## Mandala Gardens and the Fibonacci Sequence, Maya Ward

### Project Description

For my project I have created a mandala garden that is both functional and aesthetically pleasing according to mathematical principles, chiefly the Fibonacci sequence. The choice to use a mandala garden was a very intentional one. The mandala has often been regarded as one of the best representations of the intersection of math, aesthetics, and art. The mandala shown here, formed by octagonal shapes and concentric circles, helps represent some of the basic mathematical principles found in nature. The mandala garden also allows for maximum space to be used in an efficient manner, as it provides pathways to access most of the garden, so that all space can be utilized. The use of the keyholes as paths within the garden is both for aesthetic and functional purposes, it allows for the entirety of the garden to be utilized as it is now accessible by path. The keyhole paths, those that lay in-between the garden beds and are ended with either triangles or circles are a common feature seen in mandala gardens because they allow the gardener to access parts of the garden not within the circular beds. (Bittman 2016).

The first produce, found in the north, or top, circular bed has sets of the “three sisters” varieties growing. The three sisters is a type of strategy for planting that includes corn, bean, and squash. The three work together to help one another in the growing process. Corn stretches tall, which creates natural forming poles for the beans. The beans help the corn because the bean plants are nitrogen fixers, which helps corn which is a shallow rooted heavy feeder. The squash plants create living mulch which helps the heavy feeding corn and also the squash plant provides shade, killing weeds as well as warning off predators with spines. The three work in harmony, and while their harmonious relationship is intriguing for the mathematical nature in which they provide shade, surface area, and moisture for one another, they are also significant for culture. The three sisters are found in many native American communities and are seen as a gift from the great spirit to sustain life on earth  (“The Three Sisters”, 2014).

The second produce bed found on the south bottom quadrant circle bed has been divided into three sections. The sections are determined by how often the produce needs to be tended to, at the front is the produce most commonly harvested and in need of the most attention. This includes daily greens such as lettuce, kale, and chard/ the second third section has plants that are harvested slightly less, but still require somewhat frequent attention, this includes vegetables like eggplant and bell peppers. The furthest third included root crops which need little attention: ie. potatoes and other roots corps. While it may seem difficult to access this third quadrant, it can also be accessed from the keyhole paths found on either side of the garden bed.

The sunflowers and lupins were chosen to line the garden bed because they are perennial nitrogen fixers. The nitrogen fixing plants work with the bacteria in the soil and ”capture the atmospheric nitrogen and convert it to bioavailable nitrates that the plants can use to grow” (Dana 2019). Sunflowers are especially fascinating because their pattern naturally represents the Fibonacci sequence. With most sunflowers, when you count the spirals on a sunflower both clockwise and counterclockwise you will pair of numbers from the Fibonacci sequence. Most commonly is 34 and 55 or 55 and 89 spirals, but some sunflowers have 89 and 144 spirals counterclockwise and clockwise (BohannonMay). The drawing of a sunflower included in my work shows how the spiral pattern of the internal section of the sunflower actually represents a graphing of the golden ration being x=[sqrt(5)-1]/2. This naturally forming pattern is one of the best examples of how integral the principles of mathematics are in the natural formation of plants.

The flower beds in the west and east circular beds have been very intentionally planted to represent the Fibonacci sequence. The different flowers planted starting the most NW circle on the left side to the most northern circle on the right start with 2 petals, then 3, 5, 8, 13, 21, 34, and 55. Furthermore the way in which I drew the garden to have circles that go in a specific pattern are meant to represent the pattern abstractly what the Fibonacci sequence looks like graphed. The flowers include, with petal numbers: begonia (2), trillium and iris (3), buttercup (5), columbine (5), larkspur (5), delphinium (8), black eyed Susan(13) Shasta daisy (21), aster (21), gaillardia (34), pyrethrum (34), and Michaelmas daisies (55) (Knot). Together, if planted in this pattern both represent the Fibonacci sequence in their physical layout and petal amounts.

