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84-540: Dominique Pierre Perrin (b. 1946) is a French mathematician and theoretical computer scientist known for his contributions to coding theory and to combinatorics on words . He is a professor of the University of Marne-la-Vallée and currently serves as the President of ESIEE Paris . Perrin earned his PhD from Paris 7 University in 1975. In his early career, he was a CNRS researcher (1970–1977) and taught at

168-430: A broad band, or extremely low reflectivity at a single wavelength. Constructive interference in thin films can create a strong reflection of light in a range of wavelengths, which can be narrow or broad depending on the design of the coating. These films are used to make dielectric mirrors , interference filters , heat reflectors , and filters for colour separation in colour television cameras. This interference effect

252-619: A changing index of refraction; this principle allows for lenses and the focusing of light. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction n 1 and another medium with index of refraction n 2 . In such situations, Snell's Law describes the resulting deflection of the light ray: n 1 sin ⁡ θ 1 = n 2 sin ⁡ θ 2 {\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}} where θ 1 and θ 2 are

336-399: A converging lens has positive focal length, while a diverging lens has negative focal length. Smaller focal length indicates that the lens has a stronger converging or diverging effect. The focal length of a simple lens in air is given by the lensmaker's equation . Ray tracing can be used to show how images are formed by a lens. For a thin lens in air, the location of the image is given by

420-477: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include

504-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while

588-788: A political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages

672-477: A single scalar quantity to represent the electric field of the light wave, rather than using a vector model with orthogonal electric and magnetic vectors. The Huygens–Fresnel equation is one such model. This was derived empirically by Fresnel in 1815, based on Huygens' hypothesis that each point on a wavefront generates a secondary spherical wavefront, which Fresnel combined with the principle of superposition of waves. The Kirchhoff diffraction equation , which

756-522: A single point on the image, while chromatic aberration occurs because the index of refraction of the lens varies with the wavelength of the light. In physical optics, light is considered to propagate as waves. This model predicts phenomena such as interference and diffraction, which are not explained by geometric optics. The speed of light waves in air is approximately 3.0×10  m/s (exactly 299,792,458 m/s in vacuum ). The wavelength of visible light waves varies between 400 and 700 nm, but

840-437: A spectrum. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton. Some media have an index of refraction which varies gradually with position and, therefore, light rays in the medium are curved. This effect is responsible for mirages seen on hot days: a change in index of refraction air with height causes light rays to bend, creating the appearance of specular reflections in

924-465: A thickness of one-fourth the wavelength of incident light. The reflected wave from the top of the film and the reflected wave from the film/material interface are then exactly 180° out of phase, causing destructive interference. The waves are only exactly out of phase for one wavelength, which would typically be chosen to be near the centre of the visible spectrum, around 550 nm. More complex designs using multiple layers can achieve low reflectivity over

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1008-476: A variety of technologies and everyday objects, including mirrors , lenses , telescopes , microscopes , lasers , and fibre optics . Optics began with the development of lenses by the ancient Egyptians and Mesopotamians . The earliest known lenses, made from polished crystal , often quartz , date from as early as 2000 BC from Crete (Archaeological Museum of Heraclion, Greece). Lenses from Rhodes date around 700 BC, as do Assyrian lenses such as

1092-525: A wide range of scientific topics, and discussed light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, and a theology of light, basing it on the works of Aristotle and Platonism. Grosseteste's most famous disciple, Roger Bacon , wrote works citing a wide range of recently translated optical and philosophical works, including those of Alhazen, Aristotle, Avicenna , Averroes , Euclid, al-Kindi, Ptolemy, Tideus, and Constantine

1176-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics

1260-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics

1344-427: Is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams. This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics. In the absence of nonlinear effects,

1428-486: Is considered to travel in straight lines, while in physical optics, light is considered as an electromagnetic wave. Geometrical optics can be viewed as an approximation of physical optics that applies when the wavelength of the light used is much smaller than the size of the optical elements in the system being modelled. Geometrical optics , or ray optics , describes the propagation of light in terms of "rays" which travel in straight lines, and whose paths are governed by

1512-491: Is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of the application of Huygens–Fresnel principle can be found in the articles on diffraction and Fraunhofer diffraction . More rigorous models, involving the modelling of both electric and magnetic fields of the light wave, are required when dealing with materials whose electric and magnetic properties affect

1596-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing

1680-409: Is to the lens, the further the image is from the lens. With diverging lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at a spot one focal length in front of the lens. This is the lens's front focal point. Rays from an object at a finite distance are associated with a virtual image that is closer to the lens than the focal point, and on

1764-406: Is usually done using simplified models. The most common of these, geometric optics , treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically,

