Beautiful Visual Maths Stuff

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	Beautiful Visual Maths Stuff
JohnWaterman
9-7-2008 5:02 PM
703 views
tags:
mathematics, topology, visual maths
7 Comments |
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9-7-2008 5:36 PM
Kelika
Beautiful! Another great clip - thanks!
9-7-2008 5:39 PM
Kelika
but I can't see the Vortex.
9-7-2008 5:49 PM
JohnWaterman
Look at the centre of the image and focus about four inches beyond (through) your monitor. (Smoking a joint also helps
)
9-7-2008 5:51 PM
JohnWaterman
Sorry. Rude of me.

Thanks and you're welcome.
9-7-2008 9:28 PM
cakebelly
I have never been able to see those damn hidden pictures (with bio-aids or not); still, great clip
9-8-2008 2:35 AM
tanyamm
Sorry John, I only smoke what's legal at the moment.
10-19-2008 7:20 PM
jimbo1000
relax the eyes and observe a particular detail with each eye, then let the two images float together so that the detail coincides. when they 'click' together wait for a few seconds for the three D effect to appear
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<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/29e3d16e-dea3-43d8-929a-990d0ee02f40/FD0F162A-03D1-48BF-BA0A-59698C5444EC/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://bugman123.com/Math/Math.html" href="http://bugman123.com/Math/Math.html" style="font-size: 11px;">bugman123.com</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><B><A target="new" href="http://mathworld.wolfram.com/BoySurface.html">Boy’s Surface</A> <A target="new" href="http://rsp.math.brandeis.edu/3D-XplorMath/Surface/boys_bryant-kusner/boys_bryant-kusner.html">(Bryant-Kusner Parametrization)</A> - new version: <A target="new" href="http://www.povray.org/">POV-Ray 3.6.1</A>, 6/20/06
old version: <A target="new" href="http://www.wolfram.com/products/mathematica/">Mathematica 4.2</A>, <A target="new" href="http://phong.informatik.uni-leipzig.de/~kuska/mathgl3dv3/index.htm">MathGL3d</A>, <A target="new" href="http://www.povray.org/">POV-Ray 3.6.1</A>, 5/24/05
</B><TABLE width="100%"><TBODY><TR><TD><A href="http://bugman123.com/Math/BoysSurface3-large.jpg"><IMG border="1" align="left" src="http://bugman123.com/Math/BoysSurface3.jpg" /></A>
 <A target="new" href="http://bugman123.com/Math/BoysSurface1.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/BoysSurface1.gif" /></A>
 <A target="new" href="http://bugman123.com/Math/BoysSurface2.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/BoysSurface2.gif" /></A>
 <B>This one-sided surface was first parametrized correctly by <A target="new" href="http://emsh.calarts.edu/~mathart/Tactile_Optiverse.html">Bernard Morin</A>. The animation looks like it’s turning inside-out, although technically that’s impossible because it only has one side! <A target="new" href="http://www.math.duke.edu/~bryant/">Robert Bryant</A> told me that the parameters (p,q) = (0,1) give this Willmore immersion of RP<SUP>2</SUP> a trilateral <A target="new" href="http://en.wikipedia.org/wiki/Symmetry">symmetry</A>. The parameters (p,q) = (1,0) should give bilateral <A target="new" href="http://en.wikipedia.org/wiki/Symmetry">symmetry</A>. Click <A href="http://bugman123.com/Math/BoysSurface.zip">here</A> to download some POV-Ray code for this image.</B></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html">
<B><A name="HopfFibration"></A>
<A target="new" href="http://en.wikipedia.org/wiki/Clifford_torus">Clifford Torus</A> - <A target="new" href="http://www.wolfram.com/products/mathematica/">Mathematica 4.2</A>, <A target="new" href="http://www.povray.org/">POV-Ray 3.6.1</A>, 8/12/08
</B><TABLE width="100%"><TBODY><TR><TD><A href="http://bugman123.com/Math/HopfFibration1-large.jpg"><IMG border="1" align="left" src="http://bugman123.com/Math/HopfFibration1.jpg" /></A>
 <A href="http://bugman123.com/Math/HopfFibration2-large.jpg"><IMG border="1" align="left" src="http://bugman123.com/Math/HopfFibration2.jpg" /></A>
 <A target="new" href="http://bugman123.com/Math/Rzw.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/Rzw.gif" /></A>
 <A target="new" href="http://bugman123.com/Math/Rxw.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/Rxw.gif" /></A>
