Mathematics of space leoendre pdf download

It meant that the new mathematics were like machines that possessed a certain degree of autonomy from intuitive experience. A century later, nascent phenomenology would return to this gap and explore its possible philosophical signification. Among the reasons that explain such a gap, the most fundamental lies in the fact that calculus has generally to do with the consideration of time instead of dealing with purely spatial dimensions. What was the most puzzling, like the so-called infinitely small, were actually elementary dynamic processes rather than static beings.

Later on, calculus would also be instrumental in the development of economic theory by providing the means to study the circulation of goods and capitals.

Another reason explaining the growing gap between architecture and mathematics was the new relation between theory and practice involved in the transition from arithmetic and geometry to calculus. In the past, mathematical formulae were seen as approximate expressions of a higher reality deprived, as rough estimates, of absolute prescriptive power. One could always play with proportions for they pointed towards an average ratio. The art of the designer was all Figure 6. IJP, Klein Bottle about tampering with them in order to achieve a better An intriguing topological singularity.

Computer simulation is instrumental in the exploration of such complex organisational patterns. Figure 9. Michael Weinstock, Architectural and urban forms in Mesopotamia The emergence of small complex anatomical organisations makes possible the The organic property of emergence is supposed to apply to both the natural emergence of ever larger and even more complex organisations.

Complexity and the human realms. One deals also with relations. This with; they went along with the notion of restraint. Scripting and algorithmics reinforce this trend. With averages but firm boundaries that could not be tampered with.

Design was no longer involved. Theory set pervasiveness of a new kind of organicism, vitalism or, rather, limits to design regardless of its fundamental intuitions. Theorists like Viollet- Mathematics serves this new materialism, but is not seen as the le-Duc or Gottfried Semper are typical of this reorientation. This might result from the fact that Despite claims to the contrary made by architects like Le the polarities evoked earlier have not been reconstituted. Restraint and control through the establishment of standards have been lost so far.

The reconstitution of this The Ambiguities of the Present polarity might enable something like the restoration of an Today the computer has reconciled architecture with calculus. Architects need For the first time, architects can really play as much with time mathematics to embrace the contradictory longing for power as with space. They can generate geometric flows in ways and for restraint, for standardisation and for invention.

Beyond the immediate notions summoned by these simple terms, deeper meanings emerge, constantly shifting and reconfiguring one another at key moments of the history of mathematics.

Mathematics are not a stable and well-defined object, but rather a plurality of objects, practices, theories, cognitive and collective constructions, which have a history and great diversity. Scientific theories speak of neurons, atoms, celestial bodies, but also of numbers, groups or functions, most of them couched in the language of mathematics.

Hence mathematics is applied to the natural world, to its forms, to space and to reality at large. Why does the knowledge of abstract objects, such as numbers or algebraic structures, help us better understand, control and master the sensible world and its forms? Several philosophical responses are conceivable, which have more or less always coexisted, and are the focus of this introductory essay to the subject.

Geometry originated in Egypt and Mesopotamia with the survey of land, until the Greeks assumed that an interest could be taken in exact forms that are only vaguely represented by objects: thus no longer a tree trunk, for instance, but a cylinder; no longer the edge of a plank, but a truly straight line; or an absolutely flat plane without dents or bumps, and so on. Philosophy seized on this and declared that reality was an imperfect image of this exact world: the tree trunk became a defective cylinder, and the plank an adulterated plane.

With Galileo, geometry entered the world and became reality itself. The totality of nature became a geometric edifice, an edifice we could learn to see. Fdecomite, With a Little Help from my Polyhedron Building the origami version of five intersecting tetrahedra is a real puzzle.

The Wenninger model is of great help for assembling all parts in the right order. By contrast, a large number of mathematicians, notably the prestigious members of Bourbaki, the Paris-based collective which developed highly abstract and theoretical mathematics from to , defended a so-called Platonist realism, rooted in the belief that a mathematical reality exists independently of the human mind. The fact that mathematicians often reach the same results with similar objects through separate paths greatly incites them to believe in the independent existence of these objects.

A further philosophical divide opposes Platonist realism to a tradition of Constructivist philosophy of mathematics. Constructivism posits that all truth is constructed, without reference to whatever it conforms to. For Austrian philosopher Ludwig Wittgenstein, for example, the mathematical understanding of a statement does not exist outside of its proof, and in this sense the mathematician is merely an inventor as opposed to a discoverer who establishes connections and forms descriptions, but does not describe real facts.

Practitioners of mathematics frequently express a strong subjective sense of discovery or of the exploration of a terra incognita that is already extant and consistent. However, while the universality of mathematical objects does speak in favour of some kind of realism, the discoveries of logic in the 20th century have made this position difficult to defend.

