Grassmann algebra is an algebra for Geometry

© John Browne 2012.

## Pascal lines

Below is a plot of the 60 bound vectors and bivectors used to represent the 60 Pascal lines and lines
at infinity of the 60 different hexagons formed from the 6 vertices of a regular hexagon. The lack of
apparent symmetry is an artifact caused by the extra information the bound vectors and bivectors
portray.

## Hexagons in an ellipse

Given six points on a conic, there are 60 different hexagons which can be constructed: an initial
point may be joined to a second point in five ways, the second point to a third point in 4 ways and
so on, leading to 5! ways of constructing a hexagon. However for each one there corresponds the
same hexagon traversed in the reverse order, thus leading to just 60 different hexagons. Pascal's
Theorem establishes a line associated with any hexagon inscribed in a conic, so to each of these 60
hexagons will correspond a Pascal line.
Here are 20 of the hexagons.

## Pascal's Theorem

Pascal's Theorem and its converse state that
If a hexagon in inscribed in a conic, then opposite sides intersect in collinear points.
If opposite sides of a hexagon intersect in three collinear points, then the hexagon may be inscribed
in a conic.
The graphic below shows 6 vertices P defining the sides S of a hexagon whose opposite sides
intersect in points Z, the exterior product of which is T. If T is zero, the points Z lie on a line L
called the Pascal line of the hexagon. Pascal's Theorem says that if the points P lie on a conic then T
is zero. The converse says that if T is zero, the points P lie on a conic. T and L are constructed from
the points P according to the formulae below.