business casual

April 17, 2015 in Finite Geometry

two triangles are displayed here.
each has a Red, a Yellow, and a Blue
vertex.

joining the red vertices
to form a red *line*…
and forming blue and yellow
lines likewise…

the triangles are arranged in such a way
that the three lines so described
meet at a single point.

(two triangles chosen “randomly” from
triangles-in-the-plane *won’t* have
this property.
of course it’s easy to *construct*
such triangles, though:
just start with the lines
and choose the points from there
[as i’ve done here].)

now we’ve got our two red-yellow-blue
triangles in place (they are said to
be “in perspective with respect to”
the point-of-view point in the lower-
left (X marks the spot).

the next move is to “extend the sides”
of the triangles:
we can imagine that each Blue-Red edge
(for example), forms a Purple line.
the two purple lines now meet
at (what we will call) a purple *point*.

likewise, form Orange and Green points
by intersecting (respectively)
the orange and green “line pairs”
(formed by extending the red-yellow
and the blue-yellow edges [respectively])
of our Two Triangles.

desargues’ theorem: these new “secondary”
points (O, G, P) all fall on a line.
w’edia.

(when cases involving parallel lines
are taken out of the discussion.
for example,
ordinary 2-d “euclidean” space
can be enhanced with “ideal” points
in such a way as to make this
*always* true [“parallel lines
meet at infinity” becomes
*formally true* in our (enhanced)
“projective space]
).

now for bus to the office to turn in papers
and pick up other papers. life is good.

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