthy. Nevertheless, the most integrated spaces: the main hall and the main corridor, respectively on the lower and upper floors, present a strong correspondence between activity allocation, entropy and integration, all similarly significant.

The mean depth and the connectivity charts added to this conclusion. The mean depth also points out that these are the spaces that imply the least changes in direction and the connectivity indicates that these spaces are the ones with the highest number of connections towards other adjacent spaces.

The main hall is also the space with the highest connectivity, followed by the administrative services corridor and the dressing room corridor from the auditorium, the auditorium itself and the library. On the first floor besides the main corridor, which is again the most connected space, the adjacent spaces present high values of connectivity, whereas the remaining present extremely low values. This analysis corroborates the previous results indicating the main hall and the corridor as being central to the system, followed by the library and the auditorium on the ground floor, matching the entropy results in the previous chapter, explicitly for the ground floor. According to the values encountered, there are well-defined differences amongst the spaces: the ones with high connectivity from the ones with low values, which are amongst themselves similar. This is proven by an average connectivity of 2,34 on the ground floor, although it varies from 1 to 21; and on the upper floor the average connectivity of 2,15, ranges from 1 to 46.


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Figure 6. Integration map.


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The intelligibility results produced by the scattergram that correlates connectivity (in the x axis) to integration (in the y axis) indicates lower values with a R2=0,29 on the ground floor and R2=0,41 on the first floor, potentially indicating a certain dispersion of the spaces amongst the several sections in the system that hampers users’ orientation in space.

Afterwards, the same analysis was undertaken for the axial line map. The outlines of the building were drawn over the final plans and exported to depthampX, and then an axial map was produced and reduced to the fewest lines. Subsequently, a graph analysis was run primarily for the same attributes analysed for the convex map. These results can be analysed visually either by the map colour scheme and the chart’s values (Figure 6).

Clearly, the most integrated axial lines on both floors, identified in red, cross the building longitudinally paralleling the street. This matches the analysis on convex spaces, where the main hall/corridor were also the most integrated. Furthermore, all the other axial lines placed in those same convex spaces bear high levels of integration, rather than the remaining in the perpendicular sections of the building, more segregated and identified in blue. The values of mean depth and connectivity reflect the integration’s interpretation.

Additionally, the connectivity of the axial line map adds further information to this research. In the connectivity analysis of the convex spaces only the ground floor matched spaces with high connectivity with high entropy, while on the first floor the spaces with the highest entropy were placed in the extremities, so their connectivity was low. Contrarily, the analysis of connectivity for the axial map considers these spaces (the dance studio, the music studio and the orchestra) as a part of the most integrated axial line and so, the one with the highest connectivity. Thus, the gathering of all the analysis – entropy and space syntax, in both approaches – convex spaces and axial lines - provides a triangulation of results that individually adds data to a general conclusion on a potential correspondence between entropy, integration, connectivity and ultimately adaptability.

It is now possible to parallel the values from integration, connectivity and mean depth, in order to critically analyse their proximity or variance on both floors and both approaches (Figure 7). In terms of integration and connectivity the highest values are from the axial map of the ground floor, which also holds the widest scope in values. This will imply that it is possible to wander around more in this floor, amongst potentially more adaptable spaces and more effective towards the learning process. Contrarily, the convex map from the ground floor holds the lowest and the narrowest range in values and hence the highest mean depth, while both the convex and axial line map from the first floor present similar results.


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Figure 7. Integration, connectivity and mean depth comparison (table and scattergram).

This fluctuation in the values is also supported by the intelligibility analysis. Whereas the analysis of the convex spaces indicated a low intelligibility of system, the results from this analysis are higher for the ground floor with R2=0,68 and R2=0,27 for the upper floor (Figure 8). This implies that the ground floor’s intelligibility has increased significantly from the previous analysis to this one (from R2=0,29 to 0,68), leading to the hypothesis of this floor to be intelligible, when analysed through axial lines. While the value encountered for the first floor has decreased from the previous analysis on the convex space map, for a more scattered regression line (from R2=0,41 to 0,27).

Figure 8. Axial line map - Intelligibility scattergram (Connectivity, Integration).


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5. Conclusions

“Society must be described in terms of its intrinsic spatiality; space must be described in terms of its intrinsic sociality.” (Hillier and Hanson, 1984, p.26)

This paper focuses on the relevance of adaptability to educational spaces, which are requested to answer the current spatial needs and reciprocally to enhance the possibility of learning amongst the whole system and individually in each space, most reflecting Hillier and Hanson’s quote above.

