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By George K. Francis

Goals to motivate mathematicians to demonstrate their paintings and to assist artists comprehend the information expressed by means of such drawings. This ebook explains the photo layout of illustrations from Thurston's global of low-dimensional geometry and topology. It offers the foundations of linear and aerial point of view from the perspective of projective geometry.

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Extra resources for A Topological Picturebook

Sample text

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.

Complete the drawing of a cube starting from a segment and horizon line in the picture plane which represent an edge and a face plane of the cube. In a sense, all traditional perspective methods begin with a solution to this problem. My version resulted from a study of the monograph on perspective drawing by Jay Doblin [1956] and the detection of geometrical errors made for aesthetic reasons. I am grateful to Ben Halpern for convincing me of their presence. 3. Given a square and its perpendicular in a picture, discover and use its perspective information to draw other common figures, such as circular wheels with perpendicular axles.

Figure 6. A contour line drawing, 5(42) or 5(43), for the snail shaped dunce hat is too complicated to remember. Here is a more systematic way of designing a dunce hat with better symmetry as well as a simpler contour, 6(31). Moreover, it illustrates, in dimension two, a particular way of cutting and pasting topological objects which is even more useful in three dimensions. Isolate a neighborhood of the vertex by detaching three triangles, 6( 13). Assemble these separately into the cone over the vertex link, 6( 12) of the complex.

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