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Why The Circle Cannot Be Squared

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Squaring the circle using compass and straight edge in such a way that both have the same area is not possible. The question is, why not? Math logic assumes there must be an area equal to both. Presumably, there is a need to make these very different
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    Conference Proceedings in Mathematics Education A LAN R OGERSON AND J ANINA M ORSKA (E DITORS )   T HE M ATHEMATICS E DUCATION FOR THE F UTURE P ROJECT  P ROCEEDINGS OF THE 15 TH   I NTERNATIONAL C ONFERENCE   T HEORY AND P RACTICE :   A N I NTERFACE OR A   G REAT D IVIDE ? 4-9 Aug, 2019, Maynooth University, Kildare, Ireland    Why The Circle Cannot Be Squared Bradford Hansen-Smith Independent wholemovement@gmail.com  Abstract Squaring the circle using compass and straight edge in such a way that both have the same area is not possible. The question is, why not? Math logic assumes there must be an area equal to both. Presumably there is a need to make these very different 2-D shapes equal, possibly to find a geometric proof to an inverse mathematical concept about differences. To Òsquare the circleÓ gives preference to the square, four straight lines and four 90¡ angles, over a single line of the circle without angles. Maybe the emphasis more correctly is about the relationship of difference. Logically the truncation process suggests the circle is srcin to the square, meaning there can be no polygon equal to the circle. Folding the circle gives a unique perspective about the relationship of circle to square, revealing 90¡ to be an angle of change, of directional movement between two points before any construction of a fixed angle or measuring of lines and areas. Introduction The circle is one undifferentiated line without beginning or end; it is complete, whole, and self-contained. The concentric nature of the circle suggests a pre-existent unity without scale. The image of the circle, one of many 2-D shapes, is used as a unit. The radial measure defines the circle placing a distorted focus on a constructed centre point and the straight line to any point on the circumference; this is less than the full measure of the circle. To know what something is, we must know where it comes from. The circle does not come from the compass or computer programs used for drawing ideas about the circle. Traditionally a slice straight through the centre of a sphere explains the circle. Cutting the sphere apart sets up a predisposition for fragmentation. In drawing the circle the concept of spherical unity is ignored, the volume and surface area are disregarded. The circle image is then truncated into easily measured polygon parts. Geometry is based on measuring and  joining 2-D pieces to construct 3-D figures. This direction of thinking is supported by the idea of a point becoming a line that spreads out to a square plane that rises up to a cube and then projected to a hypercube. These are all 90¡ angles of change.   Understanding srcin of the circle without destroying spherical unity is through compression, bringing two opposite points on the surface of the sphere together. The horizontal movement expands out at 90¡ from the vertical line of inward compression. This is a right-angle movement that transforms the sphere to a circle disk; nothing is added or removed in this reformation of directional change. Figure 1. Unity as spherical srcin through compression forms at a right-angle the expanding concentric nature of the circle that is infinitely out and into itself. Spherical volume remains unchanged reformed to the circle disk; proportionally described by the Pythagorean theorem. Traditionally a circle disk is defined as the interior of a 2-D circumference. The 3-D circle disk has a top, bottom, a side, and two circles edges holding the three circle planes together containing unity of the srcinal spherical volume; traditionally these properties describe a cylinder. A potter understands both as the same clay moves perpendicular in direction. The first proposition in ÒThe Thirteen Books Of EuclidÕs ElementsÓ   starts with a straight line to prove the equilateral triangle by constructing two more straight lines. Two circles are used to find the relationship between the lines and angles of the triangle. 1  Circle and arcs are used for construction when straight lines and angles donÕt give enough information. This first construction shows the circle to itself to be principle before the triangle, contrary to persistence in giving greater importance to straight lines and measured angles. We have come to believe the image is the circle; using parts and pieces to calculate and construct 90¡ angles as a standard for measuring all other angles and line segments. The first movement of spherical srcin reveals a perpendicular self-referenced transformation of change without any change of inner volume. This must be considered principle before all abstracted expressions of 1 The Thirteen Books of EuclidÕs Elements . The University of Chicago, Published by Encyclopedia Britannica, Inc. 1952  a. b. c. a. b. c. d. polygons, polyhedra, measurement, or mathematical formulations; 90¡ is first an angle of change. Practice Touching two imaginary points on the circumference together and creasing will fold the circle in half. This initial straight-line crease is the result of a 90¡ movement half way between any two points on the circle. There are no visible angles, only a directional change of movement generating one line and two end points; this can happen with any shape perimeter. Using another circle, this time mark any two imagined points on the circumference, then touch them together and crease. Now there is one line with six relationships between four visible points. Draw six straight lines connecting all four points making a kite shape. There are eight triangles; six are right-angle triangles (except when the touching points are exactly furthest apart in a square relationship of eight right-angle triangles.) No two people ever pick the same two points for touching. Everyone has folded a different proportioned kite shape in the circle, each forming a uniquely placed individual diameter. Figure 2. ( a.)   Fold two points touching together. (b.) The crease divides the circle in half. (c.) Drawing lines connecting the four points forms a flat kite shape. (d.) When half of the circle is lifted from the flat plane a tetrahedron is formed with two open and two closed planes defined by the six relationships between four points. The folded diameter is the root measure for the circle, srcin for all straight-line creases, triangles, squares and endless reformations. The circumference is infinitely concentric into and out from itself making the diameter relative in length, but never equal to the circle being measured. Further, the perpendicular crease of folding the circle in half is at the same time an axis of rotation. Each half of the circle rotates 360¡ in two directions making 1440¡ of rotational movement from one right angle fold. There are four individual spherical paths of movement. These paths cannot be separated without losing the spherical envelope, thus negating spherical srcin of the circle.  a. b. c. d. e. Figure 3.  (a.) One half of the circle makes a full rotation in two directions around the diameter. (b.) The other half moves in the same way. (c.) Together both rotating halves form four congruent spherical paths of movement. This first fold decompresses spherical in-formation by expanding to a spherical envelope.  All straight lines, symmetries, and reformations take place within this spherical envelope through right-angle 90¡ movement without construction or fragmentation. This is very different than dividing the circle into four static quadrants, constructing a square to then prove isolated concepts. Every fold is a circular path that moves perpendicular between any two locations. Each folded chord in the circle is a rotational axis. This is lost in flat-plane construction. That first fold in the circle is a ratio of 1:2, a division of one whole in two parts without separation or breaking unity. The consistent progression of 1:2 is 3:6, 4:8, 5:10, 6:12, 7:14 and on through proportional folding for all symmetries (2:4 is not structurally or proportionally consistent). Following the first fold, the second and third folds in the 3:6 symmetry reveals something unexpected. The first fold is a true diameter; the next two folds are not folded chords, as one would expect. They are four reflective radial line segments that when folded accurately will appear   to be three complete diameters that bisect each other through the centre of the circle. Observing what actually happens gives a good idea about the generalizations made regarding diameters, radii, and what we see. Figure 4. ( a.) The circle folded in half. (b.) Folded in thirds; one part over two parts, a right-angle fold in the ratio of 1:2. (c.) Opened there are four end points, a kite relationship. The radii are 120¡ apart dividing the circle into thirds with one third divided in half. (d.) Refold with unfolded part (Fig.4b) behind previous fold, making all edges even. (e.) Open the circle to what appears as three diameters.
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