For the proponents of a Platonist realist view, mathematical objects are figments of the human imagination, and mathematical theories principally useful fictions. The philosopher Hartry Field went as far as proposing that mathematical theories are literally false since their objects do not really exist.

Yet what characterises a fiction is not so much its logical status whether it is true or false , but its cognitive function. As representations whose purpose is less to provide an accurate description of reality than to help us imagine possibly unreal situations, fictions are assisting our imagination, and that. To produce theories we must envision hypotheses and explore their consequences; we must build models of phenomena, and formulate idealisations.

Thus, in constant interaction with one another, representing, constructing and simulating form three inseparable moments of mathematical activity: to represent an object, other objects must be constructed from it; to construct an abstract object a function, a transformation , we must represent it; and to simulate an object, we must elaborate fictitious representations which mobilise more constructions at will.

This warm-up piece was submitted in response to a question on form and formlessness. Stefano Rabolli-Pansera, Stereotomic Self-Portrait, Diploma Unit 5 Engineering the Immaterial , Architectural Association, London, Descriptive geometry at work, submitted in response to an undergraduate design brief on self-representation.

The Mongean representation of the drawing plane demonstrates a technique unchanged since the late 18th century. Matthew Chan, Stereotomic Self-Portrait, Diploma Unit 5, Architectural Association, London, Several successive derivations of stereotomic projection projective geometry applied to stone cutting result in a highly articulated prismatic object recording, in metaphorical and narrative terms, an argument between the author and his girlfriend.

In conclusion, is mathematical realism independent of sensible reality? Is mathematics to be found in the world, or in our minds and imagination? The argument is without a doubt impossible to settle and, unless one is a philosopher, a bit of a waste of time: the question of mathematical realism is a metaphysical problem, forever unanswerable or irrefutable.

The development of the neurosciences has hardly made a difference in this regard. For the defenders of an ideal or Platonist realism, the brain may possess structures that give it access to the independent universe of mathematics.

For those who envision a realism inscribed in sensible reality, mathematics is simply in the world, and since our brains are tailored by the process of evolution to understand the world, we are naturally drawn to doing mathematics. The historian not directly immersed in creative mathematical activity nor prone to making philosophical statements will contend that the historical and cultural contexts are key to the emergence of certain concepts.

Mathematics being a human construction inscribed in a temporal dimension, it is difficult to conceive of a mathematical reality out of time and out of touch with its surroundings. And since the very essence of cultural facts is to emerge among small human groups and amalgamate in larger families while preserving their specificity, mathematics aspires to working with universal structures, which is one of its lasting characteristics.

It is also unbelievably diverse. Because of the hegemony of the form inherited from Greek mathematics, it took some time to recognise the great diversity of forms of mathematical expression. Yet, strikingly, such diverse forms are comparable, and it is possible to rigorously reinterpret the mathematical knowledge of one civilisation within the context of another. Furthermore an understanding of mathematical reality inscribed in history, space, time and culture imbues the work needed to reconstruct and analyse mathematical practices with veritable meaning.

Thinking, for example, of complex numbers as eternal members of a Platonic heaven of mathematical ideas, leads us to conceptualise the development of this key concept as a series of increasingly accurate approximations of its presentday formulation, always presumed to be its perfect embodiment. Such a deterministic history may be useful to the mathematician and the student, but it does not take into account the highly contingent aspects of the formation of mathematical knowledge, with its generally chaotic course and winding paths of discovery.

The historians of mathematics with a finite amount of time to dedicate to each phase of history cannot embrace all of. The Ancients did use some analogous curves to explore intractable problems for example, the duplication of the cube , but discounted such findings as unworthy of what they would have hoped to obtain using a ruler and a compass. The 17th century witnessed the concurrent birth of infinitesimal calculus and the science of movement, bringing radically different conceptions of mathematics to a head.

For Isaac Newton, on the other hand, all measurements were a function of time. A curve derived through the Newtonian Method of Fluxions was a reality that moved and animated. The tension opposing the two titans of the 17th century was not only a matter of temperament, or a historical fluke. It expressed the very nature of mathematics, torn between two opposed yet inseparable trends: on the one hand we have rigour, an economy of means and an aesthetic standard. For the geometers of the 19th century, a period witnessing the explosion of a realist understanding of the discipline, descriptive geometry was essentially a graphical procedure for tackling numerous real practical problems.

In , however, Jean-Victor Poncelet explored the properties of figures remaining invariant by projection with the systematic help of infinite or imaginary elements, endowing projective geometry with a far greater degree of abstraction and generality.

