The Painter\u2019s Paradox is based on the fact that Gabriel\u2019s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel\u2019s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel\u2019s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide a brief overview of the development of the notion of infinity in mathematics and the debate around Torricelli\u2019s discovery.<\/p>\n\n\n\n
Gabriel and His Horn<\/h3>\n\n\n\n
<\/figure>
\n
Gabriel was an archangel, as the Bible tells us, who \u201cused a horn to announce news that was sometimes heartening (e.g., the birth of Christ in Luke l) and sometimes fatalistic (e.g., Armageddon in Revelation 8-11)\u201d.<\/p>\n<\/div><\/div>\n\n\n\n
Torricelli’s Long Horn<\/h3>\n\n\n\n
In 1641 Evangelista Torricelli showed that a certain solid of infinite length, now known as the Gabriel\u2019s horn, which he called the acute hyperbolic solid, has a finite volume. In\u00a0De solido hyperbolico acuto<\/em>\u00a0he defined an acute hyperbolic solid as the solid generated when a hyperbola is rotated around an asymptote and stated the following theorem:<\/p>\n\n\n\n
THEOREM: An acute hyperbolic solid, infinitely long [infinite longum<\/em>], cut by a plane [perpendicular] to the axis, together with a cylinder of the same base, is equal to that right cylinder of which the base is the latus transversum<\/em> of the hyperbola (that is, the diameter of the hyperbola), and of which the altitude is equal to the radius of the base of this acute body.<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
The Queer Volume<\/h3>\n\n\n\n
Construct the surface of revolution given by rotating the function f(x) = 1\/x on [1,\u221e) around the x-axis. The volume of a surface of revolution given by rotating the function f(x), de\ffined on the interval [a; b], around the x-axis is<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
where A(x) is the cross-sectional area at x \u2208 [a, b]. Due to the construction, this is ALWAYS a circle, of radius f(x), and hence A(x) = \u03c0 (f(x))2<\/sup>, so that<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
In our case, the interval of integration is infinite, and hence the integral we define is improper. Nevertheless, we find that<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Hence, even though the Horn extends outward along the x-axis to \u221e, the improper integral does converge, and hence there is finite volume \u201cinside\u201d the Horn. One can say that one can fill the Horn with \u03c0-units of a liquid. (This is oddly satisfying)<\/p>\n\n\n\n
The Infinite Surface Area<\/h3>\n\n\n\n
The surface area of a surface of revolution is the subject. For a surface formed by revolving f(x) on [a, b] around the x-axis, the surface area is found by evaluating<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
This formula basically says that one can \fnd surface area by multiplying the circumference of the surface of evolution at x, which is a circle again, with circumference 2\u03c0f(x), by the arc-length along the original function f(x) (this is the radical part of the integrand). In our case, we get an improper integral again (call the surface area SA):<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
This integral is not such an easy calculation. However, we really do not need to actually calculate this quantity using an antiderivative. Instead, we make the following observation: Notice that on the interval [1;1), we have that<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Thus we can say that, if<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
on the interval [1;1), then<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
by the properties of integrals. And by the Comparison Theorem for improper integrals, we can conclude that, if the integral of the smaller one (with h(x) as the integrand) diverges, then so does the integral of the larger function g(x). Indeed, we fi\fnd by comparison that<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
But we have already evaluated this last integral in class. We get<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Hence this last integral diverges, and hence by comparison so does the former integral. But this implies that the surface area of Gabriel’s Horn is infi\fnite!<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
So we have a surface with infinite surface area enclosing a finite volume.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Painter\u2019s Paradox is based on the fact that Gabriel\u2019s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel\u2019s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel\u2019s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide a brief overview of the development of the notion of infinity in mathematics and the debate around Torricelli\u2019s discovery. Gabriel and His Horn Gabriel was an archangel, as the Bible tells us, who \u201cused a horn to announce news that was sometimes heartening (e.g., the birth of Christ in Luke l) and sometimes fatalistic (e.g., Armageddon in Revelation 8-11)\u201d. Torricelli’s Long Horn In 1641 Evangelista Torricelli showed that a certain solid of infinite length, now known as the Gabriel\u2019s horn, which he called the acute hyperbolic solid, has a finite volume. In\u00a0De solido hyperbolico acuto\u00a0he defined an acute hyperbolic solid as the solid generated when a hyperbola is rotated around an asymptote and stated the following theorem: THEOREM: An acute hyperbolic solid, infinitely long [infinite longum], cut by a plane [perpendicular] to the axis, together with a cylinder of the same base, is equal to that right cylinder of which the base is the latus transversum of the hyperbola (that is, the diameter of the hyperbola), and of which the altitude is equal to the radius of the base of this acute body. The Queer Volume Construct the surface of revolution given by rotating the function f(x) = 1\/x on [1,\u221e) around the x-axis. The volume of a surface of revolution given by rotating the