{"id":3006,"date":"2017-01-01T01:00:42","date_gmt":"2017-01-01T01:00:42","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=3006"},"modified":"2018-07-09T21:21:12","modified_gmt":"2018-07-09T21:21:12","slug":"chmutov-octic","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2017\/01\/01\/chmutov-octic\/","title":{"rendered":"Chmutov Octic"},"content":{"rendered":"
\n
\"Chmutov<\/a>

Chmutov Octic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

An octic<\/b> surface is one defined by a polynomial equation of degree 8. This image by Abdelaziz Nait Merzouk<\/a> shows an octic discovered by Chmutov with 144 real ordinary double points<\/b> or nodes<\/b>: that is, points where it looks like the origin of the cone in 3-dimensional space defined by<\/p>\n

$$ x^2 + y^2 = z^2 .$$<\/p>\n

The Chmutov octic does not have the largest known number of nodes for an octic. That honor currently belongs to the Endrass octic:<\/p>\n

• Endrass octic<\/a>.<\/p>\n

which has 168. It is known that an octic with only nodes and no other singularities can have at most 174.<\/p>\n

Chmutov defined a series of surfaces with many real nodes with the help of Chebyshev polynomials, in order to establish a lower bound on how many real nodes are possible for a surface of given degree. Here are the Chmutov surfaces of degrees 2 to 20:<\/p>\n

\n
\"Chmutov<\/a>

Chmutov Surfaces – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

and here is the Chmutov icosic<\/b>, that is, the Chmutov surface of degree 20:<\/p>\n

\n
\"Chmutov<\/a>

Chmutov Icosic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

Chmutov also studied surfaces with many complex<\/i> nodes, including an octic with 154 complex nodes:<\/p>\n

• S. V. Chmutov, Examples of projective surfaces with many singularities, J. Algebraic Geom.<\/i> 1<\/b> (1992), 191–196.<\/p>\n

but the Chmutov octic with real nodes shown here is discussed in this paper:<\/p>\n

• Friedrich Hirzebruch, Some examples of threefolds with trivial canonical bundle. Notes by J. Werner, in Friedrich Hirzebruch, Collected Papers II<\/i> (1985), pp. 757—770.<\/p>\n

where it is used to construct some Calabi—Yau 3-folds. <\/p>\n

I thank Juan Escudero for clarifying the difference between the two Chmutov octics. He writes:<\/p>\n

\nThe octic with 154 complex nodes belongs to the series discovered by Chmutov in the article “Examples of algebraic surfaces…” (1992). It is a complex surface with equation \\(F(u,v,w)=0\\) (\\(u,v,w \\) are complex variables). The real variant of the Chmutov surface with 154 real ordinary double points is obtained with a simple variable change: <\/p>\n

$$u=x+iy, v=x-iy, w=z $$<\/p>\n

(\\(x,y,z\\) are real variables) and has equation \\(F(x+iy,x-iy,z)=0\\), with integer coefficients.\n<\/p><\/blockquote>\n

Abdelaziz Nait Merzouk created the above pictures of the Chmutov octic and icosic and made them available on Google+<\/a> under a Creative Commons Attribution-ShareAlike 3.0 Unported<\/a> license.\ufeff He also put his animated gif of the Chmutov surfaces on Google+<\/a> under the same license.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

An octic<\/b> surface is one defined by a polynomial equation of degree 8. This image by Abdelaziz Nait Merzouk<\/a> shows an octic discovered by Chmutov with 154 real ordinary double points<\/b> or nodes<\/b>: that is, points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$.<\/p>\n

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