S S symmetry

Article Rampant and Its Optimum Geometrical Generation

Cristina Velilla 1, Alfredo Alcayde 2 , Carlos San-Antonio-Gómez 1, Francisco G. Montoya 2 , Ignacio Zavala 1 and Francisco Manzano-Agugliaro 2,*

1 Department of Ingeniería Agroforestal, Universidad Politécnica de Madrid, Av. De Puerta de Hierro, 28040 Madrid, Spain; [email protected] (C.V.); [email protected] (C.S.-A.-G.); [email protected] (I.Z.) 2 Department of Engineering, University of Almeria, ceiA3, 04120 Almeria, Spain; [email protected] (A.A.); [email protected] (F.G.M.) * Correspondence: [email protected]; Tel.: +34-950-015396; Fax: +34-950-015491

 Received: 15 April 2019; Accepted: 28 April 2019; Published: 3 May 2019 

Abstract: Gothic art was developed in western Europe from the second half of the 12th century to the end of the 15th century. The most characteristic Gothic building is the . uses well-carved stone ashlars, and its essential elements include the arch. The thrust is transferred by means of external (flying buttresses) to external buttresses that end in , which accentuates the verticality. The evolution of the flying buttresses should not only be considered as an aesthetic consideration, but also from a constructive point of view as an element of transmission of forces or loads. Thus, one evolves from a beam-type buttress to a simple arch, and finally to a rampant arch. In this work, we study the geometry of the rampant arch to determine which is the optimum from the constructive point of view. The optimum rampant arch obtained is the one with the common tangent to the two arches parallel to the slope line. A computer program was created to determine this optimal rampant arch by means of a numerical or graphical input. It was applied to several well-known and representative cases of Gothic art in ( of Saint Urbain de Troyes) and Spain (Cathedral of Palma de Mallorca), establishing if they were designs of optimal rampant arches or not.

Keywords: rampant arch; geometry; optimum; flying buttresses; cathedral

1. Introduction The rampant arch is an arch whose starts, in walls or buttresses, are located at different levels, often with a considerable difference in height [1]. It was used extensively in Gothic architecture to shape the buttress, and its function was to transmit the thrust of the vaults to the buttresses and these to the foundations [2]. In addition to this structural function, the buttress serves to carry rainwater from the vaults [3] to the exterior through the pinnacles. The most common rampant arch used in the flying buttresses of most well-known Gothic is the one formed by a single circumferential arch [4]. According to Viollet-le-Duc (1854) [5] there are two types: flying buttresses in which the center of the circumference is in the wall (Figure1A), and flying buttresses in which its center is displaced toward the interior of the building (Figure1B). The first are the oldest flying buttresses, while the second are the most used because, from a mechanical point of view, they perform better than the first [6].

Symmetry 2019, 11, 627; doi:10.3390/sym11050627 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 627 2 of 15 Symmetry 2019, 11, x 2 of 15

FigureFigure 1. 1.(A (,BA),B The) The two two types types of flying of flying buttresses buttresses according according to Viollet-le-Duc to Viollet-le-Duc [5]. ( C[5].) Section (C) Section of Durham of CathedralDurham [Cathedral7]. (D) Wooden [7]. (D props) Wooden from props the from ofthe the vault church of the of Vchurchézelay of (1) Vézelay replaced (1) byreplaced stone flyingby buttressesstone flying (2) [buttresses8]. (2) [8].

