A METHOD OF CLASSIFYING
ALL THE POLYGONS HAVING A GIVEN SET OF VERTICES
BY F. H. MURRAY
(CAMBRIDGE,
MASSACHUSETTS.)
Suppose a system of points on a plane to be such that no three are collinear; let
these be numbered in any order ai,ai...,an,
and construct the line segments
aya%, a9a3, . . . , an_^an, anax.
This set of segments forms a closed curve described in some sense, which can be
represented by the sequence of points
(a^
. . . an).
Such a closed curve will be termed a polygon corresponding to the given set of
points, and these points <z4 . . . an will be termed the vertices of the polygon. It is
immediately evident that every other sequence (a\ ,a\, ..., a'n,) in which every point
occurs just also represents a polygon with the same set of vertices, and every polygon
corresponding to the given set of points can be represented in this manner.
Polygons of this kind have been studied by L. Poinsot and others(l); but a systematic discussion and classification of the polygons with n vertices seems to be
still lacking. It is the purpose of this paper to present one method of classification
based on the theory of permutation groups; this classification is particularly interesting if the n points are equally spaced on the circumference of a circle.
The method will be applied to the cases n = i, 2, 3, 4, 5, 6, 7; certain generalizations will be indicated afterwards.
(l) Ostwald, Klassiker : Abhandlungen über die regelmässigen Sternkörper ; Max Brückner :
lieber die gleicheckig-gleichflächige diskontinuierlichen und nicht konvexen Polyeder; Abteilungen der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akademie der Naturforscher,
Band 86, Halle, 1906.
446
F. H. MURRAY.
S i . — THE GROUP OF A POLYGON.
Suppose the n points equally spaced along the circumference of a circle, and
numbered in the order in which they occur if the circumference is described in the
clockwise direction. With this notation the polygon (a^ ... an) is a particularly
simple one; it is transformed into itself by the permutation on its indices
i 2 3 . . . n\
== i a 3 . . . n.
2 3 4 . . . i]
since this permutation replaces every line segment alairi_l by the one following it,
hence leaves the set unchanged. All powers of this substitution, or the cyclic group
generated by this substitution also leave the polygon unchanged; and if the polygon
is considered as described in a definite sense this cyclic group contains all the permutations on the indices which leave the polygon unchanged. The group of permutations on the vertices of a polygon which leave it unchanged is defined to be the
group of the polygon.
The group of the polygon having a definite sense is therefore a cyclic group; if
P' is represented by the sequence (atat ... aH) its group is the cyclic group (1,2 ... n),
consisting of all powers of the substitution 1 2 3 . . . n.
The polygon V = (allan_i . . . a4) evidently has the same cyclic group; it remains
to consider the group of the polygon P to which.no sense is assigned. P can be
considered as the combination of the two polygons P', Pff; its group consists of all
substitution which either leave both P' and P" unchanged, or transform P' into P r
and Pff into P'. One substitution transforming P' into Pff is
/1 2 3
n
\n(n — i)(n — 2) . . . 1
the substitution t and 1 2 3 . . . n generate the dihedral group of order 2/1, represented by (1 2 3 . . . n)'. This particular dihedral group will be called the regular
group. The reason for this will appear immediately.
In a similar manner to any other polygon P'f corresponds a cyclic group
a
( i(O a a(0 • • • a«(0)> anc * t o t n e corresponding polygon considered as having either
sense corresponds a dihedral group (a,(O a a (0 • • • a w (0)'- Among all possible dihedral groups formed in this manner the regular group is most interesting from the
a geometrical point of view because of the significance of its substitutions; to every
A METHOD OF CLASSIFYING ALL THE POLYGONS.
447
substitution of the cyclic group (i 2 3 . . . n) corresponds a rotation through an angle
27C
which is a multiple of —, and to every substitution of the group not in the cyclic
group corresponds a reflection with respect to some diameter. If n is odd, every
such diameter passes one vertex; if n is even, there are two cases; a diameter of
this kind passes through two vertices, or it does not pass through any vertex.
