{"id":329,"date":"2013-11-01T01:00:08","date_gmt":"2013-11-01T01:00:08","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=329"},"modified":"2015-07-29T00:54:18","modified_gmt":"2015-07-29T00:54:18","slug":"enneper-surface","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/11\/01\/enneper-surface\/","title":{"rendered":"Enneper Surface"},"content":{"rendered":"
\n

\"Rotating<\/a>

Rotating Enneper Surface – Greg Egan<\/p><\/div><\/a><\/p>\n<\/div>\n

This is the Enneper surface<\/b>, as drawn by Greg Egan<\/a> using Mathematica. It’s a minimal surface<\/a>, meaning one that necessarily gets more area if you warp any small patch of it. A soap film will make a minimal surface if it doesn’t enclose any air. But the Enneper surface intersects itself: it’s immersed<\/a> in 3d space, but not embedded. So, you can’t make it with soapy water!<\/p>\n

You can describe the Enneper surface using these equations:<\/p>\n

$$ x = u – u^3\/3 + uv^2 $$<\/p>\n

$$ y = -v + v^3\/3 – vu^2 $$ <\/p>\n

$$ z = u^2 – v^2 $$<\/p>\n

As $u$ and $v$ range over all real numbers, the point $(x,y,z)$ hits every point of the Enneper surface.<\/p>\n

Alfred Enneper and Karl Weierstrass were thinking about minimal surfaces back around 1863, and they discovered every such surface that’s simply a disk mapped into $\\mathbb{R}^3$ could be described in a clever way using complex analytic functions. This is called the Enneper–Weierstrass parametrization<\/a>. The Enneper surface is distinguished by the fact that it has an extremely simple parametrization of this sort.<\/p>\n

The Enneper surface is also special because it is a complete minimal surface in $\\mathbb{R}^3$ for which the integral of the Gaussian curvature over the whole surface is $-4\\pi$. The only other surface with this property is the catenoid<\/a>, obtained by rotating a catenary. The catenoid was the first minimal surface to be found after the plane: it was proved to be minimal by Leonhard Euler in 1744.<\/p>\n

Can you figure out the symmetry group of the Enneper surface?<\/p>\n

For more, read:<\/p>\n

• Enneper surface<\/a>, Wikipedia.<\/p>\n

• Enneper–Weierstrass parametrization<\/a>, Wikipedia.<\/p>\n

• Myla Kilchrist and Dave Packard, The Weierstrass–Enneper representations<\/a>, Dynamics at the Horsetooth<\/i> 4<\/b> (2012).<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This is the Enneper surface<\/b>, as drawn by Greg Egan<\/a> using Mathematica. It’s a minimal surface<\/a>, meaning one that necessarily gets more area if you warp any small patch of it. A soap film will make a minimal surface if it doesn’t enclose any air. But the Enneper surface intersects itself: it’s immersed in 3d space, but not embedded. So, you can’t make it with soapy water!<\/p>\n

<\/div>\n","protected":false},"author":66,"featured_media":149,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2,12],"tags":[],"class_list":["post-329","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-images-library","category-surfaces"],"jetpack_featured_media_url":"https:\/\/blogs.ams.org\/visualinsight\/files\/2013\/09\/rotating_enneper_surface.gif","jetpack_shortlink":"https:\/\/wp.me\/p42Vmc-5j","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/users\/66"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/comments?post=329"}],"version-history":[{"count":11,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/329\/revisions"}],"predecessor-version":[{"id":509,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/posts\/329\/revisions\/509"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media\/149"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/media?parent=329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/categories?post=329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/visualinsight\/wp-json\/wp\/v2\/tags?post=329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}