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</o:shapelayout></xml><![endif]--></head><body lang=EN-GB link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US'>Martin:<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D;mso-fareast-language:EN-US'>No probs re mass and gravity. It’s good to talk. And if there was never any confusion, and we all agreed about everything, whatever would we talk about?<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>David:<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>I don’t have any issues with the horn torus or spindle-sphere torus. In a previous email I’ve referred to “inflating” the ring torus through these stages. See Adrian Rossiter’s <a href="http://www.antiprism.com/album/860_tori/index.html">antiprism</a> for torus animations. See Martin’s picture below? Look at the cut end, it’s a bit like a slice of onion. Mentally add more onion rings around those that are already there, and the torus gets more and more spherical. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><img border=0 width=328 height=272 id="Picture_x0020_2" src="cid:image003.jpg@01D0831D.57AAB580"></span><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>There is no charge at the centre. The charge is the twist, the winding, the knottedness, the vorticity. Undo it with the opposite twist, and instead of two field variations going round at c looking like standing fieldall, you’ve got two field variations moving linearly at c.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>Chip:<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>Is there any chance you could animate a wave in space, like this?<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><img border=0 width=640 height=157 id="Picture_x0020_3" src="cid:image004.gif@01D0831C.3AB4E140"></span><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>It’s a bit like a wave in a rubber mat, or a seismic wave. Only this is a wave of electromagnetic four-potential. The spatial derivative of this gives the sinusoidal electric waveform, and the time-derivative gives the sinusoidal magnetic waveform. See the horizontal lines? Where the tilt is steepest the electric sine wave is highest. Where the tilting is fastest is where the magnetic sine wave is highest. L</span><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#666666'>ook at </span><u><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#0033CC'><a href="http://en.wikipedia.org/wiki/Electromagnetic_radiation"><span style='color:#0033CC'>electromagnetic radiation</span></a></span></u><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#666666'> on Wiki. Read the <a href="http://en.wikipedia.org/wiki/Electromagnetic_radiation#Derivation_from_electromagnetic_theory">Derivation From Electromagnetic Theory</a> </span><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F4E79;mso-style-textfill-fill-color:#1F4E79;mso-style-textfill-fill-alpha:100.0%'>section, and you’ll see this: <em><span style='font-family:"Calibri",sans-serif'>“the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time”</span></em>. </span><span lang=EN-US style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F4E79;mso-style-textfill-fill-color:#1F4E79;mso-style-textfill-fill-alpha:100.0%'> </span><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F4E79;mso-style-textfill-fill-color:#1F4E79;mso-style-textfill-fill-alpha:100.0%'>After that I’d be very interested in an animation where two such waves pass each other and displace each other’s path, so much so that each wave ends up moving through itself. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>Regards<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D'>John D<o:p></o:p></span></p></div></body></html>