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By George K. Francis

Goals to motivate mathematicians to demonstrate their paintings and to aid artists comprehend the tips expressed by means of such drawings. This e-book explains the photo layout of illustrations from Thurston's international of low-dimensional geometry and topology. It provides the rules of linear and aerial point of view from the point of view of projective geometry.

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Hand-drawn perspective plays a special role in descriptive topology. Since perspective cubes, cylinders and cones used for framing a drawing can be copied from the computer screen, it is not absolutely essential. However, it does help the topologist to visualize and then design the object in the first place, by whatever means the finished product will be realized. Finally, I have found that teaching projective geometry, espectially to artists and designers, is best done in terms of perspective drawing.

Recall that projective space consists of ordinary 3-space augmented by an ideal point for each class of mutually parallel lines. An ideal point is joined to a real point by its representative through that point. Two ideal points, represented by two non-parallel real lines, are joined by an ideal line corresponding to the class of mutually parallel planes determined by the two lines. These ideal lines and points reside on an ideal plane, also thought of as the plane-at-infinity. In analogy to the projective plane, whose points correspond to antipodal pairs of points on a sphere under central projection, projective space has a Euclidean model in 4-space, consisting of antipodally identified objects on the 3-sphere.

Only the pen, and perusal of Escher's graphics, helped in designing the knot pair at the bottom. It was essential to depict a knot within a knot. Windows or cross sections would have detracted from the continuity. I had no choice but to try my hand at a transparent outer knot, just as Thurston had suggested in the manuscript. CHAPTER Figure 13 2 METHODS AND MEDIA LAMINATION AND EIGHT KNOT 37 38 A Figure 14 CABLE KNOT AND COMPANION TOPOLOGICAL PICTUREBOOK CHAPTER 2 METHODS AND MEDIA 39 CxC coordinates: (<2tt,t<2ir,2<2it/f) t= 0 ..

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