Whitney umbrella

File:Whitney unbrella.png

Section of the surface

In mathematics, the Whitney umbrella (or Whitney's umbrella and sometimes called a Cayley umbrella) is a self-intersecting surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line, parallel to the axis of the parabola and lying on its perpendicular bisecting plane.

Formulas

Whitney's umbrella can be given by the parametric equations in Cartesian coordinates $x(u,v) = uv$; $y(u,v) = u$; $z(u,v) = v^2$ where the parameters u and v range over the real numbers. It is also given by the implicit equation

$x^2=y^2z$

This formula also includes the negative z axis (which is called the handle of the umbrella).

Properties

File:Whitney umbrella.gif

Whitney umbrella as a ruled surface, generated by a moving straight line.

File:Whitney umbrella.jpg

Whitney umbrella made with a single string inside a plastic cube.

Whitney's umbrella is a ruled surface and a right conoid. It is important in the field of singularity theory, as a simple local model of a pinch point singularity. The pinch point and the fold singularity are the only stable local singularities of maps from R² to R³.

It is named after the American mathematician Hassler Whitney.

In string theory, a Whitney brane is a D7-brane wrapping a variety whose singularities are locally modeled by the Whitney umbrella. Whitney branes appear naturally when taking Sen's weak coupling limit of F-theory.

See also

• Cross-cap
• Right conoid
• Ruled surface

References

• (Images and movies of the Whitney umbrella.)