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Clebsch surface
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<h2 id="definition">Definition</h2> <p>The Clebsch surface is the set of points (<i>x</i>0:<i>x</i>1:<i>x</i>2:<i>x</i>3:<i>x</i>4) of <a href="/facts/Projective_space/7MSpA9cM">P4</a> satisfying the equations </p> x 0 + x 1 + x 2 + x 3 + x 4 = 0 , {\displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0,} x 0 3 + x 1 3 + x 2 3 + x 3 3 + x 4 3 = 0. {\displaystyle x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0.} <p>Eliminating <i>x</i>0 shows that it is also isomorphic to the surface </p> x 1 3 + x 2 3 + x 3 3 + x 4 3 = ( x 1 + x 2 + x 3 + x 4 ) 3 {\displaystyle x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=(x_{1}+x_{2}+x_{3}+x_{4})^{3}} <p>in P3. In ℝ3, it can be represented by<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <table><tbody><tr><td> 81 ( x 3 + y 3 + z 3 ) − 189 ( x 2 ( y + z ) + y 2 ( z + x ) + z 2 ( x + y ) ) + + 54 x y z + 126 ( x y + y z + z x ) − 9 ( x 2 + y 2 + z 2 ) − 9 ( x + y + z ) + 1 = 0. {\displaystyle {\begin{array}{c}81\left(x^{3}+y^{3}+z^{3}\right)-189\left(x^{2}\left(y+z\right)+y^{2}\left(z+x\right)+z^{2}\left(x+y\right)\right)+\\+54xyz+126\left(xy+yz+zx\right)-9\left(x^{2}+y^{2}+z^{2}\right)-9\left(x+y+z\right)+1=0.\end{array}}} </td> <td></td> <td>1</td></tr></tbody></table> <h2 id="properties">Properties</h2> <p>The symmetry group of the Clebsch surface is the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> <i>S</i>5 of order 120, acting by permutations of the coordinates (in <i>P</i>4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group. </p><p>The 27 exceptional lines are: </p> <ul><li>The 15 images (under <i>S</i>5) of the line of points of the form (<i>a</i> : −<i>a</i> : <i>b</i> : −<i>b</i> : 0).</li> <li>The 12 images of the line though the point (1:ζ: ζ2: ζ3: ζ4) and its complex conjugate, where ζ is a primitive 5th root of 1.</li></ul> <p>The surface has 10 <a href="/facts/Eckardt_point/jCvavESm">Eckardt points</a> where 3 lines meet, given by the point (1: −1: 0: 0: 0) and its conjugates under permutations. Hirzebruch (1976) showed that the surface obtained by blowing up the Clebsch surface in its 10 Eckardt points is the <a href="/facts/Hilbert_modular_surface/TmoIXYs8">Hilbert modular surface</a> of the level 2 principal congruence subgroup of the Hilbert modular group of the field <i>Q</i>(√5). The quotient of the Hilbert modular group by its level 2 congruence subgroup is isomorphic to the alternating group of order 60 on 5 points. </p><p>Like all nonsingular cubic surfaces, the Clebsch cubic can be obtained by blowing up the <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a> in 6 points. Klein (1873) described these points as follows. If the projective plane is identified with the set of lines through the origin in a 3-dimensional vector space containing an <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a> centered at the origin, then the 6 points correspond to the 6 lines through the icosahedron's 12 vertices. The Eckardt points correspond to the 10 lines through the centers of the 20 faces. </p> <h3>Explicit description of lines</h3> <p>Using the embedding (1), the 27 lines are given by ⟨<i>a</i>,<i>b</i>,<i>c</i>⟩ <i>t</i> + ⟨<i>p</i>,<i>q</i>,<i>r</i>⟩, where a, b, c, p, q, and r are all taken from the same row in the following table:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <table><tbody><tr><th>a</th><th>b</th><th>c</th><th>p</th><th>q</th><th>r</th></tr><tr><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td><td> 0 {\displaystyle 0} </td><td> 0 {\displaystyle 