# Priya proposed Math club do a fashion show

The Math club at the school met once a month and the students were encouraged to present any Math related material they desired. This time, a student named Priya raised her hand and said that she wanted to discuss if and how the Math club could participate in a Fashion Show,

Most students ridiculed the idea of fashion and Math. The general feeling was that fashion was for the airheads who don’t even care for Math. However, the Math teacher asked the students to remain open minded and at least to hear her out. Priya told the students that Math was used a lot in the fashion industry. She gave an example of the use of 3D printers in the up and coming new fabrics. Still, everyone questioned why she was interested in this, and if she knew anything about Math or for that matter even fashion. Priya told them that she worked part time at Jenna’s T-shirt shop where customers brought designs to be printed on T-shirts for males and females. So the fashion was not just for airheads but for everyone. Then she challenged them all to see if they had any logos on their T-shirts, and explain to her why they had the logos. As expected most students had the logos of their choice on their T-shirts but they still questioned how Math was involved in any of this.

# Sara supported Priya’s proposal

Priya was getting nowhere until Sara raised her hand and said: I support Priya’s idea. We should at least discuss it. If you want, Priya and I can work on something to present at our next meeting.

Almost every knew that Sara was a Math whiz. Johnny immediately supported his girlfriend Sara. Kirsten and Jonathan, who chaired the meeting, asked for opinions. Mr. Power, the Math teacher, told the students that they should always think outside the box and remain open to all ideas. No-one else said much about it but when a vote was a taken on this issue, most of the students were okay with the next meeting being on the presentation for a Math Fashion show.

Priya saw Sara in the cafeteria. They exchanged greetings and then started talking.

Priya: Sara, thanks for supporting me at the Math club meeting.

Sara: Priya, I guess I had to support my friend with the 100% on her Geography Project. Besides, I thought this would be fun but why are you so interested in it?

# Priya explained to Sara why she wanted the fashion show

Priya: Our customers are always looking for new and original images to be printed on their T-shirts. Therefore, having a large inventory of original images is important for our shop. The stock images in the market are expensive because of the royalty charges, and they also involve too much paperwork. Besides, being available to anyone in the market, the stock images are no longer original. I thought this might be one way to get such images. This fashion show may help me in finding a new line of designs. My mom has even agreed to sponsor the show by giving free shirts for the students modelling them.

Sara: Liking or not liking a design on a T-shirt is very personal. So, I don’t know if you will like my ideas.

Priya: Tell me already, please. The suspense is killing me.

# Graphs of Trig functions in polar co-ordinates

Sara: I will show you a few pictures. All of them are plots of Math functions in polar co-ordinates. Remember, we talked about them for your Geography assignments. In polar co-ordinates, the position of a point is written as: r, θ where r is the distance from the origin also called the modulus and θ is the angular location or the argument. (learn about polar co-ordinates from the story titled** ***Priya’s Geography Project*),

You need to understand two things before I show them to you. One is that angles can be measured either in degrees or in radians. There are 2π radians in a circle which is 360°. The second is that if you keep rotating something, the value of the angle does not stop after a full circle. It keeps increasing. Here are some plots of increase in radius with increasing angles.

Priya: These are all spirals (Fig. 9.1). I would assume that they can be drawn in different colors.

Sara: Yes, these are called Archimedean spirals which you can draw in any colour and line thickness. Of course, you know more about the types and colours of fabrics to use for the background. These are all plots of r = θ in radians. The first is a plot when θ goes from 0 to 6 π radians. In the second graph, it goes 0 to 12 π, and in the third one it goes 0 to 32 π. You can see that number of lines in the spiral increases with the increasing ranges of θ. In the last one, the range of θ is from 12 π to 24π. It is hollow in the middle because it is missing the initial spiral of the plot from 0 to 12 π. You can also change the hollow areas by changing the initial value of θ, and you can alter the number of spiral lines by increasing its range.

Priya: Wow, these look good.

Sara: You could draw these from any starting values of π to any final values, within reason. Each one will give you a unique image.

Priya: Do you have more?

Sara: Yes, here is another set.

Priya: These are also spirals but made from lines with different designs in them.

Sara: Have you taken Trigonometry yet?

Priya: No. I wasn’t going to but I will if it will help the business.

Sara: For a right angle triangle sin θ = height/hypotenuse, cos θ = base/hyptenuse and tan θ = height/base. I will show you pictures with all three. I will let you learn more about the Trig functions when you take the course but for now we focus on the designs. In this graph the function r = θ gives the lines for the spirals but when you add cos (2^{θ}), cos (2^{4θ}), cos (2^{8θ}), cos (2^{16θ}) (Fig. 9.2), you get the different designs on these lines. In these images, θ varies from 0 to 12π radians.

