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Blum's Cyclide

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(*x*^{2} + *y*^{2} + *z*^{2})^{2} - 8 *x*^{2} - 6 *y*^{2} + *z* ^{2} + 4 = 0

In graduate school, I began my studies of geometry by reading a text
by Porteous ([P]) and an article by Montaldi
([M3]). In his paper, Montaldi shows that a
generic surface can have six circles with five point contact. He
gives a specific example of such a surface by taking a generic
perturbation of the above surface.
A surface has the *n-circle property* if through every point of
the surface there exists exactly *n* circles passing through the
point and lying on the surface. So a sphere has the infinite circle
property, and a torus of revolution has the 4-circle property. Blum
[Bl], gave families of surfaces which had
the *n*-circle property for *n* equal to 4, 5 or 6. Blum
conjectured that there are no surfaces with the *n*-circle
property for *n* greater than 6, less than infinity. Montaldi's
arguments show that such a surface must be a torus. Takeuchi ([T]) used
homotopy arguments to show a torus cannot have the *n*-circle
property for *n* greater than 6. Combined, Blum's conjecture is
proved.

Blum's Cyclide, colored by one family of circles
(corresponding to one of the two red circles above)

To prove that this cyclide has the 6-circle property, Blum inverted
the surface with respect to a sphere centered at an arbitrary point on the
surface. All circles going through this point are inverted into lines.
The equation for the surface turns into a cubic equation, and thus has
27 (complex) lines on it, counting multiplicity. It can be shown that
17 of these lines lie at infinity, and thus there can be at most 10
lines on the real portion of the surface. Through much computation,
Blum shows that four of these lines are complex, and the other lines
are real and distinct.

The above surface after inversion.

So, what about surfaces in higher dimensions? Does the extra freedom
allow for more examples of the *n*-circle property? Most of the
arguments depend on the fact we are looking at surfaces in
**R**^{3}. What will change in higher dimensions? We have
the standard Clifford torus, and really we have any example from
**R**^{3} that is inverse stereographically projected onto
*S*^{3}. In addition, we have the self-intersecting
sphere and the Veronese surface (which is a projective plane), both of
which are covered with circles.

We can also generalize by allowing isolated points to fail the circle
property. This would allow us to consider surfaces like ellipsoids.

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Last updated 1/25/00