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Line 11: − − ===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<sup>o</sup> 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<sup>o</sup> angle from the edge of the cut (3). The stylus of the compass is then rotated 180<sup>o</sup> 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. As a single object its total surface area remains the same identical area whether it is seen as a circle or a square. − − 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 '''<big><big>π</big></big>''' ''pi'' and the [[Cylinder|formula]] '''<math> S= 2\pi r^2 + 2\pi r h \ .</math>''' − − Alternatively, the circle can simply and easily be squared, because the length of the top or bottom edge of the (hollow) flattened vertical cylinder is identical to the circumference of the circular end of the round cylinder (1), and the flattened rectangle of the side of the formerly round cylinder can be folded in half side to side, presenting two joined halves with a vertical center crease (2), and then the two joined halves together are folded again, center crease to side edges resulting in three creases, four vertical sections and two side edges (3), then opening up the folded flattened and creased vertical side of the formerly round cylinder while retaining the three resultant vertical creases, the two side edges can be joined, producing a vertical rectangular prism equal in length to the original cylinder, with four equal vertical sides, each of the four sides in horizontal width equal to one-fourth the width of the rectangle, which is identical to the flattened horizontal circumference of the formerly round cylinder joined at the two vertical edges (4), the three vertical creases each stabilized with the four vertical sides, each side set at right angles to its two adjoining sides (four sides at 90<sup>o</sup> horizontal angles), the upper horizontal edge of the four sections of the vertical side of the flattened cylinder joined at the two side edges forming a square end (5), the two resulting top and bottom square ends of the prism formed from the opened and flattened former cylinder folded and creased now each enclose with their four outer edges as a square end the same area as the circular end of the cylinder from which the flattened rectangular form was obtained. The area bounded by the outer edge of the circular configuration at the end of the cylinder is equal to the area bounded by the length of its circumference, straightened out, folded twice, creased, opened up, joined at the side edges with its folds at 90<sup>o</sup>, resulting in a square configuration. The circle has been squared, and the area of the square so formed from the circumference of the circle equals the area of the circle from which the square was derived. − − − 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]]. Line 47:
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Take out trash. It makes a mockery of one of the mathematical problems of antiquity. As a "joke", it just isn't funny. See talk.
Squaring the circle is sometimes used as an idiom in common discourse to mean taking on an impossible task.
Squaring the circle is sometimes used as an idiom in common discourse to mean taking on an impossible task.
==External links==
==External links==
[https://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_in_method Mathematical beauty: Beauty in method - Wikipedia] ''a discussion of mathematical elegance''
[https://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_in_method Mathematical beauty: Beauty in method - Wikipedia] ''a discussion of mathematical elegance''
[[category:Mathematics]]
[[category:Mathematics]]
[[Category:Idioms]]
[[Category:Idioms]]