f: A->B can look like when it is injective, surjective, or bijective while also defining all of the important concepts. 

I really wanted to help the user understand visually mapping a function that is either injective, surjective, or bijective because that helped me a lot to understand it. So I wanted to really implement that in my art piece. My creative/artistic side mostly came from the organization and building process sides whereas my final product is a bit more literal. 

I was having a hard time understanding injectivity, surjectivity, and bijectivity until I understood it as literally mapping, so while making this diagram, I was able to make the poster as space efficient and informational as possible which came with understanding the subjects as a whole.

-Tyler MacClelland

In general, proof writing requires more creativity than some other math problems, so my creative side was useful during this class in most of the proofs we did.

I don't believe that this art piece helped me understand the concept better necessarily - it was quite different from the way we used permutations, but still interesting nonetheless.

-Nathan Mann

It was confusing using the software musescore to make the sheet but after I got the hang of it I was able to work it out the way that I wanted. By choosing this art piece piece, the idea of sets is in our everday life. Everything belongs to some set, and that set to another. Seeing the world in that way, obviously with wisdom, you can work things out on how things relate to another much easier. I often find myself relating one things to another because they both belong to a larger set.

The title is supposed to be a joke FYI.

-Andres Diaz

It was difficult to find so many words that could rhyme and flow through my poem, but from the feedback given in my drafts by my peers, I decided to focus on the content and the flow of the poem instead of the rhymes. Writing this poem has expressed an artistic side of me that I rarely see. I did not know that I had it in me to write an artistic poem about a mathematical concept. The process was exciting and frustrating at times, but the final product really made it all worth it.

It made me think of writing proofs in a different way. My initial understanding of the basis of a proof by contradiction was very simple, if a statement cannot be false, then it is true. But after writing about proof by contradiction in a different context, I could visualize the flow of writing a good proof. 

-Audrey Goenanto

I drew this on a program called Krita after looking up many drawings of Ouroboros which is a symbol depicting a serpent or dragon eating its own tail.  I'm pretty bad at drawing so that's why I needed so many references so the end product looked half decent so I could show what I was meaning to relay. I chose two different elements. For the snake itself, I chose to draw ouroboros because it relates to the principle of mathematical induction and how the next case (next segment of the snakes body) is true (eaten). For the way the body looks I chose this because induction was really confusing when I first learned it so I decided to draw the snake's body in a confusing manner which is why it twists and turns a lot.

The head of the ouroboros, where it begins to consume itself, can symbolize the base case in induction. Then as the ouroboros continues to eat, it progresses along its body, similar to moving from the hypothesis that a statement is true for n to proving it true for n+1. This is as simple as induction can be explained, and this drawing helps illustrate this principle of math.

-Jay Chong

Creating this artwork is a lot harder than I thought. Mainly because I am not good at drawing so it's hard to put the image in my mind into a real art piece. But I really enjoyed drawing the spirals and the branching patterns and was fascinated by the beauty of math concepts in nature.

I think looking at real-life examples like ferns, snails, whales, etc. is like discovering the hidden patterns of nature. It helped me better understand the Fibonacci sequence because I was able to see that it actually exists, not just a blurry concept in the textbook anymore.

-Maggie Chuang

            In terms of set theory, a musical scale is a set containing seven elements—in this case, those elements are musical notes. Chords are subsets of a given scale—most often comprising three noncontiguous elements. (I find it fascinating that any three noncontiguous elements of a scale make a chord on that scale.) For two chords to be consonant with one another, either they must have a nonempty intersection with each other or they both must have a nonempty intersection with a shared, third chord. In mathematical form, I would write this as:

Let A, B and C be unique chords which are all subsets of the same scale. A and B are consonant if and only if A and B not disjoint OR there exists some C such that A & B are not disjoint with C.

            Approaching scales and chords in terms of set theory helps me better understand the relationships among chords. Previously, I struggled to determine consonance without repeatedly playing the chords and hearing them for myself. In turn, this limited my ability to improvise or create new music. As I demonstrate in this extemporaneous piece, I need only consider which chords share notes with each other or which share a third, consonant chord.

            In creating this piece, I learned that set theory has an incredibly wide scope of applications, ranging far beyond those we covered in class. I am looking forward to applying it in other real-world situations.

-Arieh Black

Essentially, the spiral is generated by drawing squares next to each other that each have a side length equal to a number in the fibonacci sequence. When you connect the farthest corners of each square, you'll generate a sort of diagonal spiral that, when "circle-fied" and drawn as curves instead of straight lines, gives you this beautiful spiral.

I've added flowers, either pink (for odd numbers) or blue (for even numbers) on the sides of each square of each number in the sequence. I thought this was a nice touch since we talked about the parity of numbers in the sequence in the class before.

