# Architectural Characteristics of Different Configurations Based on New Geometric Determinations for the Conoid

## Abstract

**:**

## 1. Introduction

#### Outline of the Problem

^{N}?

^{2}. Such a situation arises when we try to calculate the surface area of a Conoid, a ruled surface generated by parallel straight lines that project from a circumference directrix onto a linear edge (Figure 1).

## 2. Methods and Materials

#### 2.1. Resolution of the Proposed Approximation

^{2}.

^{2}, for this case, 36L

^{2}− 100 L

^{2}= −64L

^{2}

_{3}from Equation (17) is,

#### 2.2. Discussion of the Findings

#### 2.3. Definition of the Antisphera

^{2}, that is Pi squared.

#### 2.4. Comparison with Other Approximate Computing Methods

#### 2.5. Discussion for Higher Values of the Conoid Angle

_{1}would be,

#### 2.6. Calculations of the Area for an Elliptic Conoid

_{1}or κ

_{2}should be introduced in its stead.

#### 2.7. Calculations of the Volume of the Conoid

## 3. Repercussions for Radiative Heat Transfer

#### 3.1. Introduction to the Problem of Surface Factors

_{1}= 2πR

^{2}and A

_{2}= πR

^{2}.

_{11}is also ½ [8].

#### 3.2. Example 1

^{2}and the double conoid is precisely,

_{11}+ F

_{12}= 1, the self-factor F

_{11}is then,

_{11}= 1 − F

_{12}= 1 − 0.527393 = 0.4726

#### 3.3. Example 2

_{2}and θ

_{1}and being its radius r as mentioned.

_{12}= 0.1272. It represents fraction of exchange of radiative energy from the horizontal rectangle under the conoid (A

_{1}) to the semi-circular side of the figure (A

_{2}).

_{3}, following the above stated formulas (Equation (27)) is 20.9042

_{1}F

_{12}= A

_{2}F

_{21}, whence,

_{21}= (16/2π)0.1272 = 0.3239

_{13}= 1 − 0.1272 = 0.8728

_{23}= 1 − 0.3239 = 0.6761

_{3}F

_{31}= A

_{1}F

_{13}and A

_{3}F

_{32}= A

_{2}F

_{23}, which yields,

_{31}= (A

_{1}/A

_{3}) F

_{13}= (16/20.9042) 0.8728 = 0.6680

_{32}= (A

_{2}/A

_{3}) F

_{23}= (2π/20.9042) 0.6761 = 0.2032

_{31}+ F

_{32}+ F

_{33}= 1

_{33}= 1 − F

_{31}− F

_{32}

_{33}= 1 − 0.6680 − 0.2032 = 0.1288

#### 3.4. Example 3

_{2}) and (A

_{1}) the enclosing figure, (Figure 8) we obtain that, since the relation of areas is now 0.3006, the factor from the conoid to the circle is precisely this value, following Cabeza-Lainez’ second principle [8].

_{11}= 1 − 0.3006 = 0.6994

_{11}= 1 − 0.527393 = 0.4726

## 4. Generation of New Figures Based on the Previous Findings

## 5. Architectural and Engineering Significance of the Geometric Findings

#### 5.1. Historical Evolution

#### 5.2. Recent Projects of Conoids Realized by J. M. Cabeza-Lainez

#### 5.3. Future Proposals

_{x}, F

_{y}, F

_{z}) and in this case, has the value of:

## 6. Repercussions for Technology

## 7. Conclusions and Future Aims

^{2}; this is an unexpected achievement that leaves us room to speculate on the notions of π

^{3}, π

^{N}and π elevated to infinite power ∞ and their applications in future art and architecture.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Explanation of the parts of a straight conoid with circular directrix, in this case R = L = 4.

**Figure 6.**The second radiation principle of Cabeza-Lainez applied to the form-factor of a double conoid.

**Figure 8.**The same conoid of Figure 7 but closed to obtain the form-factor.

**Figure 9.**The three surfaces intervening in Example 2 whose area is calculated by computer graphic interpolation.

**Figure 17.**Architettura Civile. Camillo G. Guarini. Lastra IX. Trat. IV. Depiction of a cone ending in a line.

**Figure 23.**Interior view showing light diffusion at the central conoid. School of engineering of Seville.

**Figure 24.**Semicircular opening and projecting overhang at the extreme of the central conoid of the School of Engineering of Seville.

**Figure 26.**View of the house of the musicians. The vaults are entirely constructed in brick with occasional steel reinforcement.

**Figure 27.**Vault pieces are constructed in hollow thin brick with steel mesh for reinforcement and attached to a concrete frame.

**Figure 28.**Detail of pseudo-conoid roofs in which the sun-path produces intriguing variations through the day.

**Table 1.**Results of computing the area of several conoids by the method proposed by the author and with graphic interpolation procedures allowed by the software Grasshopper and the command Alphashape of Matlab.

Radius = L Unit | Alpha- Shape | Cabeza Approx. | Grass- Hopper | Delta Δ Alphas | Delta Δ Grassh |
---|---|---|---|---|---|

0.25 | 0.1958 | 0.1862 | 0.1842 | 0.0096 | −0.002 |

0.50 | 0.7712 | 0.7446 | 0.7369 | 0.0266 | −0.0077 |

1 | 3.0726 | 2.9784 | 2.9476 | 0.942 | −0.308 |

2 | 12.155 | 11.913 | 11.790 | 0.242 | −0.123 |

3 | 27.734 | 26.805 | 26.528 | 0.929 | −0.277 |

4 | 48.886 | 47.654 | 47.161 | 1.232 | −0.493 |

5 | 76.019 | 74.460 | 73.690 | 1.559 | −0.77 |

6 | 109.27 | 107.22 | 106.11 | 2.05 | −1.11 |

7 | 148.83 | 145.94 | 144.43 | 2.89 | −1.51 |

8 | 194.54 | 190.61 | 188.64 | 3.93 | −1.97 |

9 | 246.41 | 241.25 | 238.75 | 5.16 | −2.5 |

10 | 304.254 | 297.841 | 294.76 | 6.413 | −3.081 |

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**MDPI and ACS Style**

Cabeza-Lainez, J.
Architectural Characteristics of Different Configurations Based on New Geometric Determinations for the Conoid. *Buildings* **2022**, *12*, 10.
https://doi.org/10.3390/buildings12010010

**AMA Style**

Cabeza-Lainez J.
Architectural Characteristics of Different Configurations Based on New Geometric Determinations for the Conoid. *Buildings*. 2022; 12(1):10.
https://doi.org/10.3390/buildings12010010

**Chicago/Turabian Style**

Cabeza-Lainez, Joseph.
2022. "Architectural Characteristics of Different Configurations Based on New Geometric Determinations for the Conoid" *Buildings* 12, no. 1: 10.
https://doi.org/10.3390/buildings12010010