Reconciliation of Geometry and Perception in Radiation Physics E-Book

Benoit Beckers

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 215

Veröffentlichungsjahr: 2014

Beschreibung

Reconciliation of Geometry and Perception in Radiation Physics approaches the topic of projective geometry as it applies to radiation physics and attempts to negate its negative reputation. With an original outlook and transversal approach, the book emphasizes common geometric properties and their potential transposition between domains. After defining both radiation and geometric properties, authors Benoit and Pierre Beckers explain the necessity of reconciling geometry and perception in fields like architectural and urban physics, which are notable for the regularity of their forms and the complexity of their interactions.

Leseprobe

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s Road London SW19 4EU UK

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John Wiley & Sons, Inc.111 River Street Hoboken, NJ 07030 USA

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© ISTE Ltd 2014The rights of Benoit Beckers and Pierre Beckers to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014939768

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISSN 2051-2481 (Print)ISSN 2051-249X (Online)ISBN 978-1-84821-583-2

Contents

Introduction

1: Discovering the Central Perspective

1.1. The musical scale

1.2. The tonal system

1.3. Nomenclature of the projections

1.4. The central projection on the plane

1.5. Proportions and progressions

1.6. The eighth proposal of Euclid

2: Main Properties of Central Projections

2.1. Straight lines and conics

2.2. Coherence and cross ratio

2.3. Harmonic relation and regularity

2.4. The foreshortening

2.5. Homogeneous coordinates

3: Any Scene Carried to a Sphere and the Sphere to a Point

3.1. General concepts

3.2. Cartography of the sphere

3.3. Projection of the sphere on cylinders

3.4. Projection on the plane

3.5. Pseudocylindrical projections

3.6. Hemisphere tilling

4: Geometry and Physics: Radiative Exchanges

4.1. Geometric wave propagation

4.2. The radiosity equation

4.3. View factors

4.4. Ray tracing

4.5. Specular reflection of light and sound

Conclusion

Bibliography

Index

Introduction

If we must now reconcile geometry and perception, it is because a separation has occurred. As its etymology suggests, geometry was born in Greece about 2,500 years ago. From the figures (in particular, straight lines and conical curves) and relationships (especially parallelism and orthogonality) that structure human vision, the concern was to describe an essentially visible world. As we recall in Chapter 1 of this book, this visual geometry, first in competition with an essentially auditory arithmetic, quickly articulated with it, then took over, to the point that we have witnessed a geometrization of mechanics, of color (Newton) and, finally, of the world itself (Galileo and Kepler).

Then, in the early 19th Century, physics and mathematics turned away from this visual world. This would be, afterward, a problem for those precisely interested in the sensory world, either for itself, or to represent the results of these now so abstract sciences and mathematics. Should it be necessary to specify a date to symbolize the passage to abstraction, a particularly significant candidate would be that of 21 March 1816, when a royal decree of the recently restored French monarchy dissolved the Academy of Sciences to reinstate it without any members involved with the Revolution and the Empire, including Gaspard Monge.

With his descriptive geometry, Monge created a tool able to represent, for the first time, the infinite objects of Euclid (straight lines and planes) so that he could easily solve on the sketch plan the three-dimensional problems that are difficult to represent in space. He also gave a correct explanation of colored shadows. Indeed, he proposed a visual and mechanical interpretation of the world that the Greeks would not have discarded.

Monge was also the first director of the Ecole Polytechnique. Among his students, Jean-Victor Poncelet participated in the Russian campaign of 1812. Taken as prisoner, he spent 2 years in the prison of Saratov on the Volga. This is where he imagined the foundations of projective geometry.

Still on the Volga River, 500 km upstream, Nikolai Ivanovich Lobachevsky was then completing his studies at the University of Kazan and developing an “imaginary geometry”, which was the first of the so-called “non-Euclidean” geometries, as Carl Friedrich Gauss would qualify them later. For Lobachevsky, Euclidean geometry was a limiting case where, as he wrote in 1829, “light rays serve as straight lines; it is not a question of logic, but of physics, which cannot be answered experimentally. As our measurements show no change from Euclidean geometry, we can assume that this one governs our space” [FER 11]. To check if “penetrating extremely remote areas of the universe, our measuring devices would detect such variations”, Lobachevsky measured the interior angles of the triangle formed by the Earth, the Sun and the star Sirius. With the experimental means of the time, he was unable to verify whether their sum was equal to 180° (Euclidean space) or lower (hyperbolic space), but his approach clearly anticipated the sensory world being surpassed, realized some decades later by the combination of experimental physics and abstract mathematics.

