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href="/w/index.php?title=Rational_geometry&redirect=no" class="mw-redirect" title="Rational geometry">Rational geometry</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">2005 book reformulating plane geometry</div><style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title" style="font-size:125%; font-style:italic; padding-bottom:0.2em;">Divine Proportions: Rational Trigonometry to Universal Geometry <span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=&rft.author=Norman+J.+Wildberger&rft.date=2005&rft.pub=Wild+Egg"></span></caption><tbody><tr><th scope="row" class="infobox-label">Author</th><td class="infobox-data">Norman J. Wildberger</td></tr><tr><th scope="row" class="infobox-label">Genre</th><td class="infobox-data">Mathematics</td></tr><tr><th scope="row" class="infobox-label">Publisher</th><td class="infobox-data">Wild Egg</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Publication date</div></th><td class="infobox-data">2005</td></tr></tbody></table> <p> <i><b>Divine Proportions: Rational Trigonometry to Universal Geometry</b></i> is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> and <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, called <b>rational trigonometry</b>. The book advocates replacing the usual basic quantities of trigonometry, <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> and <a href="/wiki/Angle" title="Angle">angle</a> measure, by <a href="/wiki/Euclidean_distance#Squared_Euclidean_distance" title="Euclidean distance">squared distance</a> and the square of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for <a href="/wiki/Irrational_numbers" class="mw-redirect" title="Irrational numbers">irrational numbers</a>. </p><p>The book was "essentially self-published"<sup id="cite_ref-wiswell_1-0" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The main idea of <i>Divine Proportions</i> is to replace distances by the <a href="/wiki/Squared_Euclidean_distance" class="mw-redirect" title="Squared Euclidean distance">squared Euclidean distance</a>, which Wildberger calls the <i><b>quadrance</b></i>, and to replace angle measures by the squares of their sines, which Wildberger calls the <i><b>spread</b></i> between two lines. <i>Divine Proportions</i> defines both of these concepts directly from the <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinates</a> of points that determine a line segment or a pair of crossing lines. Defined in this way, they are <a href="/wiki/Rational_function" title="Rational function">rational functions</a> of those coordinates, and can be calculated directly without the need to take the <a href="/wiki/Square_root" title="Square root">square roots</a> or <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a> required when computing distances or angle measures.<sup id="cite_ref-wiswell_1-1" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For Wildberger, a <a href="/wiki/Finitism" title="Finitism">finitist</a>, this replacement has the purported advantage of avoiding the concepts of <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a> and <a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a> used in defining the <a href="/wiki/Real_number" title="Real number">real numbers</a>, which Wildberger claims to be unfounded.<sup id="cite_ref-gefter_2-0" class="reference"><a href="#cite_note-gefter-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wiswell_1-2" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It also allows analogous concepts to be extended directly from the rational numbers to other number systems such as <a href="/wiki/Finite_field" title="Finite field">finite fields</a> using the same formulas for quadrance and spread.<sup id="cite_ref-wiswell_1-3" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Additionally, this method avoids the ambiguity of the two <a href="/wiki/Supplementary_angle" class="mw-redirect" title="Supplementary angle">supplementary angles</a> formed by a pair of lines, as both angles have the same spread. This system is claimed to be more intuitive, and to extend more easily from two to three dimensions.<sup id="cite_ref-leversha_3-0" class="reference"><a href="#cite_note-leversha-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, in exchange for these benefits, one loses the additivity of distances and angles: for instance, if a line segment is divided in two, its length is the sum of the lengths of the two pieces, but combining the quadrances of the pieces is more complicated and requires square roots.<sup id="cite_ref-wiswell_1-4" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Organization_and_topics">Organization and topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry&action=edit&section=2" title="Edit section: Organization and topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Divine Proportions</i> is divided into four parts. Part I presents an overview of the use of quadrance and spread to replace distance and angle, and makes the argument for their advantages. Part II formalizes the claims made in part I, and proves them rigorously.<sup id="cite_ref-wiswell_1-5" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Rather than defining lines as infinite sets of points, they are defined by their <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>, which may be used in formulas for testing the incidence of points and lines. Like the sine, the cosine and tangent are replaced with rational equivalents, called the "cross" and "twist", and <i>Divine Proportions</i> develops various analogues of <a href="/wiki/Trigonometric_identity" class="mw-redirect" title="Trigonometric identity">trigonometric identities</a> involving these quantities,<sup id="cite_ref-leversha_3-1" class="reference"><a href="#cite_note-leversha-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> including versions of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, <a href="/wiki/Law_of_sines" title="Law of sines">law of sines</a> and <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>.<sup id="cite_ref-henle_4-0" class="reference"><a href="#cite_note-henle-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Part III develops the geometry of <a href="/wiki/Triangle" title="Triangle">triangles</a> and <a href="/wiki/Conic_section" title="Conic section">conic sections</a> using the tools developed in the two previous parts.