Sphericity - Revision history

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Sphericity is a measure of how spherical (round) an object is. As such, it is a specific example of a compactness measure of a shape. Defined by Wadell in 1935,[1] the sphericity, Ψ, of a particle is the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle:
<dl><dd><img class="tex" alt="\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}" src="http://upload.wikimedia.org/math/e/a/2/ea230750eba82fd8fe1b178ea651d242.png" /></dd></dl>
where Vp is volume of the particle and Ap is the surface area of the particle
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<h2>Ellipsoidal Objects</h2>
<dl><dd>See also: Earth radius</dd></dl>
The sphericity, Ψ, of an oblate spheroid (similar to the shape of the planet Earth) is defined as such:
<dl><dd><img class="tex" alt="\Psi =
\frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} =
\frac{2\sqrt[3]{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}" src="http://upload.wikimedia.org/math/f/2/6/f268c95b6388dc2002a77bc7224c190d.png" /></dd></dl>
(where a, b are the semi-major, semi-minor axes, respectively.
 
<h2>Derivation</h2>
Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.
First we need to write surface area of the sphere, As in terms of the volume of the particle, Vp
<dl><dd><img class="tex" alt="A_{s}^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac{4^2 \pi^2}{3^2} r^6\right) = 36 \pi \left(\frac{4 \pi}{3} r^3\right)^2 = 36\,\pi V_{p}^2" src="http://upload.wikimedia.org/math/8/2/7/827c80a821a4e9042325749cb7c461f7.png" /></dd></dl>
therefore
<dl><dd><img class="tex" alt="A_{s} = \left(36\,\pi V_{p}^2\right)^{\frac{1}{3}} = 36^{\frac{1}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = 6^{\frac{2}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}}" src="http://upload.wikimedia.org/math/8/6/c/86c49950a2c92504eabbd66a1336219b.png" /></dd></dl>
hence we define Ψ as:
<dl><dd><img class="tex" alt="\Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}}" src="http://upload.wikimedia.org/math/0/6/9/06986e86f9d53b2e4f3560f330909416.png" /></dd></dl>
 
<h2>Sphericity of common objects</h2>
<table style="MARGIN: 0pt auto; BORDER-COLLAPSE: collapse; TEXT-ALIGN: center" cellpadding="7" border="1">
<tbody>
<tr>
<th>Name</th>
<th>Picture</th>
<th>Volume</th>
<th>Area</th>
<th>Sphericity</th>
</tr>
<tr>
<td align="left" colspan="5">Platonic Solids</td>
</tr>
<tr>
<td>tetrahedron</td>
<td><img height="47" alt="Tetrahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/50px-Tetrahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\frac{\sqrt{2}}{12}\,s^3" src="http://upload.wikimedia.org/math/f/0/3/f03ea7115c243660a2ea99e73ad310db.png" /></td>
<td><img class="tex" alt="\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/a/6/0a6e48b85549c8b6512fdd3906ac8aa6.png" /></td>
