Elementary Surprises in Projective Geometry

Abstract

We discuss eight new(?) configuration theorems of classical projective geometry in the spirit of the Pappus and Pascal theorems. Comment: a chronological error corrected

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Elementary Surprises in Projective Geometry
Richard Evan Schwartz∗and Serge Tabachnikov†
The classical theorems in projective geometry involve constructions based
on points and straight lines. A general feature of these theorems is that a
surprising coincidence awaits the reader who makes the construction. One
example of this is Pappus’s theorem. One starts with 6 points, 3 of which
are contained on one line and 3 of which are contained on another. Drawing
the additional lines shown in Figure 1, one sees that the 3 middle (blue)
points are also contained on a line.
Figure 1: Pappus’s Theorem
Pappus’s Theorem goes back about 1700 years. In 1639, Blaise Pascal
discovered a generalization of Pappus’s Theorem. In Pascal’s Theorem, the
6 green points are contained in a conic section, as shown on the left hand
side of Figure 2.
One recovers Pappus’s Theorem as a kind of limit, as the conic section
stretches out and degenerates into a pair of straight lines.
∗Supported by N.S.F. Research Grant DMS-0072607.
†Supported by N.S.F. Research Grant DMS-0555803. Many thanks to MPIM-Bonn for
its hospitality.
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arXiv:0910.1952v2 [math.DG] 5 Nov 2009

Figure 2: Pascal’s Theorem and Brian¸con’s Theorem
Another closely related theorem is Brian¸con’s Theorem. This time, the
6 green points are the vertices of a hexagon that is circumscribed about a
conic section, as shown on the right hand side of Figure 2, and the surprise
is that the 3 thickly drawn diagonals intersect in a point. Though Brian¸con
discovered this result about 200 years after Pascal’s theorem, the two results
are in fact equivalent for the well-known reason we will discuss below.
The purpose of this article is to discuss some apparently new theorems
in projective geometry that are similar in spirit to Pascal’s Theorem and
Brian¸con’s Theorem. One can think of all the results we discuss as state-
ments about lines and points in the ordinary Euclidean plane, but setting
the theorems in the projective plane enhances them.
The Basics of Projective Geometry: Recall that the projective plane
Pis defined as the space of lines through the origin in R3. A point in
Pcan be described by homogeneous coordinates (x:y:z), not all zero,
corresponding to the line containing the vector (x, y, z). Of course, the two
triples (x:y:z) and (ax :ay :az) describe the same point in Pas long as
a6= 0. One says that Pis the projectivization of R3.
Aline in the projective plane is defined as a set of lines through the
origin in R3that lie in a plane through the origin. Any linear isomorphism
of R3– i.e., multiplication by an invertible 3×3 matrix – permutes the lines
and planes through the origin. In this way, a linear isomorphism induces a
mapping of Pthat carries lines to lines. These maps are called projective
transformations.
One way to define a (non-degenerate) conic section in Pis to say that
•The set of points in Pof the form (x:y:z) such that z2=x2+y26= 0
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is a conic section.
•Any other conic section is the image of the one we just described under
a projective transformation.
One frequently identifies R2as the subset of Pcorresponding to points
(x:y: 1). We will simply write R2⊂P. The ordinary lines in R2are
subsets of lines in P. The conic sections intersect R2in either ellipses,
hyperbolas, or parabolas. One of the beautiful things about projective ge-
ometry is that these three kinds of curves are the same from the point of
view of the projective plane and its symmetries.
The dual plane P∗is defined to be the set of planes through the origin
in R3. Every such plane is the kernel of a linear function on R3, and this
linear function is determined by the plane up to a non-zero factor. Hence
P∗is the projectivization of the dual space (R3)∗. If one wishes, one can
identify R3with (R3)∗using the scalar product. One can also think of P∗
as the space of lines in P.
Given a point vin P, the set v⊥of linear functions on R3, that vanish
at v, determine a line in P∗. The correspondence v7→ v⊥carries collinear
points to concurrent lines; it is called the projective duality. A projective
duality takes points of Pto lines of P∗, and lines of Pto points of P∗. Of
course, the same construction works in the opposite direction, from P∗to
P. Projective duality is an involution: applied twice, it yields the identity
map. Figure 3 illustrates an example of a projective duality based on the
unit circle: the red line maps to the red point, the blue line maps to the
blue point, and the green point maps to the green line.
