THE COMPACTNESS INDICATORS OF SOLIDS APPLIED TO ANALYSIS OF GEOMETRIC EFFICIENCY OF BUILDINGS

. The authors redefine and generalize the so-called relative ratio of compactness of solid with respect to sphere and cube introduced by Mahdavi and Gurtekin. The authors propose such new indicators as a relative ratio of solid compactness in the shape of a prism-shaped solid related to a cuboid with a square base and a given height (e.g. the height of the analyzed storey), a solid compactness indicator defined for the basis of the prism relative to the square. They introduce also other indicators: a relative defect of the perimeter and area. The test of indicators formulated and described in the paper was carried out in two groups of buildings: existing (spotted locally) and those whose designs can be found in the selected catalog available on the website. In addition to the compactness ratios calculated for existing and designed buildings, an analysis of the size of the indicators for the adopted building research models was made. Proposed indicators allow for a description of the compactness of the building model with an indication of the deviation from the real ideal solid. The introduced geometric determinants of solids may be useful in the initial phase of searching for satisfactory design solutions.


Introduction
At the beginning, we give an understanding of the most important terms used in this work.The geometric efficiency of a building that meets the assumed size parameters (cubic capacity, usable area) is a set of geometric features that make the building functional, economical (with low energy demand) in construction and maintenance, safe in use and aesthetic.An important geometric feature of the building is its compactness.By building compactness we mean the compactness of the solid which is an isometric geometrical model of the building envelope or its part.The geometric compactness of a rigid solid is the relationship between the enclosing surface and volume.The classical measure of compactness is defined by the dimensionless ratio (area) 3 /(volume) 2 (Bribieska, 2000).In this work as a measure of compactness, so-called compactness indicators, defined by the authors or quoted from the literature, will be used.
Already in 1934 in a monograph (Fokin, 2006) the shape of a building with a given volume was optimized on the basis of the minimum thermal energy loss criterion and the result was a building in the shape of a sphere.In the paper (Menkhoff et al., 1983) analyzed the geometry of the building structure and it was where the term geometric compactness as the quotient of the external partitions (A) and building volume (V) was introduced for the first time in literature.The analysis of the compactness was performed by studying a solid composed of four identical cubes with edges a in different configurations.Ratios from 4/a to 14.1/a were obtained.Behsh (2002) stated that the A/V ratio is not a valid indicator of thermal efficiency of buildings with complex plans.
These results: 1. Testify to the validity of the geometrical study of the compactness of the designed building, since the ratio A/V can have such a large (350%) dispersion; 2. Indicate imperfections of the A/V ratio whose values depend on the adopted units.Markus and Morris (1980) defined the so-called "Ratio of Change" (ROC).The "Ratio of Change" has been calculated by comparing the surface area to volume ratio of a building to that of a cube with the same volume (Markus & Morris, 1980).Mahdavi and Gurtekin (2001) used the socalled "Relative Compactness" (RC) with respect to sphere (cube) defined as the ratio between compactness (A/V) sph and (A/V) build ((A/V) cube and (A/V) build ).Using this indicator, objects of various shapes were studied (Mahdavi & Gurtekin, 2001;Pessenlehner & Mahdavi, 2003;Geletka & Sedláková, 2012).Bostancioğlu (2010) uses the ratio of external wall area to floor area (EWA/FA).Ourghi et al. (2007) and Tuhus-Dubrow and Krarti (2010) use the relative compactness (RC) coefficient, which express the ratio between the shape coefficient of a designed building (A/V) des and the shape coefficient of a reference (rectangular) building with an equal volume (A/V) ref .Parasonis et al. (2012) proposed several improvements the last ratio.Instead of the A/V ratio, A/S is assumed, where A is the area of the external envelope of a building and S is the useful (heated) area, showing how efficiently geometry is utilised.Authors proposed that the non-dimensional A/S ratio can be referred to as the concept of "Geometric Efficiency" (GE).As a result of the analysis of exemplary buildings, it turned out that both compact and non-compact buildings can have the same GE values.Therefore, they proposed adopting relative geometric efficiency (RGE) as a ratio GE/GE ref , where GE ref = (A/S) ref is the limit (reference) expression of geometric efficiency that is the closest to a cubic building (reference building) that accommodates a given area; RGE shows how far GE of a designed building deviates from the GE ref value of the reference building (Parasonis et al., 2012).The area research is the search for ways to assess the geometrical content of the designed building (Hemsath & Bandhosseini, 2015;Raof, 2017).Often these are quite complicated algorithms implemented as computer programs (Rodrigues et al., 2015).The search for the best shape of the building is still valid (Almumar, 2016;Lim & Kim, 2018).The authors are looking for, but in our opinion do not find a satisfactory solution.Such indicators, in contrast to the A/V ratio, are already non-quantified numbers and to a certain degree normalized, however, they do not have the characteristics to express the deviation of the compactness of a given solid from the reference (model) in percentage points.In addition, the reference to the cube still has some imperfection, namely the height of the storeys of a single-family house is basically fixed and ranges from 2.50 [m] to 3.30 [m] (N.Neufert & P. Neufert, 2012).Then the height of the multi-storey house is a particular height.Imposing the "artificial" height resulting from the modeling of the cube is a distortion of the measure of the building's compactness.The concept of the reference building also remains undetermined.
The main purpose of the work is to generalize the compactness indicators used by Mahdavi andGurtekin (2001, 2002) and to introduce new geometric characteristics of the building, which allow to improve the description of the shape of the building (compactness of the building) from the point of view of its geometrical efficiency.