Finally, the sequence itself represents a bonus question from exam 2 that asked us to prove that the limit of ratios of consecutive fibonacci numbers gears towards the golden ratio. As the spiral expands, the rectangle bounding the rectangle gets closer and closer to the rectangle the golden ratio was actually derived from. This took maybe 30 minutes to an hour to throw together on Canva, but it took much longer to explore the actual math behind it. It was a really interesting exercise to hand draw the spiral, something I encourage each of you all to do as well.

Some surprising things: first, the sequence gets big, fast. I thought I was going to have space for more, but I really didn't, which was quite shocking.

Secondly: seemingly, proofs involving recursive sequences have very little to do with the numbers themselves and much more to do with the actual recursive formula. Quite similarly to how we proved that 9.99999999 - .9999999 was equal to 9 via expression as fractions, you may do the same with recursion only using consecutive elements instead of multiplying by ten to observe some similiarity. I'm one hundred percent sure that explanation makes no sense to a reader, but it definitely made sense in my head and was a connection I wouldn't have drawn before exploring the concept.

-Roy Lee

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• The first drawing is based on the idea that theoretically, a statement is either true or false. I try to illustrate how a statement can't be both true and false with two opposing people, one representing "true" and the other "false", whose colors would never blend.

• The second one is a girl looking at her own reflection in the air, portraying the concept of negation. The girl and her shadow are the same but in opposite positions, as the negation of a mathematical statement have the same components but the opposite truth value. 

• The third concept is "biconditional" statement. I drew two people hugging to illustrate how I feel about two statements being both sufficient and necessary conditions for each other. It feels like they are inseparable and they only need each other.

• The fourth one is abstract because I want to capture the concept of tautology, the compound statement that always result in truth regardless of the truth value of its components. In the drawing, the figure representing truth will look exactly the same before, during and after going through a maze that means the truth tables.

• The fifth one means logical equivalence, as the two figures create equivalence despite looking different. 

• The last drawing is a girl finding a like-minded soul in the crowd to illustrate that like the existential quantified statement, there exists at least one among all that fits your standards, and they are ready to be found.

I really enjoyed using art to understand math, as I do love art and I think math always has a romantic and poetic side to it. Making this art piece helps me see these concepts of logic as more than just symbols and words, but with realistic meanings and even stories. 

-Diandian Shi

This was so fun and so intense. Took me about 3 or 4 hours to get all these synonyms and form simple statement like "if then" into coherent sentences. I really had to dig deep into exactly what the proof techniques were saying, direct proof in

• Figure 1, contrapositive and induction proofs in 

• Figure 2, and proof by contradiction in 

• Figure 3. The most difficult things to understand well enough to incorporate were proof by induction, which I had a lot of trouble with before, and proof by contradiction, which goes deeper than P, Q, ~P and ~Q.

This was incredible fun, and mostly based off something an English teacher said to me in junior high or high school that stuck, "A good writer does not repeat words if possible". If you can imagine entire essays written this way, you better believe I've done it. As with mathematics however, simpler can be better.

-Dylan Guzlow

I think this let me understand a slight bit of the mathematical rigor used in creating various complex proofs or even complex sets. This project took hours of making these little stars out of paper by hand this semester. I know many difficult proofs can take years. The only solved Millennium prize problem (Poincare conjecture) took 7 years to solve by one man who had over 200iq and essentially locked himself in a room for those seven years for pure dedication and focus just to solve the problem. This project is nothing compared to that, but sometimes just making one set or getting the details right on one little aspect of the problem where it's all about making sure the rigor is perfect felt a little similar to my work on this art piece.

I think it gave me a deeper appreciation and way to look at sets. Before I made this art piece, I just saw sets the same way someone sees numbers as just abstractions to be used in everyday life. I now see sets in a deeper light with how they can be made. I think it is fascinating how mathematicians can do crazy stuff with numbers like making dark numbers and transcendental numbers that non-mathematicians don't often know about since they aren't used in everyday life. It let me see something beyond sets as pure tools; it let me see the potential for beauty behind them.

-Andrew Wason

-Brenna Stewart

-Heidi Pico

I think it gives a focal line to the photos and highlights what is being centered. Its a cool way to see various shapes from certain angles and be able to pin point a specific function.

-Khalil Toubba

-Daniel Knight

-Conor Ney

-Lester Vega

"Choosing and creating this visual representation helped me better understand a geometric series by understanding that an infinite amount of terms, in this case, beer orders, may only reach one number :)" -Anaka Bamba

"While working on this piece, I gained a deeper understanding of how a series can be represented, both physically and conceptually. I also learned that math can be a little more enjoyable over a couple of beers with some friends." - Tony Hernandez

"Filming and being part of this video has definitely helped me understand what a converging geometric series looks like. After attempting many times to all come to an agreement of how we would portray this on film, it definitely solidified my understanding of an infinite sum of numbers converging to a finite number."-Cherise Lopez

"reating this piece helped me understand how an infinite number of terms can still result in a finite number which was something that felt abstract at first. By turning it into something visual and relatable, like people ordering drinks, the math became more real." - Heidi Pico

I think doing this project helped bring out my creative side within math. Being able to visualize mathematical concepts is very important for me because it helps me connect the dots between different concepts and overall better my understanding of Calculus. Coming up with this concept for my art project was not too difficult because the "squeeze" theorem already presents a visual way to understand it.