The NW outermost side of the garden has another homage to the Fibonacci sequence. It is lined with the vines of the passionflower. The passionflower which does not want direct sun will do best here, and the passionflower has in order from most inner to outer 3, 5, 55, and two sets of 5 petals as the pattern, which can be seen in my drawing. This flower is special in the way that it, although out of order, represents three of the unique numbers found in Fibonacci sequence (Iannotti 2009). Furthermore, the ratio I have used to design where my walls of passionflower vines will be is a ration of 5:3:2:1, which is also found in the Fibonacci sequence.  On the SW side of the garden there are vines for peas, I have created this in the SE quadrant because the peas need large amounts of sun (Old Farmer’s Almanac). They also create symmetry in contrast to the passionflower vines opposite of them and follow the same 5:3:2:1 ratio. While initially these may be seen as just two walls of vines, their intentionality in both symmetry, ratio, and choice of plants helps further represent the Fibonnaci sequence within my garden.

In the rest of the sections outside of the beds not covered by lupins, sunflower, passionflower, or Pease I have intentionally planted spots of clovers and alfalfas. This is because clover and alfalfa are seen as living mulch. Living mulch is classified as “Any plant that is used to cover an area of soil and adds nutrients, enhances soil porosity, decreases weeds and prevents soil erosion, among other attributes” (Libretexts 2018). Alfalfa adds nitrogen and organic matter to the soil along with Phosphorus, Potassium, Calcium, Sulfur, Magnesium, Boron, Iron, Zinc. While it may not be interlaced with the vegetables and flowers, the soil can be used for planting of potter plants and spread within the beds. White clovers are widely adapted perennial nitrogen producers with that have sallow roots mass and tough stems, helping to protects form soil erosion. These nitrogen fixers are necessary for my garden because their relationship with soil bacteria converts N2 which plants cannot use to ammonia NH3 which plants are able to use to synthesize proteins (Libretexts 2018).

The pattern of the herb spiral is meant to mimic the logarithmic spiral. For consideration it is important to note that the south is the hottest, the east dries out earlier than the west and there is wetter soil at the bottom and driest at the top. You want to put your Mediterranean herbs/herbs that like to be dry in the top of the spiral and your mint and parsley as the bottom. At the top of the herb spiral is a succulent commonly known as hens and chickens (Sempervivum tectorum) which represents a beautiful naturally formed spiral.  Hens and chickens need well drained soil and full sun, perfect for the top of the spiral (Beaulieu 2005). The herbs in the garden starting from the top and to the bottom are calendula, mint, tarragon, parsley, chives, fennel, lemon balm, lavender, basil, chamomile, sage, oregano, rosemary, lemongrass with hens and chicken (“Gaia’s Garden, Hemenway 2009”; Engels 2015). All of these herbs along with the succulent are drawn in a diagram found alongside my garden design.

Elements of symmetry, naturally forming Fibonacci sequences, use of mandalas and keyhole bed, and herb spirals all contribute to this garden. These elements not only allow for maximum efficiency of the garden, but also create aesthetically pleasing designs. While this garden may look like another garden, upon further inspection, the choices of plants, the uses of ratios, and the formation of spirals and patterns are all paying homage to the foundations of modern mathematics, the golden ratio, the Fibonacci sequence, and the logarithmic spiral. Alongside this, this garden is also meant to honor the naturally forming mathematically principles that people find so beautiful being pattern, symmetry, and bright colors. Not only is this garden both functional and intentionally created for the purposes of a multi-beneficiary ecosystem, is also made beautiful by a rooted foundation in mathematics.

### Works Consulted

Beaulieu, David. “Sempervivum: The ‘Always Live’ Plant.” The Spruce, The Spruce, 12 May 2020, www.thespruce.com/how-to-care-for-hens-and-chicks-plants-2132609.