Many more developments were to come. By the middle of that century, numerous constructions of Euclidean models of non-Euclidean geometries provided these discoveries with intuitive support and accelerated their ultimate acceptance.

In , studying the properties of figures left invariant by a given class of transformations, Felix Klein reorganised the body of all known geometries. By identifying curves in space with binary quadratic forms, Klein advocated the merger of geometry and algebra, now an established tendency of contemporary mathematics. After the prodigious development of geometry in the 19th century, the time had come to rebuild the foundations of mathematics, which the advent of non-Euclidean geometries had shaken.

A powerful research movement converged towards the axiomatisation of geometry, led by David Hilbert and his Foundations of Geometry The Foundations owes its radical novelty to its capacity to integrate the general with the technical and mathematical philosophy with practice.

By combining local and general points of view, such methods explored the relative relationships and general behaviour of trajectories, their stability and complexity, allowing one to look for mathematical solutions even when they were not quantifiable. From these new methods, other mathematicians have been able to develop current theories of complexity and chaos.

Ribbing along the other way enhances the perception that the bottle turns in on itself. The weft outlines the notional continuity between interior and exterior for which the Klein bottle is famed. Recent Developments Today the image of mathematics ruled by structures and the axiomatic method is a matter of the past.

In a technological environment characterised by the omnipresence of computers, a different set of domains — those precisely excluded by the axiomatic and structuralist programme — have acquired an increasing importance: discrete mathematics, algorithms, recursive logic and functions, coding theory, probability theory and statistics, dynamical systems and so on. George L Legendre, High-Rise Sketch, Analytic geometry at work: a parametric circle swept along a periodic curve.

The seed was eventually developed into a high-rise proposal by Richard Liu and Kazuaki Yoneda. Rob Scharein, Chart of the unpredictable states of the Lorenz attractor, The Lorenz attractor is a non-linear dynamic system simulating the two-dimensional flow of fluid for given temperature, gravity, buoyancy, diffusivity, and viscosity factors.

Similarly, the experimental method is no longer regarded as an antinomy to mathematics. In the s, under the joint influence of technological progress and the increased awareness of the social dimension of mathematical practice, questions emerged in the community that, a few years earlier, would have been deemed totally incongruous.

A new type of demonstration heavily reliant on the computer, such as the four-colour theorem by K Appel and W Haken , called into question the very nature of proof in mathematics: how could a human mind grasp a demonstration which filled nearly pages and distinguished close to 1, configurations by means of long automatic procedures?

Starting from computer-generated proofs, the discussion soon moved to other types of demonstrations, either unusually expansive or mobilising an extremely complex architecture of conjectures stemming from various domains — the work of Edward Witten on knot theory and string theory after comes to mind. A large number of conjectures emerging from extensive computational activity had been used for years before they could be rigorously proved, as was the case, for instance, with the topological properties of the Lorenz attractor.

In any field, there is a strong social standard of validity and truth …. Hence, advocated since the s by both philosophers and historians of science, the sociological dimension of mathematics eventually caught up with those thinkers so far least amenable to it.

David Hilbert, Foundations of Geometry, Architecture and Mathematics Between Hubris and Restraint The introduction of calculus-based mathematics in the 18th century proved fatal for the relationship of mathematics and architecture. As Antoine Picon, Professor of History of Architecture and Technology at Harvard Graduate School of Design, highlights, when geometry was superseded by calculus it resulted in an ensuing estrangement from architecture, an alienation that has persisted even with the widespread introduction of computation.

It is a liaison that Picon characterises as having shifted between hubris and restraint. Mathematics was sometimes envisaged as the true foundation of the architectural discipline, sometimes as a collection of useful instruments of design.

Mathematics empowered the architect, but also reminded him of the limits of what he could reasonably aim at. Inseparably epistemological and practical, both about power and restraint, the references to mathematics, more specifically to arithmetic and geometry, were a pervasive aspect of architecture. Towards the end of the 18th century, the diffusion of calculus gradually estranged architecture from mathematics.

While arithmetic and geometry remained highly useful practical tools, they gradually lost their aura of cutting-edge design techniques.

This estrangement has lasted until today, even if the computer has enabled architects to put calculus to immediate use. A better understanding of the scope and meaning that mathematics used to have for architects might very well represent a necessary step in order to overcome this indifference.

The purpose of this article is to contribute to such an understanding. Mathematics as Foundation From the Renaissance onwards, the use of mathematical proportions was widespread among architects who claimed to follow the teachings of Roman architect and engineer Vitruvius. This use was clearly related to the ambition to ground architecture on firm principles that seemed to possess a natural character. For the physical world was supposed to obey proportions, from the laws governing the resistance of materials and constructions to the harmonic relations perceived by the ear.

