<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"></head><body dir="auto"><div dir="auto">The coast of England is a classic example of a fractal, a self-similar object that has a broken dimension. A rectangle or a circle are two dimensional objects which have a one dimensional boundary, a line. The country of England has apparently the coast of England as a boundary which is neither a circle nor a rectangle. The more you zoom in, the more rugged it looks like. The coast has a fractal dimension higher than one, say 1.3 or 1.4. It is a measure how complicated the border is, how many points it contains.</div><div dir="auto"><br></div><div dir="auto">The Stokes Theorem in mathematics says roughly that everything that flows through the boundary, for instance people, leads to an increase of the population density inside. The mathematical formulation is based on integrals of manifolds and is quite complicated, but in essence this is what it says as far as I understand it.</div><div dir="auto"><br></div><div dir="auto">Your original question was about the boundary of a cloud. I think we can compare clouds to crowds at a rock concert. From the outside it looks like a unified entity, a flock of people or a cloud of droplets. From the inside you do not see the whole, just a mess of other people. If you fly with a plane through a cloud it is similar, you only see a kind of fog, and not the cloud as a whole. At the boundary there is a more or less sharp drop in the density of people in the crowd or droplets in the cloud. This sudden change in density can be considered as the boundary of the crowd or cloud. </div><div dir="auto"><br></div><div dir="auto">-J.</div><div dir="auto"><br></div><div id="composer_signature" dir="auto"><div style="font-size:85%;color:#575757" dir="auto"><br></div></div><div><br></div><div style="font-size:100%;color:#000000" dir="auto"><!-- originalMessage --><div>-------- Original message --------</div><div>From: thompnickson2@gmail.com </div><div>Date: 6/6/20 05:48 (GMT+01:00) </div><div>To: 'The Friday Morning Applied Complexity Coffee Group' <friam@redfish.com> </div><div>Subject: Re: [FRIAM] Manifold Clarification </div><div><br></div><div>So, Frank. Think of the coast of England/Scotland. It is infinitely indented. Anytime we draw a map of it, we enclose every point on that coast line and an area that is not within that coastline. So, wrap England in plastic film and pull the film as tight as we can. We have a shroud. Is there a mathematical name for that? OK, now, let the plastic be infinitely flexible, and let us suck all the air out of the space between the shroud and the coastline. What do we have now? Is there a mathematical name for that? </div></div><div class="WordSection1" dir="auto"><p class="MsoNormal">Let me give them both names. Let me call one a shroud and the other a super shrink wrap. I can imagine some mathematician, just for the hell of it, spending a life time working out what the area is between the shroud and the super shrink wrap. And then, having worked all that out, claiming, as do you, that none of these entities, shroud, ssw, or area between, exist in nature. They are mathematical objects, only. </p><p class="MsoNormal">Which is why Hywel used to say what he used to say about mathematics. Am I write about any of this? </p><p class="MsoNormal">Nick <o:p></o:p></p><p class="MsoNormal"><o:p> </o:p></p><p class="MsoNormal">Nicholas Thompson<o:p></o:p></p><p class="MsoNormal">Emeritus Professor of Ethology and Psychology<o:p></o:p></p><p class="MsoNormal">Clark University<o:p></o:p></p><p class="MsoNormal"><a href="mailto:ThompNickSon2@gmail.com">ThompNickSon2@gmail.com</a><o:p></o:p></p><p class="MsoNormal"><a href="https://wordpress.clarku.edu/nthompson/">https://wordpress.clarku.edu/nthompson/</a><o:p></o:p></p><p class="MsoNormal"> <o:p></o:p></p><p class="MsoNormal"><o:p> </o:p></p><p class="MsoNormal"><o:p> </o:p></p><div style="border:none;border-top:solid #E1E1E1 1.0pt;padding:3.0pt 0in 0in 0in"><p class="MsoNormal"><b>From:</b> Friam <friam-bounces@redfish.com> <b>On Behalf Of </b>Frank Wimberly<br><b>Sent:</b> Friday, June 5, 2020 8:07 PM<br><b>To:</b> The Friday Morning Applied Complexity Coffee Group <friam@redfish.com><br><b>Subject:</b> [FRIAM] Manifold Clarification<o:p></o:p></p></div><p class="MsoNormal"><o:p> </o:p></p><div><p class="MsoNormal">I said that no physical object is a manifold. This may be a better answer to Nick's question. The envelope of a cloud, if it could be defined, might be a manifold depending on cusps etc. Those might be handled by combining manifolds of different dimensions. This would not be a realizable project in my opinion.<o:p></o:p></p><div><p class="MsoNormal"><o:p> </o:p></p></div><div><p class="MsoNormal" style="margin-bottom:12.0pt">Frank<o:p></o:p></p><div><p class="MsoNormal">---<br>Frank C. Wimberly<br>140 Calle Ojo Feliz, <br>Santa Fe, NM 87505<br><br>505 670-9918<br>Santa Fe, NM<o:p></o:p></p></div></div></div></div></body></html>