Let us go back to Paris and the year 1816. Monge’s successor at the Academy of Sciences is Augustin Louis Cauchy, trained, as Poncelet, at the Ecole Polytechnique. Everything separates Monge and Cauchy. The first, born in 1748, spent most of his career under the Old Regime before joining the Revolution, then Bonaparte, who he followed in Egypt and to which he remained faithful until the end. Cauchy, son of the Revolution (he was born in August 1789), formed in republican schools, but from a Catholic family, would be a royalist, a legitimist, and voluntarily went into exile in 1830 after the fall of Charles X.

His works on holomorphic functions are often considered as the true break with visual geometry. During the 1820s, Cauchy in France and Gauss in Germany ruled on mathematics. In particular, they developed the study of the complex plane.

The projective plane, the complex plane, as well as spaces with a constant negative curvature (hyperbolic geometry of Lobachevsky) or a constant positive one (elliptic geometry of Bernhard Riemann, a disciple of Gauss) were the new playgrounds for geometries often competing, and moving decidedly away from perception.

At the end of the century, two great syntheses ordered these ideas. In 1872, in his “Erlangen Program”, Felix Klein described all geometry as the search for the invariants of some transformation group. In 1899, in his “Foundations of Geometry” David Hilbert axiomatically restructured Euclid’s elements.

Geometry thus seems totally abstracted from perception, looking for new physics also abstracted from the sensory world, which would occur only in the 20th Century.

In the 1800, William Herschel found that decomposed sunlight through a prism heats outside the spectrum of visible light, beyond the red. The following year, Johann Wilhelm Ritter discovered ultraviolet. These invisible waves, capable of heating more than visible light (infrared) or acting on the silver chloride (ultraviolet), are phenomena that we cannot see, but that can be measured, such as gravity, magnetism and electricity.

The physics of the 19th Century was first in the continuity of the previous two centuries, with an ever greater effort brought to measurement and algebraic formulation. In 1811, Joseph Fourier equated heat. In the second part of the century, James Clerk Maxwell unified electromagnetism. Throughout this period, evidence accumulated to the detriment of the universality of Newtonian mechanics, until the work of Max Planck on black body and quanta, presented in 1900.

These great advances in physics created a great victim: geometrization, that is to say the setting in space. Let us take the example of acoustics. The 18th century is the century of the treatise on the opera house, looking for its ideal form. Should the public be disposed in a circle, an ellipse or a horseshoe? What is the ideal size of the room? What form should the tiers, the balconies and the ceiling adopt? At the time, the sound field was not quantified, and reasoning was purely qualitative. What could not be determined for the ear was for the view: the integrator tool was still the central perspective, as in the spectacular views of Claude Nicolas Ledoux and Étienne-Louis Boullée.

However, the 19th Century was unable to apply its new equations to such complex geometric shapes. For natural light, it was even worse. Buildings of the 18th Century were expertly open to daylight so that employees could work well. In those of the next century, due to advances in lighting, openings on the sky were virtually lacking. Everywhere, industrialization was requiring simpler, more manageable, forms and progress in equipment poorly compensated the lack of shape design.

The study of perception was not neglected, quite the contrary. Many studies were performed on color (including those of Maxwell), and Hermann von Helmholtz transfigured musical acoustics (On the Sensation of Tones, 1863) with an experimental resonators device allowing him to explain the formation of the musical scale [MEU 10].

Such research found its framework in a recent discipline, physiology, soon complemented by experimental psychology. However, these disciplines, increasingly numerous and increasingly compartmentalized (musicology, physical and physiological acoustics, psychoacoustics, etc.), were less oriented to applications. Helmholtz stated that “Whoever in the pursuit of science seeks after immediate practical utility may rest assured that he seeks in vain” (Academic Discourse, Heidelberg, 1862).

In the 20th Century, some scientists developed new concepts and tools to control the energy fields in real-world projects: invention of heliodons and solar diagrams, acoustic ray tracing, etc. However, such activities were considered to be totally marginal from the viewpoint of both mathematics and physics, as they were distant from the new developments achieved far away from the human scale.