function f(x), de\ffined on the interval [a; b], around the x-axis is where A(x) is the cross-sectional area at x \u2208 [a, b]. Due to the construction, this is ALWAYS a circle, of radius f(x), and hence A(x) = \u03c0 (f(x))2, so that In our case, the interval of integration is infinite, and hence the integral we define is improper. Nevertheless, we find that Hence, even though the Horn extends outward along the x-axis to \u221e, the improper integral does converge, and hence there is finite volume \u201cinside\u201d the Horn. One can say that one can fill the Horn with \u03c0-units of a liquid. (This is oddly satisfying) The Infinite Surface Area The surface area of a surface of revolution is the subject. For a surface formed by revolving f(x) on [a, b] around the x-axis, the surface area is found by evaluating This formula basically says that one can \fnd surface area by multiplying the circumference of the surface of evolution at x, which is a circle again, with circumference 2\u03c0f(x), by the arc-length along the original function f(x) (this is the radical part of the integrand). In our case, we get an improper integral again (call the surface area SA): This integral is not such an easy calculation. However, we really do not need to actually calculate this quantity using an antiderivative. Instead, we make the following observation: Notice that on the interval [1;1), we have that Thus we can say that, if on the interval [1;1), then by the properties of integrals. And by the Comparison Theorem for improper integrals, we can conclude that, if the integral of the smaller one (with h(x) as the integrand) diverges, then so does the integral of the larger function g(x). Indeed, we fi\fnd by comparison that But we have already evaluated this last integral in class. We get Hence this last integral diverges, and hence by comparison so does the former integral. Butthis implies that the surface area of Gabriel’s Horn is infi\fnite! So we have a surface with infinite surface area enclosing a finite volume.<\/p>\n","protected":false},"author":1,"featured_media":2430,"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-2427","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\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&ssl=1","blog_images":{"medium":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=300%2C125&ssl=1","large":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&ssl=1"},"ams_acf":[],"yoast_head":"\n
GABRIEL'S HORN - 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\/gabriels-horn\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GABRIEL'S HORN - SOUL OF MATHEMATICS\" \/>\n<meta property=\"og:description\" content=\"The Painter\u2019s Paradox is based on the fact that Gabriel\u2019s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel\u2019s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel\u2019s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide a brief overview of the development of the notion of infinity in mathematics and the debate around Torricelli\u2019s discovery. Gabriel and His Horn Gabriel was an archangel, as the Bible tells us, who \u201cused a horn to announce news that was sometimes heartening (e.g., the birth of Christ in Luke l) and sometimes fatalistic (e.g., Armageddon in Revelation 8-11)\u201d. Torricelli’s Long Horn In 1641 Evangelista Torricelli showed that a certain solid of infinite length, now known as the Gabriel\u2019s horn, which he called the acute hyperbolic solid, has a finite volume. In\u00a0De solido hyperbolico acuto\u00a0he defined an acute hyperbolic solid as the solid generated when a hyperbola is rotated around an asymptote and stated the following theorem: THEOREM: An acute hyperbolic solid, infinitely long [infinite longum], cut by a plane [perpendicular] to the axis, together with a cylinder of the same base, is equal to that right cylinder of which the base is the latus transversum of the hyperbola (that is, the diameter of the hyperbola), and of which the altitude is equal to the radius of the base of this acute body. The Queer Volume Construct the surface of revolution given by rotating the function f(x) = 1\/x on [1,\u221e) around the x-axis. The volume of a surface of revolution given by rotating the function f(x), de fined on the interval [a; b], around the x-axis is where A(x) is the cross-sectional area at x \u2208 [a, b]. Due to the construction, this is ALWAYS a circle, of radius f(x), and hence A(x) = \u03c0 (f(x))2, so that In our case, the interval of integration is infinite, and hence the integral we define is improper. Nevertheless, we find that Hence, even though the Horn extends outward along the x-axis to \u221e, the improper integral does converge, and hence there is finite volume \u201cinside\u201d the Horn. One can say that one can fill the Horn with \u03c0-units of a liquid. (This is oddly satisfying) The Infinite Surface Area The surface area of a surface of revolution is the subject. For a surface formed by revolving f(x) on [a, b] around the x-axis, the surface area is found by evaluating This formula basically says that one can nd surface area by multiplying the circumference of the surface of evolution at x, which is a circle again, with circumference 2\u03c0f(x), by the arc-length along the original function f(x) (this is the radical part of the integrand). In our case, we get an improper integral again (call the surface area SA): This integral is not such an easy calculation. However, we really do not need to actually calculate this quantity using an antiderivative. Instead, we make the following observation: Notice that on the interval [1;1), we have that Thus we can say that, if on the interval [1;1), then by the properties of integrals. And by the Comparison Theorem for improper integrals, we can conclude that, if the integral of the smaller one (with h(x) as the integrand) diverges, then so does the integral of the larger function g(x). Indeed, we fi nd by comparison that But we have already evaluated this last integral in class. We get Hence this last integral diverges, and hence by comparison so does the former integral. Butthis implies that the surface area of Gabriel’s Horn is infi nite! So we have a surface with infinite surface area enclosing a finite volume.