Remarks on the Surface Area and Equality Conditions in Regular Forms Part IV: Pyramidal Forms
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Abstract
Following the same methodology and rules that were applied in the previous Parts I–III of this work, Part IV presents the mathematical remarks on the right regular pyramidal forms, either complete or truncated (a frustum of pyramid). The first remark examines the effects of θ and β on S. The second remark calculates the minimum total surface area (S_{Min}) in two cases, case of constant θ (variable β) and case of variable θ (constant β). The third remark calculates walls ratio R_{W} and the critical walls ratio R _{ Wo }. The last remark calculates the numerical equality between S and V. In conclusion, the importance of the findings of the entire work (Parts I–IV) in advanced building analysis is highlighted and discussed.
Keywords
Acoustics Pyramids Pyramidal forms Surface area Walls ratio Building analysis Minimum total surface area Numerical equalityIntroduction
Pyramidal forms are some of the oldest forms that humans have ever known. Historically, pyramids were constructed in different compositions (complete, truncated and stepped) in many places of ancient world, such as Egypt, Mexico and Latin America. The step pyramid of Djoser (Zoser, in the Saqqara necropolis) may be the first of this type (built around 2800 BC). In the realm of modern architecture, the pyramidal form still attracts architects. There are several examples for modern pyramidal buildings in different places around the world, for example the Unknown Soldier Memorial in Cairo, by Sami Rafi (1973), Louvre pyramid in Paris, by I. M. Pei (1989) and Luxor Hotel, Las Vegas, by Veldon Simpson (1993).

How do the angles θ and β affect S?

When does S become minimum (S_{Min})?

What is the ratio between walls total surface area S_{W} and S (S_{W}/S = R_{W})?

When S equals V?
Notations
 Ar

Base area, area of the lower base (truncated pyramid) (m^{2})
 Ar_{s}

Area of one side face of the pyramid (m^{2})
 Ar_{U}

Area of the upper base (truncated pyramid) (m^{2})
 β _{ o }

The critical dihedral angle, the angle at which S of a complete right regular pyramid becomes minimum
 h

The altitude of the triangle (m)
 h_{s}

Slant height (m)
 H_{R}

The altitude of the pyramid (m)
 H _{ Ro }

The critical room height, the height that fulfills (S − V) equality (m)
 n

Number of sides
 Per

Perimeter (m)
 S

Room total surface area (m^{2})
 S_{Min}

The minimum total surface area (m^{2})
 S_{W}

Walls total surface area (m^{2})
 r

Radius (m)
 r _{ o }

The critical radius, the radius that fulfills (Per − Ar) equality (m)
 R_{W}

Walls ratio, S_{W}/S (ratio)
 R _{ Wo }

The critical walls ratio, the ratio between walls total surface area and total surface area when S is minimum (S_{Min}) (ratio)
 V

Room volume (m^{3})
 ω _{ o }

The critical ratio, the ratio between H_{R} and r when S is minimum (S_{Min}) (ratio)
The other terms will be illustrated in figures according to each case as required.
Right Regular Pyramid

All sides are congruent.

All angles are congruent (thus, the angle ψ is constant and equal to 360/n).
The Mathematical Relationships of Right Regular Pyramids
In Part III of this work, the main mathematical functions between θ, L, h, r, Per and Ar have been driven. These functions are also applied in this part as long as the base of the pyramid is regular as defined and assumed. While this section will not repeat those functions of Part III, it will derive additional mathematical functions among r, S, β and V (see Fig. 1). These new functions will be utilized later to determine S_{Min} and calculate the equality conditions.

In the first, β and Ar are variables whereas n (thus θ) is constant.

In the second, β is constant whereas n and Ar (i.e., r) are variables.
Remark 1: Effects of θ and β on S

Zone [a]: where 0° < β < β _{ o }. In this zone S is a decreasing function of β.

Zone [b]: where β _{ o } < β < 90°. In this zone S is an increasing function of β.
Remark 2: The Minimum Total Surface Area, S_{Min}

Case of variable β and constant θ;

Case of constant β and variable θ.
Case I: Variable β and Constant θ
Values of ω _{ o } for the common right regular pyramids according to their bases
θ ^{ o }  60  45  36  30  22.5  0 

Shape of base  Triangle  Square  Pentagon  Hexagon  Octagon  Circle 
ω _{ o }  1.4142  2.00  2.288  2.449  2.613  2.828 

Zone [a]: where β < β _{ o }. In this zone, S is a decreasing function of H_{R} (see Fig. 4) and an increasing function of Ar (see Fig. 5). Note that the location of the zones is reversed in Fig. 5, thus any increase in pyramid height will decrease its total surface area.

Zone [b]: where β > β _{ o }. In this zone, S is an increasing function of H_{R} and a decreasing function of Ar. This means that an increase in H_{R} will increase S.
Case II: Constant β and Variable θ
Again, due to the complexity of this last equation, the program Maple^{®} for symbolic calculations was utilized to get the value of θ at which S becomes minimum. The calculations indicate that S_{Min} occurs when θ → 0.
Thus, among the different right regular pyramids, a cone has the minimum total surface area. This conclusion completely agrees with the numerical solution presented in Fig. 2 and discussed in “Remark 1: Effects of θ and β on S”.
Remark 3: Walls Ratio R_{W}
Remark 4: Case of Numerical Equality
In Part III (Elkhateeb 2014b), the numerical equality between Ar and Per was calculated. As the same condition applies here, it is not repeated again. Thus, this section considers only the numerical equality between S and V in right regular pyramids.
Equality of S and V
Truncated Right Regular Pyramid

The slant height h_{S} is the altitude of a side face.

