193
© Springer Nature Switzerland AG 2021
P. B. Roös, Regenerative-Adaptive Design for Sustainable Development, Sustainable Development
Goals Series, https://doi.org/10.1007/978-3-030-53234-5
Appendix 1
Alexander’s Mathematical Definition of Wholeness
The Structure of Wholeness
The notion of regenerative-adaptive patterns equation described in Chaps. 10 and 11, developed by
Roös and based on the fundamentals of the Borchers and Stark equations, further attempts to include
the context of ‘wholeness’. Each element of the equation is interconnected with the others, and
includes interconnected higher-level with lower-level patterns formed by the dynamics of change, and
considers the process of adaptation (Chap. 11). The regenerative-adaptive pattern language accepts
that the ecological systems process and all phenomena in the region of a specific place constitute in
the context of ‘the whole’: everything connected and interdependent on each other. Inherent to the
notion of regenerative- adaptive patterns equation are the 15 properties of wholeness. This mathematical equation of the regenerative-adaptive patterns demonstrated the wholeness of the whole region
within which the case study town of Anglesea is located, and thus Alexander’s mathematical explanation of wholeness is needed.
Mathematical Definition of Wholeness According to Alexander (2001–2005)
According to Alexander (2001–2005a, pp. 446–457),
1
wholeness (W) is a feature of a physical
space that appears everywhere, in any part of matter or space. Characterised by a clear mathematical structure, consider any region of space or a place as (R). To include boundaries on this space,
imposing a mesh or grid on the space will result in a number of points that are considered finite,
not infinite. The result is that (R) contains a number (n) of points. In the real world there is usually
some colouring or differentiation of type or character among the (n) points of (R), so that the
region (R) has a visible and identifiable structure. The simplest colouring that produces a structure
is black and white, and in the context of two-dimensional space, (R) would then be a drawing
representing a particular object. In the case where the colouring of the region is not abstract, but
concrete, the points may be assigned to physical materials, for example, solid and void, or various
other attributes. The region (R) is thus intended to represent a part of the real world in its overall
geometric form and organisation (Alexander, 2001–2005a, p. 446). The mathematical definition of
wholeness is explained as follows by Alexander:
Wholeness (W) within a region (R) can be mathematically constructed as follows:
1 Alexander, C. (2001-2005a). The Nature of Order - An Essay on the Art of Building and the Nature of the Universe,
Book One: The Phenomenon of Life. Berkeley, California, USA: The Center for Environmental Structure.
© Springer Nature Switzerland AG 2021
P. B. Roös, Regenerative-Adaptive Design for Sustainable Development, Sustainable Development
Goals Series, https://doi.org/10.1007/978-3-030-53234-5
Appendix 1
Alexander’s Mathematical Definition of Wholeness
The Structure of Wholeness
The notion of regenerative-adaptive patterns equation described in Chaps. 10 and 11, developed by
Roös and based on the fundamentals of the Borchers and Stark equations, further attempts to include
the context of ‘wholeness’. Each element of the equation is interconnected with the others, and
includes interconnected higher-level with lower-level patterns formed by the dynamics of change, and
considers the process of adaptation (Chap. 11). The regenerative-adaptive pattern language accepts
that the ecological systems process and all phenomena in the region of a specific place constitute in
the context of ‘the whole’: everything connected and interdependent on each other. Inherent to the
notion of regenerative- adaptive patterns equation are the 15 properties of wholeness. This mathematical equation of the regenerative-adaptive patterns demonstrated the wholeness of the whole region
within which the case study town of Anglesea is located, and thus Alexander’s mathematical explanation of wholeness is needed.
Mathematical Definition of Wholeness According to Alexander (2001–2005)
According to Alexander (2001–2005a, pp. 446–457),
1
wholeness (W) is a feature of a physical
space that appears everywhere, in any part of matter or space. Characterised by a clear mathematical structure, consider any region of space or a place as (R). To include boundaries on this space,
imposing a mesh or grid on the space will result in a number of points that are considered finite,
not infinite. The result is that (R) contains a number (n) of points. In the real world there is usually
some colouring or differentiation of type or character among the (n) points of (R), so that the
region (R) has a visible and identifiable structure. The simplest colouring that produces a structure
is black and white, and in the context of two-dimensional space, (R) would then be a drawing
representing a particular object. In the case where the colouring of the region is not abstract, but
concrete, the points may be assigned to physical materials, for example, solid and void, or various
other attributes. The region (R) is thus intended to represent a part of the real world in its overall
geometric form and organisation (Alexander, 2001–2005a, p. 446). The mathematical definition of
wholeness is explained as follows by Alexander:
Wholeness (W) within a region (R) can be mathematically constructed as follows:
1 Alexander, C. (2001-2005a). The Nature of Order - An Essay on the Art of Building and the Nature of the Universe,
Book One: The Phenomenon of Life. Berkeley, California, USA: The Center for Environmental Structure.