Calculus and Line Designs

Since geometry is a part of calculus because the derivative is interpreted as the slope of the line tangent to a curve at a point. We can make the conclusion that line designs and calculus are related. Mary Boole was the originator of line designs because she thought that the line designs would help children learn about the angles and spaces of geometry. Although line designs teach angles and spaces they also teach graphs and the fundamental knowledge of calculus. With tangent lines, every line in the line design is a tangent line but to the eye it does not seem like that because the designs just look like curves instead of straight lines. Even though the symmetry in line designs is not directly related to calculus, the line designs are symmetric, in every design some way or another.

The Beginning Step of a Line Design

A Completed Line Design

The Previous Design Repeated 4 Times

The Evolution of the Icosihenagon

As I created my line design, I documented the change as a few rounds were completed. Along the way though there were a few complications, as some nails came out and had to be hammered back in, and also I had to attach more string as the string I originally used ran out.

After the 7th Round

After the 15th Round

Finished !

My Line Design

So here is my actual design! Instead of just making a geometric design, I decided to create a picture. This is supposed to be a King. Let's walk through the steps...

First, I took a blank white piece of paper and drew the design. This could have been done on graph paper too. It's very important that the lines are measured out, and that the nails are placed in even increments on the line. This should be actual size! When I finished drawing, I took my piece of wood, glued black felt over the board, and then taped my drawing over the black felt. Then, I nailed in all the places where I had drawn! Make sure the nails go through the paper, the felt, and into the wood. The nails should be as straight as possible.
When all the nails were in, I tore the design paper from the board, and this is what I was left with!

Now all that's left is the stringing, to actually make the line design and to add color! Although you could see the outline of the crown in the earlier steps, you can now see the curves that make its unique shape! The eyes are now round in the center, the way eyes are supposed to be. You may be able to tell that I had difficulty with the mouth. Since it was only a half circle, it was more difficult to string evenly, which is incredibly important with line designs! Finally, I created the circle which makes up the outline of the face. This is a great example of how tangent lines can create a curve, because the circle is entirely comprised of straight lines, even though the curve is smooth.

And that's all folks!

By: Danielle Blake

Calculus in Line Designs

Yayy line designs! Okay so I'm going to explain one way that calculus plays a role in line designs. As we have learned in class, there are an infinite number of tangent lines that can be drawn to a curve. Using calculus, we can find these tangents at certain points along the curve. Line designs use this concept, but backwards. Instead, we draw many straight lines, and those lines are the tangents that create the curve. This explains why when creating a line design, the more lines drawn, the more definitive the curve. The more lines there are, the more points along the curve we are creating! By making the original triangle more acute or more obtuse, we can change the shape of the curve by changing the slopes of the tangent lines!

As you can see in the picture above, the red line is one tangent line to the green curve. At the point where they are touching, they share the same slope. By creating lots of the red lines, all of different slopes, we create the curve with a changing slope.

By: Danielle Blake

Icosihenagon

Searching for the pattern for my line design, I came across multiple patterns that were both complex and cool. There are thousands of variations of line designs that have been created but there also thousands of designs that have not been created. Continuing on my search I came across a design also known as an Icosihenagon. When looking at it, I thought it looked awesome and could not believe that the design was created with just string . The design contained 21 nails, a very long string, and a design template that was premade. After reading the instructions the pattern did not sound so complicated but after beginning it, the pattern started to get more confusing then expected because of the string.

The pattern was, start at the top peg, then
(1) Skip 0 pegs, and wrap thread around the 1st peg.
(2) Skip 1 peg, and wrap thread around the 2nd peg.
(3) Skip 2 pegs, and wrap thread around the 3rd peg.
(4) Skip 3 pegs, and wrap thread around the 4th peg.
(5) Skip 4 pegs, and wrap thread around the 5th peg.
(6) Skip 5 pegs, and wrap thread around the 6th peg.
(7) Skip 6 pegs, and wrap thread around the 7th peg.
(8) Skip 7 pegs, and wrap thread around the 8th peg.
(9) Skip 8 pegs, and wrap thread around the 9th peg.
(10) Skip 9 pegs, and wrap thread around the 10th peg.