A new approach to the geometric description of the shape of a building
Koźniewski in his work (Koźniewski, 2007) described the properties of rectangular polygons, and later in his paper (Koźniewski, 2015) introduced the concept of the relative defect of the area and perimeter as well as the span of the polygon.These results have become an inspiration for further search for other indicators, more accurately characterizing the geometric compactness of solids.
In this paper, the authors propose such new indicators as a relative ratio of solid compactness in the shape of a prism-shaped solid related to a cuboid with a square base and a given height (e.g. the height of the analyzed storey), a solid compactness indicator defined for the basis of the prism relative to the square (Koźniewski, Żaba, & Dudzik, 2015).
Then, in order to "generalize" the so-called defect of the perimeter described in the paper (Koźniewski, 2015), the authors introduce a new relative defect of the perimeter with parameter k (k -ratio of the length of the sides of the rectangle) in relation to the reference rectangle (square for k = 1).
The object of research in this study is the geometric compactness of a detached house on the projection of a rectangular polygon, hitherto understood as A/V (the quotient of the surface area of its external walls and volume).Calculations were carried out for simplified geometrical models of these buildings, mainly in relation to the rectangular part of the body of the building with the basis of a rectangular polygon, and also in relation to the projection of the building itself.The general scheme of geometric compactness of the building is shown in Figure 1.A symbolic illustration of the indicators introduced in the work and explanations of meanings of acronyms used in the paper are provided in Tables 1, 2 and 3, respectively.
Regarding the shape of the building's projection, from articles by Feather (1996) and Steadman (2006), we find out why the projections of a building in the form of a polygon are the most popular.In the catalogs of finished designs, it can be noticed that the most popular design is the one whose projection contains only concave angles (270°) and convex angles (90°).Designs that do not fit into this group usually have in their outline only some additional  elements (e.g.bay windows in the shape of a trapezoid or circular section) that do not allow them to qualify for the category in question.These deviations can be easily converted so that the outline can be considered as a rectangular polygon.Such a change will not be of significant importance in the analysis of the functional system, usable area and the amount of materials used to build the considered building.