TheThe flying flying buttresses buttresses appeared appeared for for the the first first time time in in Durham Durham Cathedral Cathedral aroundaround thethe yearyear 11001100 asas an evolutionan evolution of the of hidden the hidden buttress buttress in the ,in the trifor whereium,an where opening an opening is made thatis made allows that a longitudinalallows a routelongitudinal parallel to route the mainparallel to the [9]. main In Durham, nave [9]. theIn Durham, buttress the did buttress not yet did have no thet yet function have the of function balancing theof lateral balancing thrusts the of lateral the vaults; thrusts its of mission the vaults; was to its support mission the was roof to thatsupport covers th thee roof triforium that covers (Figure the1C). triforiumThe flying (Figure buttresses 1C). of a single circumferential arch could also have been the result of the evolution of the frameworkThe flying buttresses of auxiliary of a wooden single circumferential constructions that,arch fromcould aalso certain have moment been the on, result appeared of the to supportevolution the Romanesqueof the framework barrel of vaults auxiliary that wooden were beginning constructions to open that, up. from Over a time, certain these moment frameworks on, wouldappeared begin to to support be built the in Romanesque stone (Figure barrel1D). vaults This moment that were perhaps beginning occurred to open in up. France Over attime, the these Abbey offrameworks Vézelay [8], would where begin the buttress to be built would in stone not (Figure be an “invention 1D). This moment ex novo” perhaps but the occurred evolution in France of those woodenat the Abbey structures of Vézelay known [8], throughout where the buttress France [woul10]. d not be an “invention ex novo” but the evolution of thoseThe rampant wooden archstructures of a single known circumferential throughout France arch [10]. played an essential role in the structural and The rampant arch of a single circumferential arch played an essential role in the structural and formal conception of Gothic architecture, creating in the external volume of the cathedrals a succession formal conception of Gothic architecture, creating in the external volume of the cathedrals a of dematerialized planes composed of one, two, and even three rows of buttresses, contributing to the succession of dematerialized planes composed of one, two, and even three rows of buttresses, spatial richness of the building [11]. The successive rhythm of the flying buttresses and buttresses with contributing to the spatial richness of the building [11]. The successive rhythm of the flying buttresses their pinnacles of reinforcement makes it possible to open large with stained-glass windows and buttresses with their pinnacles of reinforcement makes it possible to open large windows with thatstained-glass give great luminositywindows that to thegive interior great luminosity of the cathedral to the interior and give of symbolicthe cathedral and and immaterial give symbolic value to theand Gothic immaterial architecture, value which to the contrasts Gothic witharchitectu the heavinessre, which of thecontrasts Romanesque with the architecture heaviness itof replaced the (FigureRomanesque1D) [12]. architecture it replaced (Figure 1D) [12]. AlthoughAlthough the the single-arch single-arch flying flying buttresses buttresses are are usually usually used used in largein large cathedrals, cathedrals, there there are are also also flying buttressesflying buttresses made up made of two up circumferentialof two circumferential arches ar (Figureches (Figure2A). In 2A). this In case, this thecase, rampant the rampant arch resultsarch fromresults the from union the of union two circumferential of two circumferential arches, arches, tangent tangent in the in key, the whose key, whose centers centers are at are di atfferent different levels, solevels, that their so that outcrops their outcrops in the walls in the or walls buttresses or buttr areesses at di arefferent at different heights. heig Exampleshts. Examples that could that could be cited include,be cited in include, France, in the France, flying the buttresses flying buttresses of the Church of the of Church Saint Urbain of Saint de Urbain Troyes de [13 Troyes] (Figure [13]2 B),(Figure where the2B), red where points the (A red and points C) are (A the and starting C) are the points starti ofng the points two arches,of the two and arches, between and which between the which unevenness the ofunevenness the flying buttress of the flying must bebuttress bridged, must and be thosebridged, of theand Cathedral those of the of ,Cathedral whichof Beauvais, alternate which a first linealternate of ramps a first of aline single of ramps arch ofof circumferencea single arch of withcircumference another ofwith two another arches of (Figure two arches3A) [ (Figure14]. In 3A) Spain, for[14]. example, In Spain, there for are example, the original there flyingare the buttresses original flying of the cathedralbuttresses ofof Palmathe cathedral de Mallorca, of Palma made de up ofMallorca, two rampant made arches up of oftwo one rampant and two arches circumferential of one and arches,two circumferential integrated into arches, a single integrated flying into buttress a (Figuresingle3 B)flying [ 15 buttress]. The rampant(Figure 3B) arches [15]. consistingThe rampant of arches two circumferential consisting of two arches circumferential are used inarches Gothic

Symmetry 2019, 11, 627 3 of 15 SymmetrySymmetry 2019 2019, 11,, 11x , x 33 of of 15 15 arearchitectureare used used in inGothic to Gothic shape architecture architecture the flying to buttresses, to shape shape the the but flying flying also buttresses, tobuttresses, build stairs. butbut also Inalso Spain, to build there stairs.stairs. are manyInIn Spain, Spain, examples there there aresuchare many as many the examples Monastery examples such such of Uclas as éthe sthe in Monastery CuencaMonastery (Figure of of Uclé Uclé3C),ss inin theCuencaCuenca palaces (Figure(Figure of the 3C), Generalitat inin thethe palacespalaces de Barcelona ofof the the GeneralitatandGeneralitat Valencia, de and,deBarcelona Barcelona in the and latter and Valencia, city,Valencia, the and, Lonja and, in in staircase the the latter latter (Figure city, city, thethe3D). LonjaLonja staircase (Figure(Figure 3D). 3D).

Figure 2. (A) Flying buttresses of Notre Dame (Paris) [5]. (B) Flying buttress of the church of Saint Figure 2. (A) Flying buttresses of Notre Dame (Paris) [5]. (B) Flying buttress of the church of Saint FigureUrbain 2. (deA) TroyesFlying [8].buttresses of Notre Dame (Paris) [5]. (B) Flying buttress of the church of Saint Urbain de Troyes [8]. Urbain de Troyes [8].

