The Bisection
of the

Yin-Yang

by Mervin B. Nogan, FPS

"At the normally perceptible level of
existence, there is nothing which re-
mains without movement, without
change. Every single thing is either com-
ing into existence, developing, decaying
or going out of existence...Change,
which is never-ending, proceeds accord-
ing to certain universal and observable
rules. (p. 39)

--John Blofeld, I Ching (1968) 6

Introduction

The true and original ancient geometry
of Euclid has been defined tersely, but
accurately, as the science of straight lines
and circles. Pre-eminently Freemasonry
displays the circle with a point at the
center in its venerable palladium of sym-
bolic treasures. The yin-yang symbol is
an impressively mature development of
this early Masonic masterpiece; with the
addition of two equal but asymmetrical
semicircles on either side of a diameter
of the bounding circle.

One of the truly magnificent, inclusive,
creative conceptions of humanity is this
intrinsically provocative oriental yin-
yang symbol. No one is likely ever to
overdo his study, analysis, contempla-
tion, and interpretation of this tre-
mendous geometrical invention.

Scattered about in the public domain
are a seemingly limitless assortment of
what are referred to as "geometric dis-
section puzzles." The vast majority of
these are polygonal figures of various
kinds; very few are of circular or curved

The Philakth~s, August, 1991

The Problem

The smaller or inner circle of Fig. 1 is
the Great Monad of Oriental origin.
This circle encloses the two comma-like
semicircles of equal area forming an
asymmetrical figure. The upper white
half is known as the YANG, while the
lower black half is identified as the YIN.
Designate as point O the unlettered cen-
ter of the circle or yin-yang figure. Such
a simple geometric figure, comprised
solely of circular arcs, with the center 0,
is the graphic illustration facing the text-
book reader whose attention is directed
to the proposition:

Point O is the center of the large circle.
The small arcs are semicircles. Show
how to draw one straight line that bisects
both areas.

This elementary curved figure and mis-
leading simple directive constitute a
near-perfect puzzle. The yang and yin
elements are obviously equal, but the
sinusoidal or serpent-like boundary be-
tween them is at once confusing and
misleading. A reference line of any na-
ture is seemingly missing, which subtly
introduces a disturbing or unsettling
emotional factor. To a high percentage of
those applying their attention to the fig-
ure, the sought for reference line persists
in being elusive.

Like all excellent puzzles, its source,
antiquity, or history is unknown. Sam
Loydl calls it "The Monad Puzzle" and
states the question " How should the
Monad be divided?" (pp. 45-46, 141-
142) H.E. Dudeney2 presents the topic
as "The Great Monad," with the direc-

tive:

Divide the Yin and the Yang into four
pieces of the same size, but different
shape, by one straight cut. (pp. 39, 174-
175)

Martin Gardner,3 4 titles the puzzle
"Bisecting the Yin and Yang" and
states: "By the way, did you know that
there is an elegant method of drawing
one straight line across the circle so that
it exactly bisects the areas of the Yin and
Yang? "

Anthony Christies points out that in
Chinese mythology the four cardinal
directions were related to the sides of
"the square earth. " (pp. 52, 56) Possibly
associated with this geometrical concept
was "a circular sky or heaven" in the
form of an annular ring or annulus. (p.
56) With these suggestions in mind, the
Great Monad of Fig. 1 calls for further
scrutiny.

Clearly the two centers of the yang and
yin semicircles, together with the origi-
nal center in Fig. 1 determine a horizon-
tal straight line. This line suggests the
orientation of the square circumscribing
the monad, and provides the sought for
reference to the yang and yin design. In
turn, the circle circumscribing the
square follows

If r is the radius of the monad, then the
radius of the outer circle is r~2, and the
area of the "heavenly" annuular ring is
the same as that of the monad.
The Solution

The veracity of this bisection is readily
and simply established. In Fig. 1 let A
denote the area of the enclosing monad
circle, which makes the area of the yang
and the yin each A/2 and the area of
circle K to be A/4. Imagine a horizontal
diameter dividing K into an upper and a
lower semicircle; making the latter A/8.
Next; included between that horizontal
diameter and the diagonal GH is a 45 ø
circular sector, the area of which is evi-
dently A/8. Hence, the area of yang
beneath the dividing line GH is A/4, and
yang is clearly bisected by line GH. The
identical situation holds for area yin.

Dudeney gives a slightly different, but
less simple solution.

Conclusion

Puzzles have attracted and retained the
concentrated attention of certain in-
dividuals for ages and they do not appear
to have lost their appeal to the minds of
men.

References

l.Martin Gardner, editor, Math~matical Puzzles of
Sam Loyd, Dover Publications, Inc., New York
Vol. 2, 1960, 177 pp.

2. Henry Ernest Dudeney, Amuscmcnts in Mathlmat-
ics, Dover Publications, Inc., New York, 1958
258 pp.

3 .Martin Gardncr 's Ncw Mathcmatical Divcrsions from
Scientific Amcrican; Univ. of Chicago Press, Chi-
cago, 1983, 253 pp. "Yang-Yin," pp. 143-146

with related references. Originally published
1966 by Simon and Schuster.

4.Martin Gardner, "The Combinatorial Basis of
the 'I Ching,' the Chinese Book of Divination
and Wisdom ;" ScimtificAmcruan, January 1974

Vol. 230, No. 1, pp. 108-113, 130, also front
cover, "Mathematics of the 'I Ching,"' and p.
4; ibid., November 1960, Vol. 203, No. 5, pp.
194-198, also previous issue.

5 .Anthony Christie, Chin~sc Mythology; Paul Ham-
lyn, New York, 1973, 141 pp.

6.Translated and edited by John Blofeld, I Ching
(The Book of Change); Dutton Paperback; E.P.
Dutton & Co., New York, 1968, 228 pp.