In his works [R1,R2] David Robbins proposed several interrelated conjectures
on the area of the polygons inscribed in a circle as an algebraic function of
its sides. Most recently, these conjectures have been established in the course
of several independent investigations. In this note we give an informal outline
of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David
  Robbins

Pak, Igor

English

arXiv

AbstractIn his works [Discrete Comput. Geom. 12 (1994) 223–236; Amer. Math. Monthly 102 (1995) 523–530] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments

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The area of cyclic polygons: Recent progress on Robbins' conjectures 

Advances in Applied Mathematics 34 (2005) 690–696
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The area of cyclic polygons:
Recent progress on Robbins’ conjectures
Igor Pak
Department of Mathematics, MIT, Cambridge, MA, 02139, USA
Received 31 July 2004; accepted 7 August 2004
Available online 22 January 2005
Abstract
In his works [Discrete Comput. Geom. 12 (1994) 223–236; Amer. Math. Monthly 102 (1995) 523–
530] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed
in a circle as an algebraic function of its sides. Most recently, these conjectures have been established
in the course of several independent investigations. In this note we give an informal outline of these
developments.
 2004 Published by Elsevier Inc.
1. Introduction
Let a1, a2, . . . , an be the side lengths of a convex polygon inscribed in a circle. What
is the area S = S(a1, a2, . . . , an) of the polygon as a function of the sides? This question
goes back to Heron of Alexandria (the case n = 3) and Brahmagupta (the case n = 4). It
seems, David Robbins was the first to address this question in full generality and suggest
the way of phrasing the answer [8,9]. We start with the general remarks on the problem
(largely due to Robbins) and then outline recent developments in an informal essay style.
Following Robbins, we call polygons inscribed in a circle the cyclic polygons. We de-
note the vertices by A1,A2, . . . ,An and the center by O .
First, observe that S(a1, a2, . . . , an) is well defined, that is there exists at most one cyclic
polygon with the given (ordered list of) side lengths. Indeed, start with a large enoughE-mail address: pak@math.mit.edu.
0196-8858/$ – see front matter  2004 Published by Elsevier Inc.
doi:10.1016/j.aam.2004.08.006
I. Pak / Advances in Applied Mathematics 34 (2005) 690–696 691circle and place n + 1 vertices A0,A1, . . . ,An at distance |Ai − Ai+1| = ai . Continuously
decreasing the radius we obtain a unique convex polygon with A0 = An, as desired.
Second, observe that S is a symmetric function in ai . Indeed, this follows from the fact
that we can interchange triangles [OAi−1Ai] and [OAiAi+1]. The details are straightfor-
ward.
Our final observation is the fact that S is an algebraic function of the side lengths ai .
First, notice that it is polynomial in the coordinates of vertices Ai = (xi, yi):
S = 1
2
∣∣∣∣x1 x2y1 y2
∣∣∣∣+ 12
∣∣∣∣x2 x3y2 y3
∣∣∣∣+ · · · + 12
∣∣∣∣xn x1yn y1
∣∣∣∣ . (◦)
Here each summand is equal to the (signed) area of the triangle [0AiAi+1], and it is easy
to see that they add up to the area of the polygon (0 denotes the origin).
Now, move the polygon in such way that A1 = (0,0) and A2 = (a1,0). There are
2(n− 2) free variables for the remaining vertex coordinates and 2 variables for the coordi-
nates of the center O . Together these give 2n − 2 variables. Similarly, the remaining side
lengths give n − 1 equations, the equality of the distance to the origin give another n − 1
equations, which total 2n − 2 equations. One can show that these equations are alge-
braically independent so all free coordinates are in fact the algebraic functions of the side
lengths ai . Thus, from the formula (◦), so is the area S.
Note that depending on the orientation of the polygon, the (signed) area S given by (◦)
is either positive or negative. Also, in the quadratic equations for the distances
(xi − xi+1)2 + (yi − yi+1)2 = a2i
only the squared edge lengths appear. Thus, for each n there exist a minimal polynomial
equation Pn(S2, a21, . . . , a
2
n) = 0 which has the squared area as its root. Changing the first
variable, we obtain αn(16S2, a21, . . . , a2n) = 0, which Robbins called the generalized Heron
polynomials.
