Posted on 24 July 2012 by

**cjf**
For me the most enticing facet of *projective geometry* is the profound way in which it treats *duality*. Duality is the notion that certain fundamental distinctions have similar structure in their complementary forms. In comparing a form with its dual, the basic structure remains even though the roles of the forms reverse. Inside and outside. Convex and concave. Yin and yang. In 2D (two-dimensional) projective geometry, point is dual with line; in 3D point is dual with plane while lines are self-dual. The relationship of duality is so penetrating and pervasive in projective geometry, that we might consider it *the geometry of fundamental duality*. It provides a geometrical stage upon which duality can be studied in a pure form.

Another profound aspect of projective geometry is its elementary treatment of *incidence* where one considers the join (∨) and meet or intersection (∧) of two basic geometrical objects such as point, line, plane, and hyperplane. The most fundamental correspondence of geometrical forms associates points and lines in dual arrangement: the points on a line form a *range* and the lines through a point form a *pencil*. The correspondence between a pencil and a range is a basic *projection*. Next a *perspective* relation joins a pencil with two ranges or a range with two pencils; that is, by combining two elementary projections. Such a *perspectivity* maps points to points, or dually, lines to lines as shown in the figure.

In the essay “Design Strategy” in Buckminster Fuller’s book *Utopia or Oblivion*, he includes projective geometry in his list of recommendations for a curriculum of design science. The connection between projective geometry and design thinking is an area that deserves more attention.

### A Catalog of Models of Projective Geometry

The rest of this essay tersely describes a broad listing of some of the more basic models of projective geometry. *Models* are a powerful tool for learning and for understanding as explained in my essay about the Importance of Model Thinking (based on Scott E. Page’s course). The models included below should provide an introduction to and an overview of projective geometry for those new to the subject (Note: some of these models require background knowledge that is not explained here. They are indicated with a . I encourage you to skim or skip such models, but to read on as later models may be more tractable.) I hope the experts will find the succinct summary and references useful. Although this list is fairly comprehensive, there are many models that are necessarily omitted. If you have a favorite model, please post a comment about it.

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