The interest in proportions was related to the intuitive content that arithmetic and geometry possessed. As we will see, the strong relation between mathematics and the intuitive understanding of space was later jeopardised by the development of calculus.

But it bore also the mark of two discrepant points of view. Proportion could be first interpreted as the very essence of the world. Envisaged in this light, proportion possessed a divine origin. Historian Joseph Rykwert has shown how influential these speculations were on the development of the architectural discipline in the 16th and 17th centuries. But proportion could also be envisaged under a different point of view, a point of view adopted by Renaissance theorist Leon Battista Alberti for whom the purpose of architecture was to create a world commensurate with the finitude of man, a world in which he would be sheltered from the crude and destructive light of the divine.

In this second perspective, proportion was no longer about the hubris brought by unlimited power, but about its reverse: moderation, restraint.

Such polarity was perhaps necessary to give proportion its full scope. Now, one may be tempted to generalise and to transform this tension into a condition for mathematics to play the role. Another way to put it would be to say that in order to provide a truly enticing foundational model for architecture, mathematics must appear both as synonymous with power and with the refusal to abandon oneself to seduction of power.

For this is what architecture is ultimately about: a practice, a form action that has to do both with asserting power and refusing to fully abandon oneself to it. To conclude on this point, one may observe that this polarity, or rather this balance, has been compromised today.

For the mathematical procedures architects have to deal with, from calculus to algorithms, are decidedly on the side of power. Nature has replaced God, emergence the traditional process of creation, but its power expressed in mathematical terms conveys the same exhilaration, the same risk of unchecked hubris as in prior times.

What we might want to recover is the possibility for mathematics to be also about restraint, about stepping aside in front of the power at work in the universe. It is interesting to note how the quest for restraint echoes some of our present concerns with sustainability. The only thing that should probably not be forgotten is that just like the use of mathematics, sustainability is necessarily dual; it is as much about power as about restraint. Our contemporary approach to sustainability tends to be as simplistic as our reference to mathematics, albeit in the opposite direction.

From an architectural standpoint, the same mathematical principle can be simultaneously foundational and practical. Vitruvian proportion corresponded both to a way to ground architecture theoretically and to a method to produce buildings.

In this domain also, a duality is at work. This second duality has to do with the fact that tools may be seen as regulatory instruments enabling coordination and. Until the 18th century, most uses of mathematics and proportion had in practice to do with coordination and control rather than with the search for new solutions.

But tools can be mobilised to explore the yet unknown; they can serve invention. From the Renaissance on, the geometry used for stone-cutting, also known as stereotomy, illustrates perfectly this ambivalence. But this ambition was accompanied with the somewhat contradictory desire to promote individual invention. Besides major architectural realisation like the castle of Anet or the Tuileries royal palace, his main legacy was the first comprehensive theoretical account of the geometric methods enabling designers to master the art of stereotomy.

For the architect, the aim was twofold. First, he wanted to achieve a better control of the building production. But the objective was also to invent. The best demonstration of this art of invention was in his eyes the variation of the Montpellier squinch that he designed for Anet. The identification of the true regulatory principles at work in a given practice could lead to truly innovative combinations.

For Monge, descriptive geometry was both about standardisation and invention. Today, what might be often lacking is not so much In contemporary cutting-edge digital practices, mathematical entities and models are most of the time mobilised in a perspective that has to do with emergence, with the capacity to amaze, to thwart received schemes. One of the best examples of this orientation lies in the way topology has been generally understood these days.

This interpretation of topology is at odds with what mathematicians consider as its principal objective, namely the study of invariance. It has to do with conservation rather than sheer emergence. It might be necessary to reconcile, or at least articulate these two discrepant takes on the role of mathematics to fully restore their status. The Calculus Breaking Point The end of the 18th century clearly marks a breaking point in the relationship between architecture and mathematics.

Until that time, arithmetic and geometry had been a constant reference for the architect, as foundational knowledge as well as practical tools, as empowerment and as incentive for restraint, as a means of control and standardisation as well as a guideline for surprising invention. Galileo Galilei, Animal bones, from Discorsi e dimostrazioni matematiche intorno a due nuove scienze, The example is used by Galileo to illustrate how strength is not proportional to size, contrary to what proportion-based theories assumed.

In all these roles, mathematics had a strong link with spatial intuition. Arithmetic and geometry were in accordance with the understanding of space.

