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Definition of Tie Knots

A tie knot is initiated by bringing the wide (active) end to the left and either over or under the narrow (passive) end, forming the triagonal basis tex2html_wrap_inline776 and dividing the space into right, centre and left (R,C,L) regions, as shown in Fig. 1 (this, and all other figures, are drawn in the frame of reference of a mirror image of the actual tie).

Once begun, the knot is continued by wrapping the active end around the triagonal basis, as indicated, for example, in Fig. 2 with the Four-in-Hand; this process may be considered a sequence of half turns from one region to another. The location and orientation of the active end are represented by one of the six states tex2html_wrap_inline828 , where R,C and L indicate the region from which the active end emanates and tex2html_wrap_inline834 and tex2html_wrap_inline836 denote the direction of the active end as viewed from in front, viz., out of the page (shirt) and into the page (shirt), respectively.

Figure 1: The two ways of beginning a knot, tex2html_wrap_inline772 and tex2html_wrap_inline774 , as viewed in a mirror. Both give rise to the triagonal basis tex2html_wrap_inline776 and divide the space into the three regions through which the wide (active) end can subsequently pass. For knots beginning with tex2html_wrap_inline772 , the tie must begin inside out.

The notational elements tex2html_wrap_inline846 , etc., initially introduced as states, may be considered moves in as much as each represents the half-turn necessary to place the active end into the corresponding state. This makes the successive moves tex2html_wrap_inline848 , for instance, impossible, and implies that tex2html_wrap_inline850 is the inverse of tex2html_wrap_inline852 . It follows that no two consecutive moves may indicate the same region or direction.

To complete a tie knot, the active end must be wrapped over the front, i.e., either tex2html_wrap_inline854 or tex2html_wrap_inline856 , then underneath to the centre, tex2html_wrap_inline858 , and finally through (denoted T) the front loop just made.

Figure 2: The Four-in-Hand, represented by the sequence tex2html_wrap_inline780 .

We can now formally define a tie knot as a sequence of moves chosen from the moveset tex2html_wrap_inline864 tex2html_wrap_inline866 , initiated by tex2html_wrap_inline774 or tex2html_wrap_inline772 and terminating with the subsequence tex2html_wrap_inline794 or tex2html_wrap_inline796 . The complete sequence is constrained such that the move direction oscillate between tex2html_wrap_inline834 and tex2html_wrap_inline836 and no two consecutive move regions be identical. Using this notation, the ubiquitous Four-in-Hand may be represented tex2html_wrap_inline780 .

next up previous
Next: Aesthetic Tie Knots Up: Tie Knots and Random Previous: Introduction

Yong Mao
Sat Nov 7 16:03:57 GMT 1998