{"id":607,"date":"2020-08-10T03:38:09","date_gmt":"2020-08-10T03:38:09","guid":{"rendered":"http:\/\/soulofmathematics.com\/?p=607"},"modified":"2020-08-10T06:11:34","modified_gmt":"2020-08-10T06:11:34","slug":"mobius-strip","status":"publish","type":"post","link":"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/","title":{"rendered":"M\u00f6bius Strip"},"content":{"rendered":"\n

The M\u00f6bius Band is an example of one-sided surface in the form of a single closed continuous curve with a twist. A simple M\u00f6bius Band can be created by joining the ends of a long, narrow strip of paper after giving it a half, 180\u00b0, twist, as in Figure 1. An example of a non-orientable surface, this unique band is named after August Ferdinand M\u00f6bius, a German mathematician and astronomer who discovered it in the process of studying polyhedra in September 1858. But history reveals that the true discoverer was Johann Benedict Listing, who came across this surface in July 1858.<\/p>\n\n\n\n

A M\u00f6bius strip embedded in Euclidean space is a\u00a0chiral<\/a>\u00a0object with right- or left-handedness. The M\u00f6bius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends.<\/p>\n\n\n\n

Definition<\/strong>– The M\u00f6bius strip Ma (with height 2a) was defined as an abstract smooth manifold made as a quotient of (\u2212a, a)\u00d7S1 by a free and properly discontinuous action by the group of order 2. Our purpose here is to work out some tangent space calculations to verify that the explicit \u201cdefinition\u201d of the M\u00f6bius strip via trigonometric parameterization is really a smooth embedding of our abstract Mobius strip of height 2a into R3.<\/p>\n\n\n\n

Using the C^\u221e isomorphism between R\/2\u03c0Z and the circle S1 \u2286 R2 via \u03b8 \u2192 (cos \u03b8,sin \u03b8) (which carries \u03b8 \u2192 \u03c0 + \u03b8 over to w \u2192 \u2212w on S1), we consider the standard parameter \u03b8 \u2208 R as a local coordinate on S1. For finite a > 0, consider the C^\u221e map f : (\u2212a, a) \u00d7 S1 \u2192 R3 defined by, <\/p>\n\n\n\n

(t, \u03b8) \u2192 (2a cos 2\u03b8 + t cos \u03b8 cos 2\u03b8, 2a sin 2\u03b8 + t cos \u03b8 sin 2\u03b8, tsin \u03b8)<\/strong>.<\/p>\n\n\n\n

Since f(\u2212t, \u03c0 + \u03b8) = f(t, \u03b8) by inspection, it follows from the universal property of the quotient map (\u2212a, a) \u00d7 S1 \u2192 Ma that f unique factors through this via a C^\u221e map f : Ma \u2192 R3. Our goal is to prove that f is an embedding and to use this viewpoint to understand some basic properties of the M\u00f6bius strip.<\/p>\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The C^\u221e inclusion S1 \u2192 (\u2212a, a) \u00d7 S1 via \u03b8 \u2192 (0, \u03b8) is compatible with the antipodal map on S1 and with the map (t, \u03b8) \u2192 (\u2212t, \u03c0 + \u03b8) on (\u2212a, a) \u00d7 S1, so we get an induced C\u221e map on quotients that is a closed C^\u221e sub-manifold (by the general good behavior of \u201cnice\u201d group-action quotients and closed sub-manifolds, as explained in the handout on quotients by group actions). Near the end of the handout on quotients by group actions, it was shown that the squaring map w \u2192 w2 from S1 to S1 gives a C^\u221e isomorphism of S1 with the quotient of S1 by the antipodal map w \u2192 \u2212w. Thus, we get a quotient circle C as a C\u221e closed sub-manifold in Ma (the image of {0}\u00d7S1 \u2286 (\u2212a, a)\u00d7S1).<\/p>\n\n\n\n