<td><img class="tex" alt="\left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671" src="http://upload.wikimedia.org/math/c/e/9/ce99e8d11ec5357af8c21c1415f40cab.png" /></td>
</tr>
<tr>
<td>cube (hexahedron)</td>
<td><img height="56" alt="Hexahedron (cube)" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Hexahedron.jpg/50px-Hexahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\,s^3" src="http://upload.wikimedia.org/math/7/9/9/7998c630c83b898c1fdb72d667936996.png" /></td>
<td><img class="tex" alt="6\,s^2" src="http://upload.wikimedia.org/math/b/4/7/b472474af030c5d9cf3620a43c5e417b.png" /></td>
<td>
<img class="tex" alt="\left(
\frac{\pi}{6}
\right)^{\frac{1}{3}} \approx 0.806" src="http://upload.wikimedia.org/math/8/b/c/8bc7f285938e1b6ff146ed239698afce.png" />
</td>
</tr>
<tr>
<td>octahedron</td>
<td><img height="50" alt="Octahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/50px-Octahedron.svg.png" width="50" border="0" /></td>
<td><img class="tex" alt=" \frac{1}{3} \sqrt{2}\, s^3" src="http://upload.wikimedia.org/math/d/5/c/d5c9af02c97d15e54291f3ddce4d2211.png" /></td>
<td><img class="tex" alt=" 2 \sqrt{3}\, s^2" src="http://upload.wikimedia.org/math/2/9/3/2935e3febfb41a617e0fea6efc2bab02.png" /></td>
<td>
<img class="tex" alt="\left(
\frac{\pi}{3\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.846 " src="http://upload.wikimedia.org/math/b/4/c/b4cb491f3e93c2bcd8f345582be18ffa.png" />
</td>
</tr>
<tr>
<td>dodecahedron</td>
<td><img height="48" alt="Dodecahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/POV-Ray-Dodecahedron.svg/50px-POV-Ray-Dodecahedron.svg.png" width="50" border="0" /></td>
<td><img class="tex" alt=" \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/c/2/7/c2750faa2f4f0b5a934390dd3d135dd2.png" /></td>
<td><img class="tex" alt=" 3 \sqrt{25 + 10\sqrt{5}}\, s^2" src="http://upload.wikimedia.org/math/1/9/b/19bcc8e5c2da1d6a9c9290e3bbb55011.png" /></td>
<td>
<img class="tex" alt="\left(
\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}
\right)^{\frac{1}{3}} \approx 0.910" src="http://upload.wikimedia.org/math/b/2/1/b21c0ccd12191d494564898dffc6daf2.png" />
</td>
</tr>
<tr>
<td>icosahedron</td>
<td><img height="48" alt="Icosahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedron.jpg/50px-Icosahedron.jpg" width="50" border="0" /></td>
<td><img class="tex" alt="\frac{5}{12}\left(3+\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/7/7/e/77ee1cd7a4858ddfa2a994c29d7d2db5.png" /></td>
<td><img class="tex" alt="5\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/f/5/0f589b4ae2f26ce5c4b29705a02d3498.png" /></td>
<td><img class="tex" alt="\left(
\frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.939" src="http://upload.wikimedia.org/math/6/f/7/6f78ec8588faabadd52bc5946ebbbe58.png" /></td>
</tr>
<tr>
<td align="left" colspan="5">Round Shapes</td>
</tr>
<tr>
<td>ideal cone 
<img class="tex" alt="(h=2\sqrt{2}r)" src="http://upload.wikimedia.org/math/f/a/0/fa0b47385960369df693112cc666e210.png" /></td>
<td> </td>
