<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Here’s a sidebar from my graphics texxtbook on shortest path:<div class=""><br class=""></div><div class=""><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">Sidebar 6.1 Finding the Angle of Reflection</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">Our derivation of the angle of reflection began with the principle that the angle of reflection equals the angle</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">of incidence. This principle was known to Euclid and Ptolemy 2000 years ago. Hero of Alexandria showed that</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">this path was the shortest from the source to the eye (see Exercise 6.13). If wasn’t until the early 17th century</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">that Fermat stated a more general version known as the principle of least time. However, none of these works</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">was based on the physics of light propagation but rather on the observation that light took the shortest (fastest)</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">path. Hence they did not explain why the angle of reflection equaled the angle of incidence nor why light didn’t</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">take multiple paths. It wasn’t until later in the 17th century that Huygens used the wave nature of light to</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">derive the angle of reflection.</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class=""><br class=""></div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">From the above, refraction follows as does the lifeguard problem.</div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class=""><br class=""></div><div style="margin: 0px; font-stretch: normal; line-height: normal;" class="">Ed</div><div style="margin: 0px; font-stretch: normal; font-size: 9.5px; line-height: normal;" class=""><br class=""></div><div style="margin: 0px; font-stretch: normal; font-size: 9.5px; line-height: normal;" class=""><br class=""></div><div class="">
<div>__________<br class=""><br class="">Ed Angel<br class=""><br class="">Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)<br class="">Professor Emeritus of Computer Science, University of New Mexico<br class=""><br class="">1017 Sierra Pinon<br class="">Santa Fe, NM 87501<br class="">505-984-0136 (home)<span class="Apple-tab-span" style="white-space: pre;"> </span> <span class="Apple-tab-span" style="white-space: pre;"> </span><a href="mailto:edward.angel@gmail.com" class="">edward.angel@gmail.com</a><br class="">505-453-4944 (cell) <span class="Apple-tab-span" style="white-space: pre;"> </span><span class="Apple-tab-span" style="white-space: pre;"> </span>http://www.cs.unm.edu/~angel<br class=""></div>
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<div><br class=""><blockquote type="cite" class=""><div class="">On Feb 16, 2020, at 10:33 PM, _ Bruno W <<a href="mailto:wbruno@gmail.com" class="">wbruno@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" class="">Not sure about the natural selection, but here are 2 small (possibly important) corrections.<div class="">1. The lifeguard is still to the left of the swimmer when the lifeguard hits the water. But by a much smaller distance than if</div><div class="">he had ran straight at the swimmer.</div><div class="">2. The most widely accepted explanation for the origin of the least action principle has to do with waves or paths, and doesn't</div><div class="">require the photon to know in advance where it should go.</div><div class="">In one version, the wave function of the photon goes out in all directions with a phase at any point given by exp(iS/hbar) where</div><div class="">S is the action. Because hbar is small, the waves oscillate rapidly and generally cancel out everywhere, except of course</div><div class="">when the derivative of the action is zero and the wave stops oscillating completely. This will happen at the path of least (or in some</div><div class="">cases greatest) action.</div><div class="">Feynman did this one better by saying you don't need waves, you integrate over all paths, and you get to include all kinds</div><div class="">of crazy paths, even ones that go in crazy loops and backwards in time and whatever. Once again the extreme action paths</div><div class="">are the only ones that survive in the classical limit (hbar -> 0).</div><div class="">Hope this helps?</div></div><br class=""><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Feb 16, 2020 at 10:05 PM <<a href="mailto:thompnickson2@gmail.com" class="">thompnickson2@gmail.com</a>> wrote:<br class=""></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div lang="EN-US" class=""><div class="gmail-m_4924567805405725653WordSection1"><p class="MsoNormal">Hi, Everybody, and Steve, and Miles<u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Please everybody forgive me for what I am about to garble. <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Steve Guerin and I have a long standing argument concerning the above question. He thinks my answer to the above question will be “natural selection.” His answer to that question is, is that as energy flows from high to low concentrations, it seeks, and finds the most efficient route. Today, steve presented to me a most astounding example, which I am sure most of you are familiar with, but which I had never quite grasped. It is illuminated by a metaphor. Imagine a beach and a lifeguard standing on the beach when a drowning swimmer calls out to the lifeguard’s right. Should the lifeguard run directly toward the drowning swimmer. No, because he can make a lot faster progress toward the swimmer while running on the beach than he can while swimming. So he should chose a path that minimizes his time to the swimmer, not the path directly toward the swimmer. That path, the path of least action, will carry him to the right of the swimmer until he reaches he water’s edge and starts to swim. Lifeguards have to be trained to do this, and lifeguards argue about which direction to head off under under which conditions. <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Well, light approaching boundary between air and water faces the same situation. And the stunning fact is that light finds the least action path. As I understand it, the light leaves the source in a direction that takes into account the boundary that it is approaching. PLEASE correct me if I have this wrong. Yet, of course, in this situation there is no trial and error. Photons of light just “know” how to do this. If this is true, I promise NEVER to yawn again when one of you is going on about quantum physics. <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Now, Steve, <span style="font-family:Wingdings" class="">è</span>IF<span style="font-family:Wingdings" class="">ç</span> I understood you, you also went on to describe ants behavior and lightening behavior as analogous processes that also find Least Action Pathways. And, I think you are also going to perhaps assert that a tornado is a structure that facilitates a least action pathway. Etc. But these are plainly “historical” processes. I.e, in all cases the process tries out options, and settles on the LAP. But with light, here is no history. Photons are like lifeguards who know instinctively what path to set out on to reach the swimmer with the least action <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Could I possibly have this right? Once we get this first bit straightened out, I have a question about its relation to natural selection. <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Nick <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal">Nicholas Thompson<u class=""></u><u class=""></u></p><p class="MsoNormal">Emeritus Professor of Ethology and Psychology<u class=""></u><u class=""></u></p><p class="MsoNormal">Clark University<u class=""></u><u class=""></u></p><p class="MsoNormal"><a href="mailto:ThompNickSon2@gmail.com" target="_blank" class="">ThompNickSon2@gmail.com</a><u class=""></u><u class=""></u></p><p class="MsoNormal"><a href="https://wordpress.clarku.edu/nthompson/" target="_blank" class="">https://wordpress.clarku.edu/nthompson/</a><u class=""></u><u class=""></u></p><p class="MsoNormal"> <u class=""></u><u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p><p class="MsoNormal"><u class=""></u> <u class=""></u></p></div></div>============================================================<br class="">
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