
835.03
We know that every point on the perimeter of the folded
semicircle is
equidistant from the point of origin. We may now go
to one end of the foldededge
diameter and fold the paper in such a manner that two
ends of the diameter are congruent.
This will fold the paper circle into four quadrants
which, by construction congruence, are
exactly equal. The legs of the 90 degree angle formed
around the origin of the circle by
this second folding are the same in length, being the
same radius as that of the circle, ergo,
of the halved diameter produced by the second folding.
The angle edges and the radii are
identical. When we open the quartercircle of four faces
folded together into the
semicircle, we find that the second fold edge, which
produced the 90degree angle, is the
radius of the diameter perpendicular to the first diameter
folded upon. The points where
this perpendicular diameter's ends intersect the circumference
of the circle are equidistant,
by construction, from the diameter ends of the first
foldededge diameter of the semicircle.
This folded semicircle, with its secondary foldmark
of verticality to its origin, can be
partially folded again on that perpendicular radius
so that the partially folded semicircle
and its partially folded, vertically impinging foldline
constitute an angularly winged unit,
with appearance similar to the outer hard covers of
a partially opened book standing
bottomless with the book's hard covers vertically perpendicular
to a table. This flying
winged, vertically hinged pair of doublethickness quartercircles
will be found to be
vertically stable when stood upon a table, that is,
allowed to be pulled vertically against
the table by gravity. In structural effect, this winged
quarterpair of open, standing "book
covers" is a tripod because the two diameter ends, A
and B, and the circle's origin point,
C, at the middle represent three points, A, B, C, in
triangular array touching the table,
which act as a triangle base for the tripod whose apex
is at the perimeter, T, of the
semicircle at the top terminal of the vertical fold.
The tripod's legs are uneven, one being
the vertical radius of the original circle, TC, and
the other two being the equidistance
chords, a and b, running from the top of the vertical
"book" column's back and leading
directly to the two wing terminals, A and B, of the
first folded diameter of the original
circle. The weight of the paper on either side of the
vertical fold extended on only one side
of any line produces weight or gravitational effect
to keep the vertical edge vertical, not
allowing it to lean farther in the direction of the
legs due to the relative structural rigidity
of the paper itself.
