Eternal Knot

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Carrick Mat
The Carrick Mat is a decorative knot, known as the 8-18 knot in knot theory.

Contents

Basic Description

Knots are everywhere, from shoe-ties to decorative bracelets. But have you ever wondered how something so simple and commonplace such as a knot can be analyzed mathematically?

The Carrick Mat shown to the right is a traditional Celtic decorative knot. Mathematically, it is made up of parts of circles that connect with four perpendicular lines which form a square in the center. The curved lines overlap, as do the straight lines in the center, forming a knot that appears to have no beginning or end.

Some basic knot theory

In knot theory, a knot is just a closed curve in 3D space. It can be a loop, as simple as this...

...Or it can be as complicated as the Carrick Mat. The unknot, in knot theory, is a knot that has no crossings, like the loop in the picture. What's interesting about the Carrick Mat is that it only has 8 crossings, but to get it back to the unknot, it has to be manipulated 18 times. Hence, its identification as the 8-18 knot in knot theory. For a more detailed explanation of aspects of knot theory, refer to Perko pair knots.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

Defining the Carrick Mat

The curves and lines of the Carrick Mat can be defined by algebraic equ [...]

Defining the Carrick Mat

The curves and lines of the Carrick Mat can be defined by algebraic equations. The way the knot appears depends on how the equations are written.

The curves

The top curve When using a unit circle, half of the top curve can be defined using the equation y=\sqrt{1-{x^2}}. When it's graphed, this equation produces a semicircle that intercepts the y-axis at 1 and intercepts the x-axis twice at 1 and -1 and has radius 1.

The bottom curve The bottom curve is defined as the opposite of the top curve, using the equation y=-\sqrt{1-{x^2}}. When this semicircle is graphed, however, it completes the circle instead of leaving room for the square and the other curves to be constructed.

So, in order to move the semicircle down the y-axis, the equation of the curve becomes -\sqrt{1-{x^2}} -1 .

Figure 1  On the y-axis, the top and bottom curves, in red.
Figure 1
On the y-axis, the top and bottom curves, in red.


The right and left curves The right and left curves are defined differently than the top and bottom curves. Since they intercept the x-axis at one point and the y-axis at two points, the equation has to be defined in terms of x instead of y. The right curve's equation is x-.5=\sqrt{(y+.5)^2}+1. When it's graphed, the curve has radius 1. The left curve's equation is the opposite of the right curve's equation. So the equation of the left curve is x+.5=-\sqrt {(y+.5)^2}+1.

Figure 2  On the x-axis, the right and left curves, in purple.
Figure 2
On the x-axis, the right and left curves, in purple.


In the Carrick mat, however, the curves are more than semicircles, so there are also eight equations for the parts of the curve that connect to the square inside. These can be found by forming a complimentary semicircle to the one that's already graphed, and then finding its intersection point with its opposite. Then graphing a linear equation that runs through that point. Basically.

So, to find the complimentary semicircle to the top curve, use the top curve's beginning, y=, with the bottom curve's ending, -\sqrt{x^2}+1 . The equation then becomes y=-\sqrt{x^2}+1. To find the complimentary semicircle to the bottom curve, use the bottom curve's beginning, y+1=, with the top curve' ending, -\sqrt{x^2}+1 . The equation then becomes y+1=\sqrt {x^2}+1.

Figure 3  Located just below the origin, on the y-axis, are the complimentary semicircles for the top and bottom curves, in grey.
Figure 3
Located just below the origin, on the y-axis, are the complimentary semicircles for the top and bottom curves, in grey.


The same concept works for the side curves. For the right curve, the complimentary semicircle's equation is x-.5=-\sqrt{(y+.5)^2}+1 , and for the left curve, the complimentary semicircle's equation is x+.5=\sqrt{(y+.5)^2}+1.

Figure 4 Located just above the origin, on the x-axis, are the complimentary semicircles for the right and left curves, in lime.
Figure 4
Located just above the origin, on the x-axis, are the complimentary semicircles for the right and left curves, in lime.


The square

The equations that make up the square are simpler. They're linear equations! So no square roots.

The four of them are:

             1) y=x+.366 
             2) y=-x+.366 
             3) y=x-1.366 
             4) y=-x-1.366 

When they're graphed, the knot looks like this:

Figure 5  The four corresponding numbered linear equations in blue. Together, they form the interior of the knot's square.
Figure 5
The four corresponding numbered linear equations in blue. Together, they form the interior of the knot's square.


After the equations are limited, and the image looks less like a jumbled blur of color, this is the knot:


Figure 6a
Figure 6a


That's the knot without the double lines, like the knot in the picture has. This one has double lines:


Figure 6b
Figure 6b




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