Figure 3: Projective duality
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Projective duality extends to smooth curves: the 1-parameter family of
the tangent lines to a curve γin Pis a 1-parameter family of points in P∗,
the dual curve γ∗. The curve dual to a conic section is again a conic section.
Thus projective duality carries the vertices of a polygon inscribed in a conic
to the lines extending the edges of a polygon circumscribed about a conic.
Projective duality takes an instance of Pascal’s Theorem to an instance
of Brian¸con’s Theorem, and vice versa. This becomes clear if one looks at
the objects involved. The input of Pascal’s theorem is an inscribed hexagon
and the output is 3 collinear points. The input of Brian¸con’s theorem is a
superscribed hexagon and the output is 3 coincident lines.
Polygons: Like Pascal’s Theorem and Brian¸con’s Theorem, our results
all involve polygons. A polygon Pin Pis a cyclically ordered collection
{p1, ..., pn}of points, its vertices. A polygon has sides: the cyclically ordered
collection {l1, ..., ln}of lines in Pwhere li=pipi+1 for all i. Of course, the
indices are taken mod n. The dual polygon P∗is the polygon in P∗whose
vertices are {l1, ..., ln}; the sides of the dual polygon are {p1, ..., pn}(con-
sidered as lines in P∗). The polygon dual to the dual is the original one:
(P∗)∗=P.
Let Xnand X∗
ndenote the sets of n-gons in Pand P∗, respectively.
There is a natural map Tk:Xn→ X ∗
n.Given an n-gon P={p1, ..., pn}, we
define Tk(P) as
{p1pk+1, p2pk+2 , . . . pn, pk+n}.
That is, the vertices of Tk(P) are the consecutive k-diagonals of P. The
map Tkis an involution, meaning that T2
kis the identity map. When k= 1,
the map T1carries a polygon to the dual one.
Even when a6=b, the map Tab =Ta◦Tbcarries Xnto Xnand X∗
n
to X∗
n. We have studied the dynamics of the pentagram map T12 in detail
in [2, 3, 4, 5, 6], and the configuration theorems we present here are a
byproduct of that study. (The map is so-called because of the resemblence,
in the special case of pentagons, to the famous mystical symbol having the
same name. See Figure 4.) We extend the notation: Tabc =Ta◦Tb◦Tc, and
so on.
Now we are ready to present our configuration theorems.
The Theorems: To save words, we say that an inscribed polygon is a
polygon whose vertices are contained in a conic section. Likewise, we say
that a circumscribed polygon is a polygon whose sides are tangent to a conic.
Projective duality carries inscribed polygons to circumscribed ones and vice
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Figure 4: The pentagram
versa. We say that two polygons, Pin Pand Qin P∗, are equivalent if
there is a projective transformation P→P∗that takes Pto Q. In this
case, we write P∼Q. By projective transformation P→P∗we mean a
map that is induced by a linear map R3→(R3)∗.
Theorem 1 The following is true.
•If Pis an inscribed 6-gon, then P∼T2(P).
•If Pis an inscribed 7-gon, then P∼T212(P).
•If Pis an inscribed 8-gon, then P∼T21212(P).
Figure 5 illustrates1the third of these results. The outer octagon Pis
inscribed in a conic and the innermost octagon T121212 (P)=(T21212(P))∗is
circumscribed about a conic.
The reader might wonder if our three results are the beginning of an
infinite pattern. Alas, it is not true that Pand T2121212 (P) are equivalent
when Pis in inscribed 9-gon, and the predicted result fails for larger nas
well. However, we do have a similar result for n= 9,12.
Theorem 2 If Pis a circumscribed 9-gon, then P∼T313(P).
Theorem 3 If Pis an inscribed 12-gon, then P∼T3434343(P).
1Our Java applet does a much better job illustrating these results. To play with it
online, see http://www.math.brown.edu/∼res/Java/Special/Main.html.