The compactness of the solid with respect to a cube and sphere
In their work Mahdavi and Gurtekin (2002), the authors examined the compactness of buildings in relation to two solids with the best compactness.As a measure, they proposed the so-called relative compactness indicator of the solid with respect to the sphere Perimeter defect (defect of perimeter of a rectangular polygon R P with respect to R) ∆A b (R P ) Area defect (defect of area of a rectangular polygon R P with respect to R)

RDA
Relative defect of area (of R P with respect to R)

RDA'
Relative defect of area (of R P with respect to R P )

RDP
Relative defect of perimeter (of R P )

R k
Reference rectangle with the ratio of sides equal to k R 1 Square (a reference rectangle with the ratio of sides equal to 1)

RDP k
Relative defect of perimeter (with respect to R k )

RDP 1
Relative defect of perimeter (with respect to a square R 1 ) and with respect to the cube (2) The numbers 6 and 4, 84, appearing in Eqns ( 1) and ( 2), directly indicate the solid relative to which the relative compactness indicator is calculated.Such a description, according to the authors of this work, does not give the possibility of a direct interpretation of the values of indicators.No wonder, why the authors Mahdavi and Gurtekin (2002) suggest transforming the value of the RC * cube indicator into a seven-point scale.They adopted the following scales of assessment (semantic differential for the subjective evaluation of the compactness shape): highly non-compact, non-compact, somewhat (fairly) non-compact, neutral, somewhat (fairly) compact, compact, highly compact.However, the indicators of relative compactness (applied for the optimization of the shape of the building) still do not give a very readable view of the degree of deviation, e.g.expressed in percentage points, relative to the reference solid.The introduction of a new indicator (RC cd ) eliminates the need for an additional scale.
In addition, comparing to highly ideal solids (sphere, cube) and as a rule strongly differing from practical, functional shapes of buildings, especially detached houses, seems to be complicated and not revealing the essence of things.Meanwhile, the building designer would prefer to have relatively accurate simple indicators, best expressed in percentage points, giving deviations from the reference pattern, considered in terms of design, economy, ecology, etc.For example, by how many percentage points the surface area of the external walls of the building with the usable area design solution will be larger than the area of the optimal solution.It is connected with one-off construction costs and many operating costs (heating, maintenance, renovation).
Admittedly, Ourghi et al. (2007), Tuhus-Dubrow and Krarti (2010), Parasonis et al. (2012) propose using the relative compactness (RC) coefficients, perfectly formulated and very well implemented, which express the ratio between the shape coefficient of a designed building and (differently defined) reference building, but this reference shape of building is not unified and requires complicated calculation software.Recent searches (Hemsath & Bandhosseini, 2015;Rodrigues et al., 2015;Raof, 2017;Lim & Kim, 2018) also do not bring simple proposals in this area.And the designers will use the building geometry indications if the proposals will be simple and will not hamper their creativity.
The authors of this paper have undertaken the task of looking for a more accurate and simple solution to this problem, so that the geometrical description of the object can be easily translated into energy costs, comfort of use as well as aesthetics.Therefore, they propose to introduce other, more realistic and intuitive, reference shapes.More general description parameters are introduced which al-low any reference solids (when defining compactness factors).

Solid compactness indicators in the aspect of application for determining the compactness of building
As we have already said, comparing the size of the surface area and volume ratio of the building body to the ratio of the same parameters for the cube has some imperfections.The building has a modular height within certain limits or a height equal to a multiple of the modular height (N.Neufert & P. Neufert, 2012).It would be rational to compare it with a "cube-like" building, but with the same height (e.g. if the prism had a square base and the same height as the building under examination).If we assume that the solid has the structure of a straight prism, with the prism having the l-connected polygon as the basis (a classic polygon with polygonal holes), and the cylinder having any l-connected basis as the base, the standard solid may be a rectangular prism with a square base.The second disadvantage of compactness indicators introduced and used by Mahdavi and Gurtekin (2002) is the way of determining the ratio.The authors of this paper propose to replace the numerator with the denominator in Eqns (1), (2).Then the modified quotients will take values greater than (or equal to) one.After multiplying by 100%, we get a result expressing the percentage increase in the surface area of the solid in relation to the area of the standard cube.We will get a readable indicator which is easy to interpret.