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/\" \/>\n<meta property=\"og:site_name\" content=\"SOUL OF MATHEMATICS\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-28T07:20:15+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-03-28T08:35:23+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&ssl=1\" \/>\n\t<meta property=\"og:image:width\" content=\"640\" \/>\n\t<meta property=\"og:image:height\" content=\"266\" \/>\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\/gabriels-horn\/\",\"url\":\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/\",\"name\":\"GABRIEL'S HORN - SOUL OF MATHEMATICS\",\"isPartOf\":{\"@id\":\"https:\/\/soulofmathematics.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&ssl=1\",\"datePublished\":\"2021-03-28T07:20:15+00:00\",\"dateModified\":\"2021-03-28T08:35:23+00:00\",\"author\":{\"@id\":\"https:\/\/soulofmathematics.com\/#\/schema\/person\/c61ee309ed66bc94ba7a27f6129b945c\"},\"breadcrumb\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/#primaryimage\",\"url\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&ssl=1\",\"contentUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&ssl=1\",\"width\":640,\"height\":266},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/soulofmathematics.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"GABRIEL’S HORN\"}]},{\"@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":"GABRIEL'S HORN - 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\/gabriels-horn\/","og_locale":"en_US","og_type":"article","og_title":"GABRIEL'S HORN - SOUL OF MATHEMATICS","og_description":"The Painter\u2019s Paradox is based on the fact that Gabriel\u2019s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel\u2019s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel\u2019s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide a brief overview of the development of the notion of infinity in mathematics and the debate around Torricelli\u2019s discovery. Gabriel and His Horn Gabriel was an archangel, as the Bible tells us, who \u201cused a horn to announce news that was sometimes heartening (e.g., the birth of Christ in Luke l) and sometimes fatalistic (e.g., Armageddon in Revelation 8-11)\u201d. Torricelli’s Long Horn In 1641 Evangelista Torricelli showed that a certain solid of infinite length, now known as the Gabriel\u2019s horn, which he called the acute hyperbolic solid, has a finite volume. In\u00a0De solido hyperbolico acuto\u00a0he defined an acute hyperbolic solid as the solid generated when a hyperbola is rotated around an asymptote and stated the following theorem: THEOREM: An acute hyperbolic solid, infinitely long [infinite longum], cut by a plane [perpendicular] to the axis, together with a cylinder of the same base, is equal to that right cylinder of which the base is the latus transversum of the hyperbola (that is, the diameter of the hyperbola), and of which the altitude is equal to the radius of the base of this acute body. The Queer Volume Construct the surface of revolution given by rotating the function f(x) = 1\/x on [1,\u221e) around the x-axis. The volume of a surface of revolution given by rotating the function f(x), de fined on the interval [a; b], around the x-axis is where A(x) is the cross-sectional area at x \u2208 [a, b]. Due to the construction, this is ALWAYS a circle, of radius f(x), and hence A(x) = \u03c0 (f(x))2, so that In our case, the interval of integration is infinite, and hence the integral we define is improper. Nevertheless, we find that Hence, even though the Horn extends outward along the x-axis to \u221e, the improper integral does converge, and hence there is finite volume \u201cinside\u201d the Horn. One can say that one can fill the Horn with \u03c0-units of a liquid. (This is oddly satisfying) The Infinite Surface Area The surface area of a surface of revolution is the subject. For a surface formed by revolving f(x) on [a, b] around the x-axis, the surface area is found by evaluating This formula basically says that one can nd surface area by multiplying the circumference of the surface of evolution at x, which is a circle again, with circumference 2\u03c0f(x), by the arc-length along the original function f(x) (this is the radical part of the integrand). In our case, we get an improper integral again (call the surface area SA): This integral is not such an easy calculation. However, we really do not need to actually calculate this quantity using an antiderivative. Instead, we make the following observation: Notice that on the interval [1;1), we have that Thus we can say that, if on the interval [1;1), then by the properties of integrals. And by the Comparison Theorem for improper integrals, we can conclude that, if the integral of the smaller one (with h(x) as the integrand) diverges, then so does the integral of the larger function g(x). Indeed, we fi nd by comparison that But we have already evaluated this last integral in class. We get Hence this last integral diverges, and hence by comparison so does the former integral. Butthis implies that the surface area of Gabriel’s Horn is infi nite! So we have a surface with infinite surface area enclosing a finite volume.","og_url":"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/","og_site_name":"SOUL OF MATHEMATICS","article_published_time":"2021-03-28T07:20:15+00:00","article_modified_time":"2021-03-28T08:35:23+00:00","og_image":[{"width":640,"height":266,"url":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/03\/LMP90V4-unscreen.gif?fit=640%2C266&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\/gabriels-horn\/","url":"https:\/\/soulofmathematics.com\/index.php\/gabriels-horn\/","name":"GABRIEL'S HORN - SOUL OF 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