This connivance was brought to an end with the development of calculus and its application to domains like strength of materials. First, calculus revealed the existence of a world that was definitely not following the rules of proportionality that architects had dwelt upon for centuries. Galileo, for sure, had already pointed out the discrepancy between the sphere of arithmetic and geometry and domains like strength of materials in his Discorsi e dimostrazioni matematiche intorno a due nuove scienze Discourses and Mathematical Proofs Regarding Two New Sciences published in But such discrepancy became conspicuous to architects and engineers only at the end of the 18th century.

The fact that some of the operations involved in calculus had no intuitive meaning was even more problematic. It meant that the new mathematics were like machines that possessed a certain degree of autonomy from intuitive experience. A century later, nascent phenomenology would return to this gap and explore its possible philosophical signification. Among the reasons that explain such a gap, the most fundamental lies in the fact that calculus has generally to do with the consideration of time instead of dealing with purely spatial dimensions.

What was the most puzzling, like the so-called infinitely small, were actually elementary dynamic processes rather than static beings. Later on, calculus would also be instrumental in the development of economic theory by providing the means to study the circulation of goods and capitals. Another reason explaining the growing gap between architecture and mathematics was the new relation between theory and practice involved in the transition from arithmetic and geometry to calculus.

In the past, mathematical formulae were seen as approximate expressions of a higher reality deprived, as rough estimates, of absolute prescriptive power. One could always play with proportions for they pointed towards an average ratio. The art of the designer was all about tampering with them in order to achieve a better result. As indicators of a higher reality, formulae were an Computer simulation is instrumental in the exploration of such complex organisational patterns.

Michael Weinstock, Drosophilia Wing Development, The emergence of small complex anatomical organisations makes possible the emergence of ever larger and even more complex organisations. Complexity builds over time by a sequence of modifications to existing forms. Michael Weinstock, Architectural and urban forms in Mesopotamia The organic property of emergence is supposed to apply to both the natural and the human realms.

In the new world of calculus applied to domains like strength of materials or hydraulics, mathematics no longer provided averages but firm boundaries that could not be tampered with. From that moment onwards, mathematics was about setting limits to phenomena like elasticity, then modelling them with laws of behaviour. Design was no longer involved. Theory set limits to design regardless of its fundamental intuitions.

Theorists like Violletle-Duc or Gottfried Semper are typical of this reorientation. Despite claims to the contrary made by architects like Le Corbusier, this indifference to mathematics was to remain globally true of modern architecture. The Ambiguities of the Present Today the computer has reconciled architecture with calculus.

For the first time, architects can really play as much with time as with space. These tools can be implemented with the help of modern perturbation methods, which, in addition to the mean element propagation, allow for the recovery of short-period effects to provide osculating elements. Within a perturbative approach, the highest frequencies of the motion, which normally have small amplitudes, are analytically filtered via averaging procedures. From the mathematical point of view, the averaging is the result of a transformation from old to new variables, which, in reference to perturbed Keplerian motion, are called osculating and mean elements, respectively.

This transformation is rarely attained in closed form, and one feels satisfied with computing the first terms of the Taylor series expansion of the transformation.

The averaged equations depend only on long period angles and, therefore, are integrated numerically with very long step sizes. The short-period effects, if required, can be recovered at any step of the integration by the simple evaluation of the analytical expressions of the averaging transformation.

The averaging transformation can be performed either directly over the variation of parameters equations, using the generalized method of averaging Bogoliuvov and Mitropolski, , or in the Hamiltonian function, using canonical perturbation theory. Analytical and semi-analytical theories based on averaging have been used in the propagation of catalogs of space objects since the beginning of the space era.

Bertachini de Almeida Prado and Martin Lara, is the design and implementation of an efficient semi-analytical theory for the fast propagation of the osculating elements of space orbits in the close Earth environment. The theory will be based on the Lie transforms method, which is specifically designed for automatic computation by machine, and will explore the possibilities of polar-nodal variables, as well as of non-singular variables based on them, for improving the performance in the evaluation 1.

The perturbation model should include the main perturbations, affecting Earth artificial satellites in the satellite regions of interest, namely the regions including low Earth orbits LEO , geo-stationary orbits GEO , mean Earth orbits MEO for traditional constellations of global navigation satellite systems, and high ellipticity orbits HEO.

This new title in the Architectural Design series updates architectural mathematics since the digital revolution With world-class contributors, this is an essential resource for anyone interested in the ways computation has transformed the discipline The book explores fascinating issues in modern design, most importantly the impact of mathematics on contemporary design creativity For students and practitioners alike, Mathematics in Space covers vital topics in a constantly changing discipline.

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