By Jay Kappraff
This e-book includes essays that stand all alone yet also are loosely hooked up. half I records how numbers and geometry come up in different cultural contexts and in nature: the traditional musical scale, share in structure, historic geometry, megalithic stone circles, the hidden pavements of the Laurentian library, the shapes of the Hebrew letters, and the shapes of organic varieties. the focal point is on how convinced numbers, resembling the golden and silver capacity, current themselves inside those structures. half II indicates what percentage of an identical numbers and quantity sequences are concerning the fashionable mathematical research of numbers, dynamical platforms, chaos, and fractals.
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Extra info for Beyond Measure: A Guided Tour through Nature, Myth, and Number
The eye projects a scene from the horizontal plane to an imaginary screen in front of the body. Rays of light can be thought to connect points on the scene to points on this imaginary plane, in which case the scene is viewed in perspective. 1. W h e n the eye is at point O, this projective mapping of scene to canvas transforms the infinitely distant line of the horizon onto a real line h on the canvas. Also, parallel lines receding from die viewer towards the horizon appear on die canvas as die oblique lines I meeting at some point on the horizon line.
One of the most valuable features of projective geometry is its ability to make the infinite accessible to human thought; points and lines at infinity have the same reality as those from the finite realm. 2 A pencil of lines indicating the point at infinity on each line. 28 Beyond Measure It is well known that depending on how a right circular cone is sliced by a plane, the boundary of the cross-section is either a circle, ellipse, parabola, or hyperbola, the so-called conic sections. Since a point source of light can be thought of as being located at the vertex, all conic sections are projectively equivalent.
4, one can see certain special points of the transformation. The intersection of x and x' is mapped to itself and is the only fixed point of the transformation. Also, the line through O parallel to x maps the point at infinity on line x to the point where this line intersects x'. Likewise, the point at infinity on x' is mapped from point the on x where the line through O parallel to x' intersects x. 4 A perspective transformation of points. 5 A perspective transformation of lines. 1) Chapter 2 The Vortex of Life 31 This relationship is known as the cross-ratio.