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1848-480: The Book of Optics ( Kitab al-manazir ) in which he explored reflection and refraction and proposed a new system for explaining vision and light based on observation and experiment. He rejected the "emission theory" of Ptolemaic optics with its rays being emitted by the eye, and instead put forward the idea that light reflected in all directions in straight lines from all points of the objects being viewed and then entered

1932-683: The Lothaire group of mathematicians that developed the foundations of combinatorics on words. He has co-authored three scientific monographs: "Theory of Codes" (1985), "Codes and Automata" (2009), and "Infinite Words" (2004), as well as the three Lothaire books . Perrin has published around 50 research articles in formal language theory . Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of

2016-617: The Nimrud lens . The ancient Romans and Greeks filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient Greek and Indian philosophers, and the development of geometrical optics in the Greco-Roman world . The word optics comes from the ancient Greek word ὀπτική , optikē ' appearance, look ' . Greek philosophy on optics broke down into two opposing theories on how vision worked,

2100-653: The Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c.  AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of

2184-676: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of

2268-399: The University of Chile (1972–1973). Later, he worked as a professor at the University of Rouen (1977–1983), Paris 7 University (1983–1993), and École Polytechnique (1982–2002). Since 1993, Perrin is a professor at the University of Marne-la-Vallée , and since 2004, he is the President of ESIEE Paris . Perrin is a member of Academia Europaea since 1989. Perrin has been a member of

2352-454: The emission theory , the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus . Some hundred years later, Euclid (4th–3rd century BC) wrote a treatise entitled Optics where he linked vision to geometry , creating geometrical optics . He based his work on Plato's emission theory wherein he described

2436-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of

2520-468: The intromission theory and the emission theory . The intromission approach saw vision as coming from objects casting off copies of themselves (called eidola) that were captured by the eye. With many propagators including Democritus , Epicurus , Aristotle and their followers, this theory seems to have some contact with modern theories of what vision really is, but it remained only speculation lacking any experimental foundation. Plato first articulated

2604-448: The superposition principle , which is a wave-like property not predicted by Newton's corpuscle theory. This work led to a theory of diffraction for light and opened an entire area of study in physical optics. Wave optics was successfully unified with electromagnetic theory by James Clerk Maxwell in the 1860s. The next development in optical theory came in 1899 when Max Planck correctly modelled blackbody radiation by assuming that

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2688-466: The surface normal , a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays and the normal lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection . For flat mirrors , the law of reflection implies that images of objects are upright and

2772-567: The African . Bacon was able to use parts of glass spheres as magnifying glasses to demonstrate that light reflects from objects rather than being released from them. The first wearable eyeglasses were invented in Italy around 1286. This was the start of the optical industry of grinding and polishing lenses for these "spectacles", first in Venice and Florence in the thirteenth century, and later in

2856-548: The Huygens–Fresnel principle states that every point of a wavefront is associated with the production of a new disturbance, it is possible for a wavefront to interfere with itself constructively or destructively at different locations producing bright and dark fringes in regular and predictable patterns. Interferometry is the science of measuring these patterns, usually as a means of making precise determinations of distances or angular resolutions . The Michelson interferometer

2940-586: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became

3024-484: The amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice versa. This results in destructive interference and a decrease in the amplitude of the wave, which for light is associated with a dimming of the waveform at that location. See below for an illustration of this effect. Since

3108-552: The angle of incidence. Plutarch (1st–2nd century AD) described multiple reflections on spherical mirrors and discussed the creation of magnified and reduced images, both real and imaginary, including the case of chirality of the images. During the Middle Ages , Greek ideas about optics were resurrected and extended by writers in the Muslim world . One of the earliest of these was Al-Kindi ( c.  801 –873) who wrote on

3192-435: The angles between the normal (to the interface) and the incident and refracted waves, respectively. The index of refraction of a medium is related to the speed, v , of light in that medium by n = c / v , {\displaystyle n=c/v,} where c is the speed of light in vacuum . Snell's Law can be used to predict the deflection of light rays as they pass through linear media as long as

3276-458: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Optics Most optical phenomena can be accounted for by using the classical electromagnetic description of light, however complete electromagnetic descriptions of light are often difficult to apply in practice. Practical optics

3360-449: The distance (as if on the surface of a pool of water). Optical materials with varying indexes of refraction are called gradient-index (GRIN) materials. Such materials are used to make gradient-index optics . For light rays travelling from a material with a high index of refraction to a material with a low index of refraction, Snell's law predicts that there is no θ 2 when θ 1 is large. In this case, no transmission occurs; all

3444-500: The earliest known mathematicians was Thales of Miletus ( c.  624  – c.  546 BC ); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c.  582  – c.  507 BC ) established

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3528-426: The exchange of energy between light and matter only occurred in discrete amounts he called quanta . In 1905, Albert Einstein published the theory of the photoelectric effect that firmly established the quantization of light itself. In 1913, Niels Bohr showed that atoms could only emit discrete amounts of energy, thus explaining the discrete lines seen in emission and absorption spectra . The understanding of