FigureFigure 3. 3.( A(A)) One- One- andand two-arch circumference circumference flying flying bu buttressesttresses of Beauvais of Beauvais Cathedral Cathedral (France). (France). (B) (B)Flying Flying buttresses buttresses of of the the Cathed Cathedralral of of Palma Palma de de Mallorca. Mallorca. (C (C) )Rampant Rampant arch arch in in the the staircase staircase of of the the FigureMonasteryMonastery 3. (A of) One-of Ucl Uclésé sand (Cuenca). (Cuenca). two-arch (D ( D)circumference) Rampant Rampant arch arch flying in in thethe bu staircasestaircttressesase ofof of La Beauvais Lonja (Valencia). Cathedral (France). (B) Flying buttresses of the Cathedral of Palma de Mallorca. (C) Rampant arch in the staircase of the Monastery of Uclés (Cuenca). (D) Rampant arch in the staircase of La Lonja (Valencia).

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The rampantrampant arches arches were were used used profusely profusely again again at the at end the of theend 19th of centurythe 19th with century the appearance with the ofappearance neo-Gothic of neo-Gothic architecture. architecture. Many churches Many werechurch builtes were throughout built throughout Europe, Europe, especially especially in France. in InFrance. America, In America, for example, for example, cathedrals cathedrals such as such New as York New or York Washington or Washington were built were in built the Unitedin the United States (FigureStates (Figure4). In the 4). sameIn the way, same in way, modernism, in modernism, these arches these werearches also were used. also used.

Figure 4. WashingtonWashington cathedral and its flyingflying buttresses.

In general, the the design design of of a arampant rampant arch arch of of a single a single arch arch does does not not present present any any difficulty, difficulty, but butthe themost most difficult difficult are arethe the rampant rampant arches arches of oftwo two arches, arches, which which throughout throughout history history were calculated graphically, and there isis nono consensusconsensus onon whichwhich thethe optimaloptimal rampantrampant archarch ofof twotwo archesarches is.is. The objectives of this work are to review the state of the art in the execution of rampant arches of two arches, toto definedefine whichwhich isis the the optimal optimal one, one, to to calculate calculate the the numerical numerical solution, solution, and and to to program program it forit for its calculationits calculation and and verification verification of rampant of rampant arches arches of already of already existing existing arches. Asarches. an additional As an additional objective, thisobjective, work seeksthis work to determine seeks to thedetermine starting the points starting of the points arches of in the the arches construction in the ofconstruction the rampant of arch. the rampant arch. 2. Rampant Arches of Two Arches Graphically Constructed 2. RampantThe Gothic Arches buildings of Two raised Arches the Graphically question of what Constructed methods were used by the builders in the design of theirThe structures. Gothic buildings The answer raised is the that question the Gothic of what builders methods had no were method, used onlyby the a greatbuilders structural in the intuitiondesign of nourished their structures. by the experience The answer of successiveis that the collapses Gothic builders of structures. had Inno classicalmethod, texts, only there a great are “structuralstructural intuition rules” for nourished dimensioning by the walls, experience pillars, of and successive abutments collapses of Gothic of structures. churches, In all classical based on texts, the there are “structural rules” for dimensioning walls, pillars, and abutments of Gothic churches, all

Symmetry 2019, 11, 627 5 of 15 Symmetry 2019, 11, x 5 of 15 proportionalitybased on the proportionality of the elements. of These the elements. rules were These applied rules were or calculated applied or using calculated geometric using or geometric graphical constructions;or graphical see, constructions; for instance, see, treaties for instance, of Derand treaties [16 of] or Derand the later [16] one or the of Blondel later one [ 17of]. Blondel [17]. Thus,Thus, the the usual usual for for the the execution execution of of rampant rampant archesarches waswas graphicalgraphical construction, construction, in in which which two two casescases were were distinguished distinguished depending depending on on which which parameters parameters werewere knownknown (see Figure Figure 55).).

FigureFigure 5. 5.Rampant Rampant arches arches of of twotwo arches graphically graphically constructed: constructed: (A) ( Aknown) known A and A andC (method C (method 1); (B) 1); (B) known A A and and C C (method (method 2); 2); (C (C) known) known radii radii of ofthe the two two arcs arcs (R1 ( Rand1 and R2).R 2). 2.1. Rampant Arch 1: Known A and C (Starting of the Two Arches) 2.1. Rampant Arch 1: Known A and C (Starting of the Two Arches) 2.1.1. Method 1 2.1.1. Method 1 In Figure5A, the graphical drawing of the rampant arch is described, with the initial (A) and final In Figure 5A, the graphical drawing of the rampant arch is described, with the initial (A) and (C) points known, where the tracing can be summarized as follows: final (C) points known, where the tracing can be summarized as follows: a. Thea. ABCD The ABCD rectangle rectangle is drawn; is drawn; b. Withb. center With center in A, circumferencein A, circumference of R =ofAD, R = AD, obtaining obtaining E; E; c. Mediatrixc. Mediatrix m of BEm of is drawn;BE is drawn; d. Thed. intersection The intersection of m withof m thewith rectangle the rectangle determines determines centers centers O and O and O’. O’.

