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<title>Angular defect</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Angular defect</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Geometry" title="Geometry">geometry</a>, the <b>angular defect</b> or simply <b>defect</b> (also called <b>deficit</b> or <b>deficiency</b>) is the failure of some <a href="Angle" title="Angle">angles</a> to add up to the expected amount of 360° or 180°, when such angles in the <a href="Euclidean_plane" title="Euclidean plane">Euclidean plane</a> would. The opposite notion is the <a href="Angle_excess" class="mw-redirect" title="Angle excess"><i>excess</i></a>.

</p><p>Classically the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while <a href="Internal_and_external_angle" class="mw-redirect" title="Internal and external angle">interior angles</a> in a triangle add up to 180°. However, on a <a href="Polyhedron" title="Polyhedron">convex polyhedron</a>, the angles of the faces meeting at a vertex add up to <i>less</i> than 360° (a defect), while the angles at some vertices of a <a href="Nonconvex_polyhedron" class="mw-redirect" title="Nonconvex polyhedron">nonconvex polyhedron</a> may add up to <i>more</i> than 360° (an excess). Also the angles in a <a href="Hyperbolic_triangle" title="Hyperbolic triangle">hyperbolic triangle</a> add up to <i>less</i> than 180° (a defect), while those on a <a href="Spherical_triangle" class="mw-redirect" title="Spherical triangle">spherical triangle</a> add up to <i>more</i> than 180° (an excess).

</p><p>In modern terms, the defect at a vertex is a discrete version of the <a href="Gaussian_curvature" title="Gaussian curvature">curvature</a> of the polyhedral surface <a href="Dirac_delta_function" title="Dirac delta function">concentrated at that point</a>. Negative defect indicates that the vertex resembles a <a href="Saddle_point" title="Saddle point">saddle point</a> (negative curvature), whereas positive defect indicates that the vertex resembles a <a href="Local_maximum" class="mw-redirect" title="Local maximum">local maximum</a> or minimum (positive curvature). The <a href="Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a> gives the total curvature as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math></span><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"></span> times the <a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi =2}">

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<annotation encoding="application/x-tex">{\displaystyle \chi =2}</annotation>

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</math></span><img src="./91ac32a862c154b73c827aa05ed431ac2d897249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.716ex; height:2.509ex;" alt="{\displaystyle \chi =2}" loading="lazy"></span>, so for a convex polyhedron the sum of the defects is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 4\pi }</annotation>

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</math></span><img src="./057444bf35a0c22b19bcae1ef06e06ecdf8abe56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 4\pi }" loading="lazy"></span>, while a <a href="Toroidal_polyhedron" title="Toroidal polyhedron">toroidal polyhedron</a> has <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi =0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \chi =0}</annotation>

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</math></span><img src="./fa2a52c29ba6859766c02e88299b3114d010e3b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.716ex; height:2.509ex;" alt="{\displaystyle \chi =0}" loading="lazy"></span> and total defect zero.

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<div class="mw-heading mw-heading2"><h2 id="Defect_of_a_vertex">Defect of a vertex</h2></div>

<p>For a <a href="Polyhedron" title="Polyhedron">polyhedron</a>, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full <a href="Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a>, as occurs in some vertices of many non-convex polyhedra, then the defect is negative.

</p><p>The concept of defect extends to higher dimensions as the amount by which the sum of the <a href="Dihedral_angle" title="Dihedral angle">dihedral angles</a> of the <a href="Cell_(geometry)" class="mw-redirect" title="Cell (geometry)">cells</a> at a <a href="Peak_(mathematics)" class="mw-redirect" title="Peak (mathematics)">peak</a> falls short of a full circle.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>The defect of any of the vertices of a regular <a href="Dodecahedron" title="Dodecahedron">dodecahedron</a> (in which three regular <a href="Pentagon" title="Pentagon">pentagons</a> meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.

</p><p>The same procedure can be followed for the other <a href="Platonic_solid" title="Platonic solid">Platonic solids</a>:

</p>

<th>Shape

</th>

<th>Number of vertices

</th>

<th>Polygons meeting at each vertex

</th>

<th>Defect at each vertex

</th>

<th>Total defect

</th></tr>

<tr>

<td><a href="Tetrahedron" title="Tetrahedron">tetrahedron</a></td>

<td>Three equilateral triangles</td>

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \pi \ \ (180^{\circ })}</annotation>

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</math></span><img src="./60fb2a00883eadfbafc5c34c357d794a84f420e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.844ex; height:2.843ex;" alt="{\displaystyle \pi \ \ (180^{\circ })}" loading="lazy"></span></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi \ \ (720^{\circ })}</annotation>

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</math></span><img src="./0c210818cf1f5c9178ec3455242c4a073b237979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle 4\pi \ \ (720^{\circ })}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Octahedron" title="Octahedron">octahedron</a></td>

<td>Four equilateral triangles</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {2\pi \over 3}\ (120^{\circ })}</annotation>