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Figure 5: If Pis an inscribed octagon then P∼T21212 (P)
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Even though all conics are projectively equivalent, it is not true that all
n-gons are projectively equivalent. For instance, the space of inscribed n-
gons, modulo projective equivalence, is n−3 dimensional. We mention this
because our last collection of results all make weaker statements to the effect
that the “final polygon”’ is cicumscribed but not necessarily equivalent or
projectively dual to the “initial polygon”.
Theorem 4 The following is true.
•If Pis an inscribed 8-gon, then T3(P)is circumscribed.
•If Pis an inscribed 10-gon, then T313(P)is circumscribed.
•(*) If Pis an inscribed 12-gon, then T31313(P)is circumscribed.
Figure 6: If Pis an inscribed decagon then T1313 (P) is also inscribed
We have starred the third result because we don’t yet have a proof for
this one. Figure 6 illustrates the second of these results. The formulation in
Figure 6 is easily seen to be equivalent to the formulation given in Theorem
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4. Looking carefully, we see that T1313(P) is not even convex. (Even though
the map T13 is well defined on the subset of convex polygons, it is not true
that T13 preserves this set.) So, even though T1313(P) is inscribed, it is not
projectively equivalent to Pnor to its dual P∗. One might wonder if this
result is part of an infinite pattern, but once again the pattern stops after
n= 12.
Discovery and Proof: We discovered these results through computer ex-
perimentation. We have been studying the dynamics of the pentagram map
T12 on general polygons, and we asked ourselves whether we could expect
any special relations when the intial polygon was either inscribed or circum-
scribed.
We initially found the 7-gon result mentioned above. Then V. Zakhare-
vich, a participant of the Penn State REU (Research Experience for Under-
graduates) program in 2009, found Theorem 2. Encouraged by this good
luck, we made a more extensive computer search that turned up the remain-
ing results. We think that the list above is exhaustive, in the sense that there
aren’t any other surprises to be found by applying some combination of di-
agonal maps to inscribed or superscribed polygons. In particular, we don’t
think that surprises like the ones we found exist for N-gons with N > 12.
The reader might wonder how we prove the results above. In several of
the cases, we found some nice geometric proofs which we will describe in
a longer version of this article. With one exception, we found uninspiring
algebraic proofs for the remaining cases. Here is a brief description of these
algebraic proofs. First, we use symmetries of the projective plane to reduce
to the case when the vertices of Plie on the parabola y=x2. We represent
vertices of Pin homogeneous coordinates in the form (t:t2: 1). Computing
the maps Tk(P) involves taking some cross products of the vectors (t, t2,1)
in R3. At the end of the construction, our claims about the final polygon
boil down to equalities between determinants of various 3×3 matrices made
from the vectors we generate. We then check these identities symbolically.
This approach has served to prove all but one of our results: the starred
case of Theorem 4. The intensive symbolic manipulation required for this
case is currently beyond what we can manage in Mathematica. We don’t
know for sure – because we can’t actually make the computation– but we
think that the relevant polynomials (in 9 variables) would have more than
a trillion terms. Naturally, we hope for some clever cancellations that we
haven’t yet been able to find.
We hope to find nice proofs for all the results above, but so far this has
eluded us. Perhaps the interested reader will be inspired to look for nice
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proofs. We also hope that these results point out some of the beauty of
the dynamical systems defined by these iterated diagonal maps. Finally, we
wonder if the isolated results we have found are part of an infinite pattern.
We don’t have an opinion one way or the other whether this is the case, but
we think that something interesting must be going on.
Additional Remarks: In this concluding section, we relate our results
to some other classical constructions in projective geometry, and also give
some additional perspective on them.
1). Let us say a few words about pentagons. The following is true:
•Every pentagon is inscribed in a conic and circumscribed about a conic.
•Every pentagon is projectively equivalent to its dual.
•The pentagram map is the identity for every pentagon: T12(P) = P.
We do not want to deprive the reader from the pleasure of discovering proofs
to the latter two claims (in case of difficulty, see [1] and [2]). Therefore one
may add the following to Theorem 1: If Pis a 5-gon, then P∼T2(P).