2.1.2.2.1.2. Method Method 2 2 In FigureIn Figure5B, the5B, graphicalthe graphical drawing drawing of the of rampantthe rampant arch arch is described, is described, with with the initialthe initial (A) and(A) and final (C)final points (C) known. points known. Note that Note the that centers the centers are no longerare no longer parallel parallel to the supportto the support pillars, pillars, where where the tracing the tracing can be summarized as follows: can be summarized as follows: a. The ABCD rectangle is drawn; a. Theb. ABCD Mediatrix rectangle m of isAB drawn; determines M; b. Mediatrixc. With m center of AB in determines M, circumference M; of diameter AC is drawn; c. Withd. center The intersection in M, circumference of this circumference of diameter with AC m is is drawn; E; d. Thee. intersection The perpendicular of this circumference by E to AC determines with m is the E; centers O and O’. e. The perpendicular by E to AC determines the centers O and O’. 2.2. Rampant Arch 2: The Radii of the Two Arcs (R1 and R2) Are Known 2.2. RampantIn this Arch other 2: Themethod, Radii the of the radii Two of Arcs the (R1 two and arcs R2) (R1 Are and Known R2) are known, which add up the horizontalIn this other distance method, between the radii both of points, the two and arcs it is (R a1 question and R2) of are determining known, which the addcenters up of the both. horizontal distance1. between R1 + R2 both = horizontal points, and distance it is a between question both of determining points (span); the centers of both. 2. In A and B, perpendicular to the pillars where they are taken, R1 and R2 are used to obtain 1. R1 + R2the= horizontalcenters O and distance O’, respectively, between both of the points arches. (span); 2. In A and B, perpendicular to the pillars where they are taken, R1 and R2 are used to obtain the 3. centersThe Optimal O and Rampant O’, respectively, Arch of the arches.

3. The OptimalIn the literature, Rampant there Arch are different points of view for the optimal rampant arch. For example, there are studies that define it from the aesthetic point of view, or from a more harmonious view, in whichIn the the literature, curvatures there of arethe di differentfferent points arches of that view compose for the optimalit were rampantmore similar; arch. here, For example, the optimal there are studies that define it from the aesthetic point of view, or from a more harmonious view, in which

Symmetry 2019, 11, x 6 of 15

Symmetryrampant2019 arch, 11, 627corresponds to the one in which the ratio between the curvatures of the lateral6 of 15 circumference and the central circumference is closer to one [18]. However, the rampant arch, apart from its aesthetic consideration, as explained in the theintroduction, curvatures has of thea structural different load arches transmission that compose function, it were as it more replaces similar; the function here, the of optimala beam [19]. rampant In archaddition, corresponds however, to theloads one can in only which be thetransmitted ratio between by compression, the curvatures as this of theis how lateral the circumferencemasonry resists and thebending central moments circumference and, therefore, is closer sets to one the [limits18]. of the resistance capacity of the arches [20]. Figure 6 showsHowever, examples the of rampant classical arch, studies apart concerned from its aesthetic with this consideration, issue, i.e., the as possible explained breaking in theintroduction, points of hasrampant a structural arches load[21,22]. transmission In this sense, function, classical asresearch it replaces attempted the function to find the of aminimum beam [19 value]. In addition,of “h” however,(the rise) loadsfor which can masonry only be transmitted is able to withstand by compression, bending moments as this is without how the any masonry tensile strength resists bending [23], momentsestimated and, to be therefore, between 3 sets and the 5 tons limits of passive of the resistance horizontal capacity thrust. The of theoptimum arches rampant [20]. Figure arch,6 fromshows examplesthe point ofof classicalview of the studies support concerned function with or the this tran issue,smission i.e., theof the possible constructive breaking loads, points is the of one rampant in archeswhich [21the,22 tangent]. In this common sense, classical to the two research circles attempted that compose to find the the arch minimum is parallel value to ofthe “h” straight (the rise) forunevenness which masonry between is ablethe points. to withstand bending moments without any tensile strength [23], estimated Note that there are examples where an inadequate intervention in the structure caused the fall to be between 3 and 5 tons of passive horizontal thrust. The optimum rampant arch, from the point of the dome or the vault, e.g., at Edinburgh in 1768, where the church of Holyrood Abbey collapsed of view of the support function or the transmission of the constructive loads, is the one in which the after an inappropriate intervention in the vaults, where the essential buttressing system was rampant tangent common to the two circles that compose the arch is parallel to the straight unevenness between arches abutting the nave vault and their pier buttresses were capped with stabilizing pinnacles [24]. the points.

