The Resonance Between Shapes of Perimeter and Core

This paper investigates the relation between the geometries of perimeter and core in architectural layouts of large buildings. The relation between perimeter and core is paramount for the schematic floor plate design of building types such as offices. This research proposes critical measures to counteract the deficiencies of earlier studies that regard only the core to perimeter metric depth. While being comprehensive, these shall include the entire geometry of the 'duo' perimeter-core by relying on the concept of equal properties of locations depicted by the offsetting of a shape. In focus, the lengths of consecutive offset polygons and their respective depths are used to measure the average perimeter depth and average core depth that are unique for two shapes of core and perimeter. These two indices are strongly linked to evaluating the entire floor plate for fitting organization types, and are related to environmental comfort and hierarchical status stratification of workspaces. The resonance between shapes has served as the basis of a comparison, among several theoretical duos with equal areas, that enables to qualify the nature of two shapes into categories of reconciled or juxtaposed. Introduction This work has started as an attempt to discover why a particular speculative office building type of ‘90s has been used in an almost prototypical manner over a number of years by architectural corporate firms in the United States of America. While the main constraints in the office building design are the workspace grid, the structural grid, fire regulations and maximizing the potential for subletting of parts of a floor, proximity to natural light and outside views and distance to corridors are of paramount importance for the comfort of workspaces in offices. Distance to windows and corridors result in hierarchical stratification of workspaces in terms of organization status of employees with preference often given to the ones close to perimeter and far away from the core. In a general level, the quality of workspaces depends primarily from the geometrical relation between the shapes of the perimeter and core. In this paper I propose two measures that are designed to capture the geometrical relation between the form of building perimeter and the one of core in a robust manner. At the current stage of this ongoing research only a few conjectures are offered on what these measures can prove useful, and most of the effort is devoted towards developing them and sorting out their variance from one complex to another. The seminal study of Duffy et al. (1976) on office building considers the metric depth between perimeter and core for analyzing the main features of the shell. Two main depths from perimeter to core can easily characterize rectangular floor plates of most speculative office buildings. However, using only such a measure is associated with two main issues. First, in cases where the depth between perimeter and core has more than one dimension, it is not easy to characterize floor plates due to the different percentages that depths might cover. This is further complicated when the perimeter has a jagged or curvilinear shape, fig. 1a. Second, even in the case of rectangular regular floor plates, perimeter to core depths account only for areas that fall in the cross-like area between core and perimeter without covering four corner areas, shown shaded in fig. 1b. Average perimeter distance and average core distance I suggest capturing the relation between perimeter and core by means of considering all floor plate locations rather than those falling in the cross-like area. Thus, I measure the average distance of any location of floor plate to perimeter or core by combining the number of potential locations equidistant to perimeter or core with their distance to them. I analyze a floor plate with a rectangular perimeter of 225’x125’ and a symmetrically positioned core of 105’x25’ similar to speculative offices of '90s in the USA. I assume a workspace with a size of 12’6”x12’6” and I offset inwards from the perimeter consecutively with an offset distance od of 12’6”. As I shall show later od does not have any effect on the overall calculation. In the first offset polygon with a length of 600’ there are 600/12’6”=48 potential workplaces that are 1 step away from perimeter; in the second one with a length of 500’ there are 40 workspaces 2 steps away from perimeter; in the third offset with a length of 400 there are 32 workspaces 3 steps away; and in the last one with a length of 300 there are 24 workspaces 4 steps away, fig. 2. Each workspace contributes its depth to the overall depth: 48x1 + 40x2 + 32x3 + 24x4. If we divide the total of these products 320 with the number of all workspaces 144, we get an average distance of all potential workspaces from the perimeter of 320/144=2.22. This shows a step or unit distance that when multiplied with od of 12’6” gives the real average depth of 27’6”. Hence, I calculate the average perimeter distance apd with the formula [1] in which po is the length of perimeter offsets, and d is the step depth from perimeter of an offset polyline. unit average perimeter distance = uapd = Σ (poi * di) / Σ (poi) [1] average perimeter distance = apd = uapd*od Another way to characterize floor plates is to consider the distance from anywhere in the floor to the edge of the core. Similar to the earlier calculation, I offset outwards from the core until all the area of the floor has been covered. I measure the average core distance acd from any workspace to the core with the formula [2] in which co is the length of core offsets, and d is the step depth from core of an offset polyline. unit average core distance = uacd = Σ (coi * di) / Σ (coi) [2] average core distance = acd = uacd*od The calculation of both measures is refined when the offset distance od is decreased. I have calculated both apd and acd for the complex cr-r1 with three offset distances od of 7’6”, 3’9” and 1’10 1⁄2”. The results show the values converge when od is made smaller. I tabulate below the calculations for cr-r1 with 1, 1/2 and 1/4 unit od: Figure 2. The concept of calculating the measure of uapd. (1) 48 workspaces alongside 1st offset line shown in bold, (2) 40 workspaces along 2nd offset line, (3) 32 workspaces located in the 3rd, (4) 24 workpdaces located 4 steps deep. 2.1