{"id":2666,"date":"2016-07-01T01:00:25","date_gmt":"2016-07-01T01:00:25","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2666"},"modified":"2016-07-30T08:35:39","modified_gmt":"2016-07-30T08:35:39","slug":"barth-decic","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/07\/01\/barth-decic\/","title":{"rendered":"Barth Decic"},"content":{"rendered":"
\n
\"Barth<\/a>

Barth Decic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

A decic surface<\/b> is one defined by a polynomial equation of degree 10. The Barth decic<\/b><\/a>, drawn above by Abdelaziz Nait Merzouk<\/a>, is currently the decic surface with the largest known number of ordinary double points<\/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

It has 345 ordinary double points, while the best known upper bound for a decic surface that’s smooth except for such singularities is 360.<\/p>\n

The Barth decic is defined by this homogeneous polynomial equation of degree 10 in four variables \\(w,x,y,z\\):<\/p>\n

$$ \\begin{array}{c} 8 (x^2 – \\Phi^4 y^2) (y^2 – \\Phi^4 z^2) (z^2 – \\Phi^4 x^2) \\left( x^4 + y^4 + z^4 – 2 x^2 y^2 – 2 x^2 z^2 – 2 y^2 z^2\\right) \\\\
\n+ (3 + 5 \\Phi) w^2 \\left( x^2 + y^2 + z^2 – w^2 \\right)^2 \\left( x^2 + y^2 + z^2 – (2-\\Phi)^2 w^2)\\right) = 0 \\end{array} $$<\/p>\n

where <\/p>\n

$$ \\Phi = \\frac{\\sqrt{5} + 1}{2} $$<\/p>\n

is the golden ratio. This equation determines a subset \\(S \\subset \\mathbb{C}^4\\) with complex dimension 2. Note that if \\((w, x, y, z \\in \\mathbb{C}^4 \\) is a solution, so is any multiple \\((cw, cx, cy, cz) \\). We may thus projectivize<\/a> \\(S\\), treating any solution as ‘the same’ as any multiple of that solution. The result is an algebraic variety<\/a> \\(X\\) in the complex projective space<\/a> \\(\\mathbb{C}\\mathrm{P}^3\\). This variety has complex dimension 2, so it is called a complex surface<\/b>. To obtain an ordinary real 2-dimensional surface we may take its intersection with a copy of \\(\\mathbb{R}\\mathrm{P}^3\\) in \\(\\mathbb{C}\\mathrm{P}^3\\). <\/p>\n

Sitting inside \\(\\mathbb{R}\\mathrm{P}^3\\) we in turn have many copies of ordinary 3-dimensional space, \\(\\mathbb{R}^3\\). The picture above shows the portion of the Barth decic living in one of these copies. Concretely, this consists of real solutions of the above equation where \\(w = 1\\).<\/p>\n

But we also have ‘points at infinity’. If you march off in either of two opposite directions in \\(\\mathbb{R}^3\\), you will approach one of these points at infinity. The points at infinity form a projective plane<\/a>, that is, a copy of \\(\\mathbb{R}\\mathrm{P}^2\\). Concretely, these points at infinity are the points in \\(\\mathbb{R}\\mathrm{P}^2\\) coming from points \\((x,y,z,w) \\in \\mathbb{R}^4\\) with \\(w = 0\\). <\/p>\n

The Barth decic has 345 ordinary double points. However, 45 of these are points at infinity, so they are not visible in the above picture, or this one:<\/p>\n

\n
\"Barth<\/a>

Barth Decic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

We can bring the double points at infinity into view by rotating \\(\\mathbb{R}\\mathrm{P}^3\\) slightly. If we slice the resulting surface to see it better, we obtain a picture like this:<\/p>\n

\n
\"Rotated<\/a>

Rotated Barth Decic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

We can also compress \\(\\mathbb{R}^3\\) into a ball, so that the points at infinity lie on the surface of this ball. More precisely, the surface of this ball is a 2-sphere, a double cover of \\(\\mathbb{R}P^2\\), so any antipodal pair of points in this 2-sphere correspond to the same point at infinity. <\/p>\n

This gives the following view of the Barth decic:<\/p>\n

\n
\"Compressed<\/a>

Compressed Barth Decic – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

You can see visually that the compressed Barth decic meets the 2-sphere in 10 great circles. To see this mathematically, we can take the equation for the Barth decic and set \\(w = 0\\):<\/p>\n

$$ \\begin{array}{c} (x^2 – \\Phi^4 y^2) (y^2 – \\Phi^4 z^2) (z^2 – \\Phi^4 x^2) \\left( x^4 + y^4 + z^4 – 2 x^2 y^2 – 2 x^2 z^2 – 2 y^2 z^2\\right) = 0 \\end{array} $$<\/p>\n

This factors into 10 linear functions:<\/p>\n

$$ (x – \\Phi^2 y)(x + \\Phi^2 y)(y – \\Phi^2 z)(y + \\Phi^2 z)(z – \\Phi^2 x)(z + \\Phi^2 x)(x-y-z)(x+y-z)(x-y+z)(x+y+z) = 0 $$ <\/p>\n

Each of these defines a plane in $\\mathbb{R}^3$ whose intersection with the unit 2-sphere is one of the 10 great circles. These 10 great circles are orthogonal to the lines going through the opposite corners of a regular dodecahedron:<\/p>\n

\n
\"Compressed<\/a>

Compressed Barth Decic with Dodecahedron – Abdelaziz Nait Merzouk<\/p><\/div>\n<\/div>\n

You can see 5 double points in each face of the dodecahedron and 1 at the midpoint of each edge, for a total of \\(5 \\times 12 + 30 = 90\\). However, antipodal points on the sphere count as the same point at infinity, so we get a total of \\(90\/2 = 45\\) double points at infinity.<\/p>\n

For more related pictures see:<\/p>\n

• Abdelaziz Nait Merzouk, Barth decic and dodecahedron<\/a>.<\/p>\n

It is worth comparing the Barth sextic:<\/p>\n

• Barth sextic<\/a>.<\/p>\n

The group of rotation and reflection symmetries of an icosahedron, \\(\\mathrm{A}_5 \\times \\mathbb{Z}\/2\\), acts as symmetries of both the Barth sextic and the Barth decic. Barth introduced these surfaces here:<\/p>\n

• Wolf Barth, Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, Journal of Algebraic Geometry<\/i> 5<\/b> (1994), 173–186.<\/p>\n

Ordinary double points are also known as nodes<\/b>. In 1984, Miyaoka proved that a decic surface in $\\mathbb{C}\\mathrm{P}^3$ with only rational double points can have at most 360 such points:<\/p>\n

• Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants<\/a>, Math. Ann.<\/i> 268<\/b> (1984), 159–171.<\/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":"

A decic surface<\/b> is one defined by a polynomial equation of degree 6. The Barth decic<\/b><\/a>, drawn here by Abdelaziz Nait Merzouk<\/a>, is the decic surface with the maximum possible number of ordinary double points<\/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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