Relative compactness indicator of a solid with respect to another solid
First, we shall introduce a relative compactness indicator related to the arbitrarily adopted reference solid.As suggested, the parameters (more accurately the parameter of the total area) of the reference solid will be in the denominator of the fraction defining the indicator.
We assume that RC Spat (Relative compactness of solid S with respect to reference solid S pat ) is expressed as the quotient of the geometric compactness of the solid S and the geometric compactness of the solid S pat where A t (S) is the total area of the solid S.
for any solid S.
Similarly we find the relative compactness indicator with respect to a sphere 36 36 V(S) . ( 6) Then we get the reversals of relationships known from the literature (Mahdavi & Gurtekin, 2002): The Eqn (10) can be recognized as the relative compactness of the prism G with the base B (and omitted in the consideration of the height h), referred to the cuboid with a square base with area equal to the base B of the prism G (and omitted in the consideration of the height h).Now let's transform the Eqn (10) and calculate the limit with h→∞: i.e.

→∞ = lim
So we have a situation in which it can be concluded that the RC sq indicator is the effective assessment of the compactness of a high building.Anyway, the result is intuitively easy to verify.Namely, a high building with an optimal compactness will have a square-like contour.It would seem that the omission of height in evaluating the compactness of the building, the property of merely highrise buildings, or rather very high.A closer analysis allows noticing that the height of a building is actually a matter of recognition of the investor and designer.In the case of a residential building in Poland, this is the size specified by law (Decree of the Minister of Infrastructure and Development, 2015), in the case of buildings for special purposes (industrial and sports halls) the height of the storey also remains outside the rational actions of the optimization analyses.The height of the building is naturally excluded from the assessment of its compactness.So we have two options when determining the compactness of a building: (A) arbitrarily take the height h of the building (storeys) and then the indicator of compactness RC cd is the best measure of compactness or (B) omit the height when determining the compactness of the building and then the best indicator of the compactness of the building is the indicator RC sq .Of course, this does not mean failing to take into account the height of the building in the calculations.The side surface of the building body will be determined by multiplying the perimeter of the contour (base polygon) by the height of the object, we will determine the floor surface (ceiling) directly by calculating the area of the contour (base polygon).Hence, the main parameter of the shape of a building is its plan.For this reason, we will discuss the so-called defect perimeter of the polygon, then generalize it for any figure (Koźniewski, 2015).

Rectangular polygons inscribed in rectangle. How to specify a reference building?
The vast majority of single-family houses and buildings in general is designed on a rectangular polygon plan (Koźniewski, 2007(Koźniewski, , 2015)).The introduced coefficients RC cd and RC sq do not exhaust the parameters characterizing the shape of the building on the plan described by a polygon with right angles, but more complex than a rectangle.It is necessary to determine natural indicators describing the properties of the so-called rectangular polygon.A rectangular polygon (Figures 4, 5) is a polygon that has only right convex angles (90 degrees) and concave angles (270 degrees).The polygon has always an even number of sides (Koźniewski, 2007) and the difference between the number of convex and concave angles is equal to 4 (Koźniewski, 2007(Koźniewski, , 2015)).Each two sides of the rectangular polygon are parallel or perpendicular to each other.For each polygon of this type, there is exactly one rectangle circumscribed on it.Let's assume that a rectangular polygon R P and a rectangle R (circumscribed on it) are given.