<td><img class="tex" alt="\frac{1}{3} \pi\, r^2 h " src="http://upload.wikimedia.org/math/c/b/b/cbb69401f7fae091004a0b025b5136fe.png" /> 
<img class="tex" alt="= \frac{2\sqrt{2}}{3} \pi\, r^3" src="http://upload.wikimedia.org/math/3/0/0/300e3da8ae4b0341c28ef5e767eed9d3.png" />
</td>
<td><img class="tex" alt="\pi\, r (r + \sqrt{r^2 + h^2}) " src="http://upload.wikimedia.org/math/d/0/3/d03ef3d12512676dbe6b51191c06be4b.png" /> 
<img class="tex" alt="= 4 \pi\, r^2 " src="http://upload.wikimedia.org/math/9/d/b/9db0629297845830f39774970ed03073.png" />
</td>
<td><img class="tex" alt="\left(
\frac{1}{2}
\right)^{\frac{1}{3}} \approx 0.794" src="http://upload.wikimedia.org/math/4/9/7/497312086e2798aba0f7232e6f3d3278.png" /></td>
</tr>
<tr>
<td>hemisphere 
(half sphere)</td>
<td> </td>
<td><img class="tex" alt="\frac{2}{3} \pi\, r^3" src="http://upload.wikimedia.org/math/d/e/5/de599428e7c43ba81b82e045513f8662.png" /></td>
<td><img class="tex" alt="3 \pi\, r^2" src="http://upload.wikimedia.org/math/7/1/d/71dce90d7880a8b7dfb4aeebf1df1b9a.png" /></td>
<td>
<img class="tex" alt="\left(
\frac{16}{27}
\right)^{\frac{1}{3}} \approx 0.840" src="http://upload.wikimedia.org/math/6/a/e/6aea0187b6dd454098ce2ee883db2c55.png" />
</td>
</tr>
<tr>
<td>ideal cylinder 
<img class="tex" alt="(h=2\,r)" src="http://upload.wikimedia.org/math/2/6/d/26d4795ee1f02913f0dd9c0f4a0a68f5.png" /></td>
<td> </td>
<td><img class="tex" alt="\pi r^2 h = 2 \pi\,r^3" src="http://upload.wikimedia.org/math/f/9/f/f9f38314067649347ddc6c18a1c93232.png" /></td>
<td><img class="tex" alt="2 \pi r ( r + h ) = 6 \pi\,r^2" src="http://upload.wikimedia.org/math/8/8/0/88001809b1d9aaeb1327584f33baa711.png" /></td>
<td>
<img class="tex" alt="\left(
\frac{2}{3}
\right)^{\frac{1}{3}} \approx 0.874" src="http://upload.wikimedia.org/math/5/c/2/5c2b5f728c566f1f1913ed4f8279d46e.png" />
</td>
</tr>
<tr>
<td>ideal torus 
(R = r)</td>
<td> </td>
<td><img class="tex" alt="2 \pi^2 R r^2 = 2 \pi^2 \,r^3" src="http://upload.wikimedia.org/math/5/c/0/5c063b1a9e01901df957ece8ace81fd0.png" /></td>
<td><img class="tex" alt="4 \pi^2 R r = 4 \pi^2\,r^2" src="http://upload.wikimedia.org/math/1/4/0/14051c34379da213ca3b815a26e82e45.png" /></td>
<td>
<img class="tex" alt="\left(
\frac{9}{4 \pi}
\right)^{\frac{1}{3}} \approx 0.894" src="http://upload.wikimedia.org/math/c/b/4/cb4dfdaded145ac88dbe0d646a39f875.png" />
</td>
</tr>
<tr>
<td>sphere</td>
<td> </td>
<td><img class="tex" alt="\frac{4}{3} \pi r^3" src="http://upload.wikimedia.org/math/9/3/a/93a32e52c9580c5f627ace8dc3ad6397.png" /></td>
<td><img class="tex" alt="4 \pi\,r^2" src="http://upload.wikimedia.org/math/1/0/9/109a7c5ee8613b982dc98a9d2f0a590d.png" /></td>
<td>
<img class="tex" alt="1\," src="http://upload.wikimedia.org/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" />
</td>
</tr>
</tbody>
</table>
 
<h2>Sphericity in Statistics</h2>
In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is the statistical test used to evaluate sphericity.