0} </td><td> − 1 3 {\displaystyle -{\frac {1}{3}}} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td><td> − 1 3 {\displaystyle -{\frac {1}{3}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> − 1 3 {\displaystyle -{\frac {1}{3}}} </td><td> 0 {\displaystyle 0} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td><td> − 1 {\displaystyle -1} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td><td> − 1 {\displaystyle -1} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td></tr><tr><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td></tr><tr><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td><td> 1 3 {\displaystyle {\frac {1}{3}}} </td></tr><tr><td> 3 {\displaystyle 3} </td><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td></tr><tr><td> 3 {\displaystyle 3} </td><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 0 {\displaystyle 0} </td><td> 3 {\displaystyle 3} </td><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> 3 {\displaystyle 3} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> 3 {\displaystyle 3} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td></tr><tr><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td><td> 3 {\displaystyle 3} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td><td> 0 {\displaystyle 0} </td><td> 1 6 {\displaystyle {\frac {1}{6}}} </td></tr><tr><td> 1 + 3 5 {\displaystyle 1+{\frac {3}{\sqrt {5}}}} </td><td> − 1 5 {\displaystyle -{\frac {1}{\sqrt {5}}}} </td><td> 1 {\displaystyle 1} </td><td> 5 + 5 30 {\displaystyle {\frac {5+{\sqrt {5}}}{30}}} </td><td> 5 + 3 5 30 {\displaystyle {\frac {5+3{\sqrt {5}}}{30}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 1 5 {\displaystyle -{\frac {1}{\sqrt {5}}}} </td><td> 1 + 3 5 {\displaystyle 1+{\frac {3}{\sqrt {5}}}} </td><td> 1 {\displaystyle 1} </td><td> 5 + 3 5 30 {\displaystyle {\frac {5+3{\sqrt {5}}}{30}}} </td><td> 5 + 5 30 {\displaystyle {\frac {5+{\sqrt {5}}}{30}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 3 − 5 {\displaystyle -3-{\sqrt {5}}} </td><td> − 5 {\displaystyle -{\sqrt {5}}} </td><td> 1 {\displaystyle 1} </td><td> 7 + 3 5 6 {\displaystyle {\frac {7+3{\sqrt {5}}}{6}}} </td><td> 3 + 5 6 {\displaystyle {\frac {3+{\sqrt {5}}}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 3 + 5 4 {\displaystyle {\frac {-3+{\sqrt {5}}}{4}}} </td><td> − 5 + 3 5 4 {\displaystyle {\frac {-5+3{\sqrt {5}}}{4}}} </td><td> 1 {\displaystyle 1} </td><td> 3 + 5 12 {\displaystyle {\frac {3+{\sqrt {5}}}{12}}} </td><td> 1 − 5 12 {\displaystyle {\frac {1-{\sqrt {5}}}{12}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 5 − 3 5 4 {\displaystyle {\frac {-5-3{\sqrt {5}}}{4}}} </td><td> − 3 − 5 4 {\displaystyle {\frac {-3-{\sqrt {5}}}{4}}} </td><td> 1 {\displaystyle 1} </td><td> 1 + 5 12 {\displaystyle {\frac {1+{\sqrt {5}}}{12}}} </td><td> 3 − 5 12 {\displaystyle {\frac {3-{\sqrt {5}}}{12}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 5 {\displaystyle {\sqrt {5}}} </td><td> − 3 + 5 {\displaystyle -3+{\sqrt {5}}} </td><td> 1 {\displaystyle 1} </td><td> 3 − 5 6 {\displaystyle {\frac {3-{\sqrt {5}}}{6}}} </td><td> 7 − 3 5 6 {\displaystyle {\frac {7-3{\sqrt {5}}}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 5 {\displaystyle -{\sqrt {5}}} </td><td> − 3 − 5 {\displaystyle -3-{\sqrt {5}}} </td><td> 1 {\displaystyle 1} </td><td> 3 + 5 6 {\displaystyle {\frac {3+{\sqrt {5}}}{6}}} </td><td> 7 + 3 5 6 {\displaystyle {\frac {7+3{\sqrt {5}}}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 5 + 3 5 4 {\displaystyle {\frac {-5+3{\sqrt {5}}}{4}}} </td><td> − 3 + 5 4 {\displaystyle {\frac {-3+{\sqrt {5}}}{4}}} </td><td> 1 {\displaystyle 1} </td><td> 1 − 5 12 {\displaystyle {\frac {1-{\sqrt {5}}}{12}}} </td><td> 3 + 5 12 {\displaystyle {\frac {3+{\sqrt {5}}}{12}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 3 − 5 4 {\displaystyle {\frac {-3-{\sqrt {5}}}{4}}} </td><td> − 5 − 3 5 4 {\displaystyle {\frac {-5-3{\sqrt {5}}}{4}}} </td><td> 1 {\displaystyle 1} </td><td> 3 − 5 12 {\displaystyle {\frac {3-{\sqrt {5}}}{12}}} </td><td> 1 + 5 12 {\displaystyle {\frac {1+{\sqrt {5}}}{12}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> − 3 + 5 {\displaystyle -3+{\sqrt {5}}} </td><td> 5 {\displaystyle {\sqrt {5}}} </td><td> 1 {\displaystyle 1} </td><td> 7 − 3 5 6 {\displaystyle {\frac {7-3{\sqrt {5}}}{6}}} </td><td> 3 − 5 6 {\displaystyle {\frac {3-{\sqrt {5}}}{6}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 1 5 {\displaystyle {\frac {1}{\sqrt {5}}}} </td><td> 1 − 3 5 {\displaystyle 1-{\frac {3}{\sqrt {5}}}} </td><td> 1 {\displaystyle 1} </td><td> 5 − 3 5 30 {\displaystyle {\frac {5-3{\sqrt {5}}}{30}}} </td><td> 5 − 5 30 {\displaystyle {\frac {5-{\sqrt {5}}}{30}}} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 1 − 3 5 {\displaystyle 1-{\frac {3}{\sqrt {5}}}} </td><td> 1 5 {\displaystyle {\frac {1}{\sqrt {5}}}} </td><td> 1 {\displaystyle 1} </td><td> 5 − 5 30 {\displaystyle {\frac {5-{\sqrt {5}}}{30}}} </td><td> 5 − 3 5 30 {\displaystyle {\frac {5-3{\sqrt {5}}}{30}}} </td><td> 0 {\displaystyle 0} </td></tr></tbody></table> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Alfred_Clebsch/NV4IyEAk">Clebsch, A.</a> (1871), "Ueber die Anwendung der quadratischen Substitution auf die Gleichungen 5ten Grades und die geometrische Theorie des ebenen Fünfseits", <i>Mathematische Annalen</i>, 4 (2): 284–345, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01442599">10.1007/BF01442599</a></li> <li><a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Hirzebruch, Friedrich</a> (1976), "The Hilbert modular group for the field Q(√5), and the cubic diagonal surface of Clebsch and Klein", <i>Russian Math. Surveys</i>, 31 (5): 96–110, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1070%2FRM1976v031n05ABEH004190">10.1070/RM1976v031n05ABEH004190</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0042-1316">0042-1316</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0498397">0498397</a></li> <li>Hunt, Bruce (1996), <i>The Geometry of some special Arithmetic Quotients</i>, Lecture Notes in Mathematics, vol. 1637, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0094399">10.1007/BFb0094399</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-61795-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1438547">1438547</a></li> <li>Klein, Felix (1873), <a href="https://zenodo.org/record/1870569">"Ueber Flächen dritter Ordnung"</a>, <i>Mathematische Annalen</i>, 6 (4), Springer Berlin / Heidelberg: 551–581, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01443196">10.1007/BF01443196</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ClebschDiagonalCubic.html">"Clebsch diagonal cubic"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Baez, John (2016-03-01). "Clebsch Surface | Visual Insight". AMS Blogs. Archived from the original on 2024-07-18. Retrieved 2024-11-24. <a href="/wiki/John_C._Baez" target="_blank">/wiki/John_C._Baez</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Masson, Paul (2017-09-17). "27 Lines on a Cubic Surface". Analytic Physics. Archived from the original on 2024-09-13. Retrieved 2024-11-24. <a href="https://analyticphysics.com/Higher%20Dimensions/27%20Lines%20on%20a%20Cubic%20Surface.htm" target="_blank">https://analyticphysics.com/Higher%20Dimensions/27%20Lines%20on%20a%20Cubic%20Surface.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>