> TVS
c .jbjb!! NKK':NNN84<Z(*@@@^``````,A!R#@@$@@^rT^Jj[NCJ0,}$}$J
j jSide-angle reciprocity a survey
by Branko Grnbaum
University of Washington, Box 354350, Seattle, WA 98195
e-mail: grunbaum@math.washington.edu
In a paper [8] published long ago in Geombinatorics, I stated that there is a kind of reciprocity between the sides and angles of convex quadrilaterals. The specifics will be described soon, but as a justification of this note I would like to say that it reports on developments of this topic, presents new problems and gives additional references to the literature and to the classification of convex n-gons.
The well-known exclusive classification by the lengths of their sides (equilateral, isosceles, scalene) has an analogue for quadrangles. For a given quadrangle, there are seven different (mutually exclusive) possibilities regarding the lengths of its sides:
{4} all sides are equal;
{3,1} three sides are equal, different from the fourth;
{2,2}A two pairs of adjacent sides are equal;
{2,2}O two pairs of opposite sides are equal;
{2,1,1}A one pair of adjacent sides are equal, the other two are different from these and from each other;
{2,1,1}O one pair of opposite sides are equal, the other two are different from these and from each other;
{1,1,1,1} all four sides are different.
The fairly self-explanatory notation by partitions of 4 has been introduced independently by Robertson [10] and Alonso [1], [2]. "A" is to remind us of "adjacent", "O" of "opposite". We shall call this classification the "side-lengths classification".
Analogous seven classification possibilities arise with respect to the angles, and now one can ask:
Which of the 49 pairs of sides and angles conditions can be realized by convex quadrangles?
The answer given in [8] was that there are precisely 20 such pairs, and examples of all 20 kinds of quadrangles were provided. The proof that the other 29 pairs are not realizable was described in [8] only in general terms. It turns out that this was a mistake: As shown in detail by Orlando Alonso in his doctoral thesis [1] and presented in condensed form in [2], the realizable pair {2,1,1}O;{2,1,1}O was missed in [8]. Independently, and without relation to [8], a construction for this pair was presented in [5] and attributed to Nathan Bowles.
A corrected table of the realizable pairs in shown in Figure 1, where the twenty realizable pairs are indicated by hollow dots, and the additional pair is signaled by a solid dot. This classification is the source of an activity described by Joseph Malkevitch [9].
Side Angle type
type {4} {3,1} {2,2}A {2,2}O {2,1,1}A {2,1,1}O {1,1,1,1}
======================================================
{4}
{3,1}
{2,2}A
{2,2}O
{2,1,1}A
{2,1,1,}O
{1,1,1,1}
Figure 1. The 21 realizable types of side-and-angle pairs.
The main point of [8] was that there is a perfect symmetry (about the main diagonal) in Figure 1 of the realizable pairs; this has not been changed by Alonso's correction of the previous enumeration. This is the "reciprocity" of the title. Paired with the observation that such a reciprocity is natural for (spherically) convex polygons on the sphere, regardless of the number of sides, and that is can be observed for triangles in the plane as well, led to the formulations of the following conjecture:
Figure 2. The example of Auroux [4] that falsifies Conjecture 1 for n = 12 and the partitions {12} and {3,3,2,2,1,1}.
Conjecture 1. For every n, there is a side-angle reciprocity of convex n-gons for paired conditions of the kind discussed above.
Chronologically the first development following [8] was the result of Denis Auroux [4], see Figure 2:
For n = 12 there exists a counterexample to Conjecture 1
Auroux expressed the opinion that the validity of Conjecture 1 may depend on the factorization of n, and that the conjecture may be valid for n prime, power of a prime, or the product of two primes. There have been no results in that direction during the 15 years that elapsed since the publication of [4]. In particular, it is not known what happens for n = 5.
One modification of the questions raised in [8] that was proposed by Malkevitch and carried out by Alonso [1] is a coarsening of the classification discussed so far. This is obtained by considering only the numbers of equal sides (or congruent angles), without attention to the order around the quadrangle. Thus (3) and (4)
Side Angle type
type {4} {3,1} {2,2} {2,1,1} {1,1,1,1} ===============================================
{4}
{3,1}
{2,2}
{2,1,1}
{1,1,1,1}
Figure 3. The 16 types of paired partition conditions realizable by convex quadrilaterals.
coalesce, as do (5) and (6); in fact, each of the resulting classes can be characterized (for arbitrary n) just by a partition of n. For n = 4 there are five partitions, namely {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}. From Figure 1, or independently, it follows that 16 pairs of 25 paired partition conditions are realizable by convex quadrilaterals. This is indicated in Figure 3.
