# Basket-Handle Arch and Its Optimum Symmetry Generation as a Structural Element and Keeping the Aesthetic Point of View

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basket-Handle Arch in Architecture

## 3. Basket-Handle Arch of Three Arches or Three Centered Oval Arches

_{2}, O

_{3}) and the point of link or tangency (M, N), are aligned. Thus, the points are on the same straight line: (O

_{1}, O

_{2}, N) and (O

_{1}, O

_{3}, M), see Figure 10. Thus, the geometrical construction has been the norm for the execution of the basket-handle arch, see Figure 10, where two different cases are identified depending on which are known parameters.

- for different angles, fixed the clear span and the radius of the lateral circumference.
- for different angles, fixed the sagita or rise and the radius of the central circumference.

#### 3.1. Basket-Handle Arch 1: Known A and B

- The distance AB is divided into four parts
- The equilateral triangle is built from the side AB/2
- The vertices of the triangle (O
_{1}, O_{2}and O_{3}) shall be the centres of the arches. See Figure 10A.

#### 3.2. Basket-Handle Arch 2: Known A, B and C

- The distance AB is divided into two parts, obtaining the half, point G, see Figure 10B.
- By point G, the circumference of radius R = GA and the perpendicular line to AB that originate at point D are traced.
- GA and the perpendicular line to AB that originate at point D are traced.
- At point M, take the height h and C is obtained. At C, the circumference of the CD radius is drawn.
- The CA and CB lines are drawn, which in their intersection with said circumference are determined E and F.
- The mediatrices of AE and BF, define the points O
_{1}, O_{2}and O_{3}centers of the three arcs that form the basket-handle arch.

## 4. The Basket-Handle Arch Proposed

#### 4.1. Graphical Geometric Construction

_{1}, O

_{2}and O

_{3}and radii O

_{1}C, O

_{2}B and O

_{3}A is obtained, which is the one with the lowest R/r of all possible basket-handle arches.

#### 4.2. Analytical Calculation

#### 4.3. Computer Software

_{1}, O

_{2}and O

_{3}) are calculated also. Finally, the results are shown graphically and mathematically.