<sup id="cite_ref-wiswell_1-6" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Well known results such as <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> for calculating the area of a triangle from its side lengths, or the <a href="/wiki/Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">inscribed angle theorem</a> in the form that the angles subtended by a chord of a circle from other points on the circle are equal, are reformulated in terms of quadrance and spread, and thereby generalized to arbitrary fields of numbers.<sup id="cite_ref-leversha_3-2" class="reference"><a href="#cite_note-leversha-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-franklin_5-0" class="reference"><a href="#cite_note-franklin-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Finally, Part IV considers practical applications in physics and surveying, and develops extensions to higher-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> and to <a href="/wiki/Polar_coordinate" class="mw-redirect" title="Polar coordinate">polar coordinates</a>.<sup id="cite_ref-wiswell_1-7" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Audience">Audience</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry&action=edit&section=3" title="Edit section: Audience"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Divine Proportions</i> does not assume much in the way of mathematical background in its readers, but its many long formulas, frequent consideration of finite fields, and (after part I) emphasis on mathematical rigour are likely to be obstacles to a <a href="/wiki/Popular_mathematics" title="Popular mathematics">popular mathematics</a> audience. Instead, it is mainly written for mathematics teachers and researchers. However, it may also be readable by mathematics students, and contains exercises making it possible to use as the basis for a mathematics course.<sup id="cite_ref-wiswell_1-8" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-barker_6-0" class="reference"><a href="#cite_note-barker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Critical_reception">Critical reception</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry&action=edit&section=4" title="Edit section: Critical reception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The feature of the book that was most positively received by reviewers was its work extending results in distance and angle geometry to finite fields. Reviewer Laura Wiswell found this work impressive, and was charmed by the result that the smallest finite field containing a regular <a href="/wiki/Pentagon" title="Pentagon">pentagon</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{19}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{19}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a143d96171f10b61bfc6322f7e2f971c3d764f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.296ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{19}}"></span>.<sup id="cite_ref-wiswell_1-9" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Michael Henle calls the extension of triangle and conic section geometry to finite fields, in part III of the book, "an elegant theory of great generality",<sup id="cite_ref-henle_4-1" class="reference"><a href="#cite_note-henle-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and William Barker also writes approvingly of this aspect of the book, calling it "particularly novel" and possibly opening up new research directions.<sup id="cite_ref-barker_6-1" class="reference"><a href="#cite_note-barker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Wiswell raises the question of how many of the detailed results presented without attribution in this work are actually novel.<sup id="cite_ref-wiswell_1-10" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In this light, Michael Henle notes that the use of <a href="/wiki/Squared_Euclidean_distance" class="mw-redirect" title="Squared Euclidean distance">squared Euclidean distance</a> "has often been found convenient elsewhere";<sup id="cite_ref-henle_4-2" class="reference"><a href="#cite_note-henle-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for instance it is used in <a href="/wiki/Distance_geometry" title="Distance geometry">distance geometry</a>, <a href="/wiki/Least_squares" title="Least squares">least squares</a> statistics, and <a href="/wiki/Convex_optimization" title="Convex optimization">convex optimization</a>. James Franklin points out that for spaces of three or more dimensions, modelled conventionally using <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, the use of spread by <i>Divine Proportions</i> is not very different from standard methods involving <a href="/wiki/Dot_product" title="Dot product">dot products</a> in place of trigonometric functions.<sup id="cite_ref-franklin_5-1" class="reference"><a href="#cite_note-franklin-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>An advantage of Wildberger's methods noted by Henle is that, because they involve only simple algebra, the proofs are both easy to follow and easy for a computer to verify. However, he suggests that the book's claims of greater simplicity in its overall theory rest on a false comparison in which quadrance and spread are weighed not against the corresponding classical concepts of distances, angles, and sines, but the much wider set of tools from classical trigonometry. He also points out that, to a student with a scientific calculator, formulas that avoid square roots and trigonometric functions are a non-issue,<sup id="cite_ref-henle_4-3" class="reference"><a href="#cite_note-henle-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and Barker adds that the new formulas often involve a greater number of individual calculation steps.<sup id="cite_ref-barker_6-2" class="reference"><a href="#cite_note-barker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Although multiple reviewers felt that a reduction in the amount of time needed to teach students trigonometry would be very welcome,<sup id="cite_ref-leversha_3-3" class="reference"><a href="#cite_note-leversha-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-franklin_5-2" class="reference"><a href="#cite_note-franklin-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-campbell_7-0" class="reference"><a href="#cite_note-campbell-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Paul Campbell is skeptical that these methods would actually speed learning.