In the last third of the century, finally, the development of electronics, computer graphics and numerical methods upset these partitioned disciplines. The main change is to visualize what is not visible, starting with infrared (thermography). Like in the Renaissance, all recovers its place in space and time.

For us, geometric projection and physical radiation are both faces of the same problem. Radiation is instantaneous (light), delayed (acoustic) or inertial (thermal). It takes place in a scene composed mainly of surfaces, which are provided with some material properties (e.g. reflection coefficients).

If we observe now, beyond the 19th Century, the history of representation, we can see an early split into two main ways. The first consists of unfolding which can be (development), cutting in volumes and, if projection is necessary, maintaining at all costs the proportions and parallels (axonometric perspective), even multiplying the projection planes to deform the objects of the scene (technical drawing) as little as possible. This trend began with prehistoric parietal paintings up until modern tomography, through Egyptian drawings, Persian or Chinese parallel perspectives, to the method of Monge. It continues to be productive today (level sets), but this is not what we discuss here.

The second way, already evident in the optics of Euclid and in Renaissance perspective works and theories, is to accept the inherent three-dimensionality of our space, as we live, hear and see it. This is the way of sensory geometry, finally reconciled with the perceptions that gave it birth, the sound and light whose serene waves radiate in it.

We will start with music.

1

Discovering the Central Perspective

1.1. The musical scale

The frequency aspect of the system of musical composition is based on a physical property; the harmonic structure of periodic sounds; an arithmetic property; almost perfect equality between the 12th term of the triple progression and the 19th power of 2; and a simple experimental device, for example, a string fixed at both ends and stretched on a graduated scale, so that it can be divided accurately, pressing on one of its graduations.

Our ears easily verify some properties: the shorter the string, the sharper its sound; the simple divisions (half, two-thirds, three-quarters, etc.) produce, with respect to the entire string, consonant intervals (respectively, the octave, the fifth, the fourth, etc.); if the same division is repeated on the shortened string, then the same interval is perceived (for example, if the string is divided into two, then into four, then into eight, three ascending octaves are successively heard).

With a very long and heavy string, whose vibration is very slow, it can be inspected visually that the perceived height is proportional to the vibration frequency and, therefore, inversely proportional to the string length. So, what the ears identify as the interval between two pitches corresponds physically to a ratio, i.e. a multiplicative factor, whose product by the frequency of the lower pitch of the interval gives the frequency of its top pitch.

With a string of length L, we can produce the following simple intervals between the entire string and its division:

Division

Ratio

Interval

L

1

unison

L/2

2/1

octave

2L/3

3/2

fifth

3L/4

4/3

fourth

4L/5

5/4

major third

5L/6

6/5

minor third

8L/9

9/8

tone

The ratio series is interrupted before including the number 7. The following intervals would be too close together, so a jump is done directly to the tone, which is already dissonant: for two singers, it is extremely difficult to sing the same melody at a distance of one tone, and the result is heard as a “friction”, i.e. dissonant. In contrast, the thirds are consonant, like further higher intervals, until the octave, the most consonant of all. When a man and a woman try to sing together in unison, an octave separates their voices. There is no doubt that the octave has always been perceived as an identifier interval: doubling the frequency of a sound only modifies its absolute pitch (that is, a jump of octave), not its relative one within a set of sounds.

Therefore, the problem of the musical scale is how to fill the octave with a maximum of consonant intervals in a logical way that will afterward allow the largest number of operations. Whenever a ratio takes us beyond the limits of the octave, we can divide by two, without changing the nature of the corresponding interval.

However, these early observations do not allow us even to boot up the composition of a full scale. To do this, we need a generalizable procedure. Around the 6th Century BC, the Pythagorean School moved to Europe a reasoning already well known in the East, possibly in Egypt, definitely in Chaldea and earlier in China.