Priya: What else do you have?

Sara: You are going to love these flowers (Fig. 9.3). The first flower is the graph of r = cos (2 θ), the next flowers are for r = cos (3 θ), r = cos (5 θ) and r= cos (8 θ). The range of θ for all of them is from 0 to 2 π radians. Again, you could put any value of n and plot a variety of r = cos (nθ) plots, n doesn’t even have to be a whole number. Here is a different set of designs. Now, these are plots for r = cos (n^{θ}) from 0 to 2π radians. The first is for cos (2^{θ}), the next for cos(2^{θ}) + 2, cos(3^{θ}) + 3, cos(4^{θ}) + 4.

Priya: Now all these look like some kind of oriental fans of different designs (Fig. 9.4). Great.

Sara: Do you want to see more?

Priya: Don’t stop. I am loving it.

Sara: Next are the images are for r = 1 + cos (2^{θ}), r = 2 + cos (2^{θ}) and r = 4 + cos (2^{θ}) and r = 5 + cos (2^{θ}) (Fig. 9.5). These are all from θ = 6π to 12π. I kept the range θ = 0 to 6π away to get a new look.

Priya: Neat, now we have rings of different sizes and thicknesses.

Sara: I am enjoying this. We could sit here and play all day but I have a class soon. I will just show you two more sets that I made last night. It should give you a good idea of what I have in mind. The next picture uses the Trig function tangent (tan) (Fig. 9.6).

Priya: Awesome! You can make all sorts of designs. Some of them look like highways with connecting cloverleaf ramps but others could be different flowers with rays. I bet these will come out beautiful in colors.

Sara: Here is the last set, and then I have to go for my class.

Priya: So you can get graphs looking like all kinds of shapes – even a heart, a butterfly and a leaf (Fig. 9.7). I love it. Let’s show these at the next Math club meeting. For me, I want to have this butterfly with one of the flowers on the same shirt.

# Mr. Power had to get a bigger room for the meeting

Before the meeting, there was a lot of buzz in the school about the fashion show. Many students asked if they could attend the meeting even though they were not in the Math club. Kirsten and Jonathan asked Mr. Power about this issue. Mr. Power thought that the open attendance could only increase the interest of students in Math. He agreed that any student from the school should be allowed to attend the next meeting. He also said that he would get a larger room for it.

At the meeting, Priya was asked by Kirsten and Jonathan if she would head that meeting and she agreed. After all, she had started this initiative.

Priya: Hello every one, welcome. I see that the attendance at this meeting is double of the normal number. I guess, it is because this meeting is about Fashion and Math. Sara developed all the designs, and she will make a presentation but first I want to make an announcement. Jenna’s T-shirt shop will sponsor this show. The shop will provide all the T-shirts free of charge, and a student will get to keep what they model. We can talk about the details after Sara’s presentation. Juan, I see you have your hand up even before the presentation.

Juan: My parents run a Mexican restaurant called La Placer and I know many of you have been there. I talked to my dad about the Math Fashion show. He liked the idea very much and said that after the show, all the participants including Mr. Power are invited for a dinner on the house at La Place.

Priya: I love the food at La Placer. So if the free shirt were not enough of an attraction, now you have a free dinner on top, if you participate. Now, Sara will show you what she had in mind.

Sara invited Mr. Power to tell the students briefly about polar co-ordinates before her presentation. Mr. Power did not know that he would have to do this, and he was not prepared. So, Sara helped by giving him the picture of a dart board for explaining the concept of the polar co-ordinates. After his explanation of this concept, Mr. Power asked Sara to resume her presentation.

Sara: All the images in the next few slides are different Trig functions in polar co-ordinates. I had a lot of fun coming up with them. We could have an image on the front of the T-shirt and description of the Trig function on the back.

Sara went through the images she had shown Priya, and then asked Priya to take over.

Priya: I hope everyone is now convinced that Math can be used for fashion designs. I think we should have an open discussion now.

# Cheerleaders wanted to participate in the show

Ashley: I am a cheerleader for the school football team but I am not a member of the Math club. Some of my cheerleader friends and I would love to model for the fashion show. Is that possible?

Mr. Power: It will be great if you and other members of your cheerleading team were to model as long as you use the T-shirts based on this Math theme. Also, I want to ask Priya and Juan about their resources.

Both Priya and Juan said that it would be fine but Priya had a concern which she asked Sara to address.

