Movement and Mirrors, Frida Graumann


Project Description

For my creative project, I decided to capture the mathematical topic of reflection through dance.

Essentially, I created a miniature music video to the song Mirrors by Justin Timberlake. The title of the song was quite fitting because I chose to use none other than the common household mirror to display the aspect of reflection in my project. With the help of a good friend and her novice videography skills, about seven mirrors of different shapes and sizes, and my dancing ability, I was able to aesthetically showcase the concept of reflection accompanied by Timberlake’s vocals.

The math behind how mirrors, meaning reflections, work is described by physics principles. Light is an essential aspect of reflection. The law of reflection explains how when light hits a surface it bounces back in a certain way, similar to a ball bouncing off of a wall. It says that the incoming angle of light, known as the angle of incidence, is always equal to the angle leaving or bouncing back from the surface, known as the angle of reflection. This is how reflection works. However, something I found interesting to ponder is that light itself is invisible until it bounces off something and hits our eyes. Meaning, a beam of light moving through space cannot be seen until it hits a surface. When the light beam runs into an object, the light is then scattered. This concept is called diffuse reflection and it represents how we see light when it hits an uneven

surface. The law of reflection is still present, but rather than the light hitting one even surface it is bouncing off of several microscopic surfaces. Because mirrors have a smooth surface, they don’t scatter light in this way. Instead, with a smooth reflecting surface, the light bounces off without disarranging the incoming image, which is known as specular reflection. This is why mirrors swap the image, turning it left to right and visa versa. A mirror image is a light-print of the image, not a reflection of the image from the perspective of the mirror.

Another aspect I wanted to focus on in my project is infinity in reflections. Meaning, whether two mirrors facing each other create infinite reflections. In my video, I tried to capture this aspect by holding up a mirror to another mirror to produce this infinite reflection looking effect. However, I learned that although they seem to create infinite reflections, it is not actually the case. The reflections get darker and darker and fade into invisibility long before they reach infinity. This is because mirrors absorb only a small fraction of the energy of the light striking them each time. There are never more than a few hundred visible reflections. Thus, when watching my video and awe-ing at the infinite reflections I had seemingly produced, remember it might not be as limitless as it appears. There is the wonderfully rhymed saying, “objects in the mirror are closer than they appear.” Perhaps, in regards to capturing infinity my music video, it should instead state, “although it appears this way to you and me, infinity is not, in fact, what we truly see.”

Of course, I also wanted my project to represent the aesthetics found in mathematics. I think dance is a very beautiful art form that has numerous connections to math. Although I wouldn’t say my project is an excellent example of aesthetic dancing, movement represents both mathematics and aesthetics. As with reflection, physics connects movement to math. Motion is the occurrence of an object changing positions over time and is thus mathematically connected to concepts and forces including velocity, displacement, distance, speed, acceleration, and time. Movement is math. Additionally, in hopes to make my video more aesthetically appealing, I wore green to match the grass and trees, and blue to reflect the sky. The bright colors in the background and reflected in the mirrors all contribute to portraying this appealing aesthetic. I wanted my project to display the mathematical aspect of reflection, but I also intended to make it enjoyable to watch. This is why I allowed my dog to make a special appearance. I think music, movement, and dance are brilliant ways to express creativity, but also to even exhibit more conceptual concepts. Mathematics and aesthetics are so much more interviewed than I think are initially presumed, and I hope my project was able to portray this beautiful connection.

References

Matthews, Robert. “Do Two Mirrors Facing Each Other Produce Infinite Reflections?” BBC Science Focus Magazine,​ www.sciencefocus.com/science/do-two-mirrors-facing-each-other-produce-infin ite-reflections/.

Flinn, Gallagher. “How Mirrors Work.” ​HowStuffWorks Science​, HowStuffWorks, 27 Jan. 2020,

science.howstuffworks.com/innovation/everyday-innovations/mirror2.htm.

