Slashdot Mirror

Most of the first responses are utterly missing the point. No one has attempted to patent an as-yet-nonexistent quantum computer, so arguing that they shouldn't do so is ludicrous. What has been patented is a number of little gadgets that do in fact work, but would only be a part of a quantum computer.

Another important point is that these components might be useful in some other invention long before a full fledged quantum computer is created.

Whether hardware patents should exist in the first place really is an utterly different issue. There must be hundreds of new patents issued every day, including design patents for new car body styles, but I don't hear people screaming of the injustice of it all every time GM cranks out yet another SUV.

Lastly, this article, and its posting on Slashdot, disappoints me for a different reason: some days ago I offered a story on a small breakthrough in quantum computing that is featured in the current issue of Nature -- a working externally controlled quantum bit.

But Slashdot prefers to run something cynical. I didn't see Paul Guinnessy interview the NEC researchers who made this latest breakthrough to see whether they thought that quantum computers were more than 20 years away or not.

I'd be surprised if someone didn't have at least a limited quantum computer in only 5-10 years, myself -- I'm a technology optimist.

Designing tie knots by random walks

This is an extract of the full article, which is online at Nature's web site. Unfortunately you'll need a subscription to see it there, so why not try your local newsagent? The equations, tables, and a substantial (the most interesting) chunk have been removed from this version.

The simplest of conventional tie knots, the four-in-hand, has its origins in late-nineteenth-century England. The Duke of Windsor, as King Edward VIII became after abdicating in 1936, is credited with introducing what is now known as the Windsor knot, from which its smaller derivative, the half-Windsor, evolved. In 1989, the Pratt knot, the first new knot to appear in fifty years, was revealed on the front page of The New York Times.

Rather than wait another half-century for the next sartorial advance, we have taken a more formal approach. We have developed a mathematical model of tie knots, and provide a map between tie knots and persistent random walks on a triangular lattice. We classify knots according to their size and shape, and quantify the number of knots in each class. The optimal knot in a class is selected by the proposed aesthetic conditions of symmetry and balance. Of the 85 knots that can be tied with a conventional tie, we recover the four knots that are in widespread use and introduce six new aesthetically pleasing knots.

A tie knot is started by bringing the wide (active) end to the left and either over or under the narrow (passive) end, dividing the space into right (R), centre (C) and left (L) regions (Fig. 1a). The knot is continued by subsequent half-turns, or moves, of the active end from one region to another (Fig. 1b) such that its direction alternates between out of the shirt ( ) and into the shirt (). To complete a knot, the active end must be wrapped from the right (or left) over the front to the left (or right), underneath to the centre and finally through (denoted T but not considered a move) the front loop just made.

[...the main body of the article was here: go and buy this week's Nature if you want to read it...]

The symmetry of a knot, which is our first aesthetic constraint, is determined by the number of moves to the right minus the number of moves to the left,

where xi=1 if the ith step is , -1 if the ith step is and 0 otherwise. Because asymmetric knots disrupt human bilateral symmetry, we consider the most symmetric knots from each class, that is, the ones that minimize s.

Whereas the centre number and the symmetry s specify the move composition of a knot, balance relates to the distribution of these moves; it corresponds to the extent to which the moves are mixed. A balanced knot is tightly bound and keeps its shape. We use this as our second aesthetic constraint. The balance b may be expressed as

[...equation elided...]

and the winding direction i(i, i+1)=1, where i represents the ith step of the walk, if the transition from i to i+1 is clockwise, say, and -1 otherwise. Of those knots that are optimally symmetric, we desire that knot which minimizes b.

The ten canonical knot classes {h, } and the corresponding most aesthetic knots are listed in Table 1. The four named knots are the only ones, to our knowledge, to have received widespread attention, either published or through tradition. Here we introduce some unnamed knots.

The first four columns of Table 1 describe the knot class {h, }, whereas the remainder relate to the corresponding most aesthetic knot. The centre fraction /h provides a guide to the shape of a knot, with higher fractions corresponding to broader knots; along with the size h, it should be used in selecting a knot.