Inside of the \u201creal world\u201d model f(Ma), the central circle is f(C), and so the assertion of interest is that f(Ma) \u2212 f(C) is connected. Since f is a homeomorphism onto its image, it is equivalent to say that the abstract complement Ma \u2212 C is connected. Note that it is crucial we worked with f and not f, since C = f({0} \u00d7 S 1 ) yet (\u2212a, a) \u00d7 S 1 \u2212 {0} \u00d7 S 1 = ((\u2212a, a) \u2212 {0}) \u00d7 S1 is disconnected. (There is no inconsistency here, since f is not even injective, let alone an embedding, so it could well carry a disconnected subset of its source onto a connected subset of its image.) To see the geometry of Ma \u2212 C, we look at the map ((\u2212a, a) \u2212 {0}) \u00d7 S 1 \u2192 Ma \u2212 C. This map is the quotient by (t, \u03b8) \u2192 (\u2212t, \u03b8 + \u03c0), so the formation of this quotient simply involved identifying (\u2212a, 0)\u00d7S 1 with (0, a)\u00d7S 1 via (\u2212t, \u03b8) \u2194 (t, \u03c0 +\u03b8) for 0 < t < a. More specifically, the connected component (0, a) \u00d7 S 1 maps onto Ma \u2212 C via a bijective C^\u221e local isomorphism, so this map is necessarily a C^\u221e isomorphism. Thus, Ma\u2212C is connected since (0, a)\u00d7S 1 is connected. Note that the subset f(Ma)\u2212f(C) is exactly f((0, a)\u00d7S 1 ), with the map f : (0, a)\u00d7S1\u2192 f(Ma)\u2212f(C) a homeomorphism (and even a C^\u221e isomorphism).<\/p>\n\n\n\n

\n
All Posts<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

The M\u00f6bius Band is an example of one-sided surface in the form of a single closed continuous curve with a twist. A simple M\u00f6bius Band can be created by joining the ends of a long, narrow strip of paper after giving it a half, 180\u00b0, twist, as in Figure 1. An example of a non-orientable surface, this unique band is named after August Ferdinand M\u00f6bius, a German mathematician and astronomer who discovered it in the process of studying polyhedra in September 1858. But history reveals that the true discoverer was Johann Benedict Listing, who came across this surface in July 1858. A M\u00f6bius strip embedded in Euclidean space is a\u00a0chiral\u00a0object with right- or left-handedness. The M\u00f6bius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends. Definition– The M\u00f6bius strip Ma (with height 2a) was defined as an abstract smooth manifold made as a quotient of (\u2212a, a)\u00d7S1 by a free and properly discontinuous action by the group of order 2. Our purpose here is to work out some tangent space calculations to verify that the explicit \u201cdefinition\u201d of the M\u00f6bius strip via trigonometric parameterization is really a smooth embedding of our abstract Mobius strip of height 2a into R3. Using the C^\u221e isomorphism between R\/2\u03c0Z and the circle S1 \u2286 R2 via \u03b8 \u2192 (cos \u03b8,sin \u03b8) (which carries \u03b8 \u2192 \u03c0 + \u03b8 over to w \u2192 \u2212w on S1), we consider the standard parameter \u03b8 \u2208 R as a local coordinate on S1. For finite a > 0, consider the C^\u221e map f : (\u2212a, a) \u00d7 S1 \u2192 R3 defined by, (t, \u03b8) \u2192 (2a cos 2\u03b8 + t cos \u03b8 cos 2\u03b8, 2a sin 2\u03b8 + t cos \u03b8 sin 2\u03b8, tsin \u03b8). Since f(\u2212t, \u03c0 + \u03b8) = f(t, \u03b8) by inspection, it follows from the universal property of the quotient map (\u2212a, a) \u00d7 S1 \u2192 Ma that f unique factors through this via a C^\u221e map f : Ma \u2192 R3. Our goal is to prove that f is an embedding and to use this viewpoint to understand some basic properties of the M\u00f6bius strip. The C^\u221e inclusion S1 \u2192 (\u2212a, a) \u00d7 S1 via \u03b8 \u2192 (0, \u03b8) is compatible with the antipodal map on S1 and with the map (t, \u03b8) \u2192 (\u2212t, \u03c0 + \u03b8) on (\u2212a, a) \u00d7 S1, so we get an induced C\u221e map