## 5. Case Studies

#### 5.1. Château de Chambord (France)

#### 5.2. Plaza de España de Vitoria (Spain)

#### 5.3. Salamanca Cathedral (Golden Chapel or Chapel of All Saints)

#### 5.4. Moscow Subway Station (Mayakovskaya)

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kimball, F.; Edgell, G.H. A History of Architecture; Harper & Brothers: New York, NY, USA, 1918. [Google Scholar]
- Cowan, H.J. A history of masonry and concrete domes in building construction. Build. Environ.
**1977**, 12, 1–24. [Google Scholar] [CrossRef] - DeLaine, J. Structural experimentation: The lintel arch, corbel and tie in western Roman architecture. World Archaeol.
**1990**, 21, 407–424. [Google Scholar] [CrossRef] - Joshi, A.M.; Deshmukh, A.M. Quality of space in Romanesque and Gothic Architecture. Int. J. Eng. Res.
**2018**, 7, 160–164. [Google Scholar] [CrossRef] - Lengyel, G. Minimum thickness of the gothic arch. Arch. Appl. Mech.
**2018**, 88, 769–788. [Google Scholar] [CrossRef] - Calvo-López, J. From Mediaeval Stonecutting to Projective Geometry. Nexus Netw. J.
**2011**, 13, 503–533. [Google Scholar] [CrossRef] - Campisi, T.; Saeli, M. Taking inspiration from nature: Rules and procedures for a good building practice. In ARCHDESIGN’16-III International Architectural Design Conference on Design and Nature; DAKAM Publishing: Istanbul, Turkey, 2016; pp. 4–17. [Google Scholar]
- Ural, A.; Oruç, Ş.; Doğangün, A.; Tuluk, Ö.İ. Turkish historical arch bridges and their deteriorations and failures. Eng. Fail. Anal.
**2008**, 15, 43–53. [Google Scholar] [CrossRef] - Campisi, T.; Saeli, M. Structural Uses of Stone and Timber in The European Historical Construction. J. World Archit.
**2018**, 2, 1–11. [Google Scholar] - Saeli, M. Mixed Stone-timber Masonry in Traditional Construction: Structural, Architectural and Anti-seismic Features across Europe. J. Archit. Res. Dev.
**2018**, 2. [Google Scholar] [CrossRef] - Byne, A.; Byne, M.S. Spanish Architecture of the Sixteenth Century: General View of the Plateresque and Herrera Styles; No. 109; GP Putnam’s Sons: New York, NY, USA, 1917. [Google Scholar]
- Jennings, N. Converso Patronage, Self-Fashioning, and Late-Gothic Art and Architecture in 15th-Century Castile. In Jews and Muslims Made Visible in Christian Iberia and Beyond, 14th to 18th Centuries; Brill: Leiden, The Netherlands, 2019; pp. 161–186. [Google Scholar]
- Rubiò Bellver, J. Dificultats per Arribar a la Síntessis Arquitectònica; Anuario de la Asociación de Arquitectos de Cataluña: Barcelona, Spain, 1913; pp. 63–79. [Google Scholar]
- Heyman, J. Beauvais cathedral. Trans. Newcom. Soc.
**1967**, 40, 15–35. [Google Scholar] [CrossRef] - Huerta, S. Mecánica de las bóvedas de fábrica: El enfoque del equilibrio. Inf. Construcción
**2005**, 56, 73–89. [Google Scholar] [CrossRef] - Huerta Fernández, S. Arcos, Bóvedas y Cúpulas. Geometría y Equilibrio en el Cálculo Tradicional de Estructuras de fábricaArcos, Bóvedas y Cúpulas. Geometría y Equilibrio en el Cálculo Tradicional de Estructuras de Fábrica; Instituto Juan de Herrera: Madrid, Spain, 2004. [Google Scholar]
- Moseley, H.L. On a new principle in statics, called the Principle of least Pressure. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1833**, 3, 285–288. [Google Scholar] [CrossRef] - Block, P.; DeJong, M.; Ochsendorf, J. As hangs the flexible line: Equilibrium of masonry arches. Nexus Netw. J.
**2006**, 8, 13–24. [Google Scholar] [CrossRef] - Ricci, E.; Fraddosio, A.; Piccioni, M.D.; Sacco, E. A new numerical approach for determining optimal thrust curves of masonry arches. Eur. J. Mech. A/Solids
**2019**, 75, 426–442. [Google Scholar] [CrossRef] - Alexakis, H.; Makris, N. Limit equilibrium analysis and the minimum thickness of circular masonry arches to withstand lateral inertial loading. Arch. Appl. Mech.
**2014**, 84, 757–772. [Google Scholar] [CrossRef] - Alexakis, H.; Makris, N. Limit equilibrium analysis of masonry arches. Arch. Appl. Mech.
**2015**, 85, 1363–1381. [Google Scholar] [CrossRef] - Tempesta, G.; Galassi, S. Safety evaluation of masonry arches. A numerical procedure based on the thrust line closest to the geometrical axis. Int. J. Mech. Sci.
**2019**, 155, 206–221. [Google Scholar] [CrossRef] - Velilla, C.; Alcayde, A.; San-Antonio-Gómez, C.; Montoya, F.G.; Zavala, I.; Manzano-Agugliaro, F. Rampant Arch and Its Optimum Geometrical Generation. Symmetry
**2019**, 11, 627. [Google Scholar] [CrossRef] - Milankovitch, M. Theorie der Druckkurven. Z. Math. Phys.
**1907**, 55, 1–27. [Google Scholar] - Heyman, J. The safety of masonry arches. Int. J. Mech. Sci.
**1969**, 11, 363–385. [Google Scholar] [CrossRef] - Alexakis, H.; Makris, N. Minimum thickness of elliptical masonry arches. Acta Mech.
**2013**, 224, 2977–2991. [Google Scholar] [CrossRef] - Boothby, T.E. Analysis of masonry arches and vaults. Prog. Struct. Eng. Mater.
**2001**, 3, 246–256. [Google Scholar] [CrossRef] - Gómez-Collado, M.C.; Roselló, V.C.; Tamborero, E.C. Mathematical modeling of oval arches. A study of the George V and Neuilly Bridges. J. Cult. Herit.
**2018**, 32, 144–155. [Google Scholar] [CrossRef] - Casado, E.A.; Sánchez, J.M.D. Geometría del arco carpanel. Suma Rev. Sobre Enseñanza Aprendiz. Matemáticas
**2015**, 79, 17–25. [Google Scholar] - San-Antonio-Gómez, C.; Velilla, C.; Manzano-Agugliaro, F. Photogrammetric techniques and surveying applied to historical map analysis. Surv. Rev.
**2015**, 47, 115–128. [Google Scholar] [CrossRef] - Al-Omari, A.; Beck, K.; Brunetaud, X.; Török, Á.; Al-Mukhtar, M. Critical degree of saturation: A control factor of freeze–thaw damage of porous limestones at Castle of Chambord, France. Eng. Geol.
**2015**, 185, 71–80. [Google Scholar] [CrossRef] - Janvier-Badosa, S.; Beck, K.; Brunetaud, X.; Al-Mukhtar, M. Historical study of Chambord castle: A basis for establishing the health record of the monument. Int. J. Archit. Herit.
**2013**, 7, 247–260. [Google Scholar] [CrossRef] - Janvier-Badosa, S.; Beck, K.; Brunetaud, X.; Al-Mukhtar, M. The occurrence of gypsum in the scaling of stones at the Castle of Chambord (France). Environ. Earth Sci.
**2014**, 71, 4751–4759. [Google Scholar] [CrossRef] - Al-Omari, A.; Brunetaud, X.; Beck, K.; Al-Mukhtar, M. Effect of thermal stress, condensation and freezing–thawing action on the degradation of stones on the Castle of Chambord, France. Environ. Earth Sci.
**2014**, 71, 3977–3989. [Google Scholar] [CrossRef] - Colding, T.H.; Minicozzi, W.P., II. Disks that are double spiral staircases. Not. AMS
**2003**, 50, 327–339. [Google Scholar] - Tanaka, H. Leonardo da Vinci, architect of Chambord? Artibus Hist.
**1992**, 13, 85–102. [Google Scholar] [CrossRef] - Dushkina, N. Russia: 20th-Century Heritage. Heritage at Risk. 2003, pp. 177–181. Available online: https://journals.ub.uni-heidelberg.de/index.php/heritage/article/download/21204/14973 (accessed on 14 August 2019).
- Gerin, A. Stories from Mayakovskaya Metro Station: The Production/Consumption of Stalinist Monumental Space, 1938. Ph.D. Thesis, University of Leeds, Leeds, UK, 2000. Available online: http://etheses.whiterose.ac.uk/6749/1/424660.pdf (accessed on 14 August 2019).
- Shilin, A.A.; Kirilenko, A.M.; Znajchenko, P.A. Complex reconstruction project of Mayakovskaya metro station in the centre of Moscow. In Structural Analysis of Historical Constructions—Anamnesis, Diagnosis, Therapy, Controls; Van Balen, K., Verstrynge, E., Eds.; Taylor & Francis Group: London, UK, 2016; pp. 1736–1741. ISBN 978-1-138-02951-4. Available online: http://znayugeo.ru/wp-content/uploads/2016/11/ch234.pdf (accessed on 14 August 2019).

