Knotwork: Advanced Techniques

Knotwork: Advanced Techniques
Edmund Peregrine
Knotwork: Advanced Techniques
A class at the Citadel A&S Collegium AS XLIII (2008)
By Lord Edmund Peregrine (Victor Singleton)
In the previous class we examined the general method of knot construction
introduced by Christian Mercat. (11) We showed how a knot can be generated
from a graph, and how a graph can be extracted from a knot.
The advantage to using graphs to plan knots is that graphs are easy to draw and
manipulate. It simplifies the process of fitting a knot into an irregular space and
allows us to classify knots in a systematic way.
In this class we will examine methods for splicing small knots into larger ones. We
will also look into some classic knot patterns and a type of knot called an
entanglement.
Encapsulation - Splicing knots together:
Larger knots can be constructed by splicing smaller knots together. This would
normally be done by placing the knots side by side and converting the loops on the
facing edges to crossings. Given two 3X3 knots, for instance, one could get:
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Knotwork: Advanced Techniques
Edmund Peregrine
However, this method requires one to have the knots drawn and spaced just so. A
simpler and more systematic method would have you splicing the graphs together
before ever having to generate the knot.
The best way to do this is to use a different form of graph called the dual graph.
Whereas the graph is extracted from a knot by connecting the bound spaces, the
dual is extracted from a knot by connecting the unbound spaces. Segments of the
dual are extended from unbound spaces to the outside and then these are connected
to make a wall going completely around the knot, an encapsulation.
Note how the outer edge of the dual is a continuous set of closures. Also note how
a closure in the graph becomes a split in the dual and visa versa.
The knot can be generated from the dual by the same method as from the graph.
The advantage of the dual graph is that one can splice together knots easily by
butting the duals together and opening one or more of the outer closures between
them to connect the knots. For instance, for the knot above, one can splice them
lengthwise,
or sideways, with different ways of linking them together.
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Knotwork: Advanced Techniques
Edmund Peregrine
The dual can also be derived directly from the
graph of the knot. First draw lines through the
center of each segment of the graph and
perpendicular to the segment. Keep the breaks in
position.
Then draw the outer line of closures around the
graph.
Classic Knot Motifs:
According to J. Romilly Allen, there are only eight basic knots that are found in
Celtic decorative art. (1) Two are based on a three cord plait and six from the four
cord plait. The plaits and the eight Allen knots are shown below with their graphs
and duals.
Three Cord Plait:
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Knotwork: Advanced Techniques
Edmund Peregrine
Knot #1:
Knot #2:
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Knotwork: Advanced Techniques
Edmund Peregrine
Four Cord Plait:
Knot #3:
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Knotwork: Advanced Techniques
Edmund Peregrine
Knot #4:
Knot #5:
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Knotwork: Advanced Techniques
Edmund Peregrine
Knot #6:
Knot #7:
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Knotwork: Advanced Techniques
Edmund Peregrine
Knot #8:
As you can see each basic knot is formed from the plait by adding a pattern of
breaks to the graph. To extend these knots simply continue the pattern.
Trefoils:
Triangular motifs are also common in Celtic design. The graph of a trefoil is a
triangle and the dual is also a triangle.
Using the dual graph one can easily splice together
trefoils in various ways. If the dual is a right triangle, it
tiles into square shapes. If it is equilateral, then it tiles
into hexagonal shapes.
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Knotwork: Advanced Techniques
Edmund Peregrine
Entanglements:
Another common motif in Celtic art is the extension of a limb, tongue, tail, or hair
into a knot. These kinds of knots have two ends. One can form these from one of
the eight braided patterns above as they have from one to four pairs of ends.
However this kind of decoration is often very freeform and squeezed into an oddly
shaped section of the page. How could we construct such knotwork?
If you take a simple knot and run a series of closures all down one side you will get
a knot with a long cord running along the outside of the knot. If you clip this cord
at its two ends you get a knot with two ends. Such a knot is called an
entanglement.
