ELTIC KNOTS

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JOHN KELLY GIRLS TECHNOLOGY COLLEGE BRENT,
LONDON Shanice, Michaela, Anuradha, Reena,
Muminatu, Baneen, Noor, Kristel, Amber.
ELTIC KNOTS
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When figuring out the rules for Celtic knots,
some of us used the internet and some of us used
hand drawn patterns. Here are some of the hand
drawn ones!
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There are 6 ROWS and 2 COLUMNS in this
pattern. When you draw according to instructions
you should end up with this.
lt ROW1
lt ROW2
lt ROW3
lt ROW4
On the next slide you will see how we analysed
the gaps and the crossings in each pattern!
lt ROW5
lt ROW 6
COLUMN1
COLUMN2
4
(6,2)
Gaps
Crossings
Row 1 2 space 1 Row 2 2 space 1 Row 3
2 space 1 Row 4 2 space 1 Row 5 2 space
1 Row 6 2 TOTAL 17
Row 1 1 column 2 Row 2 1 column 2 Row 3
1 column 2 Row 4 1 column 2 Row 5
1 column 2 Row 6 1 TOTAL 16
5
(5,4)
Gaps
Crossings
Row 1 3 column 4 Row 2 3 column 4 Row 3
3 column 4 Row 4 3 space 4 Row 5
3 TOTAL 31
Row 1 4 Space 3 Row 2 4 space 3 Row 3
4 space 3 Row 4 4 space 3 Row 5 4 TOTAL
32
6
(6,3)
Gaps
Crossings
Row 1 3 space 2 Row 2 3 space 2 Row 3
3 space 2 Row 4 3 space 2 Row 5 3 space
2 Row 6 3 TOTAL 28
Row 1 2 column 3 Row 2 2 column 3 Row 3
2 column 3 Row 4 2 column 3 Row 5
2 column 3 Row 6 2 TOTAL 27
7
(4,2)
Crossings
Gaps
Row 1 2 space 1 Row 2 2 space 1 Row 3
2 space 1 Row 4 2 TOTAL 11
Row 1 1 column 2 Row 2 1 space 2 Row 3
1 space 2 Row 4 1 TOTAL 10
8
(4,4)
Gaps
Crossings
Row 1 4 space 3 Row 2 4 space 3 Row 3
4 space 3 Row 4 4 TOTAL 25
Row 1 3 column 4 Row 2 3 column 4 Row
3 3 space 4 Row 4 3 TOTAL 24
9
(8,4)
Gaps
Crossovers
Row 1 4 space 3 Row 2 4 space 3 Row 3
4 space 3 Row 4 4 space 3 Row 5 4 space
3 Row 6 4 space 3 Row 7 4 space 3 Row 8
4 TOTAL 53
Row 1 3 column 4 Row 2 3 Column 4 Row 3
3 column 4 Row 4 3 column 4 Row 5
3 column 4 Row 6 3 space 4 Row 7
3 space 4 Row 8 3 TOTAL 52
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Here is a chart showing our results
Number of horizontal lines , R Number of vertical lines, C Number of crossings Number of spaces Pieces of strings needed
6 2 16 17 2
5 4 31 32 1
6 3 27 28 3
4 2 10 11 2
4 4 24 25 4
8 4 52 53 4
We have discovered a rule that tells us how many
strings are needed for a Celtic knot The highest
common factor of (6,2) is 2 and thats how many
strings you need. Its very simple once you get
the hang of it.
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Finding The Crossings Rule

• For the (6,2) pattern, we got (12) five times
  1
• one less than the no. of rows
• For the (6,3) pattern, we got (23) five times
  2
• For the (5,4) pattern, we got (34) four times
  3
• For the (5,6) pattern, we got (56) four times
  5
• For the (4,2) pattern, we got (12) three times
  1
• For the (4,4) pattern, we got (34) three times
  3
• (double the column minus 1) one less
  than
• the no. of columns

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Finding the rule (contd.)

• So, putting all this together gave us
• Total number of crossings
• (R-1)(2C 1) ( C 1)
• Multiplying out the brackets and simplifying
  gives
• 2CR R 2C 1 C 1
• 2CR R C
• 2CR (RC)
• Double the product of C and R, minus the sum of R
  and C.

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Finding the Rule (contd.)

• Also, we noticed that
• The number of crosses on each column
• was one less than the number of rows. This gave
  us the first part of the formula.
• Columns (R-1)C
• The number of crosses on each row was one less
  than the number of columns.
• So, the second part of the formula is
• Rows (C 1) R

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Finding the Rule (contd.)

• Putting these two parts together gives
• Total number of crossings
• (R-1)C (C 1) R
• Multiplying out the brackets and simplifying
  gives
• CR C RC R
• 2CR C R
• 2 CR (C R)
• This is the same as the first formula we got.

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Does the formula work?

• We checked it on all the patterns that we studied
  and it did.
• For example, for ( 8, 4) pattern, the total
  number of crossings 2CR (C R)
• 2 X 8 X 4 ( 8
  4)
• 64 32
• 52
• and thats
  the number of crossings if you look back to the
  drawing.

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Rule for m rows and n columns

• If you have m rows and n columns, then the rule
  is
• Total number of crossings
• 2mn (m n)
• and, the rule for the number of spaces is
• 2mn ( mn) 1 because we noticed that the
  number of spaces in every pattern is one more
  than the number of crossingsso, we just add one
  to our crossings rule.

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• The End
• Thank you for watching!
• Shanice, Michaela, Anuradha, Reena, Muminatu,
• Baneen, Noor, Kristel, Amber.

ELTIC KNOTS