Related to the second item above, is the notion of a self-polar spherical
polygon. Let p1, . . . , p5be the vertices of a spherical pentagon. The pen-
tagon is called self-polar if, for all i= 1,...,5, choosing pias a pole, the
points pi+2 and pi+3 both lie on the equator. C. F. Gauss studied the ge-
ometry of such pentagons in a posthumously published work Pentagramma
Mirificum.
2). The formulations of Theorems 1-4 are similar: if Pis inscribed,
or circumscribed, then Tw(P) is projectively equivalent to P(or is circum-
scribed). Here wis a word in symbols 1,2,3,4, that varies from statement
to statement, but in each case, wis palindromic: it is the same whether
we read it left to right or right to left. This implies that, in each case, the
transformation Twis an involution: Tw◦Tw=I d.
3). The statement of Theorem 1 can be rephrased as follows: if Pis an
inscribed heptagon then T2(P)and T12(P)are projectively equivalent. That
is, the heptagon Q=T2(P) is equivalent to its projective dual Q∗. In fact,
every projectively self-dual heptagon is obtained this way.
Similarly, Theorem 2 states: if Pis a circumscribed nonagon then T3(P)
and T13(P)are projectively equivalent, and hence Q=T3(P) is projectively
self-dual. Once again, every projectively self-dual nonagon is obtained this
way.
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For odd n, the space of projectively self-dual n-gons in the projective
plane, considered up to projective equivalence, is n−3-dimensional, see [1]
(compare with 2n−8, the dimension of projective equivalence classes of all
n-gons). The space of inscribed (or circumscribed) n-gons, considered up
to projective equivalence of the conic, is also n−3-dimensional. Thus, for
n= 7 and n= 9, we have explicit bijections between these spaces.
4). One may cyclically relabel the vertices of a polygon to deduce ap-
parently new configuration theorems from Theorems 1-4. Let us illustrate
this by an example. Rephrase the last statement of Theorem 4 as follows: If
Pis an inscribed dodecagon then T131313(P)is also inscribed. Now relabel
the vertices as follows: σ(i) = 5imod 12 (note that σis an involution). The
map T3is conjugated by σas follows:
i7→ 5i7→ 5i+ 3 7→ 5(5i+ 3) = i+ 3 mod 12,
that is, the map is T3again, and the map T1becomes
i7→ 5i7→ 5i+ 1 7→ 5(5i+ 1) = i+ 5 mod 12,
that is, the map is T5. We arrive at the statement: If Pis an inscribed
dodecagon then T535353(P)is also inscribed. Our java applet, cited above,
shows pictures of this.
5). Theorem 4 appears to be a relative of a theorem in [4]: Let Pbe a 4n-
gon whose odd sides pass through one fixed point and whose even sides pass
through another fixed point. Then the (2n−2)nd iterate of the pentagram
map T12 transforms Pto a polygon whose odd vertices lie on one fixed line
and whose even vertices lie on another fixed line. Note that a pair of lines is
a degenerate conic section. Note also that the dual polygon, Q=T1(P) is
also inscribed into a pair of lines. Thus we have an equivalent formulation:
If Qis a 4n-gon inscribed into a degenerate conic then (T1T2)2n−2T1(Q)is
also inscribed into a degenerate conic.
We wonder if this result is a degenerate case of a more general theorem,
much in the same way that Pappus’s theorem is a degenerate case of Pascal’s
theorem.
References
[1] D. Fuchs, S. Tabachnikov, Self-dual polygons and self-dual curves,
Funct. Anal. and Other Math. 2, 203–220 (2009).
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[2] R. Schwartz, The Pentagram map, Experiment. Math. 1, 71–81 (1992).
[3] R. Schwartz, The pentagram map is recurrent, Experiment. Math. 10,
519–528 (2001).
[4] R. Schwartz, Discrete monodomy, pentagrams, and the method of con-
densation, J. Fixed Point Theory Appl. 3, 379–409 (2008).
[5] V. Ovsienko, R. Schwartz, S. Tabachnikov, The Pentagram map: a
discrete integrable system, ArXiv preprint 0810.5605.
[6] V. Ovsienko, R. Schwartz, S. Tabachnikov, Quasiperiodic motion for the
Pentagram map, Electron. Res. Announc. Math. Sci. 16, 1–8 (2009).
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