Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a second-order change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam. First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig. 1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves. If points A and B are given, a wavefront expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not. If the wavefront reaches point B, it sweeps not only the ray path(s) from A to B, but also an infinitude of nearby paths with the same endpoints. Fermat's principle describes any ray that happens to reach point B; there is no implication that the ray "knew" the quickest path or "intended" to take that path. For the purpose of comparing traversal times, the time from one point to the next nominated point is taken as if the first point were a point-source. Without this condition, the traversal time would be ambiguous; for example, if the propagation time from P to P′ were reckoned from an arbitrary wavefront W containing P (Fig. 2), that time could be made arbitrarily small by suitably angling the wavefront. Treating a point on the path as a source is the minimum requirement of Huygens' principle, and is part of the of Fermat's principle. But it that the geometric construction by which Huygens tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle. Hence all the conclusions that Huygens drew from that construction — including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of "Iceland crystal" (calcite) — are also consequences of Fermat's principle. (Wikipedia).
Theory of numbers: Fermat's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Fermat's theorem a^p = a mod p. We then define the order of a number mod p and use Fermat's theorem to show the order of a divides p-1. We apply this to testing some Fermat and Mersenne numbers to se
From playlist Theory of numbers
How to prove Fermat's Last Theorem in under 7 seconds
How to prove Fermat's Last Theorem in under 7 seconds
From playlist My Maths Videos
PHYS 201 | Fermat's Principle 1 - The Path of Light
Fermat had another view on how light propgates, and it was motivated by the wisdom of God. I don't know much about God but I would think he/she is a little too busy to be worrying about stuff like that.
From playlist PHYS 201 | Geometrical Optics
The number theory revival | Math History | NJ Wildberger
After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat
From playlist MathHistory: A course in the History of Mathematics
Primality (1 of 2: Fermat's Test)
From playlist Cryptography
Number Theory | A very special case of Fermat's Last Theorem
We prove a very simple case of Fermat's Last Theorem. Interestingly, this case is fairly easy to prove which highlights the allure of the theorem as a whole -- especially given the fact that much of modern number theory was developed as part of the program that ended in the full proof. ht
From playlist Number Theory
A Short Course in Algebra and Number Theory - Fermat's little theorem and primes
To supplement a course taught at The University of Queensland's School of Mathematics and Physics I present a very brief summary of algebra and number theory for those students who need to quickly refresh that material or fill in some gaps in their understanding. This is the fifth lectur
From playlist A Short Course in Algebra and Number Theory
PHYS 201 | Fermat's Principle 3 - From Fermat to Snell
Fermat's ideas about light even jive with Snell's Law of refraction. Are Fermat and Huygens's ideas the same at some deep geometrical level? Probably. But it's not really physics so let's move on.
From playlist PHYS 201 | Geometrical Optics
Quantum Mechanics and the Principle of Least Time
In this short video I would like to tell you about the pioneering work of Pierre de Fermat, who discovered that light is the laziest object in the universe, always preferring to take the path that minimises the amount of time spent travelling between two points. But perhaps what is even mo
From playlist General Physics
L'équation du soir 6/6 - Roland Lehoucq - Univers Convergents 2018
Extrait de la séance sur le film "Premier Contact". Nouvelle année, nouvelle formule ! A chaque séance, le parrain ou marraine présente une formule mathématique en lien avec le film projeté durant la séance. Pour cette séance Roland Lehoucq nous présente une équation lumineuse ! Vous po
From playlist Ciné-Club Univers Convergents
Euler's and Fermat's last theorems, the Simpsons and CDC6600
NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) This video is about Fermat's last theorem and Euler's conjecture, a vast but not very well-known genera
From playlist Recent videos
Heptadecagon and Fermat Primes (the math bit) - Numberphile
Main (previous) video: http://youtu.be/87uo2TPrsl8 David Eisenbud from MSRI on the math behind the 17-gon and other constructible polygons. NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com
From playlist David Eisenbud on Numberphile
Oxford Mathematics Public Lectures: James Maynard - Prime Time: How simple questions about prime numbers affect us all. Numbers are fascinating, crucial and ubiquitous. The trouble is, we don't know that much about them. James Maynard, one of the leading researchers in the field explains
From playlist Oxford Mathematics Public Lectures
Is ACTION The Most Fundamental Property in Physics?
Learn More About NordVPN at: https://nordvpn.com/spacetime It’s about time we discussed an obscure concept in physics that may be more fundamental than energy and entropy and perhaps time itself. That’s right - the time has come for Action. Sign Up on Patreon to get access to the Space
From playlist The Standard Model Lagrangian Playlist
A lot of optical illusions can be explained by Fermat's principle of least time, but why does light obey it? On a fundamental level, it all comes down to quantum mechanics, specifically quantum optics, where we use the famous "Feynman path integral formulation" to explain light through pho
From playlist Optics and Light
The Bridges to Fermat's Last Theorem - Numberphile
Ken Ribet - a key player in the solution to Fermat's Last Theorem - gives a taste of how real mathematics is done... piece by piece and by human beings. More links & stuff in full description below ↓↓↓ More Fermat (with Simon Singh): http://youtu.be/qiNcEguuFSA Even more Fermat (with Simo
From playlist Fermat's Last Theorem on Numberphile
Why does light bend when it enters glass?
The motion of light depends crucially on the material in which it is traveling. When light passes from one medium to another, an unexpected thing happens: the direction of travel changes. There are many explanations for why this happens and many of those explanations are wrong. In this
From playlist Videos by Don Lincoln
In this video we introduce Fermat's little theorem and give a proof using congruences. The content of this video corresponds to Section 7.2 of my book "Number Theory and Geometry" which you can find here: https://alozano.clas.uconn.edu/number-theory-and-geometry/
From playlist Number Theory and Geometry
Snell's law part 1: Ray optics derivation
In this video, I show how to derive the scalar form of Snell's law of refraction, starting from the postulates of ray optics.
From playlist Two-part series on Snell's law