Summary: MATHEMATICAL ORIGAMI: ANOTHER VIEW OF
ALHAZEN'S OPTICAL PROBLEM
Roger C. Alperin
1. Fields and Constructions
We can solve some elementary problems from geometry using origami
foldings. Below are the axioms which guide the allowable constructible
folds and points in C, the field of complex numbers, starting from the
labelled points 0 and 1 (see  for more details and references) .
(1) The line connecting two constructible points can be folded.
(2) The point of coincidence of two fold lines is a constructible point.
(3) The perpendicular bisector of the segment connecting two constructible
points can be folded.
(4) The line bisecting any given constructed angle can be folded.
(5) Given a fold line l and constructed points P, Q, then whenever pos-
sible, the line through Q, which reflects P onto l, can be folded.
(6) Given fold lines l, m and constructed points P, Q, then whenever
possible, any line which simultaneously reflects P onto l and Q onto
m, can be folded.
The first three are the Thalian constructions which ensure that we
have a field after we have a non-real complex number z. Some proper-