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Knotwork: Advanced Techniques
Edmund Peregrine
Turning this idea around we can ask how to fill a space
like the one shown on the right with an entanglement. First
we need to construct a graph for the knot such that it has
one side all closures. The two ends of the entanglement
will be at the two ends of that line of closures. Therefore
let us decide that we need one end to be at the upper right
corner and the other to be at the upper left.
Draw a line from one corner to the other just inside the
upper limit of the space. Then construct a graph down
from this line that fills the space. Note that you can use
squares, triangles, etc. and distort them as needed. Add
some breaks to make it interesting.
Construct the knot from this graph. Start at one
end and follow the method as usual. The result
here is one single cord. This could be a single
tongue, a tail, a lock of hair, etc.
If you wanted there to be two cords, say two
tongues from two lions facing each other, then
simply cut the knot at some point to get two more ends crossing each other at that
point.
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Edmund Peregrine
Books and Articles on Celtic Art and Knotwork Construction:
(1)Allen, J. Romilly; “Celtic Art in Pagan and Christian Times”; Dover
Publications; New Ed edition (2001); ISBN-13: 978-0486416083
(2) Bain, George; “Celtic Art, the Methods of Construction”; Dover Publications,
NY, (1973); ISBN 0-486-22923-8
(3) Meehan, Aidan; “The Celtic Design Book: A Beginner's Manual, Knotwork,
Illuminated Letters”; Thames & Hudson (2007); ISBN 978-0500286746
(4) Meehan, Aidan; “Celtic Knots: Mastering the Traditional Patterns”; Thames
& Hudson (2003); ISBN-13: 978-0500283998
(5) Sturrock, Sheila; “Celtic Knotwork Handbook”; Guild of Master Craftsman
Publications, Ltd, Lewes, E. Sussex, England (1999); ISBN 1-86108-115-4
(6) Sherbring, Melinda, (aka Eowyn Amberdrake); “Interlacing Without Erasing”;
“Tournaments Illuminated”, No. 53, Winter 1979; SCA publications, Milpitas,
CA.
Websites:
(7) Abbott, Steve; “Steve Abbott's Computer Drawn Celtic Knotwork”;
http://www.abbott.demon.co.uk/knots.html as of 3-3-2008 as of 3-3-2008
(8) British Library; “Lindisfarne Gospels”;
http://www.bl.uk/onlinegallery/themes/euromanuscripts/lindisfarne.html as of 3-32008
(9) Busiak, Cari; “Knotwork Tutorials”; Aon Celtic Art & Illumination;
http://www.aon-celtic.com/cknotwork.html as of 3-3-2008
(10) Ivan, Drew; “Make Celtic Knotwork”;
http://www.thinkythings.org/knotwork/knotwork.html as of 3-3-2008
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(11) Mercat, Christian; “Celtic Knotwork: The ultimate tutorial” ;
http://www.entrelacs.net/Celtic-Knotwork-The-ultimate as of 3-3-2008
(12) Mihaloew, Reed; “Celtic Knotwork Construction Tutorial”;
http://mysite.verizon.net/mihaloew/celtic/cel_class.shtml as of 3-3-2008
(13) Sloss, Andy; “Celtic Creative Home Page”; http://www.proscribe.co.uk/ as of
3-3-2008
(14) Scharein, Robert; “The Knotplot Site”; http://knotplot.com/ as of 3-3-2008
(15) Trinity College Dublin; “The Book of Kells”;
http://www.tcd.ie/about/trinity/bookofkells/ as of 3-3-2008
(16) University College Cork; “CELT: Corpus of Electronic Texts”;
http://www.ucc.ie/celt/ as of 3-4-2008
(17) Wikipedia; “Knot Theory”;
http://en.wikipedia.org/wiki/Knot_theory#Signed_planar_graphs as of 3-4-2008
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