Newsgroup: it.hobby.enigmi
Titolo: Yin, Yang and a third one

Good morning to all!
My trip to China was a source of inspiration for my new math puzzle:
Yin and Yang. A welknown twin, to which I added a third.......
The previous cut and paste puzzle was very difficult. Only Helmut Postl
found a completely correct answer. By adding some hints (see below the
dutch version of the new puzzle) I hope to get some more good answers.
The puzzle will stay open for a little longer.
Have fun with the puzzles!
Greetings from a (still) sunny Ruurlo,
Peter 
http://home.planet.nl/~p.j.hendriks/ppvdw.htm

 
http://home.planet.nl/~p.j.hendriks/ppvdw.htm
sorry for I don't know the language of that web.
--------------------------------------------------------
    Zhu

 sorry for I don't know the language of that web.
Click on Union Jack for English

 The puzzle will stay open for a little longer.
The common representation of the Tao (yin yang) is very elemental and
keeps to geometric basics.  One begins with a string of a certain
length.  Holding one end of the "line" of string stationary, one then
uses the string like a compass, tying a stick to the other end of the
"string" and "drawing a circle" that has a radius equal to the length
of the string (line).  By "cutting" or "folding" the string (line,
radius, etc) in half, one can find the "mid-points" along the diameter
of the circle one's drawn.  Along a line representing 2r, the diameter
of the circle, the two "edge" points with the center point, mark where
the slope of a full sine cycle has a vertical tangent.  The two
"mid-points" along the same diameter line, represent where the slope of
the cycle will have horizontal tagents (parallel to the "diameter").
The Chinese, being very rational sorts of people, would no doubt wonder
why one would divide a circle into four sections in the manner
described and call it "three parts plus another part".  The initial
"length" of string, the line used to as the radius in order to draw a
circle, has a simple and proportional relationship to the full wave
cycle associated with a given circle.  The most reasonable method of
dividing a circle into three equal areas is to draw a line between the
center of the circle using the "radius" string, and then to add another
line at an angle ninety degrees plus 1/3 of ninety degrees around the
circle (take four parts of a circles area, to get three equal parts,
cut one of the four parts into three equal sub-parts, and add each
sub-part to the remaining quarters).
This said, here's the answer you're looking for: a "Reuleaux Triangle"
(turned inside out).  Here's the relevant link:
http://mathworld.wolfram.com/TrigonometryAngles.html
Here's a link to "irrational" numbers stuff.
http://mathworld.wolfram.com/IrrationalNumber.html
In terms of graphic representation of the Tao.  The ying and yang
aspects are merely threshold elements (bounds).  The grey line that
divides them is a third element ~XOR.  All of these three elements
together represent relations within a vague meta-material
superstructure (that's physical and meta-physical somehow).  Not being
very educated, nor a Taoist sorcerer, I'm probably not describing the
complexity very well.
The trinity spirals are usually Celtic.  Here's a link to a paper with
some very hot algorithms for sprial geometrical processes that conform
to Celtic spiral knots:
http://www.tug.org/TUGboat/Articles/tb24-1/moon-celtic.pdf