Some properties of knotted patterns
Let us consider a knotted pattern (chikku kOlam or melikala muggu) with r points in the row and c points in the columns. This is thus a rxc kOlam.
1) For an rxc pattern, the number of lines is the highest common denominator of r and c. If r and c are relatively prime, i.e., they are divisible by only unity (the number 1), then there is only one line. As an example, if it is a 3x6 pattern, there will be three lines as shown in the picture. If it is 3x7 pattern, there is only one line.
2) If r = c, then it is a square pattern and there will be r lines.
3) The total number of gaps between the lines is equal to rxc + (r-1) (c-1). If its is a 3x6 pattern , there will be 3x6 + 2x5 = 18+10 = 28 gaps. These are indicated by the 18 white dots and 10 blue dots. If r = c, then we have r^2 + (r-1)^2 gaps. If r = 5, then there will be 25 + 16 = 41 gaps.
4) The point where two lines (or strings) meet is called a cross-over. The number of cross-overs is one less than the number of gaps. For the 3x6 pattern, it will be 28 - 1 = 27 (5 + 6 + 5 + 6 + 5). Count these in the figure.
5) In the above figure, if we take the blue dots arranged in a staggered fashion, the dot count would be 6-5-6-5-6. Let us consider an 8x8 square. If we take the staggered dots also, the dot count will be (8-7) seven times and finally 8 dots. Now if we rotate this by 45 degrees, we get 1 to 15 to 1 straight triangular dots. That is why I always call the 1 to 15 dot pattern as 8x8 pattern. As mentioned by Mrs Mahalakshmi, the number of dots in the central row in the straight triangular arrangement is (2xr-1) where is the number of dots in any row or column of the square. For r = 8, it is 15, for r = 10, it is 19 and so on.
6) The number of patterns for a given number of cross-overs varies. For 3 and 4 cross-overs it is 1, for 5 it is 2, for 6 it is 3, for 7 it is 7, for 8 it is 21, for 9 it is 49, for 1o it is 65 and so on.
Regards! - mOhana
Comments
Dr.Rekha Shetty
Thu, 2011-04-07 21:36
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OMG 3x6 =3 lines 3x7=1line. Parameshwara please help me ........ammu,laxmi where r u ..........dum
Dr.Rekha Shetty
Thu, 2011-04-07 22:58
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Ammuuuuuuuuuuuuuuuuuuuu,laxmiiiiiiiiiiiiiiiiiiiiiiiiiii left-right ,up-down(rows-coloums)........confusion ,confusion........pl....pl.........dum
bsindhuja
Fri, 2011-04-08 03:55
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Thank you, JKM sir! The information regarding the number of lines for a particular dot matrix grid will sure help to reduce my confusion while drawing the lines.
-Sindhuja
sudhamani.ks
Fri, 2011-04-08 03:57
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Really fantastic formula for arriving 15 to 1 staggered ( i.e dots In between) pattern. Thanks a lot.JKM Sir.
lakshmiraghu
Fri, 2011-04-08 18:24
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wonderful information...it helps lot...Helo jerry ha ha ha ha ha
ammuchandhini
Fri, 2011-04-08 19:42
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Thanks for d information jkm sir....rekha mam....wait wait i am coming to hold u or else u will fall on d ground fainting....haha...omg...i heard a thud ...
Dr.Rekha Shetty
Fri, 2011-04-08 20:43
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Mai kahanhu?
vijaysowmya
Sun, 2011-04-10 23:18
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Very useful information for me ....Thanks a lot.
smahalakshmi
Mon, 2011-04-11 00:09
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JKM Sir, really very informative post.
As far as converting a square matrix, I have my own formula. Like, 8 by 8 matrix (i.e. 8 rows and 8 columns) = 15 to 1 dots pattern (i.e. in the centre row, the dots count will be (R+C)-1, so 8+8-1 = 15). A 9 by 9 matrix kolam can be rotated by 45 degrees to get a kolam with 17 - 1 dots pattern ( 9+9-1) by filling the gaps with one interlaced dot. Is my understanding is correct, JKM sir?
Mahalakshmi
Dr.Rekha Shetty
Mon, 2011-04-11 01:02
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Maha, i have not come out of rao sirs calculation now another one from u.....ammu pl.pl.help help....
jkmrao
Mon, 2011-04-11 09:03
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Thanks for your interest. You're right Mrs Mahalakshmi. I edited and included your formula too in the writeup. Also (2r-1) arrangement is not staggered, but straight triangular or straight diamond. This correction also was made.
Regards! - mOhana
smahalakshmi
Mon, 2011-04-11 09:53
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I'm really very glad JKM sir to receive this reply from you.
Mahalakshmi
smahalakshmi
Mon, 2011-04-11 21:27
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JKM sir, just wanted to add that this formula will be applicable only if it is a square matrix. Right?
Mahalakshmi