 <B>The <A target="new" href="http://mathworld.wolfram.com/HopfMap.html">Hopf map</A> is a special transformation invented by <A target="new" href="http://en.wikipedia.org/wiki/Heinz_Hopf">Heinz Hopf</A> that maps to each point on the ordinary 3D sphere from a unique circle of points on the <A target="new" href="http://en.wikipedia.org/wiki/3-sphere">4D sphere</A></B></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><B>Inside the Flat <A target="new" href="http://en.wikipedia.org/wiki/Euclidean_geometry">(Euclidean)</A> <A target="new" href="http://en.wikipedia.org/wiki/Dodecahedron">Dodecahedron</A> - <A target="new" href="http://www.povray.org/">POV-Ray 3.6.1</A>, 2/12/07
</B><TABLE width="100%"><TBODY><TR><TD><DIV>
 <A href="http://bugman123.com/Math/Dodecahedron-inside-large.jpg"><IMG border="1" align="left" src="http://bugman123.com/Math/Dodecahedron-inside.jpg" /></A>
 <A target="new" href="http://bugman123.com/Math/Dodecahedron-inside.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/Dodecahedron-inside.gif" /></A>
 <B>Here is a dodecahedron viewed from the inside with flat mirrored walls.</B></DIV>
<B><FONT color="yellow">Links</FONT></B></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><B>7
</B><TABLE width="100%"><TBODY><TR><TD><A href="http://bugman123.com/Math/Topology-large.jpg"><IMG border="1" align="left" src="http://bugman123.com/Math/Topology.jpg" /></A>
 <A target="new" href="http://bugman123.com/Math/Topology.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/Topology.gif" /></A>
 <B>H</B></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/bugman123.com/img/B74F7759-9301-47D4-882B-75FD9426A36A" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content7.clipmarks.com/blog_cache/bugman123.com/img/931D83CB-1BE4-432A-BB65-BC94E21709C5" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/B4903A55-A919-40FC-A17F-5A1B32E40D78" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/bugman123.com/img/C07F9ECB-4E83-480A-B084-F8B514E0D94A" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/bugman123.com/img/DACF7EF0-5FD5-47E2-A65E-F4780F7A1CB7" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/28A1D1EB-0982-4AFD-856E-9DC5516F595E" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/bugman123.com/img/A33354FC-C240-4256-9A37-DDE6E996B893" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content7.clipmarks.com/blog_cache/bugman123.com/img/E5EDE89D-0271-4DBE-8BCE-B038BDB532F1" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/18E90E78-2CE2-4FB0-86F6-EE11634E8F45" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/bugman123.com/img/D324CFDD-B16E-4F09-90FC-B421894030E7" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content7.clipmarks.com/blog_cache/bugman123.com/img/95736B52-741E-4068-B4A0-4648AE4D9ECA" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/20BF4204-5583-4FBA-8C25-F5376DC6C9AC" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/bugman123.com/img/84D2AE69-1B98-4E81-9EEE-0497ECA2366C" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/bugman123.com/img/CFD44C1E-A423-4A9A-930F-CC128B90E459" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content7.clipmarks.com/blog_cache/bugman123.com/img/6FACF0E9-0147-47D8-A72B-7E9AB0369D2E" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/BB3D5D14-997D-4D1E-A24A-A91484665D9A" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/bugman123.com/img/1CB27130-5E25-4678-9AB4-06790618136B" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/bugman123.com/img/789836D2-6137-49A8-BD2E-41444C095F66" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/B880B505-2DE1-4C07-AFC0-0F52EE00C5A2" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><B>6
</B><TABLE width="100%"><TBODY><TR><TD><A href="http://bugman123.com/Math/Scherk-large.jpg"><IMG border="1" align="left" src="http://bugman123.com/Math/Scherk.jpg" /></A>
 <A target="new" href="http://bugman123.com/Math/Scherk.m1v"><IMG border="1" align="left" src="http://bugman123.com/Math/Scherk.gif" /></A>
 <B>T</B></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/bugman123.com/img/4A2C4A0F-51DB-467D-9E56-0A545FC02A61" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content7.clipmarks.com/blog_cache/bugman123.com/img/82A2F7CF-25BE-4766-8FE9-8C30DA2307AD" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/bugman123.com/img/C9F013F0-2C05-42BC-BD80-D1966C451EDD" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/bugman123.com/img/B8A4625D-12DB-4628-96CE-84F657E94DF6" alt="" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://bugman123.com/Math/Math.html">This 3D vortex image is hidden in the above pictur</blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/FD0F162A-03D1-48BF-BA0A-59698C5444EC/blog/" title="blog or email this clip"><img src="http://content6.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>
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