Perimeter defect and area defect
Let us denote the perimeter (area) of the rectangular polygon R P and the rectangle by P b (R) (P b (R P )), A b (R) (A b (RP)), respectively.The following number of ∆P b (R P ) (∆A b (R P )) is expressed as follows: and called the perimeter defect (area defect) of the rectangular polygon R P (Koźniewski, 2015) (Figure 2).
Perimeter defect and area defect of the polygon defined in absolute terms do not reflect the size of the measurement deviations from perimeter and area of a rectangle.Besides, in practical applications these will depend on the accepted measurement units of length and area.Therefore, it is desirable to describe these measurement deviations (from the ideal figurea rectangle) in a relative manner.Let us introduce therefore, two concepts: the relative defect of perimeter of the rectangular polygon and the relative defect of area of the rectangular polygon Relative defect of area, with the defect of perimeter equal to zero, shows the degree of "imperfections" of the border line.The same length of perimeter takes up roughly ×100% smaller area; so there is loss in the area at the same perimeter.Because the larger the perimeter defect the larger perimeter, area defect with increased perimeter results in even greater losses of the area.
Let us assume the following shape of a rectangular polygon (Figure 3 17) we have RDA(R P ) = 27/108.We can say that the area defect is 25%.This defect is measured relative to the rectangle described on the rectangular polygon and referenced to this rectangle.Another reference may be used and in the Eqn ( 17 Then the relative area defect will be 33%. In the example described in Figure 3(b), the perimeters of comparable figures were the same.Let's take another example (Figure 3(c)).The perimeters of the figures are different, so it is advisable to use a different indicator.In the second case perimeter defect equals 8, so the relative defect of perimeter RDP(R P ) = 8/42.Thus the relative defect of perimeter is 19%.In relation to the rectangle described on the discussed rectangular polygons (Figures 3(b), 3(c)), defects are 0% and 19% respectively with the same relative area error RDA = 0.25 relative to the rectangle or the same relative field defect RDA' = 0.33 in relation to a rectangular polygon.Summing up, we can say that the RDA informs us how much space (and cubage) we have lost by giving up the rectangle as the contour of the building contour while maintaining a given surface of external partition walls (if RDP > 0).Whereas RDA' informs us how much surface of the external partition walls (RDP = 0) or how much space (and cubic capacity) we could gain by changing the surface of the external partitions by RDP (if RDP > 0).

Proposal for definition of a reference building
Let us generalize a perimeter defect by referring to a given figure (rectangular polygon R P or even any figure F with area A(F)) to a figure with the same area being a reference rectangle R k with a ratio of sides equal to k.Then R 1 (k = 1) is a square, + 1 2 R (k = 1+ 2 ) (with the different designation R s ) is a rectangle with the silver side ratio, with the different designation R g a rectangle with the golden side ratio, is a rectangle with the dimensions adopted in Figure 3(a).The area of the rectangle R k (with the ratio of one side to the other sideshorter or equal) can be written in the form where r is the length of the shorter side (1≤ k).Assume that the areas of the rectangle R k and figure F are equal.
Then the perimeter of the reference rectangle R k for the given figure F is The perimeter defect and the relative defect of perimeter are correspondingly equal Table 4. List of relative defects of perimeter depending on the ratio of sides k of rectangles Name Ratio of sides k Formula Relative defect of perimeter per rectangle with a side ratio 4: Relative defect of perimeter per "golden" rectangle Relative defect of perimeter per rectangle with a side ratio 2: List of relative defects of perimeter is provided in Table 4.
Assuming in Eqn ( 24) k = 1 we obtain a relative defect of perimeter related to the square A comparison of Eqns ( 11) and ( 25) allows noting of their relationship.Indeed, after replacing in the Eqn (25) the variable F by B we get RDP 1 (B) =RC sq (B) -1. (26)

Compactness indicators of existing or designed buildings
The test of indicators formulated and described in the paper was carried out in two groups of buildings: existingspotted locally and those whose designs can be found in the selected catalog available on the website.

Compactness indicators of existing buildings
Field research was carried out in Jastrzębie Zdrój (Poland).Six buildings have been analyzed.On the basis of dimensions (from the inventory), the roof skeleton has been reconstructed and the values of the indicators RC cube , RC cd , RDA and RDP 1 were calculated (Figures 4,5 and 6).Garages that are usually not heated are not included in the calculation of indicators.
The values of indicators for all randomly selected six examined existing buildings are shown in Table 5.The val-

Compactness indicators of designed buildings
The analysis of buildings in the design phase was made on the basis of information posted on the website.On the basis of dimensions, the values of indicators RC cube , RC cd , RDA and RDP 1 were calculated (Figures 7, 8 and 9).In all the cases, the RC cube indicator indicates quite a large deviation from the ideal cubical shape (1.29, 1.27, 1.21, or 29%, 27% and 21%, respectively).The RC cd indicator values showing the deviations from the square-shaped cuboid shape are 1.03; 1.02; 1.00, or 3%, 2%, 0%.However, the last indicator determines the ideal shape of the building, coinciding with the reference shape.It is worth  noting that this house was designed as a passive, energysaving building.The values of RDA and RDP 1 are equally interesting -both equal to zero.It is worth noting that there are buildings with real shapes and dimensions with the RC cd = 1 indicator value (Figure 9), while there are no buildings with real dimensions, with the RC cube = 1.