“Physics Tutorial: The Law of Reflection.” ​The Physics Classroom​, www.physicsclassroom.com/class/refln/Lesson-1/The-Law-of-Reflection.

“Mathematical Movement.” ​Mathematical Movement​, American Physics Society, physicsbuzz.physicscentral.com/2012/10/mathematical-movement.html.

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untitled, Stella Feuerborn

Project Description

When we were first told of the guidelines for our final project, my mind went to graphs.

Graphs, linear graphs in particular, have always helped me to better understand the math I was working with. They have created visual representations of data I couldn’t previously visualize, and helped me see clearly what I was working with. I also feel like they’re a form of math that has been pretty universally worked with, and therefore would make my project more accessible. Once I had the idea graphs, I needed a way to display them in an artistic medium. Photography is a field I’ve been working for about 3 years now, and using the human body as an art medium is a powerful form of symbolism. So, I landed on the project I just finished.

For my project, I chose to depict 5 common graphs through photography with environmental elements and the human body. The sign post and curb acted as my x and y-axis. Then, I positioned the model (the lovely Carmen) at the visual intersection of these two lines, and had her shape her body to resemble graphed lines. The five graphs I chose to show were:

= |x| , and sin(x).

After taking the photos and color-correcting them, I drew a digital axis and where the rest of the graph would’ve gone beyond the figure.

From start to finish, this project ended up going beyond just the representation of graphs in terms of mathematical engagement. I used color theory and the mathematical field of optics to form my photographs, and angle tools to draw right-angle axes. It really speaks to how much

math goes into simple aesthetic creations, and asks rather than “What is influenced by math?”, “What isn’t?”.

If I were to do this project again, I think I would either choose the forest or another more picturesque backdrop than a street corner, just to spice up the photos and make them feel like they carry more weight. I like that I stuck with the straight-on angle for all five of them, because it makes most of the lines that run through the photo appear perfectly horizontal or vertical. This maximizes the appearance of a 2-D plain, and pushes the idea that you’re looking at a flat graph rather than a piece of art with visual depth. I also thought originally about being more artistic with Carmen’s outfits, but I didn’t want to distract from her body shape and the surrounding elements that made up the axes, so I think the simple outfit was okay. One thing that would’ve been cool would be if she had worn clothes that had a grid pattern on them, to further push the graph idea.

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Ad Infinitum, Jonathan Ely

Ad Infinitum

The Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 …

___

Grapes.

Gravel.

Long ago.

Cars racing by.

It’s six in the morning.

“There goes Fred. He’s an elementary school teacher.” 

As a former teacher herself, my grandmother had quite a fondness for them.

To me, the people streaking by on the Michigan highway were more exciting –firefighters and astronauts. Maybe there was a nurse.

If we had seen her that morning, she would have been driving east, making the long, winding trip towards Marquette Hospital. It’s a two-hour drive, though. She probably lived closer to the medical center.

I never did know her name. She came and left quickly, bustling off to another calamity. An unknown variable, she shall live forever undefined to me. This is a conundrum for the defined world. We who scoff at Socrates’ musing that a wise man knows he knows nothing. The foolish Greek never encountered big data.

Roommates of mine may follow in this tradition. They are certain of their beliefs, certain of the misery of the world and are worse for it in the end. They didn’t see the cars go by that day; they didn’t eagerly munch on grapes from a Tupperware container that helped win the cold war; and they weren’t there that night. They did not rush into the hallway, hurling glances around wildly, unable to form words. And in their frenzy, they saw no savior, briskly walking towards them, scrubs swishing.

The other day, with sleep alluding me, I took a walk late at night, just after three. I came to a park bordered by a stream called Dixon Creek. The music in my earbuds was paused as I watched that bubbling torrent for a while. It was constructed of turbulent order, always there and yet never there again. But, as scientists may study their whole lives and find no answers in turbulence, I moved on, finding a large grey cat whose back I scratched. It must have been past the feline’s bedtime. Finally, I came across two wheels, the first of which was a prayer wheel like those found in Nepal. It was erected by a neighbor to commemorate a dead friend. The whole neighborhood had written prayers, placed in the wheel to be released when spun. I spun it. Why anger a ghost?