**Figure 1.**The main arch types used in architecture. (

**A**) Semicircular. (

**B**) Stilted. (

**C**) Segmental. (

**D**) Basket-handle or three-centered. (

**E**) Rampant. (

**F**) Three pointed or Gothic. (

**G**) Acute. (

**H**) Depressed. (

**I**) Keyhole. (

**J**) Ogee three-centered. (

**K**) Ogee four-centered. (

**L**) Oriental. (

**M**) Round Trefoil. (

**N**) Draped. (

**O**) Parabolic.

**Figure 4.**Basket-handle arches in France. (

**A**) Rodez Cathedral. (

**B**) Gateway to the Château de Chenonceau.

**Figure 5.**Basket-handle arch integrated with the ogee or inflected arch of the gate of Monasteries in Spain. (

**A**) Monastery of San Antonio el Real in Segovia. (

**B**) Monastery of Santa Clara in Palencia.

**Figure 6.**(

**A**) Basket-handle arch of one of the galleries of the Blenheim Palace (UK). (

**B**) Basket-handle arch of the central nave of the Metropolitan Cathedral of Mexico City.

**Figure 7.**Basket-handle arches on brick cores of the arcades of the Plaza de España in Seville (Spain).

**Figure 9.**Basket-handle arch elements: (

**1**) Keystone, (

**2**) Voussoir, (

**3**) Impost, (

**4**) Intrados, (

**5**) Rise or sagita, (

**6**) Clear span or springing line. (

**7**) Abutment, (

**8**) Springer. Red dashed line- Lower thrust line. Blue dashed line- Upper thrust line.

**Figure 10.**Basket-handle arch of three arches graphically constructed. (

**A**) Known A and B (method 1). (

**B**) Known A, B and C (method 2).

**Figure 11.**Example of several basket-handle arches for a specific clear span (A-B), and the arch proposed.

**Figure 14.**Calculation of the optimal basket-handle arch (in red) over the room giving access to the helicoidal staircase of the Château de Chambord.

**Figure 15.**Calculation of the optimal basket-handle arch (in red) over the Plaza de España of Vitoria (Spain).

**Figure 16.**Calculation of the optimal basket-handle arch (in red) over the Chapel of All Saints or Golden Chapel in Salamanca Cathedral (Spain).

**Figure 17.**Moscow subway station (Mayakovskaya). (

**A**) Original image. (

**B**) Analysis of the optimal basket-handle arch (in red).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alcayde, A.; Velilla, C.; San-Antonio-Gómez, C.; Peña-Fernández, A.; Pérez-Romero, A.; Manzano-Agugliaro, F.
Basket-Handle Arch and Its Optimum Symmetry Generation as a Structural Element and Keeping the Aesthetic Point of View. *Symmetry* **2019**, *11*, 1243.
https://doi.org/10.3390/sym11101243

**AMA Style**

Alcayde A, Velilla C, San-Antonio-Gómez C, Peña-Fernández A, Pérez-Romero A, Manzano-Agugliaro F.
Basket-Handle Arch and Its Optimum Symmetry Generation as a Structural Element and Keeping the Aesthetic Point of View. *Symmetry*. 2019; 11(10):1243.
https://doi.org/10.3390/sym11101243

**Chicago/Turabian Style**

Alcayde, Alfredo, Cristina Velilla, Carlos San-Antonio-Gómez, Araceli Peña-Fernández, Antonio Pérez-Romero, and Francisco Manzano-Agugliaro.
2019. "Basket-Handle Arch and Its Optimum Symmetry Generation as a Structural Element and Keeping the Aesthetic Point of View" *Symmetry* 11, no. 10: 1243.
https://doi.org/10.3390/sym11101243