Polynemas are the topologically distinct ways, in the plane, to join straight segments at their ends. The bidimensional character arises from the restriction to the plane, and the joining of segments implies the nondiscontinuity of the polynemas. Any two segments joined at both their ends cannot be part of a polynema — that is, a polynema contains no double segments .
The concept of polynema arose from the empirical approach: "How many different figures can be constructed with n segments?", and it is indeed a generalisation of this question. The term comes from the Greek nêma "thread".
In particular, for n
segments polynemas are said to be of order n. The names mononema,
dinema,
trinemas
etc. are conveniently employed.
Nodes are points
where two or more segments are joined.
Arms are free ends
of segments. A polynema possessing no arms is said to be armless.
Cycles are subsets
of the plane completely enclosed by segments of the polynema, without any
other segment crossing them.
The numbers of segments,
nodes, arms and cycles of a given polynema are designated respectively
by x,
y,
z and k. The set of polynemas having
the same x,
y and z is notated as [x, y,
z],
and each such set does not necessarily contain a single polynema; indeed
the reverse is commonly true.
There exist the following numbers of polynemas of the first eight orders:


















Theorem 1
Thesis: The relationship
x
+ 1 = y + z + k is valid for all polynemas.
Hypothesis: Every
polynema of order larger than 1 can be derived from the mononema through
successive addition of segments.
Proof: The proof
proceeds by complete induction: If the relationship is valid for the mononema,
and it can be proved that it will be valid for a given polynema if it is
valid for another that has one segment less, it will then be valid for
every polynema.
The relationship is valid
for the mononema, for x = 1, y = 0, z = 2, k
= 0.
Let there be a polynema
P_{0}
having x = x_{0},
y =
y_{0},
z
=z_{0},
k = k_{0}. There are five
different ways to add a new segment, producing the polynemas
P_{i}
(1 £i£ 5) having
x = x_{i},
y
= y_{i},
z =z_{i},
k = k_{i}:
1. Both ends of the new
segment are attached to nodes of P_{0}:
The five cases of segment addition treated in the above theorem may be illustrated by Figure 1, where the thicker line represents the new segment.
To enable one to refer univocally
to different polynemas, a system of nomenclature is proposed which
supposes the existence of a main segment chain (the longest continuous
chain), or of a central armless polynema, and describes the side chains
attached to that chain or armless polynema. Some examples of this nomenclature
follow::
Figure 2: 6(b1,b2)
 The central chain is composed of 6 seegments and possesses two side chains
at node b: one with 1 segment and one with 2 segments.
Figure 3: 7(2b1,c1,d1)
 The central chain has 7 segments and there are four side chains: two
at node b, both with 1 segment, and one with 1 segment at each of
the nodes c and d.
Figure 4: 9(d3(b1))
 The central chain has 9 segments; at node d there is a side chain
with 3 segments, which in turn has a subchain with 1 segment at its own
node b.
The nomenclature
of armless polynemas follows a specific system. The denomination of
the polynema in Figure 5, for instance, is C7_{2}B, containing
C
(for cyclical), 7 (the number of segments), the subscript
_{2}
(the number of cycles) and the capital B, designating it as the
second in the sequence of armless polynemas having x = 7, k
= 2.
Corollary of Theorem 1
The armless polynemas can
be interpreted as Schlegel diagrams of polyhedra. Thus, for instance, C6_{3}
is the diagram of a tetrahedron, and C8_{4}B is the diagram
of a quadrilateralbased pyramid.
By this analogy:
the number of segments x
represents the number of edges E;
the number of nodes y
represents the number of vertices V;
the number of arms z
is zero, for these polynemas are armless;
the number of cycles k
is one less than the number of faces F, for this latter includes
also the polygonal outline of the Schlegel diagram, which is not considered
a cycle in the polynema.
From Theorem 1 we have that:
Historical Remark
The idea which gave birth
to the polynemas is from 1973. Only now, more than a quartercentury later,
is it being published. I had presented it in a letter to Martin Gardner
 who at the time was still writing hiss "Mathematical Games" column in
Scientific
American magazine  and this was his response, on the 3rd of November
of 1973:
Thanks for your letter.
I had a problem very similar to yours in an early column that is reprinted
in my book, The Unexpected Hanging and other mathematical diversions [sic];
see page 79f.
I gave the topologically
distinct figures that could be formed with 1, 2, 3, 4, 5, or 6 matches.
However, I did not consider the joints as beads [I had said the segments
could be interpreted as sections of thread joined by beads at their ends
 RK], so in my interpretation a "tail"" of two matches is the same as a
tail of one match, and a square is the same as a triangle. As a result,
my lists have slightly fewer figures than yours.
I like your approach
to the problem, and will keep your letter on file. I have not seen in print
precisely the same problem, as you define it. I like your version better
than the one I used, because I think it is easier to understand.
See also: Eric
Weisstein's World of Mathematics