The second wheel belonged to a rusty old wheelbarrow with a “FREE” sign taped on. My grandmother used to tell me, “always be kind to everyone, you never know their burdens.” As I had spun the last wheel for someone whose stream had bubbled to a halt and likely did not necessarily need any prayers at the moment, thank you very much, I decided to spin the wheel of the wheelbarrow too, just in case it held prayers for some other poor fellow struggling with his own burden. Perhaps prayers from that second wheel flew straight up and east, crossing the Rocky Mountains and Mississippi River, before resting with an unknown nurse in northern Michigan. I’ve crossed the Mississippi many times by plane and car. One time, I crossed in a dust colored suburban. My grandfather drove; my grandmother sat with my sister and me in the back. We were following the car containing my parents as we travelled from Connecticut to our new life in Oregon. I was just three, but I remember the lummi sticks provided by my grandmother. Long and hard, the sticks made an impressive sound when struck together, perfect to accompany the exciting rhythms of Leroy Anderson that blared from the speakers. We also looked at the cars that we saw from our windows, much like that morning by the highway, inventing stories for everyone that streamed by.

Asked later, my grandfather said it was “the best wedding anniversary we’ve ever had.” Given the racket that we were making behind him, I am tempted to think that he was lying, but since he is the most honest man I have ever known, I must then conclude that he is, instead, slightly mad. Kant was equally mad when he turned to monstrous buildings and ferocious storms to discuss the sublime. The silly man needed only to walk up to a stranger and ask, “who are you?” It is there that we can find infinity. The enormous complexity of the human mind and spirit is enough to leave you standing in awe. All humans have lives, as infinitely complex as your own, and feelings, opinions, fears, burdens, all infinite. But it gets worse!  They just keep living every day, increasing the infinity bit by bit. Then go to an intersection, or crowded café. You will watch as infinitely complex people rush by you, like a raging flood where our own infinity is but a trickle. In this vast torrent of humanity, my grandmother’s current is almost done bubbling. When I was a senior in high school, she suffered a stroke, leaving her with extremely limited mental capability. My mother and I flew to upper Michigan in March to aid as we could. With my uncle and grandfather, we took turns sitting with her at night, making sure that in her confusion she wouldn’t wander or injure herself further. It was one night that she would not listen to me, when she insisted on getting up, that I darted into the hallway with terror in my eyes. I could not form complete sentences as I pleaded for help, but she came anyway. She talked with my grandmother patiently, helping her back to bed. She wasn’t our normal nurse who would come regularly to check in. Instead, she was a stranger who I never saw again, and I love her. I wish that I could describe all the infinities of this stranger’s life, where she grew up, if she has children, if she likes pineapple on pizza. But like a car on the highway she burst into my life, providing aid and respite, before rushing off towards the horizon. 

I wish too that I could tell you everything about my grandmother. How she was one of the first female instructors at Michigan Technological University, how she stood up to a crooked cop who abused two youthful vagabonds, or how, as a member of the league of women voters, she followed the all-male city council to a bar in order to prove that they were making deals behind closed doors and off the public record. But I cannot, just as I cannot stop the flood of time or save her from a brain’s collapse. So, I will instead leave you with a thought as your own story infinitely unfolds. You may call it the Norma Lee Stuart Conjecture if you wish. Take note of the nameless who stream around you, you know not their burdens…

Ad Infinitum 


Ad Infinitum, Author’s Description

There’s a word that I quite like, sonder. Its not a word that can be found in Webster’s dictionary, it hasn’t quite hit the mainstream. It was created in a project called the “Dictionary of Obscure Sorrows,” which aimed to put words to feelings that had been previously undefined. The following definition is given: sonder is “the realization that each random passerby is living a life as vivid and complex as your own.” When I came upon this phrase I was immediately reminded of my grandmother and our early mornings making up stories for the people that drove by on the road below us. Since she became injured, I have thought about that memory a lot. I believe that my grandmother was the first person to instill the feeling of sonder within me. It has also occurred to me that few people will ever know her incredible story, unless I told them. Unfortunately, I knew that no short story could ever contain such a meaningful life, so I instead aimed to create a broader, more applicable narrative. 

