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

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*“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.

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.**

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**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