{"id":351,"date":"2019-06-26T23:43:07","date_gmt":"2019-06-26T23:43:07","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=351"},"modified":"2019-12-20T23:33:39","modified_gmt":"2019-12-20T23:33:39","slug":"spherical-geometry","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/spherical-geometry\/","title":{"rendered":"Spherical Geometry"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Remember high school geometry? The sum of the angles of a planar\u00a0triangle\u00a0is always 180 degrees or Pi radians. However, triangles on other surfaces can behave differently!<\/p>\n\n\n\n

For instance, consider a triangle on a sphere, whose edges are “intrinsically” straight in the sense that if you were a very tiny ant living on the sphere you would not think the edges were bending either to the left or right. (Such intrinsically straight lines are called geodesics<\/em>. On spheres, they correspond to pieces of great circles whose center coincide with the center of the sphere.)<\/p>\n\n\n\n

A triangle on a sphere has the interesting property that the sum of the angles is greater than 180<\/em> degrees! And in fact, two triangles with the same angles are not just similar (as in planar geometry), they are actually congruent<\/em>! But wait, there’s more: on a UNIT sphere, the AREA of the triangle actually satisfies:<\/p>\n\n\n\n

AREA of a triangle = (sum of angles) – Pi ,<\/p>\n\n\n\n

where the angles are measured in radians. Cool!<\/p>\n\n\n\n

Another neat fact about spherical triangles may be found in\u00a0Spherical Pythagorean Theorem.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Demonstrate the assertions about angles and areas with an example: draw a picture of a sphere and then draw a triangle whose vertices are at the north pole and at two distinct points on the equator. Here’s a follow-up question for your students: are geodesic paths always the shortest paths between two points?<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
Planar geometry is sometimes called\u00a0flat<\/em>\u00a0or\u00a0Euclidean<\/em>\u00a0geometry. The geometry on a sphere is an example of a\u00a0spherical<\/em>\u00a0or\u00a0elliptic<\/em>\u00a0geometry. Another kind of non-Euclidean geometry is\u00a0hyperbolic geometry. Spherical and hyperbolic geometries do not satisfy the\u00a0parallel postulate.<\/p>\n\n\n\n

By the way, 3-dimensional spaces can also have strange geometries. Our universe, for instance, seems to have a Euclidean geometry on a local scale, but does not on a global scale. In much the same way that a sphere is “curved”, so that divergent geodesics extending from the south pole will meet again at the north pole, Einstein suggested that 3-space is “curved” by the presence of matter, so that light rays (which follow geodesics) bend near very massive objects!<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Spherical Geometry.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
J. Weeks, The Shape of Space.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"

Remember high school geometry? The sum of the angles of a planar\u00a0triangle\u00a0is always 180 degrees or Pi radians. However, triangles...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[166,8,4,121,12,44,163,165],"class_list":["post-351","page","type-page","status-publish","hentry","tag-elliptic-geometry","tag-geometry","tag-medium","tag-non-euclidean-geometry","tag-other","tag-sphere","tag-triangles","tag-universe"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/351","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=351"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/351\/revisions"}],"predecessor-version":[{"id":1643,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/351\/revisions\/1643"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=351"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=351"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}