By George K. Francis

Goals to inspire mathematicians to demonstrate their paintings and to aid artists comprehend the information expressed via such drawings. This publication explains the photograph layout of illustrations from Thurston's global of low-dimensional geometry and topology. It offers the foundations 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**

Sometimes, the time it takes to draw the picture during the lecture is also useful. The three photos of the Diapered Trefoil Knot have a little 28 A TOPOLOGICAL PICTUREBOOK story worth telling in this regard. It began when Bill Thurston asked me to draw the disc spanning a given curve in the complement of the knot. The curve linked the knot so as to represent the relator in a Wirtinger presentation of the knot group. Though I cannot review knot theory here let me remind you how to read the relator.

In this case there is no convergence: parallel lines in space remain parallel in the picture. This is often called a "parallel projection" but the term should be avoided in the company of artists and draftsmen. For them a parallel projection is a perspective projection in which the principal plane of the pictured object is parallel to the picture plane. An affine projection whose projectors are perpendicular to the picture plane is called orthographic, otherwise it is an oblique projection. This is the original meaning of "orthographic" although today this term usually refers to a composite oftwo or more "parallel" projections at right angles to each other: the plan and elevation of a building, for example.

This illustration for a paper by Thurston [1986] solves an interesting graphical problem. 12( 11) shows a triangulation of the thrice punctured sphere, or pair oj pants, covered by 12(21). In the process of pushing everything on the border lines in the covering to one side, the borders of the Riemann surface are spun about themselves an infinite number of times. 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 .