<sup id="cite_ref-campbell_7-1" class="reference"><a href="#cite_note-campbell-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Gerry Leversha keeps an open mind, writing that "It will be interesting to see some of the textbooks aimed at school pupils [that Wildberger] has promised to produce, and ... controlled experiments involving student guinea pigs."<sup id="cite_ref-leversha_3-4" class="reference"><a href="#cite_note-leversha-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, these textbooks and experiments have not been published. </p><p>Wiswell is unconvinced by the claim that conventional geometry has foundational flaws that these methods avoid.<sup id="cite_ref-wiswell_1-11" class="reference"><a href="#cite_note-wiswell-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> While agreeing with Wiswell, Barker points out that there may be other mathematicians who share Wildberger's philosophical suspicions of the infinite, and that this work should be of great interest to them.<sup id="cite_ref-barker_6-3" class="reference"><a href="#cite_note-barker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>A final issue raised by multiple reviewers is inertia: supposing for the sake of argument that these methods are better, are they sufficiently better to make worthwhile the large individual effort of re-learning geometry and trigonometry in these terms, and the institutional effort of re-working the school curriculum to use them in place of classical geometry and trigonometry? Henle, Barker, and Leversha conclude that the book has not made its case for this,<sup id="cite_ref-leversha_3-5" class="reference"><a href="#cite_note-leversha-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-henle_4-4" class="reference"><a href="#cite_note-henle-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-barker_6-4" class="reference"><a href="#cite_note-barker-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> but <a href="/wiki/Sandra_Arlinghaus" title="Sandra Arlinghaus">Sandra Arlinghaus</a> sees this work as an opportunity for fields such as her mathematical geography "that have relatively little invested in traditional institutional rigidity" to demonstrate the promise of such a replacement.<sup id="cite_ref-arlinghaus_8-0" class="reference"><a href="#cite_note-arlinghaus-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Perles_configuration" title="Perles configuration">Perles configuration</a>, a finite set of points and lines in the Euclidean plane that cannot be represented with rational coordinates</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-wiswell-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-wiswell_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wiswell_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-wiswell_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-wiswell_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-wiswell_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-wiswell_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-wiswell_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-wiswell_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-wiswell_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-wiswell_1-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-wiswell_1-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-wiswell_1-11"><sup><i><b>l</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWiswell2007" class="citation cs2">Wiswell, Laura (June 2007), "Review of <i>Divine Proportions</i>", <i><a href="/wiki/Proceedings_of_the_Edinburgh_Mathematical_Society" class="mw-redirect" title="Proceedings of the Edinburgh Mathematical Society">Proceedings of the Edinburgh Mathematical Society</a></i>, <b>50</b> (2): <span class="nowrap">509–</span>510, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0013091507215020">10.1017/S0013091507215020</a></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ProQuest" title="ProQuest">ProQuest</a> <a rel="nofollow" class="external text" href="https://www.proquest.com/docview/228292466">228292466</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Edinburgh+Mathematical+Society&rft.atitle=Review+of+Divine+Proportions&rft.volume=50&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E509-%3C%2Fspan%3E510&rft.date=2007-06&rft_id=info%3Adoi%2F10.1017%2FS0013091507215020&rft.aulast=Wiswell&rft.aufirst=Laura&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivine+Proportions%3A+Rational+Trigonometry+to+Universal+Geometry" class="Z3988"></span></span> </li> <li id="cite_note-gefter-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-gefter_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGefter2013" class="citation cs2">Gefter, Amanda (2013), "Mind-bending mathematics: Why infinity has to go", <i>New Scientist</i>, <b>219</b> (2930): <span class="nowrap">32–</span>35, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013NewSc.219...32G">2013NewSc.219...32G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0262-4079%2813%2962043-6">10.1016/s0262-4079(13)62043-6</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Scientist&rft.atitle=Mind-bending+mathematics%3A+Why+infinity+has+to+go&rft.volume=219&rft.issue=2930&rft.pages=%3Cspan+class%3D%22nowrap%22%3E32-%3C%2Fspan%3E35&rft.date=2013&rft_id=info%3Adoi%2F10.1016%2Fs0262-4079%2813%2962043-6&rft_id=info%3Abibcode%2F2013NewSc.219...32G&rft.aulast=Gefter&rft.aufirst=Amanda&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivine+Proportions%3A+Rational+Trigonometry+to+Universal+Geometry" class="Z3988"></span></span> </li> <li id="cite_note-leversha-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-leversha_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-leversha_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-leversha_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-leversha_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-leversha_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-leversha_3-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeversha2008" class="citation cs2">Leversha, Gerry (March 2008), "Review of <i>Divine Proportions</i>", <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>, <b>92</b> (523): <span class="nowrap">184–</span>186, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0025557200182944">10.1017/S0025557200182944</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27821758">27821758</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" 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