{"id":689,"date":"2019-02-24T14:17:09","date_gmt":"2019-02-24T19:17:09","guid":{"rendered":"http:\/\/josmfs.net\/?p=689"},"modified":"2021-06-12T14:15:05","modified_gmt":"2021-06-12T18:15:05","slug":"bugles-trumpets-and-beltrami","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/02\/24\/bugles-trumpets-and-beltrami\/","title":{"rendered":"Bugles, Trumpets, and Beltrami"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2233\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Bugles-Trumpets-Beltrami-Fig2.jpg\" alt=\"\" width=\"251\" height=\"153\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Bugles-Trumpets-Beltrami-Fig2.jpg 500w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/06\/Bugles-Trumpets-Beltrami-Fig2-300x183.jpg 300w\" sizes=\"auto, (max-width: 251px) 100vw, 251px\" \/>This essay began as an effort to prove Tanya Khovanova\u2019s statement in her article \u201cThe Annoyance of Hyperbolic Surfaces\u201d that her crocheted hyperbolic surface had constant (negative) curvature. I discussed Khovanova\u2019s article in my previous essay \u201c<a href=\"https:\/\/josmfs.net\/wordpress\/2019\/02\/22\/exponential-yarn\/\">Exponential Yarn<\/a>\u201d. What I thought would be a fairly straight-forward exercise turned into a more concerted effort as I concluded that her crocheted surface did not have constant curvature. However, I found additional references that supported her statement, so I was becoming quite confused. I looked at other, similar surfaces to try to understand the whole curvature situation. This involved a lot of tedious computations (with my usual plethora of mistakes) that proved most challenging. But then I realized where I had gone astray. To cover my ignorance I claimed my error stemmed from a subtle misunderstanding. Herewith is a presentation of what I found. See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/04\/Bugles-Trumpets-Beltrami-190406.pdf\">Bugles, Trumpets, and Beltrami<\/a>.<\/p>\n<p><strong>(Update 4\/6\/2019)<\/strong><!--more-->I made a cosmetic change to the presentation of the hyperbola function h(u) = 1\/u curvature in order to make it compatible with the discussions of the exponential and tractrix curvatures, namely, I expressed the equations and figures in terms of h instead of u.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This essay began as an effort to prove Tanya Khovanova\u2019s statement in her article \u201cThe Annoyance of Hyperbolic Surfaces\u201d that her crocheted hyperbolic surface had constant (negative) curvature. I discussed Khovanova\u2019s article in my previous essay \u201cExponential Yarn\u201d. What I thought would be a fairly straight-forward exercise turned into a more concerted effort as I [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[106,12,80,105,72,107],"class_list":["post-689","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-beltrami","tag-curvature","tag-differential-geometry","tag-hyperbolic-space","tag-tanya-khovanova","tag-torelli-trumpet"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/689","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=689"}],"version-history":[{"count":5,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/689\/revisions"}],"predecessor-version":[{"id":2234,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/689\/revisions\/2234"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=689"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=689"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=689"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}