Integral, Probability, and Fractal Measures

Edgar, Gerald A.

doi:10.1007/978-1-4757-2958-0

Cited by 164 publications

(35 citation statements)

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“…The definition of the s-dimensional Hausdorff measure of M, H s ðMÞ, can be found in (Falconer 1990(Falconer , 1997Edgar 1998) FðMÞ is a measure of the level of roughness of M; i.e. it is a measure of the unevenness of the structures, which determines their space-filling ability.…”

Section: Hausdorff-besicovitch Dimensionmentioning

confidence: 72%

The Fractal Pattern of the French Gothic Cathedrals

Samper

Herrera

2014

Nexus Netw J

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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.

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Section: Hausdorff-besicovitch Dimensionmentioning

confidence: 72%

The Fractal Pattern of the French Gothic Cathedrals

Samper

Herrera

2014

Nexus Netw J

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“…First, we will establish a few basic notions. More detailed mathematical references and demonstrations of these are available (Falconer 1990(Falconer , 1997Edgar 1998). …”

Section: Fractal Dimension Theorymentioning

confidence: 99%

A Study of the Roughness of Gothic Rose Windows

Samper

Herrera

2015

Nexus Netw J

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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...

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“…Fractals are studied through many tools, including topology, measure theory, and symbolic spaces; the rest of the section describes the relevant tools needed in the paper. A good reference text for the theory is [9], while [10] is a very nice introduction.…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

On the performance of Kalman filtering with intermittent observations: A geometric approach with fractals

Censi

2009

2009 American Control Conference

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Abstract-This paper describes the stationary distribution of the a-posteriori covariance matrix of a Kalman filter when the availability of measurements is subject to random phenomena such as lossy network links. If a certain non-overlapping condition is satisfied, the distribution has a fractal nature, and there exists a closed-form expression for the cdf, which is a singular function. If the condition is not satisfied, deciding whether the cdf is singular or not, even in the scalar case, is at least as hard as some open problems in measure and number theory.

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Integral, Probability, and Fractal Measures

Cited by 164 publications

References 0 publications

The Fractal Pattern of the French Gothic Cathedrals

The Fractal Pattern of the French Gothic Cathedrals

A Study of the Roughness of Gothic Rose Windows

On the performance of Kalman filtering with intermittent observations: A geometric approach with fractals

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