FigureFigure 6.6. ClassicalClassical studies studies of ofthe the rupture rupture of rampant of rampant arches: arches: (A) breakage (A) breakage of masonry of masonry by Frezier by (1737) Frezier (1737)[21]; (B [21) calculation]; (B) calculation of the trust of the line trust by lineBreymann by Breymann (1881) [22]. (1881) [22].

3.1. NoteGraphical that Design there are examples where an inadequate intervention in the structure caused the fall of the dome or the vault, e.g., at Edinburgh in 1768, where the church of Holyrood Abbey collapsed after This section describes the graphic drawing of an optimum rampant arch, i.e., the one whose an inappropriate intervention in the vaults, where the essential buttressing system was rampant arches tangent common to the two arches is parallel to the straight unevenness. The main ideas are as abutting the nave vault and their pier buttresses were capped with stabilizing pinnacles [24]. follows: 3.1. Graphical1. Once Design the gradient “d” is defined, which indicates the slope of a flying buttress or, where This sectionappropriate, describes of a staircase, the graphic the drawingstart of the of ar anches optimum is determined rampant by arch,points i.e., A and the B, one which whose tangent commonare on a to line the parallel two arches to “d”. is parallel to the straight unevenness. The main ideas are as follows: 2. Assuming the problem is solved, centers O and O’ are on the straight perpendicular to the 1. Oncegradient the gradient line “d”. “d” In isaddition, defined, O whichand O’ indicatesare on the thehorizontal slope ofstraight a flying lines buttress from A and or, whereB. appropriate,3. If from B, of the a staircase, line BC is the drawn start ofperpen the archesdicular is to determined “d”, where by logically points BC A and = OO’. B, which are on a line4. parallel At this point, to “d”. the question is reduced to place a segment OO’ on the bisectors of the angles 2. Assumingin M and the N. problem is solved, centers O and O’ are on the straight perpendicular to the gradientThus, the line graphical “d”. In construction addition, O of and Figure O’ are 7 was on theas follows: horizontal straight lines from A and B. 3. If from B, the line BC is drawn perpendicular to “d”, where logically BC = OO’. a. Any PQ line was drawn parallel to “d”. 4. At this point, the question is reduced to place a segment OO’ on the bisectors of the angles in M and N.

Thus, the graphical construction of Figure7 was as follows: Symmetry 2019, 11, 627 7 of 15

Symmetry 2019, 11, x 7 of 15 a. Any PQ line was drawn parallel to “d”. b. The bisectors b1 and b2 of the angles were in P and Q. b. The bisectors b and b of the angles were in P and Q. c. BC was traced1 perpendicular2 to “d”. c. BC was traced perpendicular to “d”. d. BC was moved (by parallelism) until the TS = BC segment was obtained on the bisectors. d. e.BC was TS was moved moved (by (by parallelism) parallelism until on the the TS perpen= BCdicular segment to was AC) obtained to points on O theand bisectors. O’, which are e. TS wasthe centers moved of (by the parallelism arches. on the perpendicular to AC) to points O and O’, which are the centers of the arches.

Figure 7. Graphical design of the optimal rampant arch. Figure 7. Graphical design of the optimal rampant arch. 3.2. Analytical Calculation 3.2. AnalyticalFor the mathematical Calculation deduction, the scheme of Figure8 was followed. For the mathematical deduction, the scheme of Figure 8 was followed.

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Figure 8. Geometric elements to define the optimal rampant arch. Figure 8. Geometric elements to define the optimal rampant arch. This scheme traces a rampant arch any ATC, of span l and rise h where the following This scheme traces a rampant arch any ATC, of span l and rise h where the following relationships exist: relationships exist: AB = l, 𝐴𝐵=𝑙, BC = HI = h, 𝐵𝐶 = 𝐻𝐼 =ℎ, OA = OT = R, 𝑂𝐴 = 𝑂𝑇 =𝑅, IC = HB = r, 𝐼𝐶 = 𝐻𝐵 =𝑟, OI = R r. 𝑂𝐼− = 𝑅 𝑟. In the triangle OMN, In the triangle OMN, h h tan α = 2 = . l 2R l R 2 tan 𝛼− = = −. In the triangle IHO, h In the triangle IHO, sin 2 α = ; R r sin 2 𝛼 = − ; therefore, it is h cos α therefore, it is 2R l = , − sin α 2𝑅 𝑙 = h , 2R 2r = , − sin α cos α 2𝑅 2𝑟 = , where  2  where h cos α 1 h sin α 2r l = − = . − sinαcos α − cos α 2𝑟 𝑙 = = . Then, we get h cos α Then, we get 2R = l + , sin α 2𝑅 = 𝑙h +sin α , 2r = l , − cosα 2𝑟 = 𝑙 , ( + ) l sin α +h cos α 2 l sin 2 α + h 1 cos 2 α R sin α ∝ ∝l sin α cos α + h cos α 2 2 l sin 2 +h cos 2 α + h = = = ∝= ∝ . r l cos α h sin α 2 l sin 2 α h(1 cos 2 α) l + h h − = l sin α =cos α h sin α = − = sin 2 α cos. 2 α cosα ∝ 2 2 − − ∝ − That is to say, in short,