Indicators for research models of building solids
In addition to the compactness ratios calculated for existing and designed buildings, an analysis of the size of the indicators for the adopted 48 building research models was made, which is shown in Figure 10.The study of the compactness of buildings was carried out on the basis of a catalog of finished projects (Projekty Muratordom, 2018).
At the beginning, it should be emphasized that the correlation coefficient calculated between the RC cube and RC cd index values for 48 building models is equal to 0,7876 (for the first 10 models it is 0.8614, for the first 22 models it is equal to 0.8748, and for the first 30 models it equals 0.8733) .In this sense, the RC cd indicator is a generalization RC cube and the RC cube indicator can be replaced by RC cd .Interestingly, the RC cd index is lower, but the graph of this indicator is more flattened than RC cube .Less sensitivity to changes is complemented by RC sq values that give greater variability between individual models.Both RC cd and RC sq indicators simultaneously present a more accurate variation of the compactness of the objects.We will see a more complete image of the relationship by analyzing the values of the other indicators: RDP, RDA, RDA' and RDP 1 (Figure 11).We have RDP = 0 in more than half of the research building models.All buildings have RDA > 0. For building No. 48, the RDA indicator is as much as 40%, and the RDA' is almost 70% (Figure 12).We will see a more complete image of the relationship by analyzing the values of the other indicators: RDP, RDA, RDA' and RDP 1 (Figure 11).We have RDP = 0 in more than half of the research building models.These indicators are dictated by, among others, that the designer wanted the building span to not exceed 10 m.This span is exactly 9.8 m.Because the RDP > 0 building material used for the exterior walls of building No. 48 would be enough for a rectangular building with dimensions of 21.3×22.9[m].
But then the span of the building would be 21.3 m.

Conclusions
The analysis of determinants of all buildings and designs selected for research indicates the rationale of introducing the RC cd indicator as definitely better reflecting the building's compactness than the RC cube indicator.It is worth noting that the passive energy-saving building, in a visual assessment having optimal dimensions (Figure 9), has the RC cube =1.21.The value of the RC cd indicator is equal to 1.00, which means that it is a building with perfect compactness.The values of RDA = 0.00 and RDP 1 = 0.00 confirm the optimal compactness of the house "Valletta Passive LDP06".In the other cases, small deviations of the RC cd indicator indicate the percentage deviation of the compactness (imperfection) from the compactness of the reference solid.They range from 2% to 4%.
All the analyzed examples confirm that the RC cd indicator is a simple good measure of the compactness of buildings.The value 1 of this indicator is a limit value and means that the building has the best compactness.According to the authors, it is the most important indicator introduced in this paper.However, all together, despite the total dependence of some (RDP 1 and RC sq ; RDA' and RDA) or limit dependence (RC cd and RC sq ), they are a tool for testing the compactness of buildings.
Although the subject of the paper is devoted to buildings constructed on a rectangular polygon plan, it should be emphasized that all indicators, except RDP and RDA(RDA'), are universal; their definitions have not been claimed to concern rectangular polygons.Proposed indicators allow for a description of the compactness of the solid (building model) with an indication (in percentage points) of the deviation from the real ideal solid (or figure in the plane).The introduced geometric determinants of solids may be useful in the initial phase of searching for satisfactory design solutions.Previously, research should be carried out by calculating the construction costs (raw shell of a closed building) and energy consumption (for the same types of building partitions) to create an appropriate information base.Such results will be adequate to the types of materials and climatic conditions.