I hoped to capture the infinity of humanity in my story. By following the form of the Fibonacci Sequence, I attempted to visually capture the unfolding process that occurs when you get to know someone. Each line could be thought of as a day, the words contained then become the lived experiences of the individual which expand into infinity. Ideally, if I have done my job right, this will instill in you a feeling of sonder. I hope too that it will encourage you to call your grandmother if you can.

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untitled, Emily Colson

My project aims to construct what my peers find to be  the most aesthetically ideal face compiled from a pre-selected group of facial features and characteristics. In order to conduct this project I had to assemble a survey for my peers, gather the results, and implement them into the project.

With my project I have had to take some concepts that would be best suited to a larger study and cater them to the size of our class. Facial features are generally difficult to confine to defining categories, so instead of asking which type of nose (for example) my classmates found most aesthetically pleasing, I provided examples. I tried to make the examples as varied as possible. I also took my examples from people who are generally predetermined to have more aesthetically pleasing features: actresses and models. This choice was also made in order to ensure an easier completion of the project on my part, as with celebrities I can find many photos of them with different angles of the facial feature so that it is easier to understand and draw it. I decided that all the features should come from the same gender for a more coherent facial structure. If I were to have included a mix of genders I would have had to determine what gender the final grouping of features would have, and I did not feel comfortable determining that based purely off appearances.

As for gathering the data, I found all the images I would use as examples of each facial feature and compiled them into an accessible multiple choice survey which was then sent out to my peers. This way, the results could be relayed to me in a statistical format, from which I could gather which facial features were most commonly agreed upon as the most aesthetically pleasing. I decided to choose the mode to determine which features I would use, as that represented the largest consensus by my peers. 

The results of my survey are as follows: 

  • The preferred nose was option B, Zazie Beetz’s, with a majority of 56.3%
  • The preferred eyes were option C, Janelle Monae’s, with a majority of 31.3%
  • The preferred mouth was option C, Naomi Scott’s, with a majority of 56.3%
  • The preferred face shape was oval, with a majority of 37.5%
  • The preferred hair texture was wavy, with a majority of 62.5%
  • The preferred hair length was medium, with a majority of 62.5%

If I were to conduct this survey again, I would primarily look for a larger sample size. I got sixteen responses on this survey, which is almost the entire class, but is still not nearly enough to eliminate bias. Another thing I would likely change would be the selection of the examples of facial features for this project. The best way to do this would be to show only the features intended for use so the participants would not let their personal beliefs about the celebrity influence their decision. It would also be better if the examples did not come from celebrities, but I chose to do this to make the execution of the project easier, as stated above. Since I did choose celebrities, my own bias is incredibly present in this project. I chose celebrities that I could think of and also like. Thus, this project would more accurately represent which of Emily Colson’s preferred celebrities’ facial features does this class prefer. Overall, though, I believe I did the best I could with the constraints that I had.

As for the execution of the project itself, I only had a few difficulties. I love to draw, and I especially love to draw people, so this project seemed the logical choice to play into what I enjoy. I originally planned to draw this project digitally, but as I went through the process of it I was unsure of the results. I tried both traditional and digital drawings and ended up sticking with the digital because the tools allowed for a more symmetrical face.