3612-586: The eye, although he was unable to correctly explain how the eye captured the rays. Alhazen's work was largely ignored in the Arabic world but it was anonymously translated into Latin around 1200 A.D. and further summarised and expanded on by the Polish monk Witelo making it a standard text on optics in Europe for the next 400 years. In the 13th century in medieval Europe, English bishop Robert Grosseteste wrote on

3696-535: The feud between the two lasted until Hooke's death. In 1704, Newton published Opticks and, at the time, partly because of his success in other areas of physics, he was generally considered to be the victor in the debate over the nature of light. Newtonian optics was generally accepted until the early 19th century when Thomas Young and Augustin-Jean Fresnel conducted experiments on the interference of light that firmly established light's wave nature. Young's famous double slit experiment showed that light followed

3780-442: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of

3864-474: The focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration . Curved mirrors can form images with a magnification greater than or less than one, and the magnification can be negative, indicating that the image is inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen. Refraction occurs when light travels through an area of space that has

3948-1060: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,

4032-411: The gloss of surfaces such as mirrors, which reflect light in a simple, predictable way. This allows for the production of reflected images that can be associated with an actual ( real ) or extrapolated ( virtual ) location in space. Diffuse reflection describes non-glossy materials, such as paper or rock. The reflections from these surfaces can only be described statistically, with the exact distribution of

4116-633: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"

4200-416: The incident rays came. This is called retroreflection . Mirrors with curved surfaces can be modelled by ray tracing and using the law of reflection at each point on the surface. For mirrors with parabolic surfaces , parallel rays incident on the mirror produce reflected rays that converge at a common focus . Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing

4284-418: The indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. In most materials, the index of refraction varies with the frequency of the light, known as dispersion . Taking this into account, Snell's Law can be used to predict how a prism will disperse light into

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4368-436: The interaction between light and matter that followed from these developments not only formed the basis of quantum optics but also was crucial for the development of quantum mechanics as a whole. The ultimate culmination, the theory of quantum electrodynamics , explains all optics and electromagnetic processes in general as the result of the exchange of real and virtual photons. Quantum optics gained practical importance with

4452-426: The interaction of light with the material. For instance, the behaviour of a light wave interacting with a metal surface is quite different from what happens when it interacts with a dielectric material. A vector model must also be used to model polarised light. Numerical modeling techniques such as the finite element method , the boundary element method and the transmission-line matrix method can be used to model

4536-483: The invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle making centres in the Netherlands. In the early 17th century, Johannes Kepler expanded on geometric optics in his writings, covering lenses, reflection by flat and curved mirrors, the principles of pinhole cameras , inverse-square law governing the intensity of light, and

4620-491: The inventions of the maser in 1953 and of the laser in 1960. Following the work of Paul Dirac in quantum field theory , George Sudarshan , Roy J. Glauber , and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light. Classical optics is divided into two main branches: geometrical (or ray) optics and physical (or wave) optics. In geometrical optics, light

4704-580: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at

4788-474: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to

4872-504: The laws of reflection and refraction at interfaces between different media. These laws were discovered empirically as far back as 984 AD and have been used in the design of optical components and instruments from then until the present day. They can be summarised as follows: When a ray of light hits the boundary between two transparent materials, it is divided into a reflected and a refracted ray. The laws of reflection and refraction can be derived from Fermat's principle which states that

4956-449: The light is modelled as a collection of particles called " photons ". Quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy , various engineering fields, photography , and medicine (particularly ophthalmology and optometry , in which it is called physiological optics). Practical applications of optics are found in

5040-422: The light is reflected. This phenomenon is called total internal reflection and allows for fibre optics technology. As light travels down an optical fibre, it undergoes total internal reflection allowing for essentially no light to be lost over the length of the cable. A device that produces converging or diverging light rays due to refraction is known as a lens . Lenses are characterized by their focal length :

5124-443: The mathematical rules of perspective and described the effects of refraction qualitatively, although he questioned that a beam of light from the eye could instantaneously light up the stars every time someone blinked. Euclid stated the principle of shortest trajectory of light, and considered multiple reflections on flat and spherical mirrors. Ptolemy , in his treatise Optics , held an extramission-intromission theory of vision:

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5208-494: The merits of Aristotelian and Euclidean ideas of optics, favouring the emission theory since it could better quantify optical phenomena. In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law. He used this law to compute optimum shapes for lenses and curved mirrors . In the early 11th century, Alhazen (Ibn al-Haytham) wrote

5292-405: The object and image distances are positive if the object and image are on opposite sides of the lens. Incoming parallel rays are focused by a converging lens onto a spot one focal length from the lens, on the far side of the lens. This is called the rear focal point of the lens. Rays from an object at a finite distance are focused further from the lens than the focal distance; the closer the object