As we mentioned above, polynomials α3, α4 were well known. In his work [8,9], Rob-
bins calculated α5, α6 and made a number of conjectures on the general form of polyno-
mials αn. By now, his conjectures have largely been established in a series of recent de-
velopment. Before we move on to outline their solutions, let us mention that Varfolomeev,
unaware of Robbins’ work, recently rediscovered some of his results and independently
made a number of advances on the subject [12,13].
As the reader will see, we do not include any technical details, nor do we present a for-
mal survey. Instead, give the reader a quick introduction to the subject and its basic ideas,
aiming to ease the entrance barrier and to simplify navigation through recent developments.
2. The first coefficient
Based on small examples, David Robbins conjectured that the polynomials αn are monic
in variable z = 16S2 (“monic” means that the highest coefficient is equal to one). This
seemingly random observation is in fact very interesting and is strongly related to Sabitov’s
692 I. Pak / Advances in Applied Mathematics 34 (2005) 690–696theory and the proof of the bellows conjecture. This connection was independently discov-
ered by Connelly [3] and Varfolomeev [12], who gave two different proofs. Let us first
elaborate on Sabitov’s work.
The story goes back to Connelly’s celebrated discovery of flexible (nonconvex) polyhe-
dra and his bellows conjecture1 as to whether the volume remains invariant under flexing
(continuous face-preserving deformations). We refer to [2] for background and references.
Later both Connelly and Sabitov conjectured that in fact the volume is integral over the ring
generated by squares of polyhedra’s edge lengths. This immediately implies the conjecture
since nonzero polynomials have only finitely many roots, and thus allow only a finite set
of possible volume values.
The bellows conjecture was established by Sabitov, who gave several consequently im-
proving expositions of his proof in a series of papers (see, e.g., [10,11]). Let i denote
the edge lengths of the polytope with edge graph G, and let V denote its volume. Sabitov
showed that there exists a nontrivial polynomial equation PG(V 2, 21, 
2
2, . . .) = 0. The dif-
ficult part in the proof is not computing this polynomial but checking that the leading
coefficient is not zero. In fact, after a change of variables P˜G(144V 2, 21, 
2
2, . . .) = 0 all
coefficients become integral, and the polynomial P˜G is monic in (144V 2).
Unfortunately, Sabitov’s proof is based on elimination theory and is more technical
than enlightening. Sabitov’s approach was later modified in [4] where the theory of places
is used to prove the bellows conjecture.
Note that when the polytope is a simplex the resulting polynomial equation can be
viewed as (a different) generalization of the Heron formula [10]. The striking similarity of
two problems led Varfolomeev to rediscovery of some of Robbins’ ideas and results. He
used Sabitov’s methods to show that polynomials αn are monic [12].
Connelly came to his proof [3] independently, after [6] advertised Robbins’ efforts. He
observed the similarity as well, and used the theory of places, to obtain a beautiful and
concise proof of this Robbins conjecture.
In the spirit of the bellows conjecture, both authors address the question as to when
cyclic polytopes are flexible. The immediate implication of the above result is the fact that
the (symplectic) area is unchanged under flexing. In fact, as was shown by Connelly much
earlier [1], the area of flexible cyclic polygons is always zero. This was also rediscovered
by Varfolomeev [12] (see also [3]).
3. The degree
The most aesthetically pleasing conjecture of Robbins is his proposed formula for the
degree of generalized Heron polynomials (in the variable z = 16S2):
degα2k+1 = ∆k, degα2k+2 = 2∆k, where ∆k = 2k + 12
(
2k
k
)
− 22k−1. ()
1 Bob Connelly declines to take credit for the bellows conjecture and wrote to me that it was communicated to
him by various people, all of whom refer yet again to other people. Therefore, the conjecture is a folklore in the
area, while Connelly deserves a great deal of credit for its advancement.
I. Pak / Advances in Applied Mathematics 34 (2005) 690–696 693Let us explain the origin of this formula for ∆n. Observe that formula (◦) works not only
for (the usual) cyclic polygons, but also for those with self-intersections, still inscribed
in a circle. Therefore, the minimal polynomial αn(z) has at least as many real roots as
the number of different areas of these self-intersecting polygons. Then, Robbins showed
that for nearly equal side lengths a1, a2, . . . , an all self-intersecting cyclic polygons have
different areas. A simple combinatorial argument gives the r.h.s. in () for the number
of different self-intersecting cyclic polygons, and implies the desired lower bound on the
degree of αn.