The idea is to use only the two more consonant intervals: the octave and the fifth. Starting at any pitch (frequency f), which defines the octave to fill (f to 2f), we ascend by the interval of a fifth to a second pitch (3/2 f). A second fifth is then applied, giving a third pitch (9/4 f). As this is beyond the octave to fill, we lower it by an octave (9/8 f). And so on, progressing by fifths and regressing by octaves, we find the 12th pitch. This is almost equal to the first one due to the following arithmetic property:

In other words, 12 fifths are almost equal to 7 octaves:

However, the approximation of the comma could not satisfy the Chinese, who considered the lu, i.e. the ordered sound, as a principle of universal equilibrium [LIE 01]. So, they tried to reduce the comma, bringing further progression of fifths. In the 1st Century BC, King-Fang showed that 53 fifths are almost equal to 31 octaves, thus reducing the comma to a value of 1.002, almost unitary. But a scale of 53 notes is unmanageable. The Arabs, lovers of arithmetic, implemented lengthy scales on their most sophisticated instrument, the Qanun. But most instruments do not allow so much subtlety and less so the human voice.

After 12 fifths, we checked in Figure 1.1. that the difference is 0.0136 (corresponding to the Pythagorean comma); it is slightly reduced after 41 fifths, and even more so after 53, as discovered by King-Fang. Khien-Lon-Ki, the 5th Century AD astronomer, discovered the similar cases of 306 and 359 fifths. Due to the computer, we can now propose the prodigious scale of 665 notes, almost perfect, but perfectly useless.

Still in China, another system was then conceived. Since the scale must have 12 notes, why not give the value of the 12th root of 2 directly to the halftone? That is already the idea of the equal temperament, where the comma is equally shared among all intervals. In the 18th Century, when the Jesuit Joseph-Marie Amiot visited the empire, the Chinese were able to evaluate the terms of the geometric progression of ratio with an accuracy of four decimals and to apply it perfectly to bamboo tubes. With the yellow bell, whose dimensions and composition were perfectly defined, they also possessed a universal diapason.

Figure 1.1.King-Fang experience

The problem with equal temperament is that all intervals are – very slightly – false. The more serious problem with all of the theories just discussed is that they are ultimately more arithmetic than musical: Plato already reproached the Pythagoreans for their excessive mathematization of music. In the mid-4th Century BC, Aristoxenus of Tarentum, using the same reproach, founded his own definition of the musical scale on another principle, more physical: the harmonic structure of sound.

So far, we have assumed two elementary mathematical entities: pure sounds (having a single frequency) and simple intervals. The problem is that their physical realization is not directly present in nature.

Most natural sounds – thunder, waterfall, etc. – have such a rich and disordered frequency structure that our ears cannot even characterize a dominant pitch. However, few sounds do have pitch: the chirping of birds, the wind through the reeds, etc., and the most important fact is that human industry can easily imitate these sounds and improve them.

The more refined sounds of music are not pure: they always cover a wide range of frequencies. However, they are periodic, i.e. they repeat their pattern in regular intervals. Fourier theory shows that any periodic waveform is decomposed into a series of harmonics that are integer multiples of the fundamental frequency. In a periodic sound, the fundamental frequency gives the root note, while the relative intensities of harmonics produce the timbre. All these frequencies are merged into a single sound, with its particular timbre, but, unlike what happens with mixing colors, mixed frequencies are perceived individually. Two physical phenomena reinforce this perception: resonance and beating.

If we produce a strong, short sound beside a harp, each string of the harp whose natural frequency is equal to any frequency emitted by the sound comes into sympathetic resonance. The vibrations of the harp inform us about the harmonic composition of the sound.

When two sounds are close in pitch but not identical, the difference in frequency generates a beating: the intensity varies, like in a tremolo, as the sounds alternatively interfere constructively and destructively. This phenomenon is used to tune instruments. In a fifth out of tune, for example, the second harmonic of the lower note will produce a beating with the top note. Setting the fifth, this beating will slow down until it disappears.

The physical properties of harmonics, resonance and beating together explain why the human ear appreciates simple intervals and why it is extremely sensitive to imperfections. But something happens.

Here is the list of the first harmonics of a periodic sound of any frequency f, with the intervals between each harmonic and the fundamental:

Harmonic

Ratio

Interval

f

1

fundamental

2f

2

octave

3f

3/2

fifth

4f

2

octave

5f

5/4

major third

6f

3/2

fifth

7f

7/4

~ diminished/minor seventh

First observation: the first six harmonics, which are consonant, form a major chord (major third, fifth and octave). When multiple instruments play this chord, they reproduce the harmonic structure of periodic sound. It is also when the triple series of harmonics best overlaps, hardly generating friction. Among all the chords of three different sounds, this is the most consonant.

