Knots and The Ellipse II

First, let me get this over with... the answer to my math problem:

The intersection of a circle centered at either end of the minor axis' radius (the intersection of lengths A and X) with a radius of the length of the major axis' radius (length B), and the major axes would be the precise locations of F1 and F2. That seems complicated (because words like to be), but graphically, what this says is X=B (!!).

How?

I + II = B

2*I+4*II = 2*X+2*II (learned from the 'string construction' method)

2*I+2*II = 2*X (learned from Algebra 1 in 7th grade)

I + II = X = B

Who cares?

It makes finding the foci of an already drawn ellipse very easy, which was my initial dilemma. Cool, I hope you enjoyed this as much as me :-)

Now, down to the point - my knot. The first thing I did was figure out how large I could make the knot with the twine I had bought. Once I figured this out, I made a to-size elliptical layout of the knot on a piece of paper. I then strung the first pass of the knot...

Getting this right took some time and was not easy. The next step entailed tracing the overs and unders with the long piece of rope. I was able to trace the first pass three more times before the long end ran out. I was left with some slack within the knot...

The last step was cleaning up the slack and finishing the knot. This task was not hard, but took a good amount of time. I was able to get one more pass in, finishing with a 5 ply knot, as seen here:

I changed the orientation of the knot for two reasons. Firstly and from the beginning, I planned on using the plat as a door mat. It might be a little small, but we will make it work. Secondly, the knot came out not quite symmetrical. This frustrated me at first, but then I concluded this enhances its functionality; the top edge is flat, which will conveniently rest against the foot of the door. For those of you who were expecting this piece to be mounted on a wall and admired as such, I am sorry. Some times art can be functional as well!

Thanks for looking and listening, and stay tuned...

Knots and The Ellipse

I have always enjoyed figuring things out, be it solving a math problem or untying a knotted bundle of jewelery. Over time, I became quite good at both tasks. Recently, I have had the drive to dive a little deeper and be the creator of such great things; yes, I am still talking about math problems and knots...

In college I worked on a ropes course where we had to learn a number of practical knots. To help expand my knowledge on the subject, I bought a book about knots. After learning to tie some of the obscure and fun to tie knots (ie. the monkey's fist), I found myself oggling over the useless but pretty decorative knots. I made one for my girlfriend, seen here:

Recently I have had my eyes on another decorative knot. The book calls it the Ocean Plat. It is a little more of a complex knot than the one above, and is elliptically shaped. Circles are simple in construction, they have a center and radius. Ellipses are harder to construct, they have two foci and a variable radius. In geometry class we learned how to construct an ellipse with two pins acting as foci and a string whose ends are tied together in a (useful) knot. For those interested, here's a video showing this procedure: http://www.youtube.com/watch?v=7UD8hOs-vaI. About a year ago I used this method to construct an ellipse, or oval, and then proceed with the method of creativity to fill in the empty space within. I called this piece 'Ovalation", seen here:

DISCLAIMER: For those of you who get queezy at the notion of math and algebra, please skip to the end of this post.

In mentally preparing for the Ocean Plat, I got distracted by the ellipse - such a dubious thing. It dawned on me that I could not find the foci of an ellipse that was already drawn. This dilemma did not really pertain to the construction of my knot, nor much else, yet I just needed to know how to do this. Here is the problem I made for myself: if you know the lengths of the radii along the major and minor axes (A & B) of an ellipse, can you find the location of the foci (F1 & F2)?

It took me a while to conclude that yes it is possible to find the foci. All you need to know is the theory behind how ellipses can be constructed, as seen in the video above. I found the measures I and II helpful. Hint: Solve for X. I'll give you (that is if there are any of you reading this) until my next post - in a week - to try to figure this out. I'd like to think an 8th grader has enough deductive reasoning and algebra skills to solve it, but do not let that hold you back...

As for the knot, last time I was at the Home Depot I bought 50 feet of sisal twine rope, 3/8 inches thick. This is all I will need - oh, and a little math :-)