Bittman, Erin. “Making Mandalas: the Ultimate Art-Math-Meditation Combo.” WeAreTeachers, 31 July 2017, www.weareteachers.com/making-mandalas-the-ultimate-art-math-meditation-combo/.

BohannonMay, John. “Sunflowers Show Complex Fibonacci Sequences.” Science, 9 Dec. 2017, www.sciencemag.org/news/2016/05/sunflowers-show-complex-fibonacci-sequences.

Dana, et al. “The Best Nitrogen Fixing Plants List for Your Garden.” Pīwakawaka Valley Homestead, 26 Apr. 2020, piwakawakavalley.co.nz/nitrogen-fixing-plants/.

Engels, Jonathon. “The Magic and Mystery of Constructing a Herb Spiral and Why Every Suburban Lawn Should Have One.” The Permaculture Research Institute, 17 Dec. 2016, www.permaculturenews.org/2015/04/17/the-magic-and-mystery-of-constructing-an-herb-spiral-and-why-every-suburban-lawn-should-have-one/.

Grant, Bonnie. “What Is Living Mulch: How To Use Living Mulch As A Ground Cover.” Gardening Know How, Gardening Know How, www.gardeningknowhow.com/edible/grains/cover-crops/living-mulch-ground-cover.htm.

“How to Grow Passion Flower.” How to Grow Passion Flower, The Sunday Gardner, www.sundaygardener.co.uk/howtogrowpassionflower.html.

Iannotti, Marie. “How to Easily Grow Exotic Passion Flowers.” The Spruce, The Spruce, 28 Sept. 2019, www.thespruce.com/passion-flowers-1403114.

Knott, R. “Fibonacci Numbers and Nature.” The Fibonacci Numbers and Golden Section in Nature – 1, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#section4.

Libretexts. “31.3A: Nitrogen Fixation: Root and Bacteria Interactions.” Biology LibreTexts, Libretexts, 19 Nov. 2019, bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/Book%3A_General_Biology_(Boundless)/31%3A_Soil_and_Plant_Nutrition/31.3%3A__Nutritional_Adaptations_of_Plants/31.3A%3A_Nitrogen_Fixation%3A_Root_and_Bacteria_Interactions.

Nationwide, SARE. “White Clover.” SARE, www.sare.org/Learning-Center/Books/Managing-Cover-Crops-Profitably-3rd-Edition/Text-Version/Legume-Cover-Crops/White-Clover.

Old Farmer’s Almanac. “Peas.” Old Farmer’s Almanac, www.almanac.com/plant/peas.

Ratnala, Sujatha. “MATHEMATICS OF A FLOWER.” Medium, Medium, 4 Feb. 2018, medium.com/@sujatha.ratnala/mathematics-of-a-flower-e507ddaf0675.

Rodney. “Living Mulch: Benefits & Uses.” Traditional Gardening, 27 Jan. 2020, traditionalgardening.com/living-mulch/.

Schmidt, et al. “Alfalfa Living Mulch Advances Biological Control of Soybean Aphid.” OUP Academic, Oxford University Press, 1 Apr. 2007, academic.oup.com/ee/article/36/2/416/402457.

“The Three Sisters.” Grow Pittsburgh, 26 Nov. 2014, www.growpittsburgh.org/the-three-sisters/.

Toby, Hemenway. “Gaia’s Garden- A Guide to Home Scale Permaculture.” Permabox, Chelsea Green Publishing Company, 2009, permabox.ressources-permaculture.fr/1-PERMACULTURE/LIVRES/BOOKS_Permaculture-(english)/BOOK_Gaia-s-Garden—A-Guide-to-Home-Scale-Permaculture_by-Toby-Hemenway.pdf.

Waddington, Elizabeth. “How To Make An Herb Spiral.” Seed To Pantry School, 8 May 2020, seedtopantryschool.com/how-to-make-an-herb-spiral/.

