Constructing a Squircle.

 

Doug Clark, 8th May 2004. [email protected]   Last revised 8th July 2004.

 

“Houdini succeeded in escaping from the box which had been submerged in the Cuyahoga.  While resting on the bank and doodling on a piece of bark with a pencil lying about he saw, to his amazement, that he had drawn a square circle.”  (See Note 1.)

 

Below is a suggestion as to what Houdini might have drawn, had some paper, a ruler and scissors also been lying about on the river bank.

 

 

Introduction.

 

A circle is a closed plane curve every point of which is equidistant from a given fixed point, the centre.

 

A square is a plane geometric figure with four straight sides of equal length and four right angles.

  

A squircle is a closed plane curve that is both a circle and the circumference of a square.

 

 

How to do it.

 

Consider a surface that consists of a two-sided disk.  This can be represented by a paper disk.  Mark the centre of one side, call this point X.  On the other side draw a square with diagonals equal to the diameter of the disk, call this curve (the square) Y.  Consider the interval between the intersection of the radius with Y and the edge of the side of the disk.  Call the midpoint of this interval P.  Cut along the curve, which consists of all possible values of P, and discard the outer portion of the disk.  Call the surface remaining S.

 

On the surface S the curve Y is a squircle, being both a circle, i.e. being a closed plane curve every point of which is equidistant from the point X, and the circumference of a square having straight sides of equal length and four right angles. 

 

 

Notes.

 

Note 1:  Unlike Houdini I first came across a square circle while reading a draft paper.  Two Arguments in Defence of Impossible Worlds, by Allan Hazlett.  http://www.cassetteradio.com/hazlett/twoarguments.pdf  Back

 

 

 

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