on quotients that is a closed C^\u221e sub-manifold (by the general good behavior of \u201cnice\u201d group-action quotients and closed sub-manifolds, as explained in the handout on quotients by group actions). Near the end of the handout on quotients by group actions, it was shown that the squaring map w \u2192 w2 from S1 to S1 gives a C^\u221e isomorphism of S1 with the quotient of S1 by the antipodal map w \u2192 \u2212w. Thus, we get a quotient circle C as a C\u221e closed sub-manifold in Ma (the image of {0}\u00d7S1 \u2286 (\u2212a, a)\u00d7S1). Inside of the \u201creal world\u201d model f(Ma), the central circle is f(C), and so the assertion of interest is that f(Ma) \u2212 f(C) is connected. Since f is a homeomorphism onto its image, it is equivalent to say that the abstract complement Ma \u2212 C is connected. Note that it is crucial we worked with f and not f, since C = f({0} \u00d7 S 1 ) yet (\u2212a, a) \u00d7 S 1 \u2212 {0} \u00d7 S 1 = ((\u2212a, a) \u2212 {0}) \u00d7 S1 is disconnected. (There is no inconsistency here, since f is not even injective, let alone an embedding, so it could well carry a disconnected subset of its source onto a connected subset of its image.) To see the geometry of Ma \u2212 C, we look at the map ((\u2212a, a) \u2212 {0}) \u00d7 S 1 \u2192 Ma \u2212 C. This map is the quotient by (t, \u03b8) \u2192 (\u2212t, \u03b8 + \u03c0), so the formation of this quotient simply involved identifying (\u2212a, 0)\u00d7S 1 with (0, a)\u00d7S 1 via (\u2212t, \u03b8) \u2194 (t, \u03c0 +\u03b8) for 0 < t < a. More specifically, the connected component (0, a) \u00d7 S 1 maps onto Ma \u2212 C via a bijective C^\u221e local isomorphism, so this map is necessarily a C^\u221e isomorphism. Thus, Ma\u2212C is connected since (0, a)\u00d7S 1 is connected. Note that the subset f(Ma)\u2212f(C) is exactly f((0, a)\u00d7S 1 ), with the map f : (0, a)\u00d7S1\u2192 f(Ma)\u2212f(C) a homeomorphism (and even a C^\u221e isomorphism).<\/p>\n","protected":false},"author":1,"featured_media":609,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-607","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-sneak-peeks"],"featured_image_src":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1","blog_images":{"medium":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=300%2C169&ssl=1","large":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1"},"ams_acf":[],"yoast_head":"\nM\u00f6bius Strip - SOUL OF MATHEMATICS<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"M\u00f6bius Strip - SOUL OF MATHEMATICS\" \/>\n<meta property=\"og:description\" content=\"The M\u00f6bius Band is an example of one-sided surface in the form of a single closed continuous curve with a twist. A simple M\u00f6bius Band can be created by joining the ends of a long, narrow strip of paper after giving it a half, 180\u00b0, twist, as in Figure 1. An example of a non-orientable surface, this unique band is named after August Ferdinand M\u00f6bius, a German mathematician and astronomer who discovered it in the process of studying polyhedra in September 1858. But history reveals that the true discoverer was Johann Benedict Listing, who came across this surface in July 1858. A M\u00f6bius strip embedded in Euclidean space is a\u00a0chiral\u00a0object with right- or left-handedness. The M\u00f6bius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends. Definition– The M\u00f6bius strip Ma (with height 2a) was defined as an abstract smooth manifold made as a quotient of (\u2212a, a)\u00d7S1 by a free and properly discontinuous action by the group of order 2. Our purpose here is to work out some tangent space