Every diameter of this kind is a line of symmetry, and the substitutions of the
regular group not in the cyclic group therefore correspond to lines of symmetry of
the polygons which have the regular group. These polygons are usually called the
regular polygons.
S 2. — TYPES OF POLYGONS.
On any given polygon PÄ = (affla^k)...
an(kf) perform the substitutions of the
cyclic group (1 2 3 . . . n); there result n polygons PÄ(l) . . . PÄ(W), which need not all
be distinct. From the geometric significance of the substitutions of the group
(1 2 . . . n) it is seen that this set of polygons is obtained from Pft by rotations
27C
through multiples of the angle —.
The groups of these polygons are also related
in a simple manner : the n groups are all of the form S _/ GS f , where
S = 1 2 3 . . . fi,
1=
î,
2, ...,
n,
as is immediately evident. This set. of n polygons is defined to be the type to which
any one of the set belongs ; two polygons therefore belong to the same type if, and
only if, one can be obtained from the other by a rotation through a multiple of the
angle —.
To every type T4 corresponds another type T'4 called the conjugate of the first,
which is constructed as follows : if PÄ is any polygon of the type T4, reflect Pft
about any one of the n diameters which are lines of symmetry for the regular polygons; the type of the polygon obtained by this reflection is defined to be the type
conjugate to the first type T4. It is seen immediately that T4 is conjugate to T'4,
and the type T'4 is the same whatever the line of reflection chosen.
It may occur that the types T4 and T'4 are the same; in this case the type T4 is
ä self-conjugate type.
It can be immediately shown that every polygon of a self-conjugate type has a
: line of symmetry, and if a polygon has a line of symmetry, its type is self-conjugate.
For if 1,8, S3 — S n_1 are the substitutions of the cyclic-group generated by
448
F. H. MURRAY.
i 2 3 . . . n, and t a substitution of the regular group not in the cyclic group, the
regular group consists of the substitutions
i, S, Sa . . . S""1,
/, ts, ts*... *sw-\
If the type of Pk is self-conjugate, then Pft operated on by some power of S is
the same polygon as PÄ operated on by some substitution tSl. Or,
P*(S4) = P*(*S').
Then
P4 = P , ( S ' ) S - \
= P 4 (»*-*).
Consequently the substitution tSl~h transforms PÄ into itself : tSl~h corresponds
to a reflection with respect to some one of the n diameters, which must therefore
be a line of symmetry for Pft. If Vk has a line of symmetry every polygon obtained
from PÄ by a rotation also has a line of symmetry, hence every polygon of T4 has
a line of symmetry.
Also, if PÄ has a line of symmetry a reflection with respect to this line leaves PÄ
unchanged, hence leaves the type of PÄ unchanged ; the theorem is therefore demonstrated.
If the n polygons of a given type are not all distinct, some power of S not the
identity transforms some polygon P of the type into itself; hence the group of P
has a sub-group in common with the cyclic sub-group of order n of the regular
group. Sub-groups of the regular group will be termed regular sub-groups.
If the group of P has a regular sub-group belonging also to the cyclic sub-group
of order n, the polygons of its type will not all be distinct; hence the theorem :
A necessary and sufficient condition that the n polygons of the type of a polygon P
be not all distinct is, that the group of P have a regular sub-group belonging to the
regular cyclic sub-group of order n.
It has already been seen that to every polygon corresponds a certain substitution
group; but to a given dihedral group on n letters several polygons may correspond.
For if S is a generating substitution of a cyclic group, every power of S prime to n
is also a generating substitution, and every such power of S corresponds to a certain
polygon. Hence if <p(n) is the number of integers less than n and prime to n,
there are -^—- polygons which have the same dihedral group.
Each of the -—- polygons has the same lines of symmetry and each type contains
2
the same number of distinct polygons, since the regular sub-group is constant.
A METHOD OF CLASSIFYING ALL THE POLYGONS.