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That is to say, in short, R 2h = 1 + , r l sin 2 +h cos 2 h ∝ ∝ − deriving R’ 2h (2l cos 2 α 2h sin 2 α) = − = 0, r (l sin 2 α + h cos 2 α h)2 − h cot 2 α = . l In other words, the optimum rampant arch is the one with the tangent parallel to the uneven line. The relationship between the radii of curvature in a rampant arch will be found when the tangent is parallel to the straight line of unevenness. Let l the distance between walls (span) and h be the rise. One has, by resemblance of triangles, l R = + x, 2 l2 + h2 R.r = , 4 √l2 + h2 h = 2 , 2 x where lh 1 x =   , 2 √l2 + h2 h − and 2 2 √ 2 2 l h l + h + h l + h R = 1 +   = ; 2 √l2 + h2 h 2l − Therefore, l2+h2   2 2 l2 + h2 2(l2+h2+h √l2+h2) √l + h l r = = =  , 4R l 2 √l2 + h2 + h where  2 2 2 R √l + h + h = . r l2  2 2 h R p 2 Dividing by l numerator and denominator and calling l = p; then r = 1 + p + p . If we R k 1 call = k, it results in p = − . r 2 √k

3.3. Computer Software The above calculus was programmed in MATLAB. Figure9 shows the programming diagram, with the calculations described in the previous section. The program can calculate the arcs either analytically, by entering the coordinates of the starting and ending points, or graphically, on a scaled image. In Figure 10, several examples of rampant arch calculations are shown, where the coordinates of the start and end points were entered. Note that they are shown in blue on the graph obtained (Figure 10A). In the first example, the coordinates were for the initial point (1, 1), and for the final point (3, 3), so that the horizontal separation or span (l) between both was 2 m and the vertical or rise (h) was also 2 m, obtaining the coordinates of the center of the major arch (5, 1) and those of the minor arch (2.33, 3). In Figure 10B, an example of calculation of rampant arches in stairs is shown, where the two circumferences and their radii are highlighted. Symmetry 2019, 11, 627 10 of 15 Symmetry 2019, 11, x 10 of 15 Symmetry 2019, 11, x 10 of 15

Figure 9. Software calculation diagram for optimal rampant arch calculation. FigureFigure 9. Software 9. Software calculation calculation diagram diagram for optimal for optimal rampant rampant arch calculation. arch calculation.

FigureFigure 10. 10. RampantRampant arches arches with with different different geometries: geometries: ( (AA)) numerical numerical data data input; input; ( (BB)) graphical graphical data data input. input. Figure 10. Rampant arches with different geometries: (A) numerical data input; (B) graphical data input. 4.4. Case Case Studies Studies 4. Case Studies InIn this this section, section, two two case case studies studies of of real real rampan rampantt arches arches were were selected, selected, representative representative cases cases of of In this section, two case studies of real rampant arches were selected, representative cases of GothicGothic art art in in France France (church (church of of Saint Saint Urbain Urbain de de Troyes) Troyes) and and Spain Spain (Cathedral (Cathedral of of Palma Palma de de Mallorca), Mallorca), Gothic art in France (church of Saint Urbain de Troyes) and Spain (Cathedral of Palma de Mallorca), wherewhere one one comes out as an optimal rampant arch and the other does not. where one comes out as an optimal rampant arch and the other does not. 4.1. Church of Saint Urbain de Troyes (France) 4.1. Church of Saint Urbain de Troyes (France) 4.1. Church of Saint Urbain de Troyes (France) The Saint Urban Church (Église Saint-Urbain) is a large medieval French church erected in the The Saint Urban Church (Église Saint-Urbain) is a large medieval French church erected in the Thecity Saint of Troyes, Urban now Church the capital(Église ofSaint-Urbain) the department is a large of Aube. medieval It was French a collegiate church church, erected endowedin the in city of Troyes, now the capital of the department of Aube. It was a collegiate church, endowed in 1262 city of1262 Troyes, by Pope now Urbanthe capital IV, and of the it isdepartment a remarkable of Aube. example It was of late a collegiate Gothic architecture. church, endowed Construction in 1262 of the by Pope Urban IV, and it is a remarkable example of late Gothic architecture. Construction of the by Popebuilding Urban probably IV, and it began is a remarkable in 1263, where example the main of late section Gothic of thearchitecture. church was Construction not completed of the until the building probably began in 1263, where the main section of the church was not completed until the building probably began in 1263, where the main section of the church was not completed until the 16th century, and the tower was not completed until 1630. One of its highlights is the collegiate , 16th century, and the tower was not completed until 1630. One of its highlights is the collegiate choir,

