A year-out student Tom Kirby emailed an office of 200 people asking what the convention was for beyond and behind dotted line styles.  No one knew. On-line searches didn’t prove very helpful so in one of those mad delightful collective moments it was decided to use Morse Code line styles. We would then be able to put any message we liked as a line, including beyond and behind or the scatological, and no one would be any the wiser.

Note that there are really 3 symbols here, a dot, a dash and a space. The definition of the space between characters and the space between words are also important in defining what is in some ways a modular system.

In AutoCAD line type definitions like this should work (followed by 7 spaces).

*BEYOND,Beyond _… . _.__ ___ _.__ _..

A,12.7, -6.35, 0, -6.35, 0, -6.35, 0, -19.0, 0, -19.0, 12.7, -6.35, 0, -6.35, 12.7, -6.35, 12.7, -19.0, 12.7, -6.35, 12.7, -6.35, 12.7, -19.0, 12.7, -6.35, 0, -6.35, 12.7, -6.35, 12.7, -19.0, 12.7, -6.35, 0, -6.35, 0, -44.5

*BEHIND,Behind _… . … .. _.__ _..

A,12.7, -6.35, 0, -6.35, 0, -6.35, 0, -19.0, 0, -19.0, 0, -6.35, 0, -6.35, 0, -19.0, 0, -6.35, 0, -19.0, 12.7, -6.35, 0, -6.35, 12.7, -6.35, 12.7, -19.0, 12.7, -6.35, 0, -6.35, 0, -44.5

But unfortunately AutoCAD only allows a maximum of 12 dots or dashes, too bad.

Yin and Yang

The I Ching consists essentially of 64 chapters with each chapter symbolised by a hexagram formed of six lines each of which is either – – (yin) or – (yang) giving 2⁶ = 64 alternatives. These can be arranged in a number of ways. (Knuth 2005)

The first arrangement is the King Wen Arrangement (c 1100 B.C.). King Wen is apparently a fairly loose period just meaning a very long time ago.

4 symmetric half hexagram symbols are recognised: Heaven, Earth, Fire and Water.

In this arrangement top-to-bottom mirror images (yin yang) are adjacent (odd and even numbers) and other important (same symbol and symmetrical) combinations are in prominent positions (boxed below), otherwise mathematically unsophisticated.

In the Shao Yung Arrangement (A.D. 1060)

4 extra asymmetric half hexagrams are recognised: Mountain, Wind, Thunder and Lake.

Each of 8 half hexagram symbols index both columns (above) and rows (below). Giving 8×8 = 64 combinations again.

For instance Mountain (column 2) above Lake (row 7) gives chapter 2 + (7 – 1) * 8 = 50

This makes a much more practical index for a book and facilitates divination of various sorts. This calculation method was known at the time.

Inversely chapter 50 gives 50 mod 8 (columns) = column 2 (Mountain)

and chapter 50 integer divide 8 (columns) + 1 = row 7 (Lake).

This method, which is less useful, was not known at the time.

So one might pick 2 sets of 3 long or short sticks and get the following result. 1 long and 2 short (Mountain) followed by 1 short and 2 long (Lake) and then go straight to chapter 50 for a reading.

Chapter 50 Extract

Sanskrit Poetics (c 2nd Century BCE)

A mnemonic for remembering all sequences developed in the Middle Ages:

Pingala‘s Chandahsutra (c. 200 BCE)

Syllables are short or long in length

1 long = 2 short

Pingala classified all the 16 different meters of four syllables like this:

1 meter of four short syllables SSSS

4 meters of three shorts SSSL, SSLS, SLSS, LSSS

6 meters of two shorts LLSS, LSSL, SSLL, SLLS, LSLS, SLSL

4 meters of one short SLLL, LSLL, LLSL, LLLS

1 meter of no shorts LLLL

Musical Sequences and Aperiodic Tiling

Taking a Penrose P2 (Kite and Dart) aperiodic tiling marked with Ammann bars leads to a set of 5 grids at 2PI/5 rotations.

Spacing between adjacent Ammann lines, short and long intervals, is in the golden ratio. Further short cannot be followed by short and two longs cannot be followed by another long as summarised in the following decision diagram.

Composing or decomposing a musical sequence gives rise to another musical sequence and an inflated or deflated tiling.

The substitution S’=L” and L’=S+L automatically give rise to the golden ratio

Bibliography