449
S 3 . — APPLICATION OF THE METHOD FOR n = 2, 3, 4, 5, 6, 7.
As a result of the discussion above it is natural to commence the classification,
for a given value of n, by a classification of the dihedral groups according to the
sub-groups which they have in common with the regular group, and a division of
the groups into sets conjugate under the powers of the substitution (12 3 — n),
corresponding to the division of the polygons into types. If a regular sub-group
occurs in a dihedral group G4, then if the powers of (1 2 3 — n) transform G4 into
Gd, Ga . . . GA, they transform the regular sub-group gk into git g± ... gk, which are
also regular sub-groups. But just as a polygon may be said to represent all the
others of the same type, it is evident that gt represents g^ .. . gk, and if all the
dihedral groups containing gi are constructed, their conjugates contain gz, ... gk.
In the classification of the dihedral groups, therefore, it is convenient to consider
first the groups containing just one of the set gt .'.. gk.
Accordingly, the first step in the classification of the dihedral groups is the classification of all the regular sub-groups into sets, the sub-groups in each set being
conjugate under the regular cyclic-group of order n; then choose one sub-group
from each set, and determine all the dihedral-groups containing each of the subgroups chosen. These dihedral groups and the others conjugate to them under the
regular cyclic group of order n will include all the dihedral groups containing any
regular sub-group.
When these groups have been constructed there will remain a certain number
of dihedral groups not containing any regular sub-group ; if P is a polygon corresponding to such a group, there are evidently n distinct polygons in the same type
as P, and each type is distinct from its conjugate type.
If n = 1, the polygon reduces to a point, its group to the identical transformation.
If n = 2, the polygon is a straight line segment, its group the group on two
letters, which can be considered as a dihedral group.
If n = 3, the polygon is a triangle, its group the symmetric group on three letters which is also a dihedral group.
If n = 4, there are three dihedral groups, the regular group and two others
conjugate under the regular cyclic group; consequently one type besides the square,
which is the only regular polygon here. Since the three dihedral groups on 4 letters have a group of order 4 in common, each of the substitutions besides the identity being of order 2, it follows that each non-regular polygon has two lines of symmetry.
57
45o
F. H. MURRAY.
Since 5 is a prime, no two cyclic groups of order 5 can have a sub-group other
than the identity in common-; hence no polygon except the regular polygons can
have more than one line of symmetry. For if P had two lines of symmetry, the
succession of two reflections is a rotation, and the corresponding group would therefore possess a regular sub-group of the regular cyclic sub-group of order 5. Since
this is impossible, each non-regular polygon can have at most one line of symmetry.
Since there are 2 4 substitutions on 5 letters of order 5, there are 6 dihedral groups,
the regular group and 5 others. It has been seen that none of these 5 dihedral
groups can have a regular sub-group contained in the regular cyclic subgroup of
order 5, hence the type of each polygon contains 5 distinct polygons. From this it
follows that the five groups must be conjugate under the cyclic group of order 5.
A construction of any one shows immediately a line of symmetry. Hence for n = 5
there are two self-conjugate types corresponding to the non-regular polygons.
For n = 6 there are 60 dihedral groups, with one polygon to each group. It
is found that all types are represented by the following substitutions; the number
of distinct polygons in each type follows the substitution :
12 3 4 5 6 (1)
12 3 4 6 5
(6);
1 2 5 6 3 4 (2);
1 3 2 4 6 5 (3)
1 4 3 2 56
(3);
1 4 6 2 5 3 (3);
14 2 3 6 5 (3)
14 6 5 2 3 (3) ;
(the last two belong to distinct conjugate types);
146235
(6);
126435
(6);
1 4 a 3 56
(6);
1 4 6 5 3 2 (6);
1 4 5 3 2 6 (6);
1 4 2 5 6 3 (6);
the last two belong to distinct conjugate types and neither has any line of symmetry.
