Talk:Squaring the circle

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I have removed the nonsense about cylinders. Even if one could manipulate solid objects in Euclidean plane geometry (you can't, obviously), the the issue seems to have been to create a square (the cylinder viewed from its side) from the circle which is its base, or vice-versa, so that the side of the square matches the diameter of the circle. That's trivial—they are the same distance. The problem of "squaring the circle" was to match areas, not lengths. That is, make a square with the same area as the circle.

The 6-step "solution" that was given in the previous version involved manipulating 3-dimensional objects (you can't; this is plane geometry), "unrolling" the side of the cylinder (you can't), folding the resulting sheet (you can't do that, though you can get the equivalent effect with compass and straightedge, but you can't call it "folding"), and so on.

Even as a "joke", this falls flat.

SamHB (talk) 13:30, 2 July 2019 (EDT)

Why do you assume a priori that the area sought is Euclidean plane geometry only. Why? The problem proposes determining the equal area which a three-dimensional object does possess of necessity, since it occupies space with length and width and height. And yes, the area of the surface of a solid can be virtually "flattened" to determine the necessary volume of a container for the solid, a boundary for an enclosed space, as in a problem in architectural engineering and construction. It is enough for me that this discussion take place on the talk page. That is the essence of peer review. I sense from the tone of your response that you are unfamiliar with boolean algebra and logic and the hyper geometries of multidimensional mathematical spaces. With all due and well-wishing respect I suspect your response is subconsciously motivated by an unintentional form of mathematical and philosophical xenophobia. Scientific prejudice is not unknown in debates over hypotheses. The "seven deadly words of committee" before rejection of a proposal are "we've never done it that way before". Think of the advances in human knowledge and potential discovery that were delayed in this way.
Although I proposed a "solution" to the problem lightheartedly, it cannot be demonstrated to be either irrational or unreasonable. I did not propose the solution as a laughable joke, but as a form of legitimate mathematical amusement, as so many mathematical puzzles and novel solutions are by their nature, and they are not mockeries of mathematical reasoning. I did say that no one takes such a solution seriously. Paradoxes have been proposed as entertainment and as serious problems at one and the same time. Challenges make many astute and competent intellectuals smile with enjoyment, and sometimes chuckle with delight, even as they seriously take up the problem proposed. Sometimes what seems insoluble is resolved when someone with sudden insight says, "Let's look at this from a different angle. Maybe we are coming at this from the wrong direction. What if instead of what we normally do, we turn the problem on its head and_____?" Remember how Archimedes shouted, "Eureka!" in solving the problem of determining specific gravity by displacement of water volume and the weight of the whole by disregarding the traditional means of measurement. Your assumption that a Euclidean plane area solution cannot be obtained by the measurement of a third-dimensional cylindrical simple solid over its whole surface area is not only untrue, but ipso facto dismisses unexamined and untried a reasonable solution to the problem which is demonstrably simple. Semper fi --Dataclarifier (talk) 19:23, 11 July 2019 (EDT)
Why do you assume a priori that the area sought is Euclidean plane geometry only. Why?
Because that's the way the classical plane geometry problems have been posed for a few thousand years. They are an important part of the history of mathematics, studied in junior high school up through college. I suggest that you read Journey Through Genius, by William Dunham, ISBN 0 14 01.4739X for a good accounting of this history.
I sense ... that you are unfamiliar with boolean algebra and logic and the hyper geometries of multidimensional mathematical spaces.
I don't think so. See the Calc3.1 course, written by me and User:JacobB some time ago.
I see no reason why the reader of this page should not immediately see the "nonsense about cylinders" that was taken out, which SamHB in his edit summary called "trash". Ergo I have posted it here beneath my response for reader evaluation. —
Dataclarifier (talk) 20:14, 11 July 2019 (EDT)
I have no problem with this material being on some talk page or mainspace page. But, where it was, it could mislead the reader into not taking seriously the topic of compass-and-straightedge constructions, a topic that occupies a serious place in the history of mathematics. See Compass_and_straightedge. SamHB (talk) 21:54, 11 July 2019 (EDT)

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 90o 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 90o angle from the edge of the cut (3). The stylus of the compass is then rotated 180o 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 π pi and the formula
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 90o 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 90o, 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.
(--Dataclarifier (talk) 20:14, 11 July 2019 (EDT))


If at the beginning of the section I had said "As a mathematical exercise" with no mention of the word "joke", or if I had simply omitted the introductory phrase and begun straight off with the statement "A circle can simultaneously present a square..." perhaps you would have been entertained with a serious-sounding proposition. I had fun with the visualization, but I was also quite serious. I repeat: Although I proposed a "solution" to the problem lightheartedly, it cannot be demonstrated to be either irrational or unreasonable.
Actually, I was holding back, because it can be achieved in 2 steps. Virtually bending the round edge of the circumference of a perfect circle (1) into four straight sections set at right angles (2) presents the same area in a square form as is bounded by the circle form. The shape has been changed but the 2-dimensional volume of the figure remains the same. This can be mentally visualized as a perfectly Euclidean plane figure using a perfect line without thickness. No mathematical proof is required. It is self-evident in its elegant simplicity.
Again, it can also be achieved physically in 4 steps as an illustration, just as perfect geometrical figures are illustrated with physical lines having visible thickness representing lines without thickness. Set a perfectly circular end of a cylinder on a flat table representing a Euclidean plane sprayed with a light mist of water (1). Tie a thin thread around it on the Euclidean plane (wet) surface of the table representing the line of circumference (2). Remove the cylinder, leaving the circle of thread unmoved (3). Arrange the circle of thread on the table into a square shape on the Euclidean plane of the (wet) table surface using an unmarked straightedge card or ruler to make the four sections of the thread straight (4). The circle of thread has been made square, enclosing the same area as the circle enclosed.
Again, inscribe a circle onto a flat surface with one of the points of a compass calipers (1). Close the points of the compass calipers and rotate the tiny interval of the closed points of the compass around the boundary of the circle point by point completing the whole circuit, counting the steps (2). Divide the number by four (3). With the straightedge as guide, using the same interval of the compass caliper points unchanged, rotate the tiny interval of the closed points of the compass calipers point by point along the straightedge on the flat surface, counting a distance equal to one-fourth of the circuit of the circumference of the circle, rotating the straightedge 90o and repeating the operation three more times, giving four straight lengths adjoined at 90o, describing a square enclosing virtually the same area as the circle (4).
This is fun, but it is at the same time a serious exercise. --Dataclarifier (talk) 00:52, 12 July 2019 (EDT)
I want to close with the fact that Paul Boisvert, professor of mathematics, Oakton Community College, Office: Room 2554 (Des Plaines Campus), claims to have a simple proof of Fermat's Last Theorem that even a child can understand. In the online article A Simple Proof of Fermat's Last Theorem (oakton.edu) he says, "Why, a child can follow the logic below." After reading it, determine for yourself if he is joking, making a mockery of a serious problem in mathematics, or if he is truly serious about his logical proof, or if on the face of it he is perhaps doing both. I am inclined to hold to the last interpretation. --Dataclarifier (talk) 02:19, 12 July 2019 (EDT)