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

$$ x^2 + y^2 = z^2 .$$

The Chmutov octic does not have the largest known number of nodes for an octic. That honor currently belongs to the Endrass octic:

which has 168. It is known that an octic with only nodes and no other singularities can have at most 174.

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:

and here is the Chmutov **icosic**, that is, the Chmutov surface of degree 20:

Chmutov also studied surfaces with many *complex* nodes, including an octic with 154 complex nodes:

• S. V. Chmutov, Examples of projective surfaces with many singularities, *J. Algebraic Geom.* **1** (1992), 191–196.

but the Chmutov octic with real nodes shown here is discussed in this paper:

• Friedrich Hirzebruch, Some examples of threefolds with trivial canonical bundle. Notes by J. Werner, in Friedrich Hirzebruch, *Collected Papers II* (1985), pp. 757—770.

where it is used to construct some Calabi—Yau 3-folds.

I thank Juan Escudero for clarifying the difference between the two Chmutov octics. He writes:

The 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:

$$u=x+iy, v=x-iy, w=z $$

(\(x,y,z\) are real variables) and has equation \(F(x+iy,x-iy,z)=0\), with integer coefficients.

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

*Visual Insight* is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!

I’ve made a 3D print : https://www.thingiverse.com/thing:2425953

Cool! I blogged about it on Google+.

> 144 real ordinary double points or nodes: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2+y^2=z^2\).

The term “double points” seems like a misnomer! That cone looks as if it’s bunching together continuum many points at the origin, not just two.

With real plane curves we say \(y = 0\) intersects \(y = x\) in one point, \(y = x^2 – \epsilon\) intersects \(y = 0\) in two points for \( \epsilon \gt 0\), and \(y = x^2\) intersects \(x = 0\) in a double point: “two points that are infinitesimally far apart”.

With surfaces I guess ‘double point’ is used by analogy.