5376-401: The optical explanations of astronomical phenomena such as lunar and solar eclipses and astronomical parallax . He was also able to correctly deduce the role of the retina as the actual organ that recorded images, finally being able to scientifically quantify the effects of different types of lenses that spectacle makers had been observing over the previous 300 years. After the invention of

5460-676: The path taken between two points by a ray of light is the path that can be traversed in the least time. Geometric optics is often simplified by making the paraxial approximation , or "small angle approximation". The mathematical behaviour then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing , which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications . Reflections can be divided into two types: specular reflection and diffuse reflection . Specular reflection describes

5544-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in

5628-511: The propagation of light in systems which cannot be solved analytically. Such models are computationally demanding and are normally only used to solve small-scale problems that require accuracy beyond that which can be achieved with analytical solutions. All of the results from geometrical optics can be recovered using the techniques of Fourier optics which apply many of the same mathematical and analytical techniques used in acoustic engineering and signal processing . Gaussian beam propagation

5712-416: The ray-based model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that light waves were in fact electromagnetic radiation. Some phenomena depend on light having both wave-like and particle-like properties . Explanation of these effects requires quantum mechanics . When considering light's particle-like properties,

5796-416: The rays (or flux) from the eye formed a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. He summarized much of Euclid and went on to describe a way to measure the angle of refraction , though he failed to notice the empirical relationship between it and

5880-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate

5964-423: The reflected light depending on the microscopic structure of the material. Many diffuse reflectors are described or can be approximated by Lambert's cosine law , which describes surfaces that have equal luminance when viewed from any angle. Glossy surfaces can give both specular and diffuse reflection. In specular reflection, the direction of the reflected ray is determined by the angle the incident ray makes with

6048-415: The same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. The law also implies that mirror images are parity inverted, which we perceive as a left-right inversion. Images formed from reflection in two (or any even number of) mirrors are not parity inverted. Corner reflectors produce reflected rays that travel back in the direction from which

6132-407: The same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens. As with mirrors, upright images produced by a single lens are virtual, while inverted images are real. Lenses suffer from aberrations that distort images. Monochromatic aberrations occur because the geometry of the lens does not perfectly direct rays from each object point to

6216-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced

6300-405: The simple equation 1 S 1 + 1 S 2 = 1 f , {\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}},} where S 1 is the distance from the object to the lens, θ 2 is the distance from the lens to the image, and f is the focal length of the lens. In the sign convention used here,

6384-464: The spectacle making centres in both the Netherlands and Germany. Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses rather than using the rudimentary optical theory of the day (theory which for the most part could not even adequately explain how spectacles worked). This practical development, mastery, and experimentation with lenses led directly to

6468-444: The superposition principle can be used to predict the shape of interacting waveforms through the simple addition of the disturbances. This interaction of waves to produce a resulting pattern is generally termed "interference" and can result in a variety of outcomes. If two waves of the same wavelength and frequency are in phase , both the wave crests and wave troughs align. This results in constructive interference and an increase in

6552-472: The telescope, Kepler set out the theoretical basis on how they worked and described an improved version, known as the Keplerian telescope , using two convex lenses to produce higher magnification. Optical theory progressed in the mid-17th century with treatises written by philosopher René Descartes , which explained a variety of optical phenomena including reflection and refraction by assuming that light

6636-440: The term "light" is also often applied to infrared (0.7–300 μm) and ultraviolet radiation (10–400 nm). The wave model can be used to make predictions about how an optical system will behave without requiring an explanation of what is "waving" in what medium. Until the middle of the 19th century, most physicists believed in an "ethereal" medium in which the light disturbance propagated. The existence of electromagnetic waves

6720-943: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in

6804-400: Was a famous instrument which used interference effects to accurately measure the speed of light. The appearance of thin films and coatings is directly affected by interference effects. Antireflective coatings use destructive interference to reduce the reflectivity of the surfaces they coat, and can be used to minimise glare and unwanted reflections. The simplest case is a single layer with

6888-540: Was emitted by objects which produced it. This differed substantively from the ancient Greek emission theory. In the late 1660s and early 1670s, Isaac Newton expanded Descartes's ideas into a corpuscle theory of light , famously determining that white light was a mix of colours that can be separated into its component parts with a prism . In 1690, Christiaan Huygens proposed a wave theory for light based on suggestions that had been made by Robert Hooke in 1664. Hooke himself publicly criticised Newton's theories of light and

6972-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support

7056-467: Was predicted in 1865 by Maxwell's equations . These waves propagate at the speed of light and have varying electric and magnetic fields which are orthogonal to one another, and also to the direction of propagation of the waves. Light waves are now generally treated as electromagnetic waves except when quantum mechanical effects have to be considered. Many simplified approximations are available for analysing and designing optical systems. Most of these use

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