geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a superegg is a

solid of revolution In geometry, a solid of revolution is a Solid geometry, solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution''), which may not Intersection (geometry), intersect the generatrix (except at its bound ...

obtained by rotating an elongated

superellipse A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows ...

with

exponent In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...

greater than 2 around its longest axis. It is a special case of

superellipsoid In mathematics, a superellipsoid (or super-ellipsoid) is a solid geometry, solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter \epsilon_2, and whose vertical sections through the center are superel ...

. Unlike an elongated ellipsoid, an elongated superegg can stand upright on a flat surface, or on top of another superegg. This is due to its

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

being zero at the tips. The shape was popularized by Danish poet and scientist Piet Hein (1905–1996). Supereggs of various materials, including brass, were sold as novelties or " executive toys" in the 1960s.

Mathematical description

The superegg is a superellipsoid whose horizontal cross-sections are circles. It is defined by the inequality :

\left, \frac\^p + \left, \frac\^p \leq 1 \, ,

where ''R'' is the horizontal radius at the "equator" (the widest part as defined by the circles), and ''h'' is one half of the height. The

''p'' determines the degree of flattening at the tips and equator. Hein's choice was ''p'' = 2.5 (the same one he used for the

Sergels Torg Sergels torg ("Sergel's Square") is a major public square in Stockholm, Sweden, constructed in the 1960s and named after 18th-century sculptor Johan Tobias Sergel, whose workshop was once located north of the square. Overview Sergels torg h ...

roundabout), and ''R''/''h'' = 6/5. The definition can be changed to have an equality rather than an inequality; this changes the superegg to being a

surface of revolution A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersec ...

rather than a solid.

Volume

The volume of a superegg can be derived via squigonometry, a generalization of

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

to squircles. It is related to the

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

V = \frac\frac \, .

References

External links

* Algebraic curves Surfaces Office toys Educational toys {{Geometry-stub

Mathematical description

Volume

See also

References

External links