A Sliceform Super-egg

See also: the posts on how to make and assemble the models and Sliceform templates and Downloads

This Sliceform surface is only available in one version, with 18 slices and takes about an hour to make. It is called a super-egg, but it is really a surface of revolution called a super-ellipsoid.

What is a Super-ellipsoid?

A superellipsoid is a surface of revolution of a super-ellipse. So what is a super-ellipse and what is a surface of revolution?

A surface of revolution is a surface formed when you take a curve and rotate it in space about a line (the axis of revolution).

  • So if you rotate a circle about a diameter as axis you get a sphere.
  • If you rotate a line about an axis which is parallel to the line, then you get a cylinder.
  • If you rotate a line about an axis which intersects the line then you get a cone. This means that the cylinder in the example above can be considered as a very special cone. Similarly, if the angle of intersection of the axis and the line is 90¡ then you get a plane.

A super-ellipse is a special curve which can look like an ellipse or it can take a variety of other shapes. The reason it is called is super-ellipse is because its equation is obtained by modification of the equation of an ellipse.

This is the shape of the super-ellipse used for making the super-ellipsoid as a surface of rotation.

Note how the shape is more rectangular than a true ellipse.

The super-ellipsoid formed when this particular super-ellipse is rotated about its long axis has been used to create the Sliceform.

What’s special about this super-ellipsoid?

This shape super-ellipse was chosen to make a super-ellipsoid by the Danish scientist, designer, inventor and poet Piet Hein (1905-1996). He called it a super-egg because of the way it balances on both ends with a kind of spooky stability.

See below for more about the super-ellipse and the inventor of the super-ellipsoid, Piet Hein including links to other web pages and to find out what is special about this super-ellipsoid.

The super-egg templates

Download the file (in the templates post below) you want for printing and use that to create the model. The following graphic is to help you understand what the slices look like in each model.

Remember, they are slices and not necessarily super-ellipses.

There is only one page of template, since the super-egg’s symmetry means that the templates in each direction are the same. One set is upside down compared to the other.

Assembling the Super-egg Sliceform

See the general instructions post on assembling Sliceforms for cutting out and cutting the slots. Take extra care with the slots.

There are 18 slices. In each direction, the largest slices are in the centre. Begin by assembling these, then fit the next smaller size either side and so on until you reach the outside.

When the model is assembled it folds flat to give the same pattern in both directions:

Since it is not solid, it does not have the same stability problems as a “real” super-egg, but it will balance in many different ways:

Piet Hein and the Super-ellipsoid

Problems worthy
of attack
prove their worth
by hitting back.

This is one of Piet Hein’s poems which he called grocks (or grucks in Danish). They are short poems with his own brand of humour which also have a wise overtone. You can find more about them in the links below.

The origin of the super-ellipse

Piet Hein did not discover the super-ellipse. That honour goes to the French mathematician Lamé who took the equation for the ellipse and asked “what if?” it was part of a wider set of curves.

The equation for the ellipse is:

Lamé wrote the equation as:

and asked what the curve would like if n was not equal to 2 but took on other values.

If n is zero then the curve becomes a pair of crossed lines. As it increases so the curve changes from being a curved star shape (you may recognise a curve called an astroid) to a rectangle with diagonals a and b when n equals 1.

Between 1 and 2 the curve turns from a curved rectangle towards an ellipse. It is an ellipse with major and minor axes a and b when n equals 2.

When n goes above 2, the curves starts to look interesting even more interesting. It becomes a rounded rectangle.


Piet Hein’s use of the super-ellipse

Piet Hein used the super-ellipse with values of n greater than 2 for designing. He made roundabouts in roads and desks and tables with this shape.

Then he took the super-ellipse with a ratio of a:b of 4/3 and with n equal to 2.5 and created a surface of revolution to give the super-ellipse used in this Sliceform model.

More about Piet Hein

You can find out more about Piet Hein with the following links, but one of the best descriptions is in Martin Gardner’s book “Mathematical Carnival”.

Notes on Piet Hein

Brief biography of Piet Hein
Piet Hein and the Soma cube
Piet Hein’s board game, Hex and here
Superellipse and Superellipsoid


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