The lateral edges are equal, and the side faces are equal isosceles trapezoids.

The two bases are similar parallel regular polygons.
The Mathematical Relationships of Truncated Right Regular Pyramids
 1.
β and Ar are variables whereas n (thus θ) is constant.
 2.
β is constant whereas n and Ar (i.e., r) are variables.
 3.
H_{R} is variable whereas both β and θ are constants.
Again, in all cases Ar is variable as long as V, β and H_{R} were determined.
Remark 1: Effects of θ, β and H_{R} on S
 S is an increasing function of θ when H_{R} and β are constants. Thus S_{Min} occurs when θ → 0 (i.e., a cone). Figure 8 represents θ–S relationship for different cases of β.
 S is a decreasing function of β when H_{R} and θ are constants. Thus S_{Min} occurs when β → 90 (i.e., a prism). Figure 9 represents β–S relationship for different cases of (n).
 S is a decreasing function of H_{R} when θ and β are constants, thus S_{Min} occurs when H_{R} is maximum. Figure 10 represents H_{R}–S relationship for different cases of θ and β.
Remark 2: Walls Ratio R_{W}
Similar to the case of the complete pyramid, Eq. 35 says that R_{W} is independent on both V and θ, but it is a function of r, r_{U} and β. It is clear that R _{ Wo } will never exist as long as the three functions (θ–S, β–S and H_{R}–S) remain constant in direction.
Remark 3: Case of Numerical Equality
As mentioned above, the condition for the numerical equality between Ar and Per is the same as the case of regular multisided right prisms. Thus, the following section considers only the case of numerical equality between S and V.
Equality of S and V
Discussion

point out special rooms that have a distinct characteristic in acoustics;

clarify why rooms that have the same volume and floor area but different shapes have different reverberation times;

establish a simple mathematical approach that can help both architects and acousticians to decide early the appropriate room dimensions. These dimensions satisfy the acoustic requirements;
 help architects and acousticians to answer two important questions:

how does S changes in θ (and/or β) and consequently how will the acoustic environment within a room be affected?

When deciding upon the appropriate room dimensions that have a given V, is it better to decrease Ar and increase H_{R} or inversely, to increase Ar and decrease H_{R}?

Upon designing a room acoustically, its T must lay within the permissible limits that depend on the acoustic function of this room. A short T (around 1 s) is recommended especially in speech rooms. As can be concluded from Eq. 44, for a given V, T is a decreasing function of S. Thus, it is recommended to increase room total surface area so as to decrease T.
Based on the findings of this work and according to the shape, remark (1) determines the conditions under which S will take its minimum value among the different rooms that have the same floor area and volume but different θ (and/or β). For example, in rectangular right prisms (or rooms), a room that has θ = 45° (square plan) has the minimum total surface area, thus the maximum T among the other rectangular rooms that have the same volume assuming that α is constant (Elkhateeb 2012).
Following the same rules, remark (2) determines the condition under which S will be minimum among the different rooms that have the same θ (and/or β) and V but different Ar. Such a room also possesses the maximum T and should be avoided (Elkhateeb 2012). Together, remarks (1) and (2) establish a clear methodology that can be applied to select the optimum room dimensions from an acoustic point of view.
In any room, walls are the typical place to install the different acoustic treatments such as absorbing and reflecting materials. Acoustically, in some applications, it is preferable to increase walls ratio so as to insure a good acoustic performance. Utilizing remark (3), R_{W} can be checked easily during the analysis and design phase.
Conclusions
Applying the same methodology, assumptions and rules that were introduced in the previous Parts I–III of this work, this final Part IV examines the case of the right regular pyramid either complete or truncated. In complete pyramids, the first remark examines the effects of θ and β on S. In the second remark, the minimum total surface area S_{Min} was calculated in two cases, case of variable β and constant θ, and case of constant β and variable θ. In the first case, the critical ratio ω _{ o } that corresponds to the critical dihedral angle β _{ o } (70.528779°) was calculated. Results showed that ω _{ o } depends entirely on θ. The values of ω _{ o } were calculated and presented for the common right regular pyramids according to their bases. In the second case, where θ is variable, results showed that S_{Min} occurs when θ → 0 (i.e., cones). In the third remark, the ratio R_{W} was calculated. Results showed that R_{W} is an increasing function of β. Results also showed that R _{ Wo } is constant (=0.75) regardless the value of (n). In the last remark, the critical room height H _{ Ro } that fulfills the numerical equality between S and V was calculated. Results indicated the limit under which this equality will never exist.
In a truncated pyramid, the first remark investigates the effects of three independent variable θ, β and H_{R} on S. When θ is variable (whereas H_{R} and β are constants), results showed that S is an increasing function of θ, thus S_{Min} occurs when θ → 0 (i.e., a truncated cone). When β is variable (whereas H_{R} and θ are constants), results showed that S is a decreasing function of β, thus S_{Min} occurs when β → 90 (i.e., a prism). When H_{R} is variable (while θ and β are constants), results showed that S is a decreasing function of H_{R}, thus S_{Min} occurs when H_{R} is maximum. In the second remark, the calculation of R_{W} indicated that it depends on the three variables r, r_{U} and β regardless the values of θ and V. In the last remark, the critical room height H _{ Ro } that fulfills the numerical equality between S and V was calculated. In this remark also, the limit under which this equality will never exist was presented. Finally, the importance of the findings of this entire work (Parts I–IV) in room acoustics as a branch of the advanced building analysis was presented.
Notes
Acknowledgments
The author would like to express his gratitude and sincere appreciation to EditorinChief Kim Williams for her valuable advice and support. Thanks also go to Arch Zinub Najeeb for her continuous help during the preparation of this work.
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