Squaring the circle

Squaring the Circle refers to a classic mathematical challenge originated in classical antiquity:

Construct a square equal in area to a circle using only a straightedge and compass.

This was one of the three geometric problems of antiquity, and was perhaps first attempted by Anaxagoras. It was finally proved to be an impossible problem when pi was proven to be transcendental by Carl Louis Ferdinand von Lindemann in 1882.

The other two problems from antiquity are:
"Duplicating the cube"

Construct the length of the side of a cube that would have half the volume of a cube of a given side, effectively constructing the number cube root of 2, and

"Trisecting an angle"

Construct an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge, and a compass.

These three problems were finally shown to be unsolvable by straightedge and compass with the algebraic tools of the 19^th century.

Squaring the circle is sometimes used as an idiom in common discourse to mean taking on an impossible task.

The section of a cylinder as a squared circle

As a mathematical joke, a circle can simultaneously present a square if a perfectly round, solid cylinder is produced with a sectional length equal to its measured diameter. Using only a straightedge ruler and a compass, a solution is obtained in six simple steps.

The straightedge determines the exactitude of the first straight smooth cut across the round cylinder at a right angle to the straight body of the cylinder (1). The compass point is placed at what is determined to be the exact center of the diameter of the round crosscut end of the solid cylinder, and the stylus is adjusted to the outer edge of the cut and rotated around the edge to determine with exactitude the center of the crosscut and the length of the radius of the cut (2). The span of the compass is then fixed and it is then moved 90^o to the straight side of the cylinder and positioned with the stylus at the exact edge of the crosscut, with the point of the compass positioned exactly at a 90^o angle from the edge of the cut (3). The stylus of the compass is then rotated 180^o away from the edge of the first crosscut end to determine with exactitude where to make the second right angle cut across the cylinder (4). The second straightedge smooth crosscut of the cylinder makes a finished cylinder section with a measured exact length identical to the measured exact diameter of the cylinder, a cylindrical section exactly as long as it is wide (5).

Looked at from the side it presents a square configuration. Looked at from the end it presents a circular configuration. The perfect circle formed here by crosscut exists as an exactly squared object and the exact square formed here by the same crosscut (twice) exists as a perfectly circular object. The circle has been squared and the square has been circled. What is perfectly round is also exactly square and what is exactly square is also perfectly round. The perfectly round exact square and the exactly square perfect round together are both in essence one object having the same single surface area in toto. This one square and round object has identically the same single volume as itself, in itself, by itself, by the exactly squared shaping of one perfectly round cylinder.

If the external surface of the straight side of the cylinder only is opened up and laid flat, its area can be determined by measuring the four sides of the resultant rectangle with a straightedge. The lengths of the top edge and the bottom edge only of the opened and flattened area of the straight sides of the cylinder are identical to the circumference of the circular ends, each length taken and divided by four and rearranged as four equal straight sides of a square, the area of the resultant square determined for one end of the cylinder is then multiplied by two for both ends, and the resulting combined area of the two ends added to the area of the opened and flattened straight side of the cylinder, give the area of the squared-off circular surface of the single cylindrical section with both round ends, the area of this one single squared circle object determined by straightedge measurement (6).

This simple solution using basic arithmetic and practical common sense in six simple steps elegantly does away with any need for the exact value of the transcendent number π pi and the formula $S=2\pi r^{2}+2\pi rh\ .$

No one takes this seriously as being properly related in any way to an actual solution of the classical problem posed by the ancient geometers, as a real solution to an insoluble problem in only six simple steps.

See Paradox and Parody.

External links

Circle Squaring (mathworld.wolfram.com)

Squaring the circle - Wikipedia

What Does Squaring the Circle Mean? by Catherine Beyer (learnreligions.com)

Who called pi transcendental (answers.com)

(PDF) Pi is Transcendental: Von Lindemann's Proof Made Accessible to Today's Undergraduates, Randy K. Schwartz (academia.edu)

Mathematical beauty: Beauty in method - Wikipedia a discussion of mathematical elegance