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Copied images to imgur.com, as they were not being displayed because of new https rule. Added links to original image sources.
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edit approved Sep 30, 2017 at 2:11
jeq
edit approved Sep 30, 2017 at 2:11
jeq
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By a drawing of the Fano plane I mean a system of seven simple curves and seven points in the real plane such that

  • every point lies on exactly three curves, and every curve contains exactly three points;
  • there is a unique curve through every pair of points, and every two curves intersect in exactly one point;
  • the curves do not intersect except in the seven points under consideration.

The familiar picture

Traditional Fano plane Traditional Fano plane http://math.haifa.ac.il/%7Eseva/MathOverflow/Fano-1.jpg(source)

does not count as a drawing, since the last requirement is not satisfied: there are two "illegal" intersections. In fact, this is easy to fix:

Non-intersecting Fano plane Non-intersecting Fano plane http://math.haifa.ac.il/%7Eseva/MathOverflow/Fano-2.jpg(source)

However, this drawing is degenerate in the sense that two of the curves just "touch" each other, without crossing, at some point. And here, eventually, my question goes:

Is every drawing of the Fano plane degenerate?

(Although I can give a topological definition of degeneracy, it is a little technical and, may be, not the smartest possible one, so I prefer to suppress it here.)

By a drawing of the Fano plane I mean a system of seven simple curves and seven points in the real plane such that

  • every point lies on exactly three curves, and every curve contains exactly three points;
  • there is a unique curve through every pair of points, and every two curves intersect in exactly one point;
  • the curves do not intersect except in the seven points under consideration.

The familiar picture

Traditional Fano plane http://math.haifa.ac.il/%7Eseva/MathOverflow/Fano-1.jpg

does not count as a drawing, since the last requirement is not satisfied: there are two "illegal" intersections. In fact, this is easy to fix:

Non-intersecting Fano plane http://math.haifa.ac.il/%7Eseva/MathOverflow/Fano-2.jpg

However, this drawing is degenerate in the sense that two of the curves just "touch" each other, without crossing, at some point. And here, eventually, my question goes:

Is every drawing of the Fano plane degenerate?

(Although I can give a topological definition of degeneracy, it is a little technical and, may be, not the smartest possible one, so I prefer to suppress it here.)

By a drawing of the Fano plane I mean a system of seven simple curves and seven points in the real plane such that

  • every point lies on exactly three curves, and every curve contains exactly three points;
  • there is a unique curve through every pair of points, and every two curves intersect in exactly one point;
  • the curves do not intersect except in the seven points under consideration.

The familiar picture

Traditional Fano plane (source)

does not count as a drawing, since the last requirement is not satisfied: there are two "illegal" intersections. In fact, this is easy to fix:

Non-intersecting Fano plane (source)

However, this drawing is degenerate in the sense that two of the curves just "touch" each other, without crossing, at some point. And here, eventually, my question goes:

Is every drawing of the Fano plane degenerate?

(Although I can give a topological definition of degeneracy, it is a little technical and, may be, not the smartest possible one, so I prefer to suppress it here.)

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Seva
Seva
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Fano plane drawings: embedding PG(2,2) into the real plane

By a drawing of the Fano plane I mean a system of seven simple curves and seven points in the real plane such that

  • every point lies on exactly three curves, and every curve contains exactly three points;
  • there is a unique curve through every pair of points, and every two curves intersect in exactly one point;
  • the curves do not intersect except in the seven points under consideration.

The familiar picture

Traditional Fano plane http://math.haifa.ac.il/%7Eseva/MathOverflow/Fano-1.jpg

does not count as a drawing, since the last requirement is not satisfied: there are two "illegal" intersections. In fact, this is easy to fix:

Non-intersecting Fano plane http://math.haifa.ac.il/%7Eseva/MathOverflow/Fano-2.jpg

However, this drawing is degenerate in the sense that two of the curves just "touch" each other, without crossing, at some point. And here, eventually, my question goes:

Is every drawing of the Fano plane degenerate?

(Although I can give a topological definition of degeneracy, it is a little technical and, may be, not the smartest possible one, so I prefer to suppress it here.)