Robbins’ conjecture for the degree, the formula (), was established in [5], and later by
a simpler but related argument in [7]. Both proofs first obtain the corresponding formula for
the degree of polynomial equations on the radius of the circle, and then move to the degree
of αn. The study in [5] goes much deeper, as the authors establish formal connections with
Sabitov’s theory, which we outline below.
It was observed by Sabitov that not only the volume, but other “polynomial invariants”
of polytopes are roots of polynomial relations with coefficients being polynomial in the
squared side lengths. It is a natural general question to compute the degree of these minimal
polynomials. We should emphasize that here we discuss only convex polytopes, so certain
technical difficulties of Sabitov’s approach do not appear in this case. Now, for convex
polytopes not only the volume, but all diagonals are the roots of polynomial equations and
thus one can think of them as of an extension of the Cauchy rigidity theorem. Following
this logic, in [5] we refined a known algebraic proof of the Cauchy rigidity theorem and
added an argument from algebraic geometry. We obtained a general upper bound on the
degree of minimal polynomial relations for all polynomial invariants of convex polytopes,
including the volume, (squared) diagonal lengths, etc. In this special case the upper bound
we obtain for PG is in terms of complex realizations of the graph G (realizations in C3) and
in the worst case gives 2m, where m is the number of edges in G (= the number of edges
in the polytope).
Now, after we learned from [6] about Robbins’ conjectures, we discovered a formal
connection between our work (Sabitov’s theory) and that of Robbins. Consider a bipyramid
with (large enough) equal length edges leaving north or south poles, and the edge lengths
a1, a2, . . . , an in the middle. Clearly, the middle edges form the desired cyclic polygon, and
in fact different (real) realizations of this bipyramid produce different (self-intersecting)
cyclic polygons. Also, the main (north to south pole) diagonal is related by Pythagoras
theorem to the radius. It may seem like our main upper bound is directly applicable in this
case to obtain the degree of the minimal polynomial relation for the radius. The problem is
that the number of complex realizations is a difficult quantity to compute in most cases. In
fact, our logic moves backwards and is more convoluted.
First, we use an ad hoc combinatorial argument to compute explicitly the minimal poly-
nomial relation for the radius and its degree. The corresponding polynomial relation turns
out to have a nice closed formula amenable to direct calculation. Then we use the rela-
tionship described above to obtain the polynomial relation for the main diagonal, and thus
bound the number of complex realizations, which we show is equal to the number of real
realizations (self-intersecting cyclic polygons). Finally, we apply our upper bound theo-
rem to obtain the upper bound on the degree of αn, the minimal polynomial for the area of
cyclic polygons. With Robbins’ matching lower bound we obtain the result.
694 I. Pak / Advances in Applied Mathematics 34 (2005) 690–696It is interesting to note that we never actually obtain any useful formula for αn. In fact,
the only polynomial invariant that stands out in this case is the radius—all others play a
supporting role. For example, instead of the area we could be proving formula () for the
degree of the minimal polynomial relation on the sum or squares of all diagonals in a cyclic
polygon.
Let us say a few words on the proof in [7, §5]. This work started out by David Robbins
and Julie Roskies just a few month before Robbins’ premature death, and continued later
with the help of Miller Maley. Their proof of the degree formula starts with the use of
Möbius formulas for the radius of the cyclic polygon, which are essentially equivalent to
our formulas. Rather than utilize the general theory we develop in [5], the authors use
a simplified algebraic geometry argument adjusted in this particular case to obtain the
result. Basically, their argument is the same as our argument for the special case of a
bipyramid.
Finally, let us note that Varfolomeev also studies explicit formulas for the radius in
cyclic polygons [12]. He also guesses the answer in terms of self-intersecting polygons,
but never obtains a general formula nor even calculates their number beyond few small
cases.
4. Explicit formulas
One more Robbins’ conjecture concerns the form of polynomials of α2k−1 versus that
of α2k . Roughly speaking, he claimed that given the formula for α2k−1 one can easily
obtain the formula for α2k as a product of the formula for α2k−1 and its variation. This
conjecture was established by Varfolomeev [12] by a direct argument (see also [7]). As a
corollary, calculations of Robbins et al. for cyclic pentagons and heptagons immediately
translate to give the formulas for cyclic hexagon and octagons [7,8].