In conclusion, this project has been a great test of both my mathematical and drawing skills. It was invigorating to create a survey and compile data; it is incredibly rewarding to see results. It was also a nice change of pace to draw for my homework instead of typical homework tasks. Overall I am happy with the results of this project.

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Exploring Physics in Music, Aazaad Burn

Abstract

There are many ways of making sound on an instrument, which we take for granted, and seldom precisely understand why each different technique yields a different sound.  In my creative project, I attempt to explain the physics behind the techniques that yield different sounds on the violin through a composition. The composition showcases different methods of making unique sounds on the violin using techniques such as col legno, pizzicato, natural harmonics, sul tasto, and chords.  I also included a few measures designed for the musical saw because of the interesting properties of its sound.

Project Description

The first step in my project was to make a list of the elements that make violin music aesthetically appealing, allowing me to pick the ones on which to focus.  The elements were timing/time signature, tempo, dynamics, key, timbre, and technique/effects.  I decided to choose technique/effects as my focal point because they are used much less frequently than the other five elements, and hence are often overlooked. The techniques I integrated into my piece are col legno, pizzicato, sul tasto and harmonics.  I also decided to include a few measures to be played on the musical saw because of its particularly interesting sound. 

The intersection of mathematics with these effects isn’t obvious.  However, these effects are made using physics, which is a branch of study that is closely involved with mathematics.  Violin music is created by vibrating a string, which creates standing waves, and by manipulating the waves in different ways, different effects are created.  

To play the violin in the most traditional way, a bow is drawn across a string.  The bow creates sound in the same way that plucking a string repeatedly creates a sound.  Drawing a bow across a string creates a slip-stick cycle where the bow grips the string and then releases it.  Rosin is generally applied to the bow hair to increase its coefficient of static friction.  Bowing the string too slowly makes it so you can hear the sticks exclusively and bowing too fast makes it so you can hear the slips exclusively.  When bowing a string, the entire range of high, low, and the fundamental harmonic are maintained and only dissipate when the bow stroke ends. There must be an equation or slip-stick frequency in relation to bow speed, though I do not know if it is linear, exponential, logarithmic… and I have not succeeded in finding it.  It can be determined that “the cycle of stick and slip on the bow has the same period as the vibration of the string” (Wolfe).  In my project, we will work out an equation to describe the time of a slip-stick cycle, and one for the amplitude of the motion of the string at the bowing point based on variables such as string length, distance from the bridge, and period of the vibration of the string.  

Having thus laid the foundation for understanding the physics behind bowing the violin in the traditional way, we transition to a discussion of the effects mentioned above.  

Sul Tasto refers to the technique in which the bow is drawn over the fingerboard to create a finer, more muffled sound.  The sound here is more muffled because the area where the string is vibrating most is positioned away from the f-holes.  The string vibrates most where the bow interfaces with the string, so positioning it away from the f-holes gives the impression of a muffled sound.  

The pizzicato technique yields a short burst of sound that dissipates quickly, in contrast with the long, sustained sound that a bow drawn across the string produces.  This is because drawing a bow across the string sustains the vibration without letting it dissipate, whereas a pizzicato only vibrates the string for an instant.  When a bow is drawn across a string, a note is maintained.  However, after a pluck, the high harmonics fade away quickly, leaving only the fundamental and some weak lower harmonics. Bowing maintains the rich harmonic spectrum.

Col Legno refers to tapping the string with the back of the bow that produces an effect similar to pizzicato, where we perceive a short burst of sound that dissipates quickly.  The sound dissipates quickly because striking the wood is not a continuous input of energy.  This sound has a different quality than pizzicato; it sounds drier and more staccato.  This is because we hear the collision of the stick and the string, which is percussive.