The methodology for assessing adaptability engaged entropy and space syntax and considered both convex spaces and axial lines. Hence, the results aim to provide a more complete outlook on educational spaces and the possibility to triangulate partial results in order to reach a more supported overall conclusion on the potential relation between space syntax and entropy, and ultimately how it potentially affects adaptability in schools.

The results encountered correspond to its application on the case study, but may possibly differ when applied to other schools with different entropy and building morphology.

To undertake the entropy calculation, activities and feasibility matrices were produced that traced the allocation potential of each space to a range of activities. Two different distributions were considered, concluding that the most disaggregated one provided higher levels of entropy, although the hierarchy between spaces was fairly similar between them both.

Another significant conclusion, while comparing both distributions, was the tendency for the entropy stabilisation after about 20 activity allocations to a space. This information can be introduced in future designs as operative input for the architect, while pondering between the cost and benefit of the design solutions.

Besides, the concepts of “axial line entropy” and “average axial line entropy” were also introduced as a novelty. This calculation established these values changed according to the sequence of spaces and their respective individual entropy.

In the space syntax analysis the highest integration was found in the axial line map of the ground floor and the lowest on the same floor’s convex space map.

By establishing a correlation between integration and entropy for both convex spaces and axial lines it is possible to conclude upon their compared results. The correlation for convex spaces is not straightforward because it links spaces identified by function with spaces identified by their position in an overall system. Hence, spaces identified in the entropy approach (Figure 5) as restroom, generic classroom, or storage (in eg.), which are frequent and placed differently in the building, will present diverse values of integration. This leads to a correspondence of the same entropy to multiple integration values.

Spaces for specific activities, whose uncertainty degree in activity allocation is very low and hence with the lowest entropy, were recognised as fairly integrated and connected, mainly on the ground floor. This can be explained by their functional utility, because although spaces such as the entrance, ticket office, reception desk, restroom and vertical accesses are supportive spaces, their proximity is required so that the main active environments can be productive. Particularly in this school, spaces that support external events need to be integrated and intelligible for the external user.

The correlation between integration and entropy for convex spaces is scattered and the regression line not significant (Figure 9), but according to the same conclusion for the entropy approach, the domain IV’s distribution presents a higher correlation than domain III’s.


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Figure 9. Convex spaces correlation scattergram – Integration, Entropy.

Following the demonstration shown on how to calculate the axial line entropy and the average axial line entropy, this has been applied to all axial lines recognised by DepthampX, through the identification of the convex spaces each axial line intersected and then estimating their axial entropy. Then, a chart and graphs were produced with the axial line entropy and integration correspondence to study their possible correlations (Figure 10).

Some spaces with high entropy values are not necessarily the most integrated in the system per se, but when analysed within an axial line, that correlation is meaningful. Thus, there is a higher correlation between integration and entropy for axial lines rather than for convex spaces, proven by the regression lines and respective determination coefficients (R2) of both graphs. Pedagogically this is crucial, because it supports the fact that this school considers not only formal learning spaces, but also spatial sequences as potential moments and environments for active learning.

There is also a higher correlation on the ground floor than on the upper floor and for the entropy distribution for domain IV rather than for domain III. The average axial line entropy is less related to the integration than the axial line entropy, because many of the spaces analysed have zero or non-meaningful values of entropy, which diminished the average axial line entropy. Also the connectivity bears less relevance for the entropy than the integration. Finally, it was also remarked that for similar values of integration there was a correspondence to similar values of entropy, which implies that spaces with analogous functional purposes are placed in analogous positions on the school’s layout.

The highest correlation between integration and entropy was found on the axial line map for the ground floor, with the most significant R2=0,35, fairly distant from any of the other values (Figure 10). This implies that integration, as the independent variable, explains in 35% the entropy values in the y axis. Hence, the ground floor is a more representative interface for interaction and learning, by acknowledging more adaptable spaces, with higher entropy and which also has more syntactic representativeness in terms of axial lines.


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Figure 10. Axial line correlation between Integration and Entropy: a) Table sample from an extended list. b) Scattergram for both floors and both domains.

Consequently, in terms of relating integration with adaptability, it is possible to conclude on the potential of learning in lines rather than in circumscribed spaces, much remembering Hertzeberger “learning street”: “There are school buildings where learning and instruction are not confined to the classroom, where there is as much going on outside the classrooms as inside […]” (2008, p.113).

This values the school overall as an active learning environment, whose morphology and sequential layout play a relevant role towards adaptability and ultimate learning in the diverse ways considered today.


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