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Saturday, October 22, 2011

After innumerable failed attempts, the impossibility to square the circle has been established in the 19th century

Nowadays, finding the area of curvilinear shapes falls in the purview of calculus. But the problem of finding areas draw much interest in antiquity and preoccupied mathematicians ever since. One of the acknowledged results by Hippocrates of Chios (470-410 B.C.) is the Squaring of a Lune. The problem of squaring a shape refers to a construction of a square (by means of straightedge and compass) equal in area to that shape. Not all shapes could be squared. After innumerable failed attempts, the impossibility to square the circle has been established in the 19th century, although the formula for the area of a circle has been known yet to Archimedes.

For some shapes, squaring can be done by a visually appealing dissection. The shape is getting cut into smaller peaces that are then reassembled into a square.

We have a dynamic illustration of such a dissection of a vase-like shape:

Here the vase has been cut into four pieces that when stuck together a little differently produced a square. For this particular shape three pieces would suffice [Alsina & Nelsen, p. 157]:

Here is another dissection in which the pieces reassemble into a rectangle:

A rectangle can be decomposed into a square.

As many before and after him, the Renaissance polymath Leonardo da Vinci has been fascinated with the problem of squaring the circle and other curvilinear shapes.

Here are two simple examples found in his notebooks.

Leonardo's dissection of a two-sided axe (or a mallet):

Leonardo's dissection of an "umbrella":

A wonderful master-piece became known as "Leonardo's Mirror" [Moskovich, pp. 24, 105]

The shape could be obtained by XOR-ing (or taking the symmetric difference - which is the same - of) two differently oriented Yin (or Yang) symbols. The comma-like shape is bounded by three arcs: one 180° arc, and two 90° arcs of half the radius of the big one. In Leonardo's Mirror the two commas intersect so as to cut off a quarter of the big circle.

Cut as shown, the pieces recombine into a 2×1 rectangle which is easily dissected into a square (already Socrates knew how to do that and so did Japanese masters.)

An intimidating name of "Leonardo's Claw" is associated with another of his discoveries.

The claw consists of what remains of a big circle after the removal of a smaller circle and a lense. The latter is formed by two 90° arcs of the big circle. (This provides enough information to determine the ratio of the radii of the two circles.) Leonardo has shown that the area of the claw equals that of the maximal square it may hold. There does not appear to be a solution by dissection.


  1. C. Alsina, R. B. Nelsen, Charming Proofs, MAA, 2010
  2. I. Moskovich, Leonardo's Mirror and Other Puzzles, Dover, 2011

Related posts:

  1. Isoperimetric Theorem for Rectangles
  2. Engaging math activities for the summer break - Day 6

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