One advantage of the partitions-type conditions over the conditions underlying Figure 1 is that there are fewer cases to be distinguished. This makes it reasonable to consider extension of the classification to polygons with more than four sides. Since there are only seven partitions of n = 5, there are just 49 paired conditions a situation much more manageable than the 144 cases that need to be examined for Conjecture 1 for n = 5.
Figure 4. An equiangular realization of the paired partitions {3,3,2,2,1,1};{12} by a convex 12-gon.
It is easily seen what happens with Auroux' example under the coarsened conditions; the partitions involved are {12} and {3,3,2,2,1,1}, and a realization of the equiangular case is shown in Figure 4. Alonso and Malkevitch [3] mention that it is an open question whether the side-angle reciprocity holds for polygons with more than four sides. In general it would seem reasonable to propose:
Conjecture 2. For every n, there is a side-angle reciprocity of convex n-gons for paired partitions conditions.
Clearly, the side-lengths conditions {4}, ..., {1,1,1,1} listed earlier, as well as the analogous conditions for angles of quadrilaterals, introduce classifications of quadrilaterals. For example, as formulated, {4} by itself describes rhombi (in the inclusive meaning, covering squares as well), while {2,2}O characterizes parallelograms (again in the inclusive meaning). However, if these conditions, or the paired conditions, are understood in the exclusive sense (so that {4} characterizes non-square rhombi), then we obtain two classifications (by conditions on sides, or on angles) into seven classes, or one into 21 (non-empty) classes. Similarly, Alonso's classification leads to two partitions of the quadrangles into five classes, or one into 16 (non-empty) classes.
These classifications refine the traditional ones, which in most writings distinguish squares, rectangles, rhombi, kites, parallelograms, and various other kinds of quadrilaterals. Some of the traditional classes depend on symmetry properties [10], or are restricted to quadrangles inscribed into circles (or circumscribed to circles), or concern other restricted classes. A detailed discussion of the definitions used appears in [11], where also the usage by authors of various texts are compared. Other aspects of classification of restricted kinds of polygons are considered in [7], with some attention given to pairs of dual classifications.
Many Internet sites describe more-or-less clearly defined classifications of quadrangles, most closely related to the traditional kinds.
Alonso [1] considers several refinements of the classification. Although these are of interest, most of them do not lead to any side-angle reciprocity.
Some discussions and classifications related to the reciprocity appear to have been considered by de Villiers [6]. This work has not been available to me. But his other works, as well as those of some other writers, seem to be limited to stating that for any side-lengths classification there is a corresponding angle-sizes classification without considering paired conditions.
References.
[1] O. B. Alonso, Making sense of definitions in geometry: Metric-combinatorial approaches to classifying trangles and quadrilaterals. Doctoral thesis, Teachers College, Columbia University, New York.
[2] O. B. Alonso, Grnbaum's convex quadrangles enumeration and an extension of the angle-side reciprocity of quadrangles. Geombinatorics 20(2010), 45 47. MR2732515
[3] O. B. Alonso and J. Malkevitch, Enumeration via partitions. Consortium, The Newsletter of the Consortium for Mathematics and its Applications, 98(2010), 17 21.
[4] D. Auroux, Nonsymmetric rigid polygons and the angle-side reciprocity conjecture. Geombinatorics 6(1996), 6 14. MR1392792
[5] A. Bogomolny, A quadrilateral with equal opposite sides and angles, from Interactive Mathematics Miscellany and Puzzles,
HYPERLINK "http://www.cut-the-knot.org/Geometry/NonParallelogram.shtml" http://www.cut-the-knot.org/Geometry/NonParallelogram.shtml
(Accessed July 17, 2011)
[6] M. de Villiers, Some Adventures in Euclidean Geometry. Univ. of Durban-Westville, 1996. 3rd ed., Dynamic Math. Learning, 2009.
[7] M. de Villiers, Equiangular cyclic and equilateral circumscribed polygons. Math. Gazette 95(2011), 102 107.
[8] B. Grnbaum, The angle-side reciprocity of quadrangles. Geombinatorics 4(1995), 115 118.