Figure 1 .
Figure 1.General research scheme of the geometric compactness of a building be a prism on base B and height h.Let A b (B) denote the area of the base B and P b (B) the perimeter of the base B. Let us assume that S pat is a cuboid (a rectangular prism with a base of a square) with an edge length a and a height h.Using (3), we will define the relative compactness of a solid S with respect to a cuboidRC cd = RC Spat (S).(9) Because V(S) = A b (B)h due to (4) we have A b (B)h = a 2 h, hence A b (B) = a 2 .Then a = b A (B) .Therefore A t (S) = 2A b (B) + P b (B)h.Simultaneously A t (S pat ) = 2a 2 + 4ah.Due to a= b A (B) we get A t (S pat ) = 2A b (B)+4b A (B) h.After substituting into the Eqn (3) we get up the height of the prism and the surface of the top and bottom bases of the solids.The reference will be only the shape of the solid's base B. Let P b (B) be the perimeter and A b (B) area of a planar figure B. Then relative compactness indicator of figure B with respect to the square has a form

Figure 2 .
Figure 2. Rectangular polygons (hatched) inscribed in a rectangle with dimensions of x×y, x = 9u, y = 12u: a) with a big area defect (∆Ab = 88u 2 ), RDA = 0.81 (81%), and zero's perimeter defect, RDP = 0 (0%); b) with a positive perimeter defect (∆Pb = 24u, RDP = 0.57 (57%)), with area defect (∆Ab = 38u 2 , RDA = 0.35 (35%)); c) with a positive perimeter defect (∆Pb = 40u, RDP = 0.95 (95%)), with defect (∆Ab = 44u 2 , RDA = 0.41 (41%)) Figure 3.The base rectangle and a rectangular polygon inscribed therein: (a) reference rectangle; (b) rectangular polygon with a perimeter defect equal to zero; (c) rectangular polygon with a perimeter defect different from zero Figure 3(b), we will have RDA'(R P ) = 27 .81Then the relative area defect will be 33%.In the example described in Figure3(b), the perimeters of comparable figures were the same.Let's take another example (Figure3(c)).The perimeters of the figures are different, so it is advisable to use a different indicator.In the second case perimeter defect equals 8, so the relative defect of perimeter RDP(R P ) = 8/42.Thus the relative defect of perimeter is 19%.In relation to the rectangle described on the discussed rectangular polygons(Figures 3(b), 3(c)), defects are 0% and 19% respectively with the same relative area error RDA = 0.25 relative to the rectangle or the same relative field defect RDA' = 0.33 in relation to a rectangular polygon.Summing up, we can say that the RDA informs us how much space (and cubage) we have lost by giving up the rectangle as the contour of the building contour while maintaining a given surface of external partition walls (if RDP > 0).Whereas RDA' informs us how much surface of the external partition walls (RDP = 0) or how much space (and cubic capacity) we could gain by changing the surface of the external partitions by RDP (if RDP > 0).

Figure 4 .
Figure 4. Characteristics of the shape of an existing building and its selected indicators of compactness

Figure 7 .
Figure7.Characteristics of the shape of an existing building and its selected compactness indicators on the basis of the design Denver DCB112 MC.In the calculation, a garage that is not heated is not included.The design was downloaded from LipinscyDomi (2018)

Figure 8 .
Figure8.Characteristics of the shape of an existing building and its selected compactness indicators on the basis of the design Bastia DCB111.In the calculation, a garage that is not heated is not included.The design was downloaded from LipinscyDomi (2018)

Figure 9 .
Figure 9. Characteristics of the shape of an existing building and its selected compactness indicators on the basis of the design Valletta Pasywny LDP06.The design was downloaded from Lipinscy Domi (2018)

Figure 10 .
Figure 10.List of RC cube , RC cd and RC sq indicators for 48 research models

Table 1 .
Symbolic illustration of geometric compactness indicators for RC cube and RC cd building solids

Table 3 .
List of acronyms used in paper Mahdavi & Gurtekin, 2002)s of solid S with respect to a sphere (in the senseMahdavi & Gurtekin, 2002)RC SphereRelative compactness of solid S with respect to a sphere A b (B) Area of the base B of the solid S P b (B) Perimeter of the base B of the solid S RC cd Relative compactness indicator of solid S with respect to the cuboid RC sq Relative compactness indicator of figure B with respect to the square

Table 6 .
Parameters of the six examined buildings, calculated on the basis of the dimensions of the building designh [cm] DP b (…) [cm] RC cube RC cd RDA RDP 1

Table 5 .
Parameters of the examined existing buildings h [cm] DP b (…) [cm] RC cube RC cd RDA RDP 1