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## untitled, Lauren Van Horn

### Project Description

My creative project is focused on documenting how the sun travels across the sky and how lighting changes throughout the day. In order to do so, I took photos of the same seven locations at different times during the day. I took photos at dawn, in the afternoon, and at dusk. Furthermore, I wanted to ensure that each of the seven locations I chose to photograph offered a unique perspective. With this goal in mind, I chose a wide variety of locations to explore how sunlight is affected by different factors. There are four indoor locations and three outside ones. Three of the indoors locations have north-west facing windows. The last indoor location has a south-west facing window. The outside photos are taken facing south, south-east and east. To continue, taking the photos facing different directions allows a deeper insight to how the sun’s path interacts with earth.

Through the photos, I better understand what parts of my house receive sunlight and when. Some locations where I had thought there would be a large change in appearance only had a slight adjustment. There are certain factors that have influenced this outcome. For one, there were obstructions to the sunlight that I had not originally taken into account. Because of these interferences in sunlight, some parts of the house were never fully illuminated. As a result, some photos rely heavily on the hue of light in the photo to indicate the sun’s position. In regards to the sun’s light temperature, cloudy, overcast skies presents the bluest hues. On the other side of the spectrum, the warmest lighting occurs during sunrise and sunset. Right in the middle, neutral lighting is due to daylight at noon. In the photos, the viewer can notice the slight variations in lighting temperature as the day progresses, as well as between the sunny and cloudy day.

In a sense, the photos are acting as a modern, not very accurate, sundial. Sundials work by measuring the sun’s movement across the sky based off of shadows. By a similar manner, the way the light interacts with these locations can become an indicator of what time of day it is. This idea is most noticeable for the photos taken outside. Opposed to tracking the time of day due to the shadow cast by the sun, the viewer can track the time of day through how the patch of sun moves across the yard. Again, this is due to the obstructions of sunlight that were present in my backyard.

The path that the sun takes as it crosses the sky can be influenced by multiple factors. Throughout the year, the sun’s path varies slightly. The Earth has an axial tilt of around 23.5​° which affects the sun’s placement in the sky. ​Therefore, if I were to repeat this project during a different month but still photographing during the same hours, the locations would appear dramatically different. Another intriguing project would be to photograph the same location at the same time throughout the year to track the sun’s movement at varying times of the year. Furthermore, weather can influence the way light travels as well. Clouds disperse light, creating even lighting throughout the day. Because of these factors, it was important to photograph the locations both on a sunny day and on a cloudy day to see how the different lighting affects the image. Notably, there is less variation in both shadows and coloration in photos taken on the cloudy day than compared to photos taken on the sunny day.

During this process, there were a couple things that I noticed. First, capturing the subtle differences in lighting with a camera can be difficult. There are multiple factors that could have influenced this difficulty. The lack of variety in lighting on the cloudy day plays a significant role. Throughout the day, there was little need to change the camera settings. In comparison, the camera settings had to be altered almost every time a new photo was taken on the sunny day. When more light is introduced, the camera settings need to be more specific. The uniformity of the diffused lighting, on the other hand, did not need drastic camera alterations.

Overall, this project helps explain how lighting interacts with the appearance of a location. There are times of day and weather conditions that make photography easier and more “beautiful.” For consistent lighting and no harsh shadows, a cloudy day is perfect. Warm, sun-soaked images require a sunny day, preferably around golden hour. Lighting can affect the mood of photography in dramatic ways, as seen through this project.

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## Artist Statement: The Mathematics of Irish Dance

In first grade, I was jumping around waiting for my teacher to check my work, and she told me I should look into Irish dance. The K-8 school had several dancers that competed internationally—like with actual Irish citizens—and some even won. While I never got past advanced beginner (because I didn’t like to go to competitions), Irish dance was so important to me in a lasting way.