[9] J. Malkevitch, Classifying quadrilaterals.
HYPERLINK "http://www.york.cuny.edu/~malk/geometricstructures/quadrilateral.html" http://www.york.cuny.edu/~malk/geometricstructures/quadrilateral.html (Accessed on July 18, 2011)
[10] S. A. Robertson, Classifying triangles and quadrilaterals. Math. Gazette 61(1977), 38 49.
[11] Z. Usiskin, J. Griffin, E. Willmore, and D. Witonsky, The Classification of Quadrilaterals: A Study of Definition. Information Age Publishing, Charlotte, NC 2008.
Page PAGE 62
GEOMBINATORICS 21(2011), 55 62
"%&.
A
X
Y
cdƼƼƳƨƳۅwhjh{6NHOJQJhjh{6OJQJhrh{>*OJQJh{6NHOJQJhW`h{OJQJh{6OJQJh{NHOJQJh{OJQJh*Jh{NHOJQJh*Jh{OJQJhrh{OJQJhrh{5CJOJQJ+"#6n.0 L uZ$
88hd,]8^8`ha$gd{$
88hd,]8^8`ha$gd{$88hd,x]8^8`ha$gd{$88hd,]8^8`ha$gd{$88d]8^8a$gd{$88d]8^8a$$88]8^8a$$88]8^8a$
-. L
@v
^gd{$88dh]8^8a$gd{$88hd]8^8`ha$gd{$88hd]8^8`ha$gd{$88hdx]8^8`ha$gd{$
88hd,]8^8`ha$gd{
!"_`}~./BCVWXķķķķķįğh*Jh{5OJQJh{jh{Uh*Jh{NHOJQJh*Jh{OJQJh>Fh{CJ$EHOJQJh>Fh{CJ$EHOJQJh{OJQJhBh{OJQJ<2p"]V$88d]8^8a$$88dx]8^8a$gd{
$ x^dh^gd{
T$ d^
$ ^gd{
^VXOY~i\88]8^8gd{$88d,x]8^8a$gd{$88hd,xx]8^8`ha$gd{$88d]8^8a$gd{$88dx]8^8a$gd{$88d]8^8a$$88d]8^8a$gd{$88dh]8^8a$gd{NO#$)0puUVXYdefghiqrstuvwx[\oxLMnoLMh>Fh{CJ$EHOJQJh{OJQJhW`h{6OJQJh*Jh{NHOJQJh*Jh{OJQJh*Jh{5OJQJh*Jh{6OJQJDP[jzvp]]$88d,]8^8a$gd{
$ dh^gd{
$ 88dh]8^8gd{
$ 88dh]8^8gd{
$ 88]8^8gd{
8v]8^v`gd{
,-.(ST / . / N W T!U!""Y"Z"""####$$W$X$$Źhjh{6OJQJhW`h{6OJQJh*Jh{6NHOJQJh*Jh{6OJQJh*Jh{5OJQJh*Jh{OJQJh{j*h{Uh*Jh{OJQJh*Jh{NHOJQJ5v,.!$$<%&&&&&}}88]8^8gd{$
88hd,x]8^8`ha$gd{$88d,]8^8a$gd{$88d,x]8^8a$gd{$88d]8^8a$gd{$88hd,x]8^8`ha$gd{$$$$2%3%<%&&&C'D'''((r(s(((((")#))))(*)***e*f****?+@+K+L+++++,,W,X,Y,ĵ굪y#jhq^h{OJQJUhW`h{0JOJQJ#j@hW`h{OJQJUhW`h{OJQJjhW`h{OJQJUh*Jh{6OJQJhBh{NHOJQJh*Jh{NHOJQJh*Jh{OJQJhBh{OJQJ/&':((a))g**+u++,,----....$
88]8^8a$gd{$
88x]8^8a$gd{Y,,,,,,,------------...ƹƹƵh{h{0JCJmHnHuh{h{0JCJjh{h{0JCJUh{h{CJh{h*Jh{NHOJQJh*Jh{OJQJhW`h{OJQJh{OJQJjhW`h{OJQJUhq^h{0JOJQJ3
000P7:p{/ =!"#$x%|HH@ Rc(hh@d-*DdFx@NSN
C*AFigure 2BBJ/H{\K9dD0T\ˠy֍BJ/H{\K9ƪExWklU>[j-T #<˾(`yѠ@ہvf4MV
DE4$hh FD@" D?0MFfeHg:+حr@Z30ɵa(PEPHM(b+b&ҫf7qf[AHYޢ*ά/]ZBAUɒa3,[|S2䘬cqq҆Ew(NTv2v*
0^h0:#en-a8Tr~_7`889ሸ:"1:"h>җxso/ƟWe<>p;WGB;nP?ujS
l`i6( H%~j(E/8uVR&fE}#w,|c;)#Cίp/Sot?C)g;>M~{W)|+_8gM+&?pW~+l_Gw*QxA~+*u:H(튔`I3%LRsdYW.:)l5nwa+gpJ>9;
&P.i0OQ|sR:O9ri1\-%ԙIxdG-nDd(R
C.AUntitled 1Bp
Li_$ڠVLn0TD)yJݽ׃ל-