The word “harmonics” denotes the overtones that are present in addition to the fundamental frequency.  The standing wave pattern of a fundamental frequency contains only two nodes.  As we add nodes, we get to the second harmonic, then the third, etc..  When we know the number of waves in a string, we can derive an equation relating the wavelength of the standing wave to the string length.  On the violin, we can play a “true harmonic” without its fundamental frequency by placing a finger lightly onto a string in specific places, and drawing the bow across the string as usual.  To play an “artificial harmonic”, one simply places a finger as if to play a regular note, and places an additional finger lightly on the string four whole steps away.

Our final “effect” isn’t really an effect; it is a peculiar sound made by a peculiar instrument.  A sound by a musical saw is clearer than on most instruments because a saw usually only creates one single harmonic along with the fundamental frequency, instead of a broad range of harmonics, which is the reason why the sound carries so well on the wind and sounds much more clear. A sound is made when the bow is drawn across the saw which is curved into the S shape; this can also occur when the saw is in a C shape.  The player must find the “sweet spot” of the saw in order to make a sustained pitch.  The “sweet spot” is a flat part of the metal, and it can be moved up and down based on the curvature of the saw.  When the player drags the bow across the saw, they cause the saw to vibrate, which creates the sound.  Like bowing a violin, a slip-stick motion occurs here as well.  

This paper has attempted to explore some of the physics behind violin effects.  I wonder if this exploration will change the way I perceive my own playing, and I ask if my playing will be improved with this new knowledge.  I am curious if violin students would benefit from the knowledge of this physics and if it would improve their playing.  I wonder if I could put together a booklet on this. But that is a project for another time!

Works Consulted

Cox, Trevor. “Musical Saws and Harmonics.” The Sound Blog, 31 Dec. 2009, acousticengineering.wordpress.com/2009/12/31/musical-saws-and-harmonics/.

McNamee, David. “Hey, What’s That Sound: Musical Saw.” The Guardian, Guardian News and Media, 7 June 2010, www.theguardian.com/music/2010/jun/07/musical-saw.

“Physics Tutorial: Fundamental Frequency and Harmonics.” The Physics Classroom, 2000, www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics#:~:text=The%20frequencies%20of%20the%20various,frequency%20of%20the%20first%20harmonic.

“Tonal Effects.” Strings – Orchestration Skills-Step 8, inmusica.fr/SC/Orchestration_Skills_-_Strings-Step_8.htm.

Wolfe, Joe. “Bows and Strings.” The Bowed String, The University of New South Wales, newt.phys.unsw.edu.au/jw/Bows.html.

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Umbilical Torus, Gavin Barrett, 2020

Overview of Project

With this piece, I set out to create an umbilical torus through the carving of a block of wood. While I recognized that this would be a very difficult endeavor, I found that after breaking the process down into more basic steps, it came together quite nicely. This leap into complex geometry lead me to encounter a variety of issues, including an utter failure of my differential geometry skills that resulted in having to restart entirely; however, after overcoming some of the obstacles, I was very happy with the piece I created and learned a great deal about woodworking, art, and geometry in the process.

Mathematics

In this piece, a variety of mathematics are at play, both in the final product and the creation of said product. The name of this shape is an umbilical torus, and it has become a relatively popular tangible example of calculus over the years, whether on the covers of calculus textbooks or as a statue (a rather famous one is located at Stony Brook University). The simple explanation of this piece is that it is a single edged three-dimensional shape, something which at first thought might sound impossible. Naturally, to go along with this lone edge that wraps around the body of the figure three times uniquely (and infinitely over the same track) there is only a single side to this figure as well. These traits make it a particular fascination in the field of singularity theory.

The mathematics that apply to this rather complex shape are equally complex if not more. While I can take no credit for the personal discovery of such things, the math behind the figure is as follows:

x = sin u (7 + cos(u/3 – 2v) + 2cos(u/3 + v)

y = cos u (7 + cos(u/3 – 2v) + 2cos(u/3 + v)

z = sin(u/3-2v) + 2sin(u/3-v)

for the interval:

-3.14 < u < 3.14

-3.14 < v < 3.14

These mathematics work together to form what is (in layman’s terms) a quadrilateral traveling through a three-dimensional space while orbiting a point while rotating at 90° for every orbit, meaning that each edge of the quadrilateral lines up perfectly with its adjacent side’s path following a full orbit around the center point.

