Friday, April 20, 2012

Perfect partnerships

The previous post led to the creation of projective geometry by extending "normal" geometry by an ideal plane filled with ideal points and lines.  In this post I want to explore the consequences of this extension in one particular direction, revealing a startling symmetry permeating projective geometry.

It is characteristic of mathematics that it begins with simple elements and combines them to create compound objects.  In geometry, these simplest elements are points, lines, and planes.  These elements in turn have incidence relations to one another: points can lie on lies, planes can contain or pass through points (or lines), etc.

Consider the pairs of incidence statements from planar Euclidean geometry:
A. Every two points have a unique joining line.
B. Every two lines have a unique intersection point.
The truth of Statement A is immediately obvious.  Statement B, on the other hand, is only "almost always" true:   if the two lines are parallel, then they by definition do not intersect.

It turns out that in planar projective geometry, statement B is also always true, since two parallel lines have an ideal point in common.  In fact, in projective geometry, every true statement has a partner which is also true.  This partner is called its dual, and it is obtained from the original statement by exchanging a set of words and phrases with their dual partners.  

For example, statement B can be obtained from statement A by replacing the blue text: by switching "point" and "line", and by switching "joining" and "intersection". These word-pairs are said to be dual in planar projective geometry. Other such pairs include the incidence properties   "contain" and "lie on" (or "passes through").  This set of words and phrases can be thought of as a dictionary for translating any statement into its dual.  One can also dualize configurations of geometric elements, without regard for the truth content.  For example, "3 points and their joining lines" is dual to "3 lines and their intersection points". Finally, notice that the dual of the dual is the original statement, so the two statements (or configurations) are really like a pair of twins.

It's natural to ask, Why should the principle of duality apply to all of projective geometry?  The simplest way to understand this is to observe that the axioms of projective geometry all exhibit this property: statements A and B are examples of two such axioms.  All the other statements in projective geometry can be derived from the axioms by logical necessity; any statement derived from a set of statements exhibiting duality will also exhibit duality, since I can apply the dictionary of duality to any proof to obtain a valid proof of the dual statement.

Duality for planar (2-dimensional) projective geometry is slightly different than for spatial (3-dimensional) projective geometry, which just means that the dictionary of duality for the two cases is slightly different.   Here we first focus on the 2-dimensional case, then discuss and give  an example of 3D duality.

We focus here on the existence of  perfect partnerships in projective geometry, arising as a result of duality.  Start with a point P and a line m which are not incident. Consider the set of all the lines passing through P (called a line pencil centered at P), and the set of all points lying on m (called the point range of m).  See figure on the left. Then, the perfect partnership is established by associated to every line through P, its intersection point with m; and conversely, every point on m is associated to its joining line with P.  It's clear that this partnership is perfect only because the red line in the figure parallel to m intersects m in an ideal point.  In euclidean geometry the partnership is not perfect.

A perfect partnership of this form (between the elements incident with two simple forms which are themselves not incident) is called a perspectivity.  By chaining together such perspectivities that share a common element, one can construct further partnerships.  For example, chaining together two such perspectivities that share a common line pencil establishes a perfect partnership between the points of the two point ranges.   A future post on this blog will take up this theme further.

In 3D duality, points and planes are dual; lines are self-dual.  To see why this is so, consider the statement:  "Two points have a unique joining line."  The spatial dual reads "Two planes have a unique intersection ____."  Clearly the only reasonable choice for the missing term is line.  Hence, a line is self-dual.  To be precise, "a line and all the points it contains" is dual to "a line and all the planes it lies in."  The former is called a point range (as above); the latter is called a plane pencil.

We close today's post by considering a simple example of 3D duality.

Begin with the cube, one of the five Platonic solids, a regular polyhedron consisting of 6 faces, 8 edges, and 8 vertices (image on left).  In order to simplify the procedure, we simplify by thinking of the faces as infinite planes, and the edges as complete lines.  (Dualizing the finite faces and edges is a more difficult task which we'll postpone for later, see exercise below.)   The dual polyhedron will then have 6  vertices, 12 edges, and 8 faces.  By considering the other Platonic solids, one sees that the octahedron satisfies these conditions (image on right).  In order to make sure that this is really the dual of the cube, attempt to translate descriptions of one solid into their dual form, and see whether they in fact are true.  Only when the two figures are dual in all their detailed incidence properties is one justified in calling them dual partners.
 3 faces and 3 edges meet at each vertex of the cube, and 4 faces and 4 edges meet at each vertex of the octahedron.
translates to:
3 vertices and 3 edges lie on each face of the octahedron, and 4 vertices and 4 edges lie on each face of the cube.
It's simple to verify that the second statement indeed is true.  Let's try something a bit more difficult:
The three joining lines of pairs of opposite vertices of the octahedron intersect in a point, the center point of the octahedron.
This translates as:
The three intersection lines of pairs of opposite planes of the cube lie in a plane, the center plane of the cube.
Verify that the dual statement makes sense: pairs of opposite faces of the cube lie in parallel planes, whose intersection line is therefore an ideal line.  Hence all three lie in the ideal plane.  Furthermore,  duality implies that this plane should be considered the center of the cube!  As Dorothy said, "We're not in Kansas anymore."   Rather than trying to explain how to think about this middle plane, we leave the reader to ponder it.  Those who are interested in further exploration in this direction  are invited to try the following exercise.

Exercises. 1. Devise a reasonable definition to decide when a point is inside the octahedron.  (Hint: start by defining that the center point is inside.) Then dualize this to a definition to decide when a plane is inside the cube.  Extend or generalize this result to dualize the actual faces and edges of the cube (as finite pieces of infinite planes and lines).  What is dual to moving a point along an edge of the cube between the two endpoints of the edge?
2.  We could have begun by defining the center point of the cube as the intersection of the 4 space diagonals of the cube.  What is the dual of this point in the octahedron?
P. S. If you're interested in meeting more dual polyhedra, try out this interactive application for exploring the platonic and archimedean solids and their duals.  A screenshot is shown below.

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