Fresnel Integral
Greetings all!<p>&gt;[tanek]: Why, though? Just because it’s pretty?<p>Isn&#x27;t that reason enough? &lt;g&gt;<p>Usually in Math&#x2F;Science&#x2F;Physics -- if something is pretty -- then usually it has a lot of purposes&#x2F;functions around it that are employed by Nature in various ways...  One example of this might be Phi, AKA The Golden Ratio: <a href="https:&#x2F;&#x2F;www.goldennumber.net&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.goldennumber.net&#x2F;</a><p>Both beautiful and purposeful!<p>But back to Fresnel Integrals!<p>&gt;[sidlls]: They have applications all over physics.<p>Indeed they do!<p>&gt;[exmadscientist]: Strongly agreed. Why are these more interesing than the Airy functions?<p>Well, I don&#x27;t know! &lt;g&gt;<p>I do know that prior to your comment, I didn&#x27;t know what Airy functions were!<p>So, here&#x27;s a link for the folks back home, just tuning in!:<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Airy_function" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Airy_function</a><p>Note the similar features (at least in part!) of the graph of Fresnel Integrals (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Fresnel_Integrals_(Unnormalised).svg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Fresnel_Integrals_(Unnorm...</a>) and Airy Functions (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Airy_Functions.svg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Airy_Functions.svg</a>).<p>It&#x27;s sort of like, if you took one graph and flipped it over 180 degrees, and cut out the part where the two functions start converging or start diverging, and ignore the units of measure, then the two graphs <i>almost</i> look similar...<p>So maybe they&#x27;re related in some way... maybe there&#x27;s something there...<p><i>Airy functions -- seem like they would be very interesting to study alongside Fresnel Integrals!</i><p>So, thank you for suggesting them!<p>The rest of my comments are for everyone in general -- so back to the Fresnel Integrals!<p>OK, so here&#x27;s what my intuition tells me...<p>We&#x27;ve all seen a hologram right?<p>OK, so a hologram is basically a 2D space -- masquerading as a 3D space.<p>First question (discovered easily by inverting&#x2F;reversing the above idea):<p><i>Is it possible to have a 3D space -- which is actually a 2D space, in disguise?</i><p>?<p>Sort of like a &quot;reverse-hologram&quot; -- for lack of better terminology...<p>In other words, is it possible, for an actual, real, 3D space in reality -- to have similar characteristics to a flat, 2D hologram?<p>?<p>Well, I for one don&#x27;t know! &lt;g&gt;<p>But what I do know is that <i>IF</i> such a thing is possible, <i>IF</i> it is (again, this is highly speculative!) -- then <i>IF</i> so -- then on the 2D holographic plane -- <i>ALL</i> of the 3D information -- must exist!<p>So now the question is (again, <i>IF</i> such a thing is possible, which is highly speculative!) -- <i>How&#x2F;where is that INFORMATION -- which is in the 3D view of reality -- encoded into the 2D plane, which is a basically hologram of that view of reality?</i><p>The answer for that question, is simply: &quot;I don&#x27;t know!!!&quot;<p>But Fresnel Integrals (and Airy Functions!) -- seem like they might have something to offer here -- some way of encoding extra (depth!) information -- into what is basically a 2D frame!<p>Now <i>IF</i> that&#x27;s true (and I should point out that we&#x27;re beyond speculative at this point -- we&#x27;re in <i>&quot;super-crackpot-they&#x27;ve-gone-to-plaid&quot;</i> land now, just for reference! &lt;g&gt;), then Fresnel Integrals do not just generically have applications all over physics -- but they&#x27;re also intimately, intimately related to such things as the double-slit experiment, gravity (&quot;A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration.&quot; -- well, this is information, and this information encoded in a 2D image would determine gravity (gravity = acceleration)), and of course, diffraction -- which is sort of like the measure of how fuzzy distant objects in a 2D image are, depending upon where the camera lens is <i>focused</i>...<p>So, that&#x27;s <i>why</i> Fresnel Integrals (and Airy Functions!) -- might be interesting for Future Physicists (and people who are not physicists -- but just like &quot;pretty&quot; things! &lt;g&gt;.  I should point out that I am in the latter camp! &lt;g&gt;)<p>Anyway, since you asked! &lt;g&gt;