Though the Danish poet and scientist Piet Hein (1905–1996) is often credited with the discovery of the super-ellipse, it was the geometer Gabriel Lamé (1818) who first generalized the concept of an ellipse to a super-ellipse by generalizing the exponents of the standard equation of an ellipse as shown below:

This super-ellipse formula can be converted to the following
equivalent, polar coordinate form by substituting *x* = *r*.cos
θ and *y* = *r*.sin θ into the above equation
and solving for *r* to obtain:

More recently, the Belgian Botanist, Johan Gielis (1996, 2003) from
Antwerpen generalized the polar coordinate version further as follows
by considering possibly different values of the exponents and also
including a factor *m*/4 to divide the polar coordinate plane
into *m* sectors:

**References**

Gielis, J. (1996) Wiskundige supervormen bij bamboes. *Newsletter
of the Belgian Bamboo Society* 13 (December 2, 1996), 20-26.

Gielis J. (2003) A generic geometric transformation that
unifies a large range of natural and abstract shapes. *American
Journal of Botany* 90(3), Invited Special Paper, 333-338.

Lamé, G. (1818) *Examen de differentes
méthodes employées pour résoudre les
problèmes de géometrie*. M. V. Courcier imprimeur
Libraire. (New Edition 2008 from Editions Gabay).

**Note**: In the dynamic *JavaSketchpad* sketch below, the
above formula is
represented. Drag the **red
points** *m*, *n _{1}*,

The Gielis Super Shape Formula

A fascinating aspect of the Gielis Super Shape formula is that it can be
used to model the shape of numerous natural objects from plants and
flowers to snowflakes and crystals. For example, the shape shown above,
produced by the parameter values (6, 100, 38, 38, 0.5, 1) - in the
order (*m*, *n _{1}*,

FootNote 1: A snowflake hexagon like the first one above can be called a *semi-regular angle-gon* since all the angles are equal and the two pairs of alternate sides are equal. Read more about semi-regular polygons at *Generalizations of rectangles and rhombi*.

Use the above dynamic sketch to further explore the effect of the parametersor use the animation buttons. Many other examples of shapes produced by the Gielis formula can be found in the online encyclopedia, Wikipedia, at Super Formula.

The parameter *m* basically determines the number of
'sectors', 'bulges', 'hollows' or 'corners' of the shape. When *m*
= 0, the shape is a circle, but with *m* = 4, depending on the
value of the other parameters, we can get a square with four vertices,
or a shape like the cactus cross section with four 'corners' and four
'hollows', or a standard ellipse (which has two narrow 'bulges' and two
wide ones). If *n*_{1} > *n*_{2} = *n*_{3}
the sides can tend to straight lines and with appropriate values,
regular polygons can be produced. The parameters *a* and *b*
determine the lengths of the major axes and therefore the size of the
shape. With higher even values of *m*, different values of *a*
and *b*, will produce alternate sides of different length.

The Gielis super shape formula can be generalized to 3D in at least
two different ways, for example, by rotating the 2D version around one
of the axis, or by using polar coordinates in 3D. The 2D formula itself
can obviously be further generalized by considering different
parameters for *m* in the two parts of the formula it occurs.
Though the creation of closed figures are important if one wants to
model closed shapes such as flowers or snowflakes, mathematically
nothing prevents us to modify the Gielis super shape formula by instead
of adding the two absolute values, we could simply subtract the second
one from the first to produce a Super Hyperbola formula as shown below:

In the Link in the dynamic WebSketchpad sketch above, the *Super Hyperbola* formula is
represented. Have FUN dragging the **red
points** *m*, *n _{1}*,

**Note**: The slider values in the
*JavaSketchpad* sketch unfortunately change discreetly; so only approximate values for whole numbers can
mostly be obtained. Therefore, because of these approximations, each hyperbolic branch of the Java sketch for (5, 2, 2, 2, 1, 1) appears to 'split' into two as it goes off to infinity. Also be aware, that the *JavaSketchpad* sketch, for some values of the parameters, does not always correctly display asymptotes and branches going off to infinity. For more accurate explorations, it is better to use either *Sketchpad* or some other similar graphing software. Accurate *Sketchpad* 5 sketches for both the Gielis formula and the Super Hyperbola formula are available for downloading at *Gielis Super Shape Formula and Super Hyperbola*.

*Back to "Dynamic Geometry Sketches"*

*Back to "Student Explorations"*

Created by Michael de Villiers, 19 June 2011; updated 3 June 2019.