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

Some properties of knotted patterns - 3x6-1.JPG

Comments

Dr.Rekha Shetty's picture

OMG 3x6 =3 lines 3x7=1line. Parameshwara please help me ........ammu,laxmi where r u ..........dum

Dr.Rekha Shetty's picture

Ammuuuuuuuuuuuuuuuuuuuu,laxmiiiiiiiiiiiiiiiiiiiiiiiiiii left-right ,up-down(rows-coloums)........confusion ,confusion........pl....pl.........dum

bsindhuja's picture

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's picture

Really fantastic formula for arriving 15 to 1 staggered ( i.e dots In between) pattern. Thanks a lot.JKM Sir.

lakshmiraghu's picture

wonderful information...it helps lot...Helo jerry ha ha ha ha ha

ammuchandhini's picture

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's picture

Mai kahanhu?

vijaysowmya's picture

Very useful information for me ....Thanks a lot.

smahalakshmi's picture

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's picture

Maha, i have not come out of rao sirs calculation now another one from u.....ammu pl.pl.help help....

jkmrao's picture

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's picture

I'm really very glad JKM sir to receive this reply from you.

Mahalakshmi

smahalakshmi's picture

JKM sir, just wanted to add that this formula will be applicable only if it is a square matrix. Right?

Mahalakshmi