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</math></span><img src="./15d00b62618b173cea6aac0a196a3ebd9a300f91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.262ex; height:5.176ex;" alt="{\displaystyle {2\pi \over 3}\ (120^{\circ })}" loading="lazy"></span></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi \ \ (720^{\circ })}</annotation>

</semantics>

</math></span><img src="./0c210818cf1f5c9178ec3455242c4a073b237979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle 4\pi \ \ (720^{\circ })}" loading="lazy"></span>

</td></tr>

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<td>Three squares</td>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\pi \over 2}\ \ (90^{\circ })}</annotation>

</semantics>

</math></span><img src="./a88258e4ca28fac41807e69007ea6927194a39d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.518ex; height:4.676ex;" alt="{\displaystyle {\pi \over 2}\ \ (90^{\circ })}" loading="lazy"></span></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi \ \ (720^{\circ })}</annotation>

</semantics>

</math></span><img src="./0c210818cf1f5c9178ec3455242c4a073b237979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle 4\pi \ \ (720^{\circ })}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Icosahedron" title="Icosahedron">icosahedron</a></td>

<td>Five equilateral triangles</td>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\pi \over 3}\ \ (60^{\circ })}</annotation>

</semantics>

</math></span><img src="./87722824fb47ef7218a081f586135ea930628e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.518ex; height:4.676ex;" alt="{\displaystyle {\pi \over 3}\ \ (60^{\circ })}" loading="lazy"></span></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi \ \ (720^{\circ })}</annotation>

</semantics>

</math></span><img src="./0c210818cf1f5c9178ec3455242c4a073b237979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle 4\pi \ \ (720^{\circ })}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Dodecahedron" title="Dodecahedron">dodecahedron</a></td>

<td>Three regular pentagons</td>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\pi \over 5}\ \ (36^{\circ })}</annotation>

</semantics>

</math></span><img src="./8cafea520cf6341453404a31825f3f29f66dc472.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.518ex; height:4.676ex;" alt="{\displaystyle {\pi \over 5}\ \ (36^{\circ })}" loading="lazy"></span></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi \ \ (720^{\circ })}</annotation>

</semantics>

</math></span><img src="./0c210818cf1f5c9178ec3455242c4a073b237979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle 4\pi \ \ (720^{\circ })}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Descartes's_theorem">Descartes's theorem</h2></div>

<p>Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a> to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4<span class="texhtml mvar" style="font-style:italic;">π</span> radians). The polyhedron need not be convex.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>A generalization says the number of circles in the total defect equals the <a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of the polyhedron. This is a special case of the <a href="Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a> which relates the integral of the <a href="Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a> to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the curvature is zero (the surface is locally <a href="Isometry" title="Isometry">isometric</a> to a Euclidean plane) and the integral of curvature at a vertex is equal to the defect there (by definition).

</p><p>This can be used to calculate the number <i>V</i> of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect (which is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math></span><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"></span> times the Euler characteristic). This total will have one complete circle for every vertex in the polyhedron.

</p><p>A converse to Descartes' theorem is given by <a href="Alexandrov's_uniqueness_theorem" class="mw-redirect" title="Alexandrov's uniqueness theorem">Alexandrov's uniqueness theorem</a>, according to which a metric space that is locally Euclidean (hence zero curvature) except for a finite number of points of positive angular defect, adding to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi }</annotation>

</semantics>

</math></span><img src="./057444bf35a0c22b19bcae1ef06e06ecdf8abe56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 4\pi }" loading="lazy"></span>, can be realized in a unique way as the surface of a convex polyhedron.

</p>

<div class="mw-heading mw-heading2"><h2 id="Positive_defects_on_non-convex_figures">Positive defects on non-convex figures</h2></div>

<p>It is tempting to think that every non-convex polyhedron must have some vertices whose defect is negative, but this need not be the case if the Euler characteristic is positive (a topological sphere).

</p>

<caption>Polyhedra with positive defects

</caption>

</td>

</td></tr></tbody></table>

<p>A counterexample is provided by a <a href="Cube" title="Cube">cube</a> where one face is replaced by a <a href="Square_pyramid" title="Square pyramid">square pyramid</a>: this <a href="Elongated_square_pyramid" title="Elongated square pyramid">elongated square pyramid</a> is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same and so are all positive.

</p><p>Two counterexamples which are self-intersecting polyhedra are the <a href="Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a> and the <a href="Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a>, with twelve and twenty convex points respectively, all with positive defects.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a>, <i>Progymnasmata de solidorum elementis</i>, in <i>Oeuvres de Descartes</i>, vol. X, pp. 265–276</span>

</li>

</ol></div>

<div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3></div>

<ul><li><a href="David_Richeson" title="David Richeson">Richeson, D.</a>; <i><a href="Euler's_Gem" title="Euler's Gem">Euler's Gem: The Polyhedron Formula and the Birth of Topology</a></i>, Princeton (2008), Pages 220–225.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/defect" class="extiw external" title="wiktionary:defect">defect</a></b></i> in Wiktionary, the free dictionary.</div></div>

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AngularDefect.html">"Angular defect"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul></div><div><div class="zim-footer">

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