causes them to occur?
(iii) Occurrences of tracksets having three different periods are also rare.
Three occur in the column headed (3,2), and one in column (2,1). They all fit the formula 2 ^ ) (Eyi)2( p - l )( p - 1 ). What is the underlying reason for this?
(iv) Three trackset-spectra in column (1,2) are equal to the three cor- responding ones (when p = 17,23,37) in column (1,3). Why does this happen? We hope that further study will shed light on this question.
Next we show briefly how tracksets of the vector forms of sequences treated above may be said to correspond to knots in Z3.
C\ is a degenerate null knot (as defined).
C2 has a knot which is the disjoint union of the Ci knot (all group knots have this degenerate knot as a component) and the null knot obtained from the triangle on points (0,1,1), (1,1,0), (1,0,1). Note that the first and third points are in plane x + y = z, and the second point is in plane x + y — 2 = z.
C3 has only two tracks. One gives the degenerate knot. The other is [0,1,1,2,0,2,2,1]g which provides eight points, of which 5 are in plane x + y = z and 3 are in x + y — 3 = z. The vector polygon intersects itself at one point at which we change x, y, z to x, y, —Sz. The closed polygon is then equivalent to a null knot.
Ci has four tracks. One provides the degenerate knot, and the knot resulting from the other three is a 3-link which is commonly known as the Russian Wedding Rings. In the topological knot tables it is the link 6|. Its diagram is shown below.
The only non-cyclic group with n < 4 is the Vier group:
V4, the Vier group, has five tracks other than [0]i, each of which has a 3- point vector polygon. Their points lie in a stack of four parallel honeycomb planes, with parameters m = 0, —2, —4, —6. The equivalent knot is the disjoint union of a degenerate knot, a null knot, and a chain of four linked rings. Its diagram is shown below.
(a) The C4 Knot (b) The V4 Knot 7.8 On Aesthetics and Applicability of Tracksets
In this Chapter the concept of a Fibonacci trackset, denoted by the symbol T, has been defined and illustrated. The author has attempted to show
not only that tracksets are elegant mathematical objects, having many interesting properties that beg for deeper study, but also that they are useful tools for investigating a wide variety of algebraic structures. Indeed, he has shown how tracksets actually define binary operation algebras, and in themselves display, or hint at, many of their properties.
To review examples of trackset use, and to discuss associated aesthetic values, let us consider first the tracksets of the cyclic groups (see the table in Section 3). We wish to do this in the spirit of J. P. King's plea [16, p.
181], in his book The Art of Mathematics. He says that mathematicians should attempt to assess^ the aesthetic values of any mathematical con- cepts which they create or discover. He suggests that we use two principles as standards by which the aesthetic quality of a mathematical notion can be gauged, namely the principles of minimal completeness and of maximal
applicability. The closer the new notion meets these two standards, the higher is its aesthetic quality.
For tracksets (say T) to score highly on these tests, it must be demon- strated [quote] (1) that T contains within itself all properties necessary to fulfil its mathematical mission, with T containing no extraneous proper- ties (i.e. minimal completeness), and (2) that T contains properties which are widely applicable to mathematical notions other than T (i.e. maximal applicability).
The following paragraphs give evidence for a good rating of T against both of these standards.
Minimal completeness
Tracksets are exactly equivalent to operation tables; the one can be derived from the other, and vice versa. Thus a trackset completely defines a binary algebra. Its 'mathematical mission' is to enable lists of properties of the algebra to be discovered; that is, to develop and analyse the algebra.
This it can do, or at least it provides a complete basis for doing so. It is minimal in the sense that with any less information than the trackset contains, the binary algebra would not be completely defined.
[It is conceded that there are other ways of presenting algebras, each of which provides its own benefits for further study of its algebra: for example, representations of groups by means of generators and relations.]
^ T h e noted American painter Robert Henri has said of 'telling': "Low art is just telling things, for example 'There is the night.'; High art gives the feel of the night."
The next two paragraphs compare and contrast the use of tracksets and operation tables for studying groups.
The operation table of a cyclic group is merely a square matrix with a top row of {0,1,2, • • • , zn_ i } , followed by n—1 rows which are progressively cycled permutations of the top one. The result is an n x n matrix having all forward (upward) diagonals sporting constant elements: a serene, simple, block of n2 integers - pretty but bland. Moreover, with regard to the whole class of cyclic groups, if you've seen one such table you have seen them all (likewise for the generator/relation presentations). There is no hint in them that cyclic groups have different properties amongst themselves, with interesting (sometimes surprising) actions and subgroups to explore.
By contrast, even a cursory examination of the tracksets of groups (cyclic and non-cyclic; and also of other binary operation structures) re- veals all manner of patterns worthy of study. Subgroups can be readily discovered as subsets of tracks, and checked against smaller tracksets; and the period- and identity-spectra vary widely, with interesting properties presenting themselves for study; the periodic behaviour of elements within tracks, related to Fibonacci sequence entry-point theory, also holds much fascination. If work were to be done to understand and classify trackset properties, then new developments in the algebraic structures which they define would be bound to follow. The contrasts in aesthetic appeals of op- eration tables versus tracksets seem to the author to be 'static and bland' versus 'dynamic, vigorous and revealing'.
Maximal applicability
With regard to standard (2), the Venn diagram below shows adequately that the notion T has a huge range of 'mathematical applicability'. It was a pleasant surprise to the author to realise that Fibonacci tracksets (i.e. T) could be used to underpin - indeed define - every binary algebraic structure.
A good case has surely been made for tracksets to be highly rated by King's two aesthetic-value principles.
Plus-minus
algebras ^/_~ Latin square algebras^v
Binary algebras
Groups \
- Cyclic ^ /
Fibonacci Track Mathematics
Figure 1. Venn diagram showing the embrace of Fibonacci Trackset mathematics
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Part B, Section 1
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