It was Robbins’ wish to obtain a concise formula for α7 and although he did not live to
finish the project, such a formula was recently obtained in [7]. Of course, Robbins already
showed that some kind of formula exists, but given the large number of terms one can ask
if there is a way to simplify it. In view of our earlier impression (see above) that a nice
formula may exist only for the radius, we find it amazing that the authors were able to
obtain a concise formula for the area.
Let us mention that already the Robbins’ formula for α5 is very interesting as it ex-
presses the area as a discriminant of a certain “mystery cubic” [9]. It remains unclear
where this cubic comes from and what is its role in the grand scheme of things. For ex-
ample, Varfolomeev [12] does not notice this formula and uses rather elaborate explicit
formulas for α5.
Now, in [7] the authors obtain a closed formula for α7 in terms of a resultant of two
concise, but not generalizable polynomials. The resulting formula is nice but again very
mysterious. It remains open whether this work can be extended to obtain concise formulas
for αn, where n 9.
I. Pak / Advances in Applied Mathematics 34 (2005) 690–696 6955. Final remarks and open problems
From the point of view of Sabitov’s theory and our paper [5], it would be natural to
ask for the minimal degree polynomial relations for the volume or the diagonal lengths in
various families of convex polytopes. As noted in [5], even for relatively small polytopes
this problem is computationally intractable and new ideas are needed even to obtain the
exact asymptotic behavior. At the moment, the precise formulas that we found for regular
bipyramids are clearly beyond reach in most cases.
An interesting twist on cyclic polygons was proposed in [7] where the authors define
what they call “semicyclic polygons”, where one side is forced to be a diameter and its
length is not specified. It seems that much of the work extends to this case with little
difficulty. We propose to consider an equivalent model of the centrally symmetric cyclic
polygons with given edge lengths. This version has the advantage of being possible to
generalize to cyclic polygons with Z/kZ cyclic symmetry. It would be interesting to see
if the analysis extends to this case. In general, one can consider general polytopes with a
given symmetry group. Developing the corresponding “equivariant Sabitov theory” seems
like a fruitful direction.
When it comes to the area and generalized Heron polynomials αn, it is probably too
much to ask for a concise general formula. Still, we remain optimistic of other research
venues. In his latest work [13], Varfolomeev calculated the Galois group of α5 and showed
that it is the full group of permutations S7 (he did this also for the radius). There seem to be
no immediate implications of this result except perhaps the impossibility of “construction”
of the cyclic pentagon with a ruler and a compass, given the generic lengths of edges (such
construction of a regular pentagon is well known). In any case, it would be nice to extend
these calculations for general αn.
Finally, further connections to rigidity theory are waiting to be explored. We refer to
final remarks in [3] for directions and motivation. Also, an intriguing construction of a
finitely generated infinite-dimensional Lie algebra was announced in [13] and promised to
be the subject of the future investigations. We are anxious to see how this theory further
develops.
Acknowledgments
I am very grateful to Lynne Butler, Miller Maley, Julie Roskies, Izhad Sabitov and Vi-
talij Varfolomeev for informing me of their results and the interest in my work. Special
thanks to Bob Connelly for his help, encouragement and for reading the previous ver-
sion of this mini survey. Finally, I would like to thank my coauthor and advisee Maksym
Fedorchuk for his bravery in choosing to work in an unfamiliar area with an unfamiliar
advisor.
References
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[2] R. Connelly, in: Handbook of Convex Geometry, North-Holland, Amsterdam, 1993, pp. 223–271.
696 I. Pak / Advances in Applied Mathematics 34 (2005) 690–696[3] R. Connelly, Comments on generalized Heron polynomials and Robbins’ conjectures, preprint, available at
http://www.math.cornell.edu/~connelly, 2004.
[4] R. Connelly, I. Sabitov, A. Walz, The bellows conjecture, Beiträge Algebra Geom. 38 (1997) 1–10.
[5] M. Fedorchuk, I. Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. (2004), in
press; available at http://www-math.mit.edu/~pak/research.html.
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[8] D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (1994) 223–236.
[9] D.P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (1995) 523–530.
[10] I.Kh. Sabitov, The generalized Heron–Tartaglia formula and some of its consequences, Sb. Math. 189 (1998)
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