These are the first ten steps. To complete the design, I have to continue on through the 10 steps a total of 21 times. So after I complete the 10 steps once I still have 20 more times to complete the 10 steps, and so on.

A Completed Icosihenagon Line Design

Graphing Line Designs

I bet you didn't think that line designs had anything to do with calculus...but they do!

Look carefully at this parabola. Do you see the curve it forms as it approaches the origin? Well this shape formed by the graph is very similar to that of the shape formed in line designs! Specific line designs actually have mathematical formulas that can be related to graphing in calculus!

Equation:

x^2/3 + y^2/3 = L^2/3

Equation:

x²/a² - y²/b² = 1

Equation:

x^2/3 + y^2/3 = L^2/3

^{every line has an equal length of L}

Equation:

x² - 2xy + y² - 2Lx - 2Ly + L² = 0

x-intercept: c

y-intercept: L - c

Erin Appenzoller

Hexagon Line Design

To complete my line design I needed a small piece of wood with black felt glued on top, colored string to create my design, and small nails to hold the string in place. For my line design I decided to use a hexagon shape, so I started by taking one triangle and measuring 8 equal segments on each side. I repeated the process until I had 6 equal triangles to create my line design shape. As you can see in the photo below I taped the triangles together to form a hexagon, then I placed my shape on top of my piece of wood. As I mentioned before each side of the triangle had 8 markings equally spaced out; this indicated where I would hammer my nails.

Here you can see the beginning of hammering the nails into the wood.

This triangle shows the equal markings where the nails were hammered.

After I was finished hammering ever nail into the wood, I began to make my design using orange and purple string. I tied one end of the string around the first nail of the lower side of the triangle. Then I crossed the string across the middle of the triangle to the eighth nail of the upper side of the triangle. I continued in a pattern: first nail to eighth nail, eighth nail to second nail, second nail to seventh nail, and so on. I stopped when I had wrapped the string around each nail, tied the end of the string, and then moved on to the next triangle.

As you can see above, the line design is starting to take shape!

I alternated colors with purple and orange next to each other for the outsides of each triangle until my line design was complete.

Close up of the final product.

This was the end result of the line design. It is a hexagon with a clover design in the middle!

Erin Appenzoller

Line designs applied to Calc

Since it’s Math Thursday, we’ll discuss line designs! Line designs directly relate to Calculus is multiple ways and are tons of fun to make! Line designs create curves, which in turn help create the design, from straight line segments! Then, each line segment serves as the tangent line of the curve being created. While doing this project, I learned that you can create curves that are circles, parabolas, ellipses, hyperbolas, and even spirals! Making tangent lines applies directly to our Calculus topics at the beginning of the year, and played a major role throughout the year. Tangent lines are everywhere!

Cara Brennan

Line Design Process

Line designs! While making my line design, I needed a black foam board, wood, string, a hammer and nails. First, I glued the black foam onto the wood. Then, I designed a template of the design I wanted to use. To make the template, I used a ruler and a protractor, and drew even lines to resemble a star. In creating my template, I made even marks where I hammered in the nails later on. Once I had hammered all of the nails into the template, I started stringing the string around the nails. I secured each end of the string to a nail and wrapped the string around each individual nail. This produced my line design.

Let's talk about my pathetic looking line design. Well, I've always loved to draw stars, however, I felt that this was not the best line design representation. However, in order to incorporate my love of stars into my calc project, I made an abstract star!! My star does not have equal spacings between each point, because I believe it added to the uniqueness of my line design. I then picked out my favorite colors of strings to use for my line design. All of this contributed to my line design.

Cara Brennan