From external to internal subtraction
As the 1916 law implemented one mode of division of the maximum mass, so the 1961 law established another subtly different one. While the former gave birth to iconic stepped pyramidal masses, so the latter initiated the tower and plaza model. The intended outcome of both, however, was to effect greater separation between forms by subtraction from the outside of the maximum mass. An objective to achieve the opposite result, namely proximity of adjacent forms, suggests the adoption of an inverse strategy; subtraction from the inside.
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Menger Sponge; product or process
Being the result of a sequence of subtractive operations, the condition of the Menger Sponge is less one of product than of process. The form shown is at only the third 'level' in the hierarchy of volumes which can be removed from the primitive mass. The image represents an artificially frozen state in a process where the elements removed are, ultimately, infinitely large in number yet each is of infinitely small volume. The Menger Sponge is, therefore, a dynamic and endless process and potentially a method for the production of urban form.
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Menger Sponge; subtracted and residuum
The properties of the sponge are revealed through juxtaposing the subtracted volume (a superposition of the elements removed) with the residual form. Even though one is the reciprocal of the other, both appear equally solid.
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Menger Sponge; comparison with external subtraction
At the third level of removal, the volumes of the subtracted and the residual (expressed as percentages of the primitive cube) are approximately 60% and 40% respectively. The internal subtracted form is compared to a cube which has been subject to external subtraction which possesses the same relative proportions between subtracted and residual.
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Menger Sponge; layer types A and B
Further dimensions of the sponge are revealed by slicing it into layers of a thickness equal to the dimension of the smallest removed module. Eight morphological types are generated by this analysis. The different types exhibit diverse combinations of form and space. As these evoke associations with familiar and conventional urban structures, they simultaneously suggest possibilities for radically different urban environments.
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Menger Sponge; layer types C and D
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Menger Sponge; layer types E and F
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Menger Sponge; layers types G and H
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Menger Sponge; deconstruction and reconstruction
The dynamic and successive removal of the layers exposes the interior of the sponge, concealed when the object is in its complete form.
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Menger Sponge; a model for mixed use development
The layers of the sliced up sponge are arbitrarily allocated with colours suggesting possible combinations of use.
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Menger Sponge as subtractive model
The Menger Sponge is a fractal form generated by such a process of 'internal' subtraction. A primitive cube is subject to the sequential removal of its volume. At each subsequent stage the elements removed are smaller in size but greater in quantity. This hierarchical process is conceptually infinite and tends toward an 'object' of zero volume but infinite surface area. The resulting form is, therefore, counter-intuitive; its outward appearance is one of considerable bulk whilst its volume is actually minimal. Conversely, the volume of space which it encloses is considerable, hierarchically infused through the interior of the cube.
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Zoning as subtraction
Before the introduction of zoning, each lot possesses a maximised development envelope - a virtual, vertical extrusion from the lot boundary to theoretically unlimited height. After 1916 this maximum 'mass' is divided into a space within which development is permitted and, its reciprocal, a space within which development is precluded. The 'setback' and 25% tower rules prescribe a series of slicing planes which divide the maximum mass into these two volumes, the negative and the positive, the subtracted and the residual.
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Copyright © 2004, 2005 Chris Burrows
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