New PDF release: A Topological Picturebook

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 booklet explains the picture layout of illustrations from Thurston's international of low-dimensional geometry and topology. It offers the rules of linear and aerial standpoint from the point of view of projective geometry.

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Extra resources for A Topological Picturebook

Example text

In this way any word in the free group generated by the segment labels determines a closed curve in the knot complement. Since knot groups can be presented with a single relator there will be one word whose corresponding curve is spanned by a disc embedded in the complement of the knot. The solution to the problem is shown on the right of the first photo. On the left is an intermediate stage of the disc which I drew to help visualize the former. Serendipity, however, intervened. The helper picture already solves the problem if you use a conjugate of the given relator.

TRIPRONG. Figure 3. In designing a line pattern for a surface one is sometimes able to place the border curves with confidence, only to wonder where the contours should go. To obtain a plausible answer I smoothe a piecewise flat and cornered version of the surface. Each line on such a polyhedral swface is an edge bordering one face, or separating two or more faces. Because it is flat, each 18 Figure 3 A TOPOLOGICAL PICTUREBOOK TRIPRONG face of the polyhedron is either wholly visible or wholly invisible.

All but two of the triangles collapse to edges. The trick was to let the eventual position of the rear simplices, labeled B, occupy the space left over by the eventual position of the from simplex A . It was important to suggest that the surface continues beyond the belts but not so as to detract attention from the pants themselves. Tim Poston pointilliste style seemed most appropriate here. LAMINATION AND EIGHT KNOT. Figure 13. The hyperbolic plane routine was indispensible for designing this illustration for another Thurston paper [1982].

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