In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a superegg is a solid of revolution

Solid of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line that lies on the same plane....

 obtained by rotating an elongated super-ellipse

Superellipse

A superellipse is a geometric figure defined in the Cartesian coordinate system as the set of all points withwhere n, a and b are positive numbers....

 with exponent

Exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

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

Superellipsoid

In mathematics, a super-ellipsoid or superellipsoid is a solid whose horizontal sections are super-ellipses with the same exponent r, and whose vertical sections through the center are super-ellipses with the same exponent t....

.

Unlike an elongated ellipsoid

Prolate spheroid

A prolate spheroid is a spheroid in which the polar axis is greater than the equatorial diameter. Prolate spheroids stand in contrast to oblate spheroids...

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

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 being zero at the tips.

The shape was popularized by Danish poet and scientist Piet Hein

Piet Hein (Denmark)

Piet Hein was a Danish scientist, mathematician, inventor, designer, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone"...

 (1905–1996). Supereggs of various materials, like the one shown, were sold as novelties or "executive toy

Executive toy

An executive toy is a novelty item that is usually a small mechanical gadget placed on the desk of a corporate executive or other office workers...

s" in the 1960s. A 1-ton superegg made of steel

Steel

Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...

 and aluminum was placed outside Kelvin Hall

Kelvin Hall

The Kelvin Hall in Glasgow, Scotland, is a mixed-use arts and sports venue that opened as an exhibition centre in 1927. It has been a music hall, indoor arena and barrage balloon factory, and is currently home to the Kelvin Hall International Sports Arena and from 1987 to 2010, Glasgow's Museum of...

 in Glasgow

Glasgow

Glasgow is the largest city in Scotland and third most populous in the United Kingdom. The city is situated on the River Clyde in the country's west central lowlands...

 in 1971, on occasion of a lecture by Piet Hein.

Mathematical description

The superegg is a super-ellipsoid whose horizontal cross-sections are circles. It is defined by the implicit equation

Implicit function

The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...

where r is the horizontal radius at the "equator" (the widest part), and h is one half of the height. The exponent p determines the degree of flattening at the tips and equator. Hein's choice was p=2.5 (the same he used for the Sergels Torg

Sergels Torg

Sergels torg is the most central public square in Stockholm, Sweden, named after 18th century sculptor Johan Tobias Sergel, whose workshop was once located north of the square.- Overview :...

 roundabout), and r/h = 3/4.

The definition can be changed to have a strict equality rather than inequality, this changes the superegg to being a surface of revolution

Surface of revolution

A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane ....

rather than a solid.
.