frame, Section of the surface In

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Whitney umbrella (or Whitney's umbrella, named after American mathematician

Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integrati ...

, and sometimes called a Cayley umbrella) is a specific self-intersecting

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directri ...

placed in three dimensions. It is the union of all

straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment ...

s that pass through points of a fixed

parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...

and are

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane.

Formulas

Whitney's umbrella can be given by the

parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ...

s in

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

\left\{\begin{align}
 x(u, v) &= uv, \\
 y(u, v) &= u, \\
 z(u, v) &= v^2,
\end{align}\right.

where the parameters ''u'' and ''v'' range over the

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s. It is also given by the

implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit fun ...

x^2 - y^2 z = 0.

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

Properties

Whitney's umbrella is a

and a right conoid. It is important in the field of

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

, 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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. 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 In theoretical physics, F-theory is a branch of string theory developed by Iranian physicist Cumrun Vafa. The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in the for ...

References

* {{cite web , url=http://www.geom.uiuc.edu/zoo/features/whitney/ , title=Whitney's Umbrella , work=The Topological Zoo , publisher=The Geometry Center , access-date=2006-03-08 (Images and movies of the Whitney umbrella.) Differential topology Singularity theory Surfaces Algebraic geometry

Formulas

Properties

See also

References