calculations to verify that the explicit \u201cdefinition\u201d of the M\u00f6bius strip via trigonometric parameterization is really a smooth embedding of our abstract Mobius strip of height 2a into R3. Using the C^\u221e isomorphism between R\/2\u03c0Z and the circle S1 \u2286 R2 via \u03b8 \u2192 (cos \u03b8,sin \u03b8) (which carries \u03b8 \u2192 \u03c0 + \u03b8 over to w \u2192 \u2212w on S1), we consider the standard parameter \u03b8 \u2208 R as a local coordinate on S1. For finite a > 0, consider the C^\u221e map f : (\u2212a, a) \u00d7 S1 \u2192 R3 defined by, (t, \u03b8) \u2192 (2a cos 2\u03b8 + t cos \u03b8 cos 2\u03b8, 2a sin 2\u03b8 + t cos \u03b8 sin 2\u03b8, tsin \u03b8). Since f(\u2212t, \u03c0 + \u03b8) = f(t, \u03b8) by inspection, it follows from the universal property of the quotient map (\u2212a, a) \u00d7 S1 \u2192 Ma that f unique factors through this via a C^\u221e map f : Ma \u2192 R3. Our goal is to prove that f is an embedding and to use this viewpoint to understand some basic properties of the M\u00f6bius strip. The C^\u221e inclusion S1 \u2192 (\u2212a, a) \u00d7 S1 via \u03b8 \u2192 (0, \u03b8) is compatible with the antipodal map on S1 and with the map (t, \u03b8) \u2192 (\u2212t, \u03c0 + \u03b8) on (\u2212a, a) \u00d7 S1, so we get an induced C\u221e map on quotients that is a closed C^\u221e sub-manifold (by the general good behavior of \u201cnice\u201d group-action quotients and closed sub-manifolds, as explained in the handout on quotients by group actions). Near the end of the handout on quotients by group actions, it was shown that the squaring map w \u2192 w2 from S1 to S1 gives a C^\u221e isomorphism of S1 with the quotient of S1 by the antipodal map w \u2192 \u2212w. Thus, we get a quotient circle C as a C\u221e closed sub-manifold in Ma (the image of {0}\u00d7S1 \u2286 (\u2212a, a)\u00d7S1). Inside of the \u201creal world\u201d model f(Ma), the central circle is f(C), and so the assertion of interest is that f(Ma) \u2212 f(C) is connected. Since f is a homeomorphism onto its image, it is equivalent to say that the abstract complement Ma \u2212 C is connected. Note that it is crucial we worked with f and not f, since C = f({0} \u00d7 S 1 ) yet (\u2212a, a) \u00d7 S 1 \u2212 {0} \u00d7 S 1 = ((\u2212a, a) \u2212 {0}) \u00d7 S1 is disconnected. (There is no inconsistency here, since f is not even injective, let alone an embedding, so it could well carry a disconnected subset of its source onto a connected subset of its image.) To see the geometry of Ma \u2212 C, we look at the map ((\u2212a, a) \u2212 {0}) \u00d7 S 1 \u2192 Ma \u2212 C. This map is the quotient by (t, \u03b8) \u2192 (\u2212t, \u03b8 + \u03c0), so the formation of this quotient simply involved identifying (\u2212a, 0)\u00d7S 1 with (0, a)\u00d7S 1 via (\u2212t, \u03b8) \u2194 (t, \u03c0 +\u03b8) for 0 < t < a. More specifically, the connected component (0, a) \u00d7 S 1 maps onto Ma \u2212 C via a bijective C^\u221e local isomorphism, so this map is necessarily a C^\u221e isomorphism. Thus, Ma\u2212C is connected since (0, a)\u00d7S 1 is connected. Note that the subset f(Ma)\u2212f(C) is exactly f((0, a)\u00d7S 1 ), with the map f : (0, a)\u00d7S1\u2192 f(Ma)\u2212f(C) a homeomorphism (and even a C^\u221e isomorphism).\" \/>\n<meta property=\"og:url\" content=\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/\" \/>\n<meta property=\"og:site_name\" content=\"SOUL OF MATHEMATICS\" \/>\n<meta property=\"article:published_time\" content=\"2020-08-10T03:38:09+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-08-10T06:11:34+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/i2.