 
<h2>References</h2>
<ol class="references">
<li id="cite_note-0">^ <cite style="FONT-STYLE: normal">Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". Journal of Geology 43: 250–280.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Volume%2C+Shape+and+Roundness+of+Quartz+Particles&rft.jtitle=Journal+of+Geology&rft.date=1935&rft.volume=43&rft.aulast=Wadell&rft.aufirst=Hakon&rft.pages=250%E2%80%93280"> </li>
</ol>
 
<h2>See also</h2>
<ul>
<li>Rounding (sediment)</li>
</ul>
<a id="External_links" name="External_links"></a>
<h2>External links</h2>
<ul>
<li><a class="external text" title="http://www.howround.com/" href="http://www.howround.com/" rel="nofollow">How round is your circle?</a></li>
<li><a class="external text" title="http://people.uncw.edu/dockal/gly312/grains/grains.htm" href="http://people.uncw.edu/dockal/gly312/grains/grains.htm" rel="nofollow">Grain Morphology: Roundness, Surface Features, and Sphericity of Grains</a></li>
</ul>

@@ Line 154: / Line 154: @@
 </ol>
 <p>&nbsp;</p>
-<h2><span class="mw-headline">See also</span></h2>
+<p><span class="mw-headline"><font size="5">See also</font></span></p>
 <ul>
      <li>Rounding (sediment) </li>
 </ul>
-<p><a id="External_links" name="External_links"></a></p>
+<p>&nbsp;</p>
-<h2><span class="mw-headline">External links</span></h2>
+<p><span class="mw-headline"><font size="5">External links</font></span></p>
 <ul>
      <li><a class="external text" title="http://www.howround.com/" rel="nofollow" href="http://www.howround.com/">How round is your circle?</a> </li>
      <li><a class="external text" title="http://people.uncw.edu/dockal/gly312/grains/grains.htm" rel="nofollow" href="http://people.uncw.edu/dockal/gly312/grains/grains.htm">Grain Morphology: Roundness, Surface Features, and Sphericity of Grains</a> </li>
 </ul>

@@ Line 2: / Line 2: @@
 <dl><dd><img class="tex" alt="\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}" src="http://upload.wikimedia.org/math/e/a/2/ea230750eba82fd8fe1b178ea651d242.png" /></dd></dl>
 <p>where <span class="texhtml"><em>V</em><sub><em>p</em></sub></span> is volume of the particle and <span class="texhtml"><em>A</em><sub><em>p</em></sub></span> is the surface area of the particle</p>
 <p>&nbsp;</p>
-<h2><span class="mw-headline">Ellipsoidal Objects</span></h2>
+<p><span class="mw-headline"><font size="5">Ellipsoidal Objects</font></span></p>
 <p>The sphericity, <span class="texhtml">&Psi;</span>, of an oblate spheroid (similar to the shape of the planet Earth) is defined as such:</p>
 <dl><dd><img class="tex" alt="\Psi =
@@ Line 12: / Line 10: @@
 <p><em>(where a, b are the semi-major, semi-minor axes, respectively.</em></p>
 <p>&nbsp;</p>
-<h2><span class="mw-headline">Derivation</span></h2>
+<p><span class="mw-headline"><font size="5">Derivation</font></span></p>
 <p>Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.</p>
 <p>First we need to write surface area of the sphere, <span class="texhtml"><em>A</em><sub><em>s</em></sub></span> in terms of the volume of the particle, <span class="texhtml"><em>V</em><sub><em>p</em></sub></span></p>
@@ Line 21: / Line 19: @@
 <dl><dd><img class="tex" alt="\Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}}" src="http://upload.wikimedia.org/math/0/6/9/06986e86f9d53b2e4f3560f330909416.png" /></dd></dl>
 <p>&nbsp;</p>
-<h2><span class="mw-headline">Sphericity of common objects</span></h2>
+<p><span class="mw-headline"><font size="5">Sphericity of common objects</font></span></p>
 <table style="MARGIN: 0pt auto; BORDER-COLLAPSE: collapse; TEXT-ALIGN: center" cellpadding="7" border="1">
      <tbody>
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 </table>
 <p>&nbsp;</p>
