Space and Counterspace I

The principle of duality in projective geometry creates a partner for any configuration or statement.  It is based on a dictionary of duality that allows any description to be translated into its dual partner.

For 2D geometry, this dictionary starts with the entries:

• point             line
• join               meet
• lies on           passes through
• move along.  rotate around

So for example, "Two points have a unique joining line" is dual to "Two lines have a unique meeting point." (Only true in PG since parallel lines meet in ideal points). The dictionary is symmetric, so that you can look up a word in either column and replace it with its partner.  Words that don't appear in the dictionary are left alone. The above dictionary applies to projective geometry in general. These blog posts [xxx] deal with this basic version of duality.

In this post we want to explore a special extension of this dictionary that allows us to dualize euclidean geometry to produce dual euclidean geometry. Restricted to 2D this leads to the euclidean plane (EP) and the dual euclidean plane (DEP).   We then begin to explore and compare these two partner geometries.

This blog isn't the place to give the details of this construction.  We can give some hints however.  The euclidean plane is derived from the projective plane by identifying a special line called the ideal line (sometimes called the line at infinity by careless speakers).  The points of this line are ideal points and can be identified with directions or vectors, that form the foundation of euclidean measurement.  In DEP, there is correspondingly a single ideal point, along with all the lines passing through it, the ideal lines of the point, which play an analogous role in dual euclidean measurement.

While the ideal line of the euclidean plane appears to be built into the nature of the world, the ideal point of each DEP can be chosen independently.  

We want to compare EP and DEP on the basis of a simple geometric pattern. The following image shows a regular hexagon containing within it a hexagonal pattern of colored dots centered on the origin of the coordinate system.  We choose the ideal line of the DEP to be this origin.  It's marked by a star icon.  What will the dual partner of this pattern look like in DEP?  The simple answer is (ignoring the six colored segments bounding the hexagon for the moment): each colored point in the central grid corresponds to a line of the same color in the outer figure. For example, the red point on the right side of the hexagon corresponds to the vertical red line on the left hand side close to the cyan vertex. Why: the point coordinate (.9, 0, 1) produces the line coordinate [.9, 0, 1] which corresponds to the line x = -1.11... The dual line appears on the opposite side of the origin from the original point, with a slope perpendicular to the line joining the origin to the given point.  The center point (0,0,1) corresponds to the ideal line z = 0. This equation reflects the fact that ideal points have z-coordinate 0. 

One characteristic feature of the line pattern is that there are many points where several different-colored lines meet.  To find the pendant in the original pattern, we have to look for lines upon which several different-colored points lie.  These are easy to find; the hexagon grid is criss-crossed with such lines -- although the lines themselves haven't been drawn.  For example the point on the left where red, yellow, magenta, and 2 olive green lines meet corresponds to the vertical row of dots running from the yellow dot on the upper right of the hexagon boundary to the red dot on the lower right.