Second observation: the order of the harmonics is not exactly the same as that of the simple intervals previously established: here, the major third (ratio 5/4) appears long before the fourth (ratio 4/3). In medieval polyphony, chords of fourths were considered to be very consonant. In the Renaissance, they disappeared completely, being substituted by chords of fifths and thirds. With little exaggeration, we can say that with this mutation, music went from being essentially arithmetic to essentially harmonic.

Third observation: Aristoxenus theory, based on the harmonic structure of sound, explains much better than that of the Pythagoreans, based on simple intervals, why certain intervals naturally sound consonant to the ears.

The double generation of the scale by arithmetic and physics properties is not sufficient to give rise to a composition system based on the consonant/dissonant contrast. On the one hand, the harmonic structure explains consonances, but remains static, because it cannot raise a logical development of complementary dissonances. Moreover, the progression by fifths offers a full 12 semitones scale, but this scale is also very static because it remains identical whatever the note from which we start to play it.

To make the harmony dynamic, it is necessary to reduce the scale in order to deploy two types of intervals instead of the single semitone. This is achieved by limiting the progression of fifths to five notes (e.g. F – C – G – D – A) or seven notes (e.g. F – C – G – D – A – E – H). Once the notes within the octave are rearranged, we obtain, in the first case, the pentatonic scale, which consists of only two tones and minor thirds, and in the second case, the heptatonic scale, which consists only of tones and two semitones.

The first system, which was adopted in China, India, and probably in pre-Columbian America, offers five modes of playing the scale because, according to the starting note, there are five possible positions for the minor thirds within the octave, characterizing the five different modes:

Similarly, the second system, which was adopted in India, Chaldea, Greece and then across Europe, offers seven modes of playing the scale, with seven possible positions for the semitones E – F and B – C. The Greeks considered each of these modes as capable of arousing a particular emotion (happy, funeral, pompous, etc.).

Incidentally, the description of the heptatonic scale allows us to explain the names of the intervals that we have been using throughout this chapter. European languages have names for only seven notes, the five other ones are described using alterations (F sharp or E flat, for example). The interval between two consecutive notes of the heptatonic scale is called second. It contains one tone (major second: C – D) or one semitone (minor second: E – F). The other names correspond to the notes to be counted in the interval, including the ends. So, C – E is a third (C, D, E) and C – G is a fifth (C, D, E, F, G). A third is called major when it contains two tones (C – E) and minor when it contains one tone and one semitone (D – F). With alterations, augmented intervals can be formed (as the fourth C – F #) so as diminished ones (as the fourth C # – F).

1.2. The tonal system

In a 1962 paper, collected in the book Music, Architecture, Xenakis [XEN 71] offered a very useful distinction for understanding the systems of musical composition, and particularly the tonal system. He distinguished between:

– an “out-of-time” structure (relations and operations that are independent of time);

– a “temporal algebra” (sound events forming a set of durations);

– a “within-time” structure (correspondence between the previous two).

The composition system that Greeks left to Byzantines and Arabs was essentially an out-of-time structure, with highly developed and subtle frequency scales, formed by a large number of intervals. Instead, by their aspiration to develop polyphonic music, composers of medieval Europe sacrificed all these subtleties and moved to a within-time structure characterized by harmony and counterpoint.

Under the rule of the modal system, until the early 17th Century, they only used the seven modes of the heptatonic scale constructed with tones and semitones.

To enter the polyphonic reasoning, let us consider the work of Thomas Tallis (Spem in Allium, 1573), written for 40 voices divided into eight choirs. The composer Luigi Nono wrote: “with eight choirs, you can only use the tonic and dominant. But the extraordinary thing is that, with this presumed reduction of possibilities, Tallis explores space, makes living space, makes the space itself become cantabile” [NON 93].

Starting from an arbitrary note that we can consider as the ground of the harmonic construction, and therefore called tonic, we ascend by the interval of a fifth to a second note, which is called dominant because it is the second most important note of the scale composed on the tonic: the dominant and the tonic form together a fifth, the most consonant among all chords of two sounds (the octave is considered as an identifier interval).