Along with the dance lessons, we also learned about the Celtic culture we were practicing from Breda Yeates, an Irish immigrant. I’m a McElligott on my mom’s side which is very obviously an Irish name but I didn’t feel any connection to the culture in my own right; I never considered until now that my great-great-great-so on relatives might have practiced this dance that Yeates taught us. The dances developed when the Irish were enslaved and not allowed to celebrate their culture. The stiffness of the dancers made their movements unrecognizable as dance (or so I was told). The dancers’ hallmark look includes tight curls, a stiff paneled dress with elaborate embroidered decorations, and poodle socks. The Celtic knots are thought to have been popularized by the spread of Christianity but the designs that predate Christianity are hard to trace if they don’t depict an animal or recognizable symbol.

Celtic knotwork is often one continuous looping ribbon with distinct repeating patters. The circular and square knots are usually centered on a quadrant system and are exactly the same in each quadrant rotated. The unorthodox shaped ap knots are reflections over an axis. On the dresses, the skirt is paneled with the most elaborate design on the front of the dress. For the sake of design, most dresses use knots with 3 or more colorful ribbons. The adornments add a uniqueness to the stage. No two girls (from different schools) wear the same dress in the competitive stage.

For my creative project, I wanted to design a pattern that could be found on a typical dress. The way that Celtic knots curl on themselves and cross over and under in a specific pattern. By far, the hardest part of the design process was making the crossings evenly numbered; often, I would find myself at an impasse to determine what to do next because if I went one way I would put two “overs” or “unders” in a row which throws off the whole design. The design aspect is so intrinsically tied into mathematics, that there are computer programs specifically for designing Celtic knot patterns. Unfortunately, many links to the academic work with these programs are broken or inaccessible to the public but one free program, Knots3D is available on Windows. Unfortunately, I have a Mac and couldn’t run the program but from test pictures supplied on the download site, I was intrigued by the intricacy of the knots and their weaving. The nature of computer generation, of turning a 2D image into a 3D image, and furthermore, of a repeating loop image requires knowledge of topological spaces and geometry.

In my initial development of the knot, I focused too much on the drawing of the physical ribbon instead of controlling or containing the negative space. I was unsatisfied with the design but then, read up on the geometric patterns in Celtic knot work. Each knot has a “skeleton” with vertices in the negative spaces. Then, the midpoint of the side lengths is where the ribbon crosses over or under. The instructor describes the rule for crossing over or under akin to the right hand rule: if you were to physically pick up the ribbon to cross the lines, you would always cross left over right as you follow the path from am arbitrary point. The instructor pointed out that, ignorer to make a large knot, you first have to find a small knot pattern you like and tile it out through the entire area.

Regarding some basic knot theory from introductory documents from class, the definition of a knot is a (closed) tangled string. While the art form exists in a 2D place, the assumption for knots is that they are three dimensional. Without a loss of generality, we can view the 2D image as if it were 3D in order to apply the knot theory terms. Already, I have mentioned that Celtic knots are alternating knots. We can see that the classic Celtic trefoil is a knot and is the simplest nontrivial knot. At first glance, I would have assumed most Celtic knots are trivial but after revisiting some of the knot theory documents, I think that the knot in the outer ring of my design is nontrivial. There are sixteen figure-eight knots (thank you girl scouts) which would not pull apart. As for the inner design, I have no intuition about whether or not it would pull apart by looking at it alone. I presume that the majority of celtic knots are nontrivial because they develop so much upon themselves. The trefoil knot is the simplest knot topologically as mentioned previously, and the Celtic motif of the trefoil appears within much of the art that decorates dresses.

This project was a lot of fun; the nostalgia for me was so inspiring. I hope that in a few years, I can restart and join an academy. The class was asked what we would have changed about the project if we had chosen our topics at the end of the class instead of the beginning, and I will say that while I was inspired by the images in the knot theory papers we read for my topic, I would have decided to focus on the music and dance aspect of Irish culture. The difference between Reels, Slip Jigs, Light Jigs, Treble Jigss, and the Blackbird are so subtle but so profoundly tied into math. I think back to the only dance I remember in full:

O’er-2-3, o’er-2-3, o’er-2-3, o’er-2-3, hop-2-3-4-5-6-7, hop-step-step, o’er-2-3. Repeat. Hop- step-step, hop-step-step, hop-2-3-4-5-6-7, o’er-2-3-4-5-6-7, hop-step-step-and-a-point-hop-back. Repeat.

