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.6 [214].This interesting dissection was found by superimposing twodifferent strips of triangular elements.Incidentally, this is not a minimal-piecedissection.Recreation 7 (Polygonal Dissections [109]).Mathematicians appetite forpolygonal dissections never seems to be sated [109]. Mathematical Recreations 107(c)(b)(a)Figure 4.7: Polygon-to-Triangle Dissections; (a) Pentagon.(b) Hexagon.(c)Nonagon (Enneagon).[109]Witness Figure 4.7: (a) displays Goldberg s six piece dissection of a regularpentagon; (b) displays Lindgren s five piece dissection of a regular hexagon; (c)displays Theobald s eight piece dissection of a regular nonagon (enneagon).Allthree have been reassembled to form an equilateral triangle and all are believedto be minimal dissections [109].Figure 4.8: Dissections into Five Isosceles Triangles [135]Recreation 8 (Dissection into Five Isosceles Triangles [135]).Figure4.8 shows four ways to cut an equilateral triangle into five isosceles triangles[135].The four patterns, devised by R.S.Johnson, include one example of noequilateral triangles among the five, two examples of one equilateral triangleand one example of two equilateral triangles.H.L.Nelson has shown thatthere cannot be more than two equilateral triangles. 108 Mathematical Recreations(b)(c)(a)Figure 4.9: Three Similar Pieces: (a) All Congruent.(b) Two Congruent.(c)None Congruent.[138]Recreation 9 (Dissection into Three Parts [138]).It is easy to trisectan equilateral triangle into three congruent pieces as in Figure 4.9(a).It is much more difficult to dissect it into three similar parts, just two ofwhich are congruent as has been done in Figure 4.9(b).Yet, to dissect thetriangle into three similar pieces, none of which are congruent, is again easy(see Figure 4.9(c)) [138].Figure 4.10: Trihexaflexagon [121]Recreation 10 (Trihexaflexagon [121]).Flexagons are paper polygons whichhave a surprising number of faces when flexed [121]. Mathematical Recreations 109To form a trihexaflexagon, begin with a strip of paper with ten equilateraltriangles 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 triangleand Voila! The assembled trihexaflexagon is a continuous band of hinged trian-gles 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.Thisremarkable geometrical construction was discovered by Arthur H.Stone whenhe was a Mathematics graduate student at Princeton University in 1939.AFlexagon Committe consisting of Stone, Bryant Tuckerman, Richard P.Feyn-man and John W.Tukey was formed to probe its mathematical propertieswhich are many and sundry [244].Figure 4.11: Bertrand s Paradox [122]Recreation 11 (Bertrand s Paradox [122]).The probability that a chorddrawn at random inside a circle will be longer than the side of the inscribed1 1 1equilateral triangle is equal to , and [122]!3 2 4With reference to the top of Figure 4.11, if one endpoint of the chord is fixedat A and the other endpoint is allowed to vary then the probability is equal1to.Alternatively (Figure 4.11 bottom left), if the diameter perpendicular to3the chord is fixed and the chord allowed to slide along it then the probability1is equal to.Finally (Figure 4.11 bottom right), if both endpoints of the2chord are free and we focus on its midpoint then the required probability is1computed to be.Physical realizations of all three scenarios are provided in4[122] thus showing that caution must be used when the phrase at random isbandied about, especially in a geometric context. 110 Mathematical RecreationsFigure 4.12: Two-Color Map [123]Recreation 12 (Two-Color Map [123]).How can a two-color planar mapbe drawn so that no matter how an equilateral triangle with unit side is placedon it, all three vertices never lie on points of the same color [123]?A simple solution is shown in Figure 4.12 where the vertical stripes areclosed on the left and open on the right [123]
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