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1\" \/>\n\t<meta property=\"og:image:width\" content=\"500\" \/>\n\t<meta property=\"og:image:height\" content=\"281\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/gif\" \/>\n<meta name=\"author\" content=\"Rajarshi Dey\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Rajarshi Dey\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/\",\"url\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/\",\"name\":\"M\u00f6bius Strip - SOUL OF MATHEMATICS\",\"isPartOf\":{\"@id\":\"https:\/\/soulofmathematics.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1\",\"datePublished\":\"2020-08-10T03:38:09+00:00\",\"dateModified\":\"2020-08-10T06:11:34+00:00\",\"author\":{\"@id\":\"https:\/\/soulofmathematics.com\/#\/schema\/person\/c61ee309ed66bc94ba7a27f6129b945c\"},\"breadcrumb\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/#primaryimage\",\"url\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1\",\"contentUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1\",\"width\":500,\"height\":281,\"caption\":\"Mobius Strip\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/soulofmathematics.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"M\u00f6bius Strip\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/soulofmathematics.com\/#website\",\"url\":\"https:\/\/soulofmathematics.com\/\",\"name\":\"SOUL OF MATHEMATICS\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/soulofmathematics.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/soulofmathematics.com\/#\/schema\/person\/c61ee309ed66bc94ba7a27f6129b945c\",\"name\":\"Rajarshi Dey\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/soulofmathematics.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/14acfcec71e13078f5b322bb6adfd1f6579c091317d0e0077c2311511263a8b0?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/14acfcec71e13078f5b322bb6adfd1f6579c091317d0e0077c2311511263a8b0?s=96&d=mm&r=g\",\"caption\":\"Rajarshi Dey\"},\"sameAs\":[\"http:\/\/soulofmathematics.com\"],\"url\":\"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"M\u00f6bius Strip - SOUL OF MATHEMATICS","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/","og_locale":"en_US","og_type":"article","og_title":"M\u00f6bius Strip - SOUL OF MATHEMATICS","og_description":"The M\u00f6bius Band is an example of one-sided surface in the form of a single closed continuous curve with a twist. A simple M\u00f6bius Band can be created by joining the ends of a long, narrow strip of paper after giving it a half, 180\u00b0, twist, as in Figure 1. An example of a non-orientable surface, this unique band is named after August Ferdinand M\u00f6bius, a German mathematician and astronomer who discovered it in the process of studying polyhedra in September 1858. But history reveals that the true discoverer was Johann Benedict Listing, who came across this surface in July 1858. A M\u00f6bius strip embedded in Euclidean space is a\u00a0chiral\u00a0object with right- or left-handedness. The M\u00f6bius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends. Definition– The M\u00f6bius strip Ma (with height 2a) was defined as an abstract smooth manifold made as a quotient of (\u2212a, a)\u00d7S1 by a free and properly discontinuous action by the group of order 2. Our purpose here is to work out some tangent space calculations to verify that the explicit \u201cdefinition\u201d of the M\u00f6bius strip via trigonometric parameterization is really a smooth embedding of our abstract Mobius strip of height 2a into R3. Using the C^\u221e isomorphism between R\/2\u03c0Z and the circle S1 \u2286 R2 via \u03b8 \u2192 (cos \u03b8,sin \u03b8) (which carries \u03b8 \u2192 \u03c0 + \u03b8 over to w \u2192 \u2212w on S1), we consider the standard parameter \u03b8 \u2208 R as a local coordinate on S1. For finite a > 0, consider the C^\u221e map f : (\u2212a, a) \u00d7 S1 \u2192 R3 defined by, (t, \u03b8) \u2192 (2a cos 2\u03b8 + t cos \u03b8 cos 2\u03b8, 2a sin 2\u03b8 + t cos \u03b8 sin 2\u03b8, tsin \u03b8). Since f(\u2212t, \u03c0 + \u03b8) = f(t, \u03b8) by inspection, it follows from the universal property of the quotient map (\u2212a, a) \u00d7 S1 \u2192 Ma that f unique factors through this via a C^\u221e map f : Ma \u2192 R3. Our goal is to prove that f is an embedding and to use this viewpoint to understand some basic properties of the M\u00f6bius strip. The C^\u221e inclusion S1 \u2192 (\u2212a, a) \u00d7 S1 via \u03b8 \u2192 (0, \u03b8) is compatible with the antipodal map on S1 and with the map (t, \u03b8) \u2192 (\u2212t, \u03c0 + \u03b8) on (\u2212a, a) \u00d7 S1, so we get an induced C\u221e map on quotients that is a closed C^\u221e sub-manifold (by the general good behavior of \u201cnice\u201d group-action quotients and closed sub-manifolds, as explained in the handout on quotients by group actions). Near the end of the handout on quotients by group actions, it was shown that the squaring map w \u2192 w2 from S1 to S1 gives a C^\u221e isomorphism of S1 with the quotient of S1 by the antipodal map w \u2192 \u2212w. Thus, we get a quotient circle C as a C\u221e closed sub-manifold in Ma (the image of {0}\u00d7S1 \u2286 (\u2212a, a)\u00d7S1). Inside of the \u201creal world\u201d model f(Ma), the central circle is f(C), and so the assertion of interest is that f(Ma) \u2212 f(C) is connected. Since f is a homeomorphism onto its image, it is equivalent to say that the abstract complement Ma \u2212 C is connected. Note that it is crucial we worked with f and not f, since C = f({0} \u00d7 S 1 ) yet (\u2212a, a) \u00d7 S 1 \u2212 {0} \u00d7 S 1 = ((\u2212a, a) \u2212 {0}) \u00d7 S1 is disconnected. (There is no inconsistency here, since f is not even injective, let alone an embedding, so it could well carry a disconnected subset of its source onto a connected subset of its image.) To see the geometry of Ma \u2212 C, we look at the map ((\u2212a, a) \u2212 {0}) \u00d7 S 1 \u2192 Ma \u2212 C. This map is the quotient by (t, \u03b8) \u2192 (\u2212t, \u03b8 + \u03c0), so the formation of this quotient simply involved identifying (\u2212a, 0)\u00d7S 1 with (0, a)\u00d7S 1 via (\u2212t, \u03b8) \u2194 (t, \u03c0 +\u03b8) for 0 < t < a. More specifically, the connected component (0, a) \u00d7 S 1 maps onto Ma \u2212 C via a bijective C^\u221e local isomorphism, so this map is necessarily a C^\u221e isomorphism. Thus, Ma\u2212C is connected since (0, a)\u00d7S 1 is connected. Note that the subset f(Ma)\u2212f(C) is exactly f((0, a)\u00d7S 1 ), with the map f : (0, a)\u00d7S1\u2192 f(Ma)\u2212f(C) a homeomorphism (and even a C^\u221e isomorphism).","og_url":"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/","og_site_name":"SOUL OF MATHEMATICS","article_published_time":"2020-08-10T03:38:09+00:00","article_modified_time":"2020-08-10T06:11:34+00:00","og_image":[{"width":500,"height":281,"url":"https:\/\/i2.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/gi5c94B.gif?fit=500%2C281&ssl=1","type":"image\/gif"}],"author":"Rajarshi Dey","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Rajarshi Dey","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/","url":"https:\/\/soulofmathematics.com\/index.php\/mobius-strip\/","name":"M\u00f6bius Strip - SOUL OF 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