Li_$ڠV xkA_tE>fP*(RۊfwMv ēbAEы7O_G^I(6IRd}̛UE{85h!no=9p1ox+n,aW%3qdtZ.X\ϕUVL]9OUs晖!0
*g+逪"]q0^炕^`M'wf6Ñ41gf0ryzE19֬(M1Xy5&:g&J[w*еjČ6%Q##-2j\@Lb
n NgI⼤.88+#'#}9zt UnU%ʞFs|J87r;tؔ1mFwTWơ$*rёC/dR/;t?;7]8q_rBmZ2WmænяqqISǇԡ55<˩?H &L
{v:}uhkr.p(-[f3WXp$yMHD:axGEH@{}jQV'-Pmiz?.eDyK<http://www.cut-the-knot.org/Geometry/NonParallelogram.shtmlyKxhttp://www.cut-the-knot.org/Geometry/NonParallelogram.shtmlDyKFhttp://www.york.cuny.edu/~malk/geometricstructures/quadrilateral.htmlyKhttp://www.york.cuny.edu/~malk/geometricstructures/quadrilateral.htmlD@DNormalCJOJQJkH'mH sH tH DA@DDefault Paragraph FontZiZTable Normal :V4
l4a_H(k(No List4 @4Footer
!4@4Header
!:@:
Footnote TextCJNB@"N Body Text$pd,]pa$OJQJ0U@10W` Hyperlink>*B*@V@A@[=FollowedHyperlink>*B*.)@Q.{Page Number(N5N 7z 8z 9z :z ;z <z =z >z@]
$(_|"#6n.0LL@ 2
p
"]V
X
OYP[jzv,.< !:""a##g$$%u%%&&''''(((0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0ʀ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0ʀ0ʀ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0Ѐ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0Ѐ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ00ɀY܂0ɀd"#6n.0LL@ 2
p
"]V
X
OYP[jzv,.< !:""a##g$$%u%%&&'(0ʀ00000@0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ
4W0P
4W00ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ@0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ0ʀ(99<$Y,. "$& Vv&.!#%.#)$e$&X&&(XX<!867@60(
B
S ?((, q
w
gqdh3!;!!!R"\"""""h#q#$$
%%%%''.'?'G'P'X''(#%no7:RWJKMS[JQ")3
7
,\y{P!!$$$$'''(::::::::::::::::::::::::::::::::{&U808^8`0o(.^`.pLp^p`L.@@^@`.^`.L^`L.^`.^`.PLP^P`L.{ (i.@!@a;ug(0@UnknownGTimes New Roman5Symbol3Arial3Times9New York5Geneva"AhBfDG) E+xx>dH({`+#EConnected (n4) configurations exist for almost all n an updateGalya DimentBranko Grnbaum
Oh+'0( @L
lx
'HConnected (n4) configurations exist for almost all n an updateGalya DimentNormalBranko Grnbaum208Microsoft Word 11.6.6@ς@notD@66@"q-
՜.+,D՜.+,\`hpx
'EH(FConnected (n4) configurations exist for almost all n an updateTitleL 8@_PID_HLINKS'AqFhttp://www.york.cuny.edu/~malk/geometricstructures/quadrilateral.htmlr<http://www.cut-the-knot.org/Geometry/NonParallelogram.shtmlKSV Figure 2>0,Untitled 1
!"#$%&')*+,-./123456789:;<=>?@ABDEFGHIJLMNOPQRURoot Entry F96-WData
(1Table0$WordDocumentNSummaryInformation(CDocumentSummaryInformation8KCompObjX FMicrosoft Word DocumentNB6WWord.Document.8