There absolutely has to be some fun mathematical connection in there!

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## Objects Warped by the Shape of Glass, Liza Richards

### Project Description

For my creative project, I chose to demonstrate through photography how transparent glasses filled with water warp the appearance of the object behind them when looking through the glass. Differently shaped glass warps objects in a variety of ways. They may cause the appearance of the object to be smaller, larger, stretched, twisted, magnified, etc. The photograph I produced is designed to show exactly this and highlights their differences. While taking this photo I chose to use as plain of a background as I could find to emphasize the true focus of the photograph, the glasses and the distortions of the objects behind them. In order to make the ground more plain, I placed down pieces of blank white paper to avoid the patterns on the ground to take away from the flowers on display. I placed 7 flowers of different shapes, sizes, and colors behind the 4 glasses to use as the objects to be warped by the glass. My initial set up plan was to place each flower directly adjacent to one another, but I quickly figured out that with the ways the glasses warped the shapes, some of the flowers were not reflected through the glasses. To fix this, I had to move some flowers farther away from the others so that they would actually be seen through the glass which you can see with the photos I provided of the set up and the flowers without the glasses in front of them. I also struggled with the angles at which to take the photograph. I provided two somewhat different views of the glasses to allow the viewer to see how the angle at which you view the glasses changes the shape of the objects as well. You may notice that the the outermost right glass is the one that changed the appearance of the flowers the most with the different angle.

When comparing the photograph of the original flowers to the one of them behind the glasses, it can be observed that the flowers in the glass on the far left are pushed to the left side of the glass more and appear to have shrunken a little. This is due to the convex shape and curve of the glass. In the tallest glass, it is seen that there are actually two images of the same flower reflected in it. The one on the top is the same as the bottom except it has been flipped upside down. This is due to the concave curve of the glass near the top. In addition, the lower concave curve leads to the lower flower being shrunk. The glass in the middle right has multiple edges and corners, leading to multiple different refractions of the same flower through each section. The glass on the farthest right has an octagonal pattern engraved the glass while being concave and therefore magnifies and bends the flower into a combination of octagonal shapes.

We use glass in a multitude of ways similar to how they are used in this photograph. It is used in magnifying glasses, binoculars, and microscopes to allow the viewer to have a close up view of what they are observing. The way the curves of the glasses change the shapes of the flowers can also be seen in fun house mirrors, where the shape of the mirror warps the image reflected off of it. For these reasons, this photograph has mathematical connections to geometry, optics, and refraction.

Geometry is concerned with the properties of surfaces, solids, lines, etc. Geometry is an important part of mapping like navigation and astronomy. It can also be used in animations and designing. The geometry of the glasses changes how the object is viewed through the glass. The convex or concave curves will effect if an image is magnified or shrunk. The shapes in the glass lead to the object to be distorted to fit those same shapes as well. In addition, the appearance of the object through the glass depends on the original geometry of the object as well. Each of the different geometries present in these glasses changes the appearance of the object in their own unique way.

Optics is the study of sight and behavior of light and deflection. The geometry of the glasses deflects the original object, changing the appearance of the it in multiple different ways. The sole change in the appearance itself of the object through warped glass involves the study of optics. The study of optics is most known in our society through the creation of contact lenses and glasses to allow better eyesight. When we view the object through the water filled glasses, we are studying optics because we are observing how the image of the object changes as it is deflected through the glass.

