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Architects and historians commonly claim that when designing the Palau Güell, Gaudí experimented with some architectural elements which would influence his later works, with the arches being prevalent in the building. But, to date, no mathematically thorough study has been conducted in order to determine the type of each of such arches. Furthermore, literature contains subjective and intuitive claims which create ambiguity and contradictions. Owing to the methods used by Gaudí, the arches in Palau Güell are best fit by conical curves (ellipse, parabola, hyperbola) or hyperbolic-cosine curves (catenary, Rankine). Using photogrammetrical and mathematical techniques, in this paper we will classify the arches of this building according to their types, we will eliminate the inconsistencies which currently exist in literature, and we will obtain the corresponding analytical equations, which may be helpful in future studies.

The rose window is one of the most representative elements of Gothic art and architecture. In this work we analyze fifteen rose windows from fifteen Gothic cathedrals using fractal geometry. Specifically, we examine the texture and roughness of these rose windows focusing on three factors, their designs, glass areas and solid areas. In this investigation we generate parameters which provide a measure of roughness of the rose windows in order to find out if they show a general non-random fractal pattern. The paper concludes that statistically, there is a characteristic fractal pattern in the solid and glass areas of the rose windows of the Gothic style, but not necessarily in their overall design.Keywords Rose windows Á Gothic architecture Á Fractal geometry Á Fractal parameter Historical Background: Light in the Gothic CathedralThe Gothic style signaled a veritable revolution in architecture as a result of its technical innovations which challenged the conventional concepts of construction at the time and also changed the manner in which large indoor spaces were conceived. Louis VI, King of France, aspired to rule all of the vast territories which had formed the Carolingian Empire three centuries before. To achieve this aspiration he was assisted in this task by Abbot Suger of Saint-Denis, an advisor with both intelligence and diplomatic skills. In gratitude for the services rendered, Louis VI granted the abbey of Saint-Denis great privileges so that they would have a competitive advantage and be able to hold their prosperous annual fair, which gathered both the faithful and merchants providing major benefits for the abbey. As a result of this, Saint-Denis became the richest Benedictine monastery in France after Cluny. Influenced by the Neoplatonic thoughts which spread amongst scholars of the early twelfth century in the Paris region, Suger was fascinated by light as a means to connecting with God. With this concept as an intellectual guide, Suger inferred that the House of God, the Christian church, had to become a temple of light and even more: ''a city bathed in the light of God'' (Berger 1906;Panofsky 1970). In order to achieve this ideal, it was necessary to modify and improve the construction system of the great Romanesque churches. That is, it was imperative to remove some walls and tear others from top to bottom in order to place large windows in them which would capture the sunlight. The name of the architect who found the solution to Suger's problem is unknown. Perhaps it was the master builder who directed the construction of the Romanesque structure in the abbey of Saint-Denis. Regardless, shortly before finishing this work, Suger ordered the following inscription to be places, in Latin verses on the main door of the abbey church. ''Portarum quisquis attolere quaeris honorem, Aurum nec sumptus, operis mirare laborem. Nobile claret opus, sed opus quod nobile claret, Clarificet mentes, ut eant per lumina vera, Ad verum lumen, ubi Christus jaunua vera. Quale sit intus in his determinat aurea p...

<p><strong>Abstract.</strong> This research work proposes a methodology to statistically determine the geometric configuration of a masonry cross vault. Within Cultural Heritage it is possible to find architectural elements with absent or scarce historical sources about design approach or construction techniques. The cross vault case study belongs to a partially destroyed vaulted system distributed along the aisles of ancient Assunta’s Cathedral which is part of the Aragonese Castle on Ischia island, near Naples (Italy). Using photogrammetrical data acquisition, standard geometric analysis, numerical processes, computing and statistics this paper shows a method to objectively determine the geometric shape which best fits one of the existing Cathedral vault according to critical interpretation about stylistic and cultural contents linked to specific geographical and temporal contexts. This paper provides explanations, methods and objective calculation algorithms to find the best-fitting shape for a generic given point cloud and it is aimed at demonstrating the complementarity between descriptive geometry and algorithmic mathematical approaches. The final product of this multidisciplinary workflow is a 3D model deriving from the comparison between an ideal automatic model built thanks to the translation of traditional geometric rules in visual scripting language and an automatic model deriving from the mathematical analysis of survey data, curves and surface, of the architectural element. This experimentation generates 3D models to perform in-depth multidisciplinary tests and AR and VR applications to promote the communication of destroyed or inaccessible cultural heritage.</p>

The classic patterns of Euclidean Geometry were used in the construction of the Gothic cathedrals to provide them with proportion and beauty. Still, there is also another complex concept related to them: the un-evenness of their structures, which determines their space-filling ability, that is, their level of roughness. In this paper we use the techniques of Fractal Geometry to generate parameters which provide a measure of roughness. In this way we show that the French Gothic cathedrals do not only follow Euclidean geometric patterns, but also have a general non-random fractal pattern.