MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Recreations 109
To form a trihexaflexagon, begin with a strip of paper with ten equilateral
triangles numbered as shown in Figure 4.10. Then, fold along ab, fold along cd,
fold back the protruding triangle and glue it to the back of the adjacent triangle
and Voila! The assembled trihexaflexagon is a continuous band of hinged triangles
with a hexagonal outline (“face”). If the trihexaflexagon is “pinch-flexed”
[244], as shown, then one face will become hidden and a new face appears.This
remarkable geometrical construction was discovered by Arthur H. Stone when
he was a Mathematics graduate student at Princeton University in 1939. A
Flexagon Committe consisting of Stone, Bryant Tuckerman, Richard P. Feynman
and John W. Tukey was formed to probe its mathematical properties
which are many and sundry [244].
Figure 4.11: Bertrand’s Paradox [122]
Recreation 11 (Bertrand’s Paradox [122]). The probability that a chord
drawn at random inside a circle will be longer than the side of the inscribed
equilateral triangle is equal to 1 1 , 3 2
and 1
4 [122]!
With reference to the top of Figure 4.11, if one endpoint of the chord is fixed
at A and the other endpoint is allowed to vary then the probability is equal
to 1.
Alternatively (Figure 4.11 bottom left), if the diameter perpendicular to
3
the chord is fixed and the chord allowed to slide along it then the probability
is equal to 1.
Finally (Figure 4.11 bottom right), if both endpoints of the
2
chord are free and we focus on its midpoint then the required probability is
computed to be 1.
Physical realizations of all three scenarios are provided in
4
[122] thus showing that caution must be used when the phrase “at random” is
bandied about, especially in a geometric context.