Refraction is the phenomenon of light and objects being deflected in passing through an interface. It is mostly seen in rainbows, camera lenses, and binoculars, all involving how light is reflected and seen. When taking the photograph, I tried to avoid as much outside light as possible in order to allow only natural light to effect the appearance of the object. The deflection of the object can be considered a part of refraction. The natural light shining through the glasses allows the object to be warped in the way that it is. As the light passes through the glass, the portrayal of the object behind it changes.

In conclusion, as light shines through the water filled glasses, the shape of the glass effects the appearance of the flowers viewed through the transparent glass. The convex and concave shapes in the glasses shrink or magnify the flowers behind them. The edges and shapes in the glass lead to different distortions of the flowers through the glass. The changes in the appearance of the flowers are a result of the geometry of the flowers and the glasses, the optics and how the appearance of the flowers changes through the glass, and the refraction of light through the glass changing the shape of the flowers in the first place. A combination of these different types of mathematics is what explains the altered appearance of the flowers resting behind the glasses.

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## The Path of a Frisbee, Carmen Resnick

The project I will conduct will observe the path taken by a frisbee in flight from a “backhand huck.” There are two facets to my project, the aesthetics of the path taken and the math behind the aerodynamics of a frisbee.

A backhand huck is used in ultimate frisbee at many of the most vital times in a game. An official game of ultimate frisbee begins with a “pull” where the frisbee is sent from one end of the field to the other by the defending team. Often, a backhand throw is used for this because it is the easiest to put enough power behind using the rotation of the torso and hips. To throw a backhand, the thrower steps with their dominant foot across their body and the disc starts at the non-dominant side. This movement will be demonstrated in the final project through a visual presentation.

In a study done by Kathleen Baumback at the University of South Florida in 2010, the path of a frisbee was recorded in relationship to its initial angle. A series of calculations and formulas was used to understand the aerodynamics of a frisbee. Frisbees have multiple elements of their shape, as well as components such as spin, that enable them to maintain “lift” after the disc has left the hand of the thrower. This experiment was done to reproduce a similar study in 2005 which sought to “predict the path of a frisbee” (Baumback 3). This study recognizes the mathematic derivations to determine how the angle of initial release impacts the forces of lift and drag on the frisbee’s flight. All calculations and determinations were done through Java programming.

The mathematics in my project are tailored more towards the aesthetics of a frisbee’s flight rather than the practicality of predicting it. I am interested also in taking the derivative of the determined graph (if possible) to understand the change in the frisbee’s angle over time. Two graphs will be created, one is similar to Baumback’s study of distance where the height reached by the frisbee will be observed over time and the other will study the angle of the disc with respect to the ground.

Though not mathematically profound, this project serves to represent the beauty of seeing mathematical concepts in practice. Much more research could be done to comprehend how and why the physical shape and size of a frisbee impacts its flight. Additionally, the visualization of how a disc moves through the air will provide some clarity for me as a thrower to understand the disc’s momentum.

There are many variables that influence the graphing of a frisbee in flight, both in the movement of the frisbee as well as the camera set-up and post-filming production content. I will video a frisbee in flight, ensuring that the camera is positioned behind the thrower. The angle behind the thrower will show the angle of the disc with relation to the ground, and the camera will be set up in a way that ensures that the bottom of the screen is parallel to the ground. From this angle, I will record both the degrees of inclination from the horizontal at various points in time, as well as the height of the disc from the ground.

The original plan to observe the distance traveled by the disc was thrwarted when we took to the fields and realized that the disc’s distance would be impossible to capture in one frame of a video, and similarly difficult to measure on a screen. The height measurement is still accurate with regards to the screen because the camera was set up perfectly perpendicular to the ground.