Some readers may notice the use of knots whose sequences are equivalent to those shown in Table 1 apart from transpositions of , groups, such as the use of LRCRLCT in place of the half-Windsor (T. P. Harte and L. S. G. E. Howard, personal communication); some will argue that this is the half-Windsor. Such ambiguity follows from the variable width of conventional ties (the earliest ties were uniformly wide). This makes some transpositions arguably favourable, namely the last , group in the knots {5, 2}, {6, 2}, {7, 2}, {8, 3} and {9, 3} in Table 1. We do not attempt to distinguish between these knots and their counterparts; this much we leave to the sartorial discretion of the reader.

Thomas M. Fink, Yong Mao
Cavendish Laboratory, Cambridge CB3 0HE, UK
e-mail: tmf20@cus.cam.ac.uk

--
W.A.S.T.E.

Designing tie knots by random walks

This is an extract of the full article, which is online at Nature's web site. Unfortunately you'll need a subscription to see it there, so why not try your local newsagent? The equations, tables, and a substantial (the most interesting) chunk have been removed from this version.

The simplest of conventional tie knots, the four-in-hand, has its origins in late-nineteenth-century England. The Duke of Windsor, as King Edward VIII became after abdicating in 1936, is credited with introducing what is now known as the Windsor knot, from which its smaller derivative, the half-Windsor, evolved. In 1989, the Pratt knot, the first new knot to appear in fifty years, was revealed on the front page of The New York Times.

Rather than wait another half-century for the next sartorial advance, we have taken a more formal approach. We have developed a mathematical model of tie knots, and provide a map between tie knots and persistent random walks on a triangular lattice. We classify knots according to their size and shape, and quantify the number of knots in each class. The optimal knot in a class is selected by the proposed aesthetic conditions of symmetry and balance. Of the 85 knots that can be tied with a conventional tie, we recover the four knots that are in widespread use and introduce six new aesthetically pleasing knots.

A tie knot is started by bringing the wide (active) end to the left and either over or under the narrow (passive) end, dividing the space into right (R), centre (C) and left (L) regions (Fig. 1a). The knot is continued by subsequent half-turns, or moves, of the active end from one region to another (Fig. 1b) such that its direction alternates between out of the shirt ( ) and into the shirt (). To complete a knot, the active end must be wrapped from the right (or left) over the front to the left (or right), underneath to the centre and finally through (denoted T but not considered a move) the front loop just made.

[...the main body of the article was here: go and buy this week's Nature if you want to read it...]

The symmetry of a knot, which is our first aesthetic constraint, is determined by the number of moves to the right minus the number of moves to the left,

where xi=1 if the ith step is , -1 if the ith step is and 0 otherwise. Because asymmetric knots disrupt human bilateral symmetry, we consider the most symmetric knots from each class, that is, the ones that minimize s.

Whereas the centre number and the symmetry s specify the move composition of a knot, balance relates to the distribution of these moves; it corresponds to the extent to which the moves are mixed. A balanced knot is tightly bound and keeps its shape. We use this as our second aesthetic constraint. The balance b may be expressed as

[...equation elided...]

and the winding direction i(i, i+1)=1, where i represents the ith step of the walk, if the transition from i to i+1 is clockwise, say, and -1 otherwise. Of those knots that are optimally symmetric, we desire that knot which minimizes b.

The ten canonical knot classes {h, } and the corresponding most aesthetic knots are listed in Table 1. The four named knots are the only ones, to our knowledge, to have received widespread attention, either published or through tradition. Here we introduce some unnamed knots.

The first four columns of Table 1 describe the knot class {h, }, whereas the remainder relate to the corresponding most aesthetic knot. The centre fraction /h provides a guide to the shape of a knot, with higher fractions corresponding to broader knots; along with the size h, it should be used in selecting a knot.

Some readers may notice the use of knots whose sequences are equivalent to those shown in Table 1 apart from transpositions of , groups, such as the use of LRCRLCT in place of the half-Windsor (T. P. Harte and L. S. G. E. Howard, personal communication); some will argue that this is the half-Windsor. Such ambiguity follows from the variable width of conventional ties (the earliest ties were uniformly wide). This makes some transpositions arguably favourable, namely the last , group in the knots {5, 2}, {6, 2}, {7, 2}, {8, 3} and {9, 3} in Table 1. We do not attempt to distinguish between these knots and their counterparts; this much we leave to the sartorial discretion of the reader.

Thomas M. Fink, Yong Mao
Cavendish Laboratory, Cambridge CB3 0HE, UK
e-mail: tmf20@cus.cam.ac.uk

--
W.A.S.T.E.