Blog 3: Moving away from R for a second...

 She won't read this so there's no risk of me saying I bought my girlfriend an engagement ring last weekend! She sort of knows that this is happening because I took her ring shopping but she doesn't know that I've got it on order. My ring on the other hand has been a collaborative process that she's spearheaded, but I've had A LOT of opinions on. 

For this Genius Hour, let me walk you through the week of mathematics, Desmos, and communication I had to subject her and the designer to.

After looking at men's wedding rings for a day, I quickly knew I wanted to have an inset engraving design that goes around the circumference of my ring. While in Vancouver I bought a cheap $18 ring that has the basics of my design which I'm wearing on my right hand for now. I also knew I wanted it to be medium grey with purple insets. Basically it'll look like this around the outside:

The wave is an interpretation of my girlfriend recording herself saying "I love you". Pretty sweet, right? 

Now, this image seems to be pretty straightforward. Make a curve that'll fit within the width and circumference of the ring for the engraver to use. But all of a sudden words like circumference and width show up, and all of a sudden there's math involved. Before I could even start on the design, I had to know how much space I needed. I began by measuring in millimeters the width of the ring that I had and deciding how much flat space there will be for the engraving. Those grey lines at the top and bottom denote my bounds where the ring chamfers off. Then i had to measure the diameter of the ring and calculate the usable circumference that I had. The ratio of width to length was 1 : 12.55. 

At this point, I was ready to start making the curves that I had roughed out on paper. However, I didn't want to work in radians so I scaled everything up to degrees. The length of my space became 1680 (degrees) versus a height (width) of 1. My certainty with these dimensions were only found when I took a ruler to my computer screen and made sure what I saw was 1 : 12.55. I was satisfied, so next came the hard part of making the curves. 

At this point, I began fiddling with different types of sine and cosine curves -- very much in the same way I have been fiddling with code to get it to do what I want. The first curve was tricky because it ended up being a complex sin(x) + cos^2(x) that had had the slightly sharper peaks. I didn't know that would be the result until I started changing exponents and operations between my trig functions. Eventually I got the first one to where I liked it and I cut it's bounds to be between [0, 360]. I tried longer and shorter by adding a-s-d-f transformations to my functions but it was smarter for me to end on 360 where I can loop from a height of 0 again (hence the - 3/16). 

y = 1/2sin(x) + 1/4cos^2(x+30) - 3/16 {0 <= x <= 360}

The next wave was a lot more simple, it was a simple cosine curve going from [360, 840], going a distance of one and a half wavelengths. That's why I had to use cosine. The last two curves are a sine and a cosine, the short one being her pause between "love" and "you". 

One thing I definitely had to take into consideration was the height of my waves, which I adjusted at the end using the vertical scaling term we all have learned in grade 10 math. It wasn't until I was adjusting the ways enough so their transitions were smooth that I realized I was making piecewise functions. Obviously I was using techniques like liming the domain of each wave, I just didn't define it in my head until the end. 

y = 3/8cos(3/4x) {360 <= x <= 840}

y = -1/8sin(3/2x) {840 <= x <= 1080}

y = 1/3cos(3/5(x-30)) {1080 <= x <= 1680}

Desmos was an absolutely amazing tool for this whole project because I was able to see all my adjustments in real-time and check to see if my curves were ending a y = 0 without having to calculate it on paper. Plus, the visual simplicity and customization allowed me to see the shape in the colours I wanted. 

You can see the Desmos "Rings of Power Lines" file here: https://www.desmos.com/calculator/a8doveibu6

I really loved this project and why I was doing it. I'm not the best at trig equations, so starting this was a little daunting, but with some trial and errors, and some exemplars, I was able to do it and I couldn't have been happier with the result. And whenever I look at my wedding ring I'll be able to see the equations that shaped it. It's not just art, but art and science put together. 

STEAM!

Coda: 

For my exemplars, I have saved a lot of Desmos's tweets that share work done by students using a piecewise function style. Like this Star Wars one! So yeah, examples are very clearly how I learn. 

I also spent a lot of time making sure my viewport was locked and showing the amount of screen I needed so that it matched when I took a ruler to my screen. Locking down my perspective was crucial.

And finally, clicking and holding on the icons for each equation allows you to change the colour and thickness of each line, or even shade in with transparency areas between bounds. That's how I got my final product to look exactly what I think it will when the ring is forged (in the fires of Mount Doom).