Symmetry 2019, 11, 627 11 of 15 Symmetry 2019, 11, x 11 of 15 long16th acclaimed century, as and a themasterpiece tower was of not French completed Gothic until architecture, 1630. One and of its demonstrates highlights is thethe collegiate advanced choir, degreelong of acclaimed sophistication as a masterpiece of of French Gothic in the architecture, third quarter and of demonstratesthe 13th century the advanced[25,26]. This degree churchof sophistication was the object of of French numerous architecture architectural in the studies third quarter as an example of the 13th of centurythe French [25 ,Gothic26]. This style church [13]. was Figurethe object11 shows of numerous the calculation architectural of the studies optimu asm an rampant example ofarch the calculated French Gothic with style the [ 13software]. Figure 11 overlappedshows the over calculation the original of thearch optimum of the Church rampant of archSaint calculated Urbain de with Troyes. the software The starting overlapped (A) and over endingthe (C) original points arch are ofmarked the Church in the of section Saint Urbainof the ra dempant Troyes. arch. The Note starting that the (A) actual and ending arch is (C) far points from are the optimalmarked rampant in the section arch calculated of the rampant with arch.the software. Note that the actual arch is far from the optimal rampant arch calculated with the software.

FigureFigure 11. Calculation 11. Calculation of the of optimal the optimal rampant rampant arch arch(in red) (in red)over overa rampant a rampant arch archof the of flying the flying buttress buttress of theof church the church of Saint of Saint Urbain Urbain de Troyes de Troyes (France). (France).

4.2. 4.2.Cathedral Cathedral of Palma of Palma de Mallorca de Mallorca (Spain) (Spain) The TheCathedral- Cathedral-Basilica of Santa of Santa María Mar deía Palma de Palma de Mallorca, de Mallorca, also alsoknown known as the as theCathedral Cathedral of of Mallorca,Mallorca, is isone one of of the mostmost significant significant monuments monuments of Spanishof Spanish Gothic Gothic architecture architecture [27]. Its construction[27]. Its constructionstarted around started the around year 1300 the andyear was1300 completed and was completed around the around year 1600 the [28 year]. Its 1600 vault [28]. reaches Its vault a height reachesof 44 a m,height only of surpassed 44 m, only by surpassed the Cathedral by the of Cathedral Beauvais (48of Beauvais m), the highest (48 m), Gothicthe highest cathedral Gothic in the cathedralworld, in and the slightlyworld, and lower slightly than lower the height than the of the height Cathedral of the Cathedral of Milan (45of Milan m). Moreover, (45 m). Moreover, it surpasses it surpassesthe Cathedral the Cathedral of Cologne of Cologne 43 m, and 43 m, it is, and therefore, it is, therefore, considered considered as one as of one those of withthose a with nave a of nave greater of greaterheight amongheight theamong cathedrals the cathedrals of European of European Gothic style Gothic [29]. Instyle Figure [29]. 12 ,In the Figure optimal 12, rampant the optimal arch was rampantcalculated arch was on onecalculated of the archeson one of the arches façade of of the the façade cathedral of the of cathedral Palma de of Mallorca. Palma de Here, Mallorca. you can Here, you can see how the arch calculated with the software and the real arch match. Indeed, a recent study on the structure of the cathedral of Palma de Mallorca showed the horizontal displacements of

Symmetry 2019, 11, 627 12 of 15 see how the arch calculated with the software and the real arch match. Indeed, a recent study on the Symmetry 2019, 11, x 12 of 15 structure of the cathedral of Palma de Mallorca showed the horizontal displacements of the façade [27], wherethe façade it was [27], observed where that it was the observed rampant archesthat the do rampant not have arches any displacement,do not have any as adisplacement, result of its proper as a constructionresult of its andproper design. construction and design.

FigureFigure 12. 12.Calculation Calculation ofof the optimal rampant rampant arch arch over over a flying a flying buttress buttress (in (inred) red) of the of Cathedral the Cathedral of ofPalma Palma de de Mallorca Mallorca (Spain). (Spain). Note Note that that the the yellow yellow line line highlights highlights the the initial initial and and final final points points of ofthe the rampantrampant arch. arch.