Sara: Priya has asked me to address the issue that I have shown you only a limited number of designs. We may need more unique designs if many students participate. There are different ways to go about. The first is that Priya might decide to combine some of the images. For example. you can get a shirt with a butterfly, a flower or both. The second possibility is that I can create some more images. But some of you who want to model and also like to create the designs can also come up with your own images the same way I did. It will work better if you have some background in Trig. I had a lot of fun doing this. I played with a program on this website: http://www.wolframalpha.com/widgets/view.jsp?id=8d8e2c27bcaa121d6ee0de4b98774bb4.

If some of you want, you can do it yourself and bring the designs but we will have to make sure that there are no duplicates.

Mr. Power: That will be great. Priya, what do you want to add?

Priya: Please, send me a phone message or an e-mail within next 7 days if you want to model. If you are creating your own designs, please talk to Sara about it. The time and the date for the Fashion show will be announced soon by Mr. Power. I hope the venue will be the school auditorium but I am not sure. It will also be announced.

## The fashion show was a big success

The Fashion Show was held in the school auditorium about six weeks after this meeting of the Math Club. The auditorium was full and many students were even standing in the back because all the seats were taken. The principal and Mr. Power took pride in this unique event. There were fifty models – school students ranging from cheerleaders some of whom had previously given up on Math to the Nerds who took pride in that they had made their own designs and were also models. Name of each student was announced, after which they came as if doing a catwalk. Johnny who loved bikes, wore a shirt with two Archimedian spirals connected by a line so it gave the look of a fancy custom bike made. All the models were applauded. Then Priya took over the stage to thank Sara for all the creations and the concept of this theme. This was followed by a long thunderous applause that would make you deaf if it had gone any longer or louder.

# A week later Priya met Sara.

They both agreed that the show was a big success. Priya wanted to use this design concept for Jenna’s T-shirt shop.

Sara: You could write a program using these functions in different ways.

Priya: Could you do that for Jenna’s T-shirt shop? We will pay you.

Sara: This will take a fair amount of time. I have to focus on my studies. Why don’t you ask Joe? I have seen you talking to him. He knows Trig and programming.

Priya: You mean Joe the Nerd?

Sara: Why do you call him Joe the Nerd? I wonder what you call me behind my back.

Priya: You have a different reputation. We call you a lovebird. You care so much for your boyfriend, and are also kind to others. Look at how much you’ve helped me and the Math Club.

Priya and Sara said bye to each other.

# Two years later

Two years went by. Priya was still in school but Sara had finished and gone to attend a university. When Sara came home for the summer, she got a call from Priya who wanted to come to say hello. Sara agreed for Priya to come over.

Sara: Hi Priya. I haven’t seen you for two years. How have you been?

Priya: I came by before but you were away at the university. Your Nana told me that you would come home for the summer. So, I am here now. I brought you some flowers.

Sara: Thanks for the flowers. So what have you been up to?

Priya: Here’s my business card.

Sara: Jenna and Priya’s T-shirt shop? Congratulations for becoming a partner in the business.

Priya: We opened a store near the bridge three months after we talked last time. We also hired Joe the Nerd to write a program. The store got really busy and I was working hard at the store plus making sure that Joe did everything to my liking. We paid him two thousand dollars for it. Mom couldn’t pay me for a while. So she made me an equal partner and that’s why the store is now Jenna and Priya’s T-shirt shop.

Sara: How is the business now?

Priya: Great. Like you said, “All you can hope is that it will work out in the long run.” In the last 6 months, our profits were what they used to be for two years at the old location. We have 5 laptops running. Customers use them to make their own designs. Joe had set it up for 10 different functions like the ones you showed me. A customer can get a unique design by entering a value of range from 1 to 10 for each of the functions. When they don’t want a function, they enter zero. They can also pick between five different line colors and five different fabric colors.

Sara: Looks, impressive. It looks like there is a choice of more than a billion designs. Johnny and I will have to come and see it.

Priya: Customers are happy. We charge them extra for this feature and there are no royalties to be paid. That adds to the profit. No one else has this feature. We have even filed a utility patent on it.

Sara: That’s great. I am really happy for you.

Priya: Here, a small token of appreciation.

Sara: It is a cheque for a thousand dollars. I cannot accept it. I did not do much to help you.

Priya: I love your modesty. You will make me happy by accepting it because I want to honor you for doing me this great favor.

Priya’s phone rang. She was needed at the store.

Priya: Bye, got to go. Oh, come by sometime with Johnny, I want to show you the set up.

Sara: Bye Priya. Thanks for coming by, and for your appreciation with flowers and the cheque.