As for the mathematics of creating it, there were many complications that ought to be noted. First, a proficiency in the way that the shape operates was fundamental to planning this project as it is a remarkably difficult shape to imagine when drawn two-dimensionally. However, after viewing enough angles, I felt I had a good grip on the process I would need to take (it should be noted that this shape was extremely difficult to orient without the back side). I had to take measurements of the wood block I used in order to ensure that the thickness of the “band” I was creating would be circular and devoid of serious irregularities, and then I simply had to begin carving until I had created a donut shaped piece. Following this, I had to find the circumference of the circle orbiting the center point and find the arc length of one fourth of it. Knowing that this arc length would constitute one orbit around the center point, I had figure out how much the rotation would be at smaller intervals, for example, how much rotation would occur in 1/12 of an orbit around the center point. Once I had this figured out, I connected the lines and continued this process three times around the circle until I had come back to my beginning point. After this, I grinded, whittled, carved, and sanded off the wood until the lines I had created represented the edge.

This shape has gained popularity due to its bizarre characteristics, but also its aesthetic draw, as it leads the viewer to engage in the mathematics of it as well. Upon my first viewing, I pondered whether it was even possible to create such a figure in a three-dimensional form, or whether it was simply an illusion. However, as this piece, and hundreds of other recreations of this geometric marvel have shown, it is possible, fascinating, and a remarkable example of the interaction between mathematics and aesthetics.

Last Thoughts

It is deeply unfortunate that we find ourselves in a set of circumstances wherein the three-dimensionality of this shape is lost. One of the most interesting aspects of this shape is tracing the side around the object, and simply viewing it from a variety of different angles. This being said, I hope that the curiosity of this shape is not lost in the finitude of a photograph, and I hope you can enjoy it visually and mathematically as much as I do.

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untitled, Olivia Banks, 2020

My creative project is a somewhat abstract exploration of how the idea of ​recursion​ can be explored through an artistic lens.

-Elizabeth Banks

What is recursion in a mathematical sense?

According to the Cambridge Dictionary, recursion is “the practice of describing numbers, expressions, etc. in terms of the numbers, expressions, etc. that come before them in a series.” In other words, it is when something is defined by itself. This is quite broad, and recursion has many diverse applications in the field of mathematics and computer science. For example, one of the most familiar applications of recursion is the Fibonacci sequence: ​F​(​n)​ = ​F(​ ​n​ − 1) + ​F​(​n​ − 2). The reason it is recursive is because it uses numbers within its own set to define subsequent numbers within the set.

How does recursion present itself in the artistic world?

While recursion can seem quite complex mathematically, it actually presents itself quite simply in an artistic sense. One somewhat famous example of recursion in art is called the Droste effect. It originated from this packaging design:

In the image on the package, there is an image of the same package.

The Droste effect can be seen widely throughout the artistic world in the form of images containing their own image. 

How can I apply recursion to my own art-making?

While the Droste effect is an easily recognizable and succinct application of the idea of recursion, I really wanted to push myself further into how I could represent this idea in my own way. As an artist, I am always drawn to self-portraiture, especially in oils. I thought it would be fun to do my creative project in my own artistic style, as if it could simply be part of my existing portfolio. With this in mind, I chose my subject and my medium: a self-portrait in oils.

In order to really manipulate the idea of recursion into a more abstract representation, I chose to do my portrait on a mirror. This represents recursion because, when I look at my self-portrait, me, the subject, stares back in the mirror. In this way, it is almost like an image is depicted with an image; the viewer defines the content of the piece. While this is not the most straightforward way to present recursion in art, I thought it best suited my artistic style, and it was a unique and abstract way to present a mathematical topic.

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