-<h2><span class="mw-headline">Sphericity in Statistics</span></h2>
+<p><span class="mw-headline"><font size="5">Sphericity in Statistics</font></span></p>
 <p>In statistical analyses, sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. This is an assumption of an ANOVA with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. Mauchly's sphericity test is the statistical test used to evaluate sphericity.</p>
 <p>&nbsp;</p>
-<h2><span class="mw-headline">References</span></h2>
+<p><span class="mw-headline"><font size="5">References</font></span></p>
 <ol class="references">
      <li id="cite_note-0"><strong>^</strong> <cite style="FONT-STYLE: normal">Wadell, Hakon (1935). &quot;Volume, Shape and Roundness of Quartz Particles&quot;. <em>Journal of Geology</em> <strong>43</strong>: 250&ndash;280.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Volume%2C+Shape+and+Roundness+of+Quartz+Particles&amp;rft.jtitle=Journal+of+Geology&amp;rft.date=1935&amp;rft.volume=43&amp;rft.aulast=Wadell&amp;rft.aufirst=Hakon&amp;rft.pages=250%E2%80%93280">&nbsp;</span> </li>

@@ Line 2: / Line 2: @@
 <dl><dd><img class="tex" alt="\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}" src="http://upload.wikimedia.org/math/e/a/2/ea230750eba82fd8fe1b178ea651d242.png" /></dd></dl>
 <p>where <span class="texhtml"><em>V</em><sub><em>p</em></sub></span> is volume of the particle and <span class="texhtml"><em>A</em><sub><em>p</em></sub></span> is the surface area of the particle</p>
-<script type="text/javascript">
 <p>&nbsp;</p>
 <h2><span class="mw-headline">Ellipsoidal Objects</span></h2>
@@ Line 40: / Line 36: @@
          <tr>
              <td>tetrahedron</td>
-             <td><img height="47" alt="Tetrahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/50px-Tetrahedron.jpg" width="50" border="0" /></td>
+             <td><img height="47" alt="Tetrahedron" width="50" border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tetrahedron.jpg/50px-Tetrahedron.jpg" /></td>
              <td><img class="tex" alt="\frac{\sqrt{2}}{12}\,s^3" src="http://upload.wikimedia.org/math/f/0/3/f03ea7115c243660a2ea99e73ad310db.png" /></td>
              <td><img class="tex" alt="\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/a/6/0a6e48b85549c8b6512fdd3906ac8aa6.png" /></td>
@@ Line 47: / Line 43: @@
          <tr>
              <td>cube (hexahedron)</td>
-             <td><img height="56" alt="Hexahedron (cube)" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Hexahedron.jpg/50px-Hexahedron.jpg" width="50" border="0" /></td>
+             <td><img height="56" alt="Hexahedron (cube)" width="50" border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Hexahedron.jpg/50px-Hexahedron.jpg" /></td>
              <td><img class="tex" alt="\,s^3" src="http://upload.wikimedia.org/math/7/9/9/7998c630c83b898c1fdb72d667936996.png" /></td>
              <td><img class="tex" alt="6\,s^2" src="http://upload.wikimedia.org/math/b/4/7/b472474af030c5d9cf3620a43c5e417b.png" /></td>
@@ Line 58: / Line 54: @@
          <tr>
              <td>octahedron</td>
-             <td><img height="50" alt="Octahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/50px-Octahedron.svg.png" width="50" border="0" /></td>
+             <td><img height="50" alt="Octahedron" width="50" border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Octahedron.svg/50px-Octahedron.svg.png" /></td>
              <td><img class="tex" alt=" \frac{1}{3} \sqrt{2}\, s^3" src="http://upload.wikimedia.org/math/d/5/c/d5c9af02c97d15e54291f3ddce4d2211.png" /></td>
              <td><img class="tex" alt=" 2 \sqrt{3}\, s^2" src="http://upload.wikimedia.org/math/2/9/3/2935e3febfb41a617e0fea6efc2bab02.png" /></td>