-Annie Chiappe

For me, the p-series test stood out as a good & present example to use. With learning so many series tests, it definitely helped stick the p-series test into my brain. Additionally, I think it overall was great to represent convergence and divergence in a new way that helps my brain process something so abstract into a concrete or semi-practical environment.

-Jazz Beecham

Drawing the nesting dolls just helped me remember the concept of shells better, as I think out of all the things we’ve learned I have already understood this most. However, seeing my own understanding of the shell method actually drawn out in front of me just reminded me about where exactly the radius, height, and thickness from the original integral/formula came from, helping me picture all the original shells not as just formulas, but actual layers (thanks to the dolls). It helped the whole thing feel more like an actual thinking process rather than just mindless integration.

-Aerin Ganatra

Choosing this concept for the art piece has helped me with the process of U-sub. I found that showing the steps that need to be done in order, while also using basketball as a visual to understand the subbing process has cleared confusion when it comes to setting up an integration problem that uses U-sub.

-Trevor Grosso

Making this tornado helped me understand the washer method a lot better. Before, I just plugged numbers into the formula. But by drawing it, I could see that each washer represents a small layer of volume and when those are all added up through integration, they build the whole tornado. It really helped me grasp how calculus can turn something complex and irregular into something measurable.

-Edwin Hernandez-Avina

I sometimes find it challenging to conceptualize the essence of mathematical formulas and theorems. I often feel that I can “apply them” as needed, but at times, I lack a deep understanding of why the property works. In the process of creating this piece, I found that I was able to better grasp the mathematical concept by trying to make it simple enough for kids to understand. This forced me to unravel the theory of the Geometric Series Test into digestible pieces.

-Joshua Matz

Creating this helped me better understand why the work equation involves integration. It represents the total effort as the force changes. By drawing it, I realized that the spring gets harder to push the more it's compressed, which I always kind of knew, but now I fully get why.  By drawing it, I understood that the spring gets harder to push. The visual helped solidify how calculus explains real-world motion.

-Anthony Musnit

Main concept is rotating around a curve and following along a curve. By following the main curve and rotating around it (main curve = red line) it is forming a tube around it.

Then I’m subdividing the tubes represented by the elipses. It helps me understand better how to find the areas with axis or curve rotations.

-Jasmine Aye Chan San

By portraying the process of determining which series test to utilize artistically through my art piece, I'm able to visualize this operation mentally and, therefore, better understand how this concept works.

-Naila Surahmat

My experience in expressing my creativeness with mathematics and understanding it was  puzzling at first. I was stuck in thinking how I could express math through art, but then I thought about what mathematical concept stuck with me the most, which was whether or not an integral/series converges or diverges.  I later figured that this could be related to destinations and roads. This correlation gave me a bit more understanding with how convergence and divergence can work.

-Miguel Vasquez

Initially, I found it difficult to tie a mathematical concept to my creative/artistic side because I usually don’t associate those two groups together. When I got the idea to use a video game to explain the mathematical concept I chose, I found it really fun to connect the two things and to just have fun playing a game I like. It challenged me to think of ways to explain convergence/divergence in limits which helped me understand this concept further.

-Eliana Yee

Title: Visual Exploration of Related Rates
Mathematical Topic: Related Rates (Calculus 1)
Prepared by: Alim Kham, Zeqian Mo

Key Ideas:
Our project focuses on Related Rates, a topic in Calculus that helps us understand how changing one quantity affects another over time when both are connected. Using derivatives, we can find the rate at which one variable changes based on the rate of change of another. This idea applies to many real-life scenarios where motion or growth happens at the same time across different quantities.

For this project, we created a short video showing three everyday examples that demonstrate Related Rates visually: pouring water into bottles, the growth of a shadow, and the motion of clock hands. Each scene shows how a change in one measurable value causes another to change at a related but different rate.

-Angelo Volokh

Working on this project pushed me to use my creative side in a way I normally do not when studying math. Usually, I think of math as solving problems step by step, but creating art around a math concept helped me look at it from a different perspective. It made the idea feel less intimidating and more connected to imagination and everyday life.

Having to use my "creative" side to understand a math concept was a bit odd at first. I didn't understand the correlation between the two, but now, after looking at some math art pieces and having to create my own, I see how they are intertwined. So having to think of a math concept artistically was weird at first, but overall, sort of fun and rewarding in the end.

-Troy Ortega

I found this project very fun and enjoyable. It's rare that I'm able to engage in creative content like this in college, so creating a comic-book-like story to explain a mathematical concept was quite exhilarating. I initially struggled to incorporate an in-depth explanation of how quadratics work, but I think I ultimately did a pretty good job of explaining it and weaving it within the story well.

-Daniel Mendelson

-Yasmine Morsy