For n = 7 there are 3 polygons corresponding to each dihedral group. Since 7
is a prime, no polygon except the regular polygons can have more than one line of
symmetry. The regular sub-groups are therefore of order 2, and there is just one
conjugate set. There are 5! dihedral groups in all, with 7 polygons in each type.
It is found that the polygons having lines of symmetry belong to the types represented by the following 8 substitutions and their powers :
1234567;
1423675;
1342756;
1235467;
1325476;
1243657;
1375426;
1324576.
Methods of constructing the polygons having any given regular sub-group can
easily be found.
A METHOD OF CLASSIFYING ALL THE POLYGONS.
451
S 4. ,— SOME GENERALIZATIONS.
For any value of n, suppose the classification of dihedral groups completed.
Then suppose any n points given in the plane, no three of wrhich are collinear.
Corresponding to each order numbering these points at, a2 . . . an there is a polygon
(a4, aa . . . an), and a closed curve G can be drawn, passing through each point ak
just once, and passing through the n points in the order a4, a2 . . . an. The polygon
(a4 . . . an) is one such curve.
The classification of all the polygons having the n points as vertices, with respect to the polygon (a4 . . . an) can be made in such a way that the curve G plays a
part analogous to that of the circle in the case discussed above. For a permutation
which is a power of i 2 ... n corresponds to a cyclic interchange of vertices along G ;
a permutation corresponding to a reflection with respect to a diameter in the case
of a circle, corresponds here to a division of G into two arcs G,, Ca, by two points
Q4, Qf, such that there is the same number of vertices at ... aa on each arc (Q4 or
Q2 may be points of the set at ... an), and the substitution for a vertex ak of the
vertex ak' which is the same number of vertices away, along G, from Q4, or Qf,
but on the opposite arc. If Qf is a point of the set it is left unchanged.
With this geometric interpretation of the permutations of the regular group a
geometric classification qf the polygons can be given in a manner analogous to that
given above. This classification is especially interesting if the curve G is convex,
in such a way that no line intersects it in more than two points. It is easily demonstrated that any two lines öTöT, akal of a polygon P intersect in the interior of G
if and only if ai and a5 divide ak and at on G ; that is, if ai lies on one arc, ai on
the other of the two arcs of which ak and at are end-points. If ai and a^ divide
ak and av ai+4 and aj+l divide ak_hi and al_hi ; from this fact it follows that if a
given pair of lines intersects in the interior of G, every pair of lines corresponding
to this pair either in the type of P or in the conjugate type intersects in the interior of G.
Still another generalization can be obtained as follows.
If any finite number of points in a space of three dimensions are represented by
at, a2, . . . an, a line can be represented by a pair of points Ö7a7, a plane by a triplet
aiajak. Consequently any configuration consisting of points, lines through two
points and planes through three points of the set can be represented as a combination
S = (ai ... ak, . . . ataj ... alanam
...).
452
F. H. MURRAY.
For every such representation there exist some permutations, perhaps only the
identity, which transform this representation of the set of points, lines and planes
into itself. The set of all such permutations evidently forms a group and is defined
to be the group of the configuration. This group is evidently dependent on the particular sequence in which the points are numbered.
Given the n points and any configuration S, on these points, the substitutions
of a given permutation group G formed with the letters a, . . . an transform S, successively into S a . . . SN; the configuration
S = (S t ,S i . . . . S . )
is transformed into itself by all the permutations of G. Hence the group of S contains G as a sub-group.
From this point of view the polygons discussed above are particular linear configurations in the plane. Other interesting configurations can be obtained by superposing all the polygons which have the same dihedral group; this group is evidently
a sub-group of the group of the resulting configuration. Similarly the regular group
is a sub-group of the group of the configuration obtained by superposing all the polygons of a given type, together with those of the conjugate type.
In the plane it was seen that those polygons are most interesting whose dihedral
groups contain regular sub-groups; in space those configurations are the most interesting whose groups contain sub-groups corresponding to rotations about a fixed
point, or reflections with respect to planes through this fixed point. The regular
polyhedra and regular star-shaped bodies are special cases of these.