Geometrical Figures III: Squares

"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"The circle is the synthesis of the greatest oppositions. It combines the concentric and the eccentric in a single form and in equilibrium. Of the three primary forms [triangle, square, circle], it points most clearly to the fourth dimension." (Wassily Kandinsky, [letter] 1926)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"The Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible represents world. When a near-equality is drawn between the circle and square, the infinite is able to express its dimensions or qualities through the finite." (Robert Lawlor, "Sacred Geometry", 1982)

"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems - the determination of the diagonal of a square and that of the circumference of a circle - revealed the existence of new mathematical beings for which no place could be found within the rational domain." (Tobias Dantzig)