5.5. Discussion Discussion InIn Gothic Gothic constructions, constructions, thethe flyingflying buttressesbuttresses of of the the first first constr constructionsuctions were were generally generally circular, circular, wherewhere the the shape shape of of the the line line of of thrustthrust diddid notnot match, normally, with with the the geometric geometric forms forms of of traditional traditional constructionconstruction formed formed by by circumferential circumferential archesarches [[4].4]. In In addition, addition, when when the the line line of of thrust thrust reaches reaches or or touchestouches the the edge edge of of the the arch, arch, an an articulation articulation isis formedformed that can can cause cause the the arch arch to to collapse collapse [21]. [21 ].Enough Enough archarch thickness thickness will will easily easily accommodate accommodate anan infiniteinfinite number of of thrust thrust lines. lines. By By progressively progressively reducing reducing thethe thickness thickness of of the the arch, arch, the the moment moment willwill comecome whenwhen there there will will be be only only one one line line of of thrusts thrusts and and the the thicknessthickness of of the the arch arch cannot cannot be be reduced reduced withoutwithout itit exitingexiting the the line line of of the the arch. arch. Note Note that that load-bearing load-bearing structuresstructures in in Gothic Gothic cathedralscathedrals account account for for 85% 85% to to 90% 90% of ofthe the building building materi materialal [30]. [Therefore,30]. Therefore, the thedesign design of ofthe the rampant rampant arches arches of oftwo two arches arches tried tried lightening lightening the the structure structure while while maintaining maintaining or or improvingimproving the the thrust thrust resistance resistance of of the the flying flying buttresses.buttresses. InIn general, general, the the only only action action to to bebe consideredconsidered in constructions made made with with ashlars, ashlars, such such as as Gothic Gothic cathedrals,cathedrals, is is the the position position ofof thethe lineline ofof thruststhrusts byby its own weight. Scientific Scientific calculation, calculation, therefore, therefore, justifiesjustifies the the use use ofof geometricgeometric or or graphical graphical patterns. patterns. Therefore, Therefore, the theancient ancient builders builders understood understood the essence of building with ashlars: the proper placement of weights, the balance of thrusts, and counterbalances. This knowledge of the decisive importance of geometry is perhaps the secret of

Symmetry 2019, 11, 627 13 of 15 the essence of building with ashlars: the proper placement of weights, the balance of thrusts, and counterbalances. This knowledge of the decisive importance of geometry is perhaps the secret of Gothic constructions. In this research, it was shown how the rampant arches of cathedrals like the Palma de Mallorca are optimum from the geometric point of view. This work may have limitations from a practical point of view. Firstly, if it is analyzed numerically, the precision of the calculation of centers and radii will depend on the precision of the coordinates of the starting and finishing points of the arc. Secondly, from the point of view of analyzing existing constructions, historical plans were analyzed to scale, which are not always available, and can sometimes have distortions or errors [31–33]. In order to analyze real existing cases, a centimetric precision survey would be needed. Although, currently, with the technology of unmanned aerial vehicles, it could be solved in a relatively economic way by realizing a photogrammetric survey [34]. In the literature, several examples can be found [35,36]. On the other hand, if the rampant arches to be analyzed are close to the ground, such as those commented on for staircases or inside buildings, other techniques can be used, such as close-range photogrammetry [37–39] or terrestrial laser scanning [40].

6. Conclusions In this work, the geometry of the rampant arch was studied from different approaches, the graphical point of view known in classical literature and the current mathematical point of view, with its subsequent programming. An optimal geometrical solution was defined from the constructive point of view and was calculated and programmed. The optimum rampant arch obtained is the one with the common tangent to the two arches parallel to the slope line. Additionally, knowing the radii of the two arcs, the software allows calculating the starting points of both arches on the pillars or walls. It was applied to several well-known and representative cases of Gothic art in France (church of Saint Urbain de Troyes) and Spain (Cathedral of Palma de Mallorca), establishing that the first was not an optimal design while the second was. The conception and study of certain architectural elements, such as the rampant arch, not only help identify and describe existing aspects of our traditional architecture, but can also serve as a starting point for the verification of existing structures, which, despite standing for many years, may require intervention. This research opens new perspectives for the geometric study of buttresses of cathedrals in general and of rampant arches in particular. It was possible to analyze the geometries used in the design of cathedrals, and specifically through the proposed case studies. It was verified how the constructors reached the optimum rampant arch from a structural point of view, defined as the one whose tangent between the two arches is parallel to the straight unevenness between the initial and final points of the rampant arch.

Author Contributions: C.V., A.A., C.S.-A.-G., F.G.M., I.Z., and F.M.-A. conceived the study, designed the method, and wrote the manuscript. Funding: This research received no external funding. Acknowledgments: The authors would like to thank CIAIMBITAL (University of Almeria, CeiA3) for its support. Conflicts of Interest: The authors declare no conflict of interest.

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