@@ Line 69: / Line 65: @@
          <tr>
              <td>dodecahedron</td>
-             <td><img height="48" alt="Dodecahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/POV-Ray-Dodecahedron.svg/50px-POV-Ray-Dodecahedron.svg.png" width="50" border="0" /></td>
+             <td><img height="48" alt="Dodecahedron" width="50" border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/POV-Ray-Dodecahedron.svg/50px-POV-Ray-Dodecahedron.svg.png" /></td>
              <td><img class="tex" alt=" \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/c/2/7/c2750faa2f4f0b5a934390dd3d135dd2.png" /></td>
              <td><img class="tex" alt=" 3 \sqrt{25 + 10\sqrt{5}}\, s^2" src="http://upload.wikimedia.org/math/1/9/b/19bcc8e5c2da1d6a9c9290e3bbb55011.png" /></td>
@@ Line 80: / Line 76: @@
          <tr>
              <td>icosahedron</td>
-             <td><img height="48" alt="Icosahedron" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedron.jpg/50px-Icosahedron.jpg" width="50" border="0" /></td>
+             <td><img height="48" alt="Icosahedron" width="50" border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Icosahedron.jpg/50px-Icosahedron.jpg" /></td>
              <td><img class="tex" alt="\frac{5}{12}\left(3+\sqrt{5}\right)\, s^3" src="http://upload.wikimedia.org/math/7/7/e/77ee1cd7a4858ddfa2a994c29d7d2db5.png" /></td>
              <td><img class="tex" alt="5\sqrt{3}\,s^2" src="http://upload.wikimedia.org/math/0/f/5/0f589b4ae2f26ce5c4b29705a02d3498.png" /></td>
@@ Line 157: / Line 153: @@
 <h2><span class="mw-headline">References</span></h2>
 <ol class="references">
-     <li id="cite_note-0"><strong>^</strong> <cite style="FONT-STYLE: normal">Wadell, Hakon (1935). &quot;Volume, Shape and Roundness of Quartz Particles&quot;. <em>Journal of Geology</em> <strong>43</strong>: 250&ndash;280.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Volume%2C+Shape+and+Roundness+of+Quartz+Particles&amp;rft.jtitle=Journal+of+Geology&amp;rft.date=1935&amp;rft.volume=43&amp;rft.aulast=Wadell&amp;rft.aufirst=Hakon&amp;rft.pages=250%E2%80%93280">&nbsp;</span></li>
+     <li id="cite_note-0"><strong>^</strong> <cite style="FONT-STYLE: normal">Wadell, Hakon (1935). &quot;Volume, Shape and Roundness of Quartz Particles&quot;. <em>Journal of Geology</em> <strong>43</strong>: 250&ndash;280.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Volume%2C+Shape+and+Roundness+of+Quartz+Particles&amp;rft.jtitle=Journal+of+Geology&amp;rft.date=1935&amp;rft.volume=43&amp;rft.aulast=Wadell&amp;rft.aufirst=Hakon&amp;rft.pages=250%E2%80%93280">&nbsp;</span> </li>
 </ol>
 <p>&nbsp;</p>
 <h2><span class="mw-headline">See also</span></h2>
 <ul>
-     <li>Rounding (sediment)</li>
+     <li>Rounding (sediment) </li>
 </ul>
 <p><a id="External_links" name="External_links"></a></p>
 <h2><span class="mw-headline">External links</span></h2>
 <ul>
-     <li><a class="external text" title="http://www.howround.com/" href="http://www.howround.com/" rel="nofollow">How round is your circle?</a></li>
+     <li><a class="external text" title="http://www.howround.com/" rel="nofollow" href="http://www.howround.com/">How round is your circle?</a> </li>
-     <li><a class="external text" title="http://people.uncw.edu/dockal/gly312/grains/grains.htm" href="http://people.uncw.edu/dockal/gly312/grains/grains.htm" rel="nofollow">Grain Morphology: Roundness, Surface Features, and Sphericity of Grains</a></li>
+     <li><a class="external text" title="http://people.uncw.edu/dockal/gly312/grains/grains.htm" rel="nofollow" href="http://people.uncw.edu/dockal/gly312/grains/grains.htm">Grain Morphology: Roundness, Surface Features, and Sphericity of Grains</a> </li>
 </ul>