Ronald Kyrmse                                 kyrmse@yahoo.com.br

POLYNEMAS
Ronald E. Kyrmse

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 non-discontinuity 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:

 x 1 2 3 4 5 6 7 8 Number 1 1 3 5 12 30 79 227
A listing of these polynemas is given.

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 P0 having x = x0, y = y0, z =z0, k = k0. There are five different ways to add a new segment, producing the polynemas Pi (1 £i£ 5) having x = xi, y = yi, z =zi, k = ki:
1. Both ends of the new segment are attached to nodes of P0:

x1 = x0 + 1; y1 = y0; z1 = z0; k1 = k0 + 1
2. One end of the new segment is attached to a node of P0 and the other remains free:
x2 = x0 + 1; y2 = y0; z2 = z0 + 1; k2 = k0
3. One end of the new segment is attached to the free end of an arm of P0 and the other remains free:
x3 = x0 + 1; y3 = y0 + 1; z3 = z0; k3 = k0
4. One end of the new segment is attached to the free end of an arm of P0 and the other to a node of P0:
x4 = x0 + 1; y4 = y0 + 1; z4 = z0 - 1; k4 = k0 + 1
5. Both ends of the new segment are attached to free ends of arms of P0:
x5 = x0 + 1; y5 = y0 + 2; z5 = z0 - 2; k5 = k0 + 1
Thus, each increase of one unit in x implies an equal increase in the sum y + z + k. Therefore the mentioned relationship remains valid in every case.
Q. E. D.

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 C72B, 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, C63 is the diagram of a tetrahedron, and C84B is the diagram of a quadrilateral-based 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:

E + 1 = V + 0 + (F - 1)
therefore:
E + 2 = V + F
This is the familiar form of Euler's Polyhedral Formula, here proved through the properties of polynemas.

Historical Remark
The idea which gave birth to the polynemas is from 1973. Only now, more than a quarter-century 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.