### Works Cited

Baumback, K. (2013). The Aerodynamics of Frisbee Flight. Undergraduate Journal of Mathematical Modeling: One + Two, 3(1). doi:10.5038/2326-3652.3.1.31

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## untitled, Laurel O’Brien

In geometry, regularity is often given great value, since it makes things simple and easily comprehensible. The symmetry and regularity of polygons tends to translate there into an aesthetic value. For this creative project, the main concept was to take an interesting 3-D shape drawn in 2-D, and change it by applying to it the idea of limits and to make it irregular, but still beautiful. The inspiration for this project was simply the curiosity and desire to look at and tinker with an interesting shape and take aspects of both the geometric and the calculus and mash them together into their own aesthetic fusion. Essentially, I wanted to do project synthesizing mathematical representations with the medium of embroidery, using an increasing number of stitches per sector of the structure going counter-clockwise. The desired effect was that it would slowly become more circular, the number of stitches increasing and more and more compressed in the manner of the values of a curve approaching a limit or bound.

The shape in question that I tinkered with is best described as a sort of 3-D hexagonal mobius strip, a structure that folds in on itself endlessly. In the drawing I found, it looked almost Escher-esque (we then covered Escher much later in the term), and studying it, I found that the 2-D representation, being quite hexagonal in nature, could be split up into six primary sectors. It has six arms spiraling out from the center, six inner corners, and 12 sides made up of 18 congruent lengths. To construct the sectors, then, two sides, or 3 of the congruent lengths, would comprise each sector. Taking that into account, the congruent lengths within each successive sector would each have an additional stitch than those in the last, meaning that the stitch count for the outer border would increase per sector by 3 ( (#of sector)*3= sector outer border stitch count). Using this, I overlaid a circle onto the outer border and then plotted points on the circle, split the sector of the circle for each congruent length into its allotted amount of stitches. Then, for the spiraling sections, I took the spiral that operated largely within a sector and paired them, overlaying a circle onto each one, and giving each congruent length the same number of stitches as the outer and inner borders of its sector. These would have the result of creating an increasingly circular shape going around the structure counterclockwise.

Unfortunately, the complexities of the pattern turned out less visible as I had hoped. First, I think the shape, already being hexagonal, did not leave much room for a visual comparison between angularity and circularity. Had I done this with a similar structure that was pentagonal, the results might have been more satisfyingly defined. Since the idea is to make the irregular beautiful, I had hoped to make the irregularity of its shape clear, because if it had a clear pattern, it could simultaneously be irregular and retain a sort of regularity in that pattern.

The project as a whole was a bit of a battle for visibility. I switched from white cotton fabric to painted black canvas to ensure that stitches would stand up and pop out from the embroidered medium, and had used 12-count (an homage of sorts to the recurring theme of base 6) white thread to try and bolden the outlines and raise the ends of the stitches away from each other. I wanted to make each individual stitch clearly defined and countable. I used four shades of green thread to shade the spiraling sections, trading thick 12-stranded thread for 3-stranded thread to try and keep the shaded areas (which ended up meaning a bit more than four times the stitches overall for smooth coverage). Also, I settled on green for the color of the shading after our unit on color theory, wanting to keep the color itself one of the base three, and when I had digitally prototyped the different colors, it looked much more natural than blue or red (In retrospect, this was probably subconsciously due to the existence of the wreath). Using red thread, I also made six French knot points on both the inner and outer borders of the structure to visually help separate and contain the individual sectors, twelve points in total. Despite these attempts for clarity and visibility, things ended up less defined than I would like. Working with painted canvas embroidery for the first time, I did not expect that each time a hole was poked in the canvas, the rupturing canvas would bring back up the white underside, making things messy. So, after all of this, what I seem to have created, visually, is a mathematical wreath with mathematically-placed berries. Perhaps, if we were ever able to display our works in a room on campus somewhere, we could hang it on the door.

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## Patterns and Symmetries in Music, Chloe Miller

For my project I did a video presentation about the patterns and symmetries within music, specifically focusing on musical fractals. I decided this format was best so I could share both my own musical fractal as well as already existing musical fractals that exist in other composers’ music. While my video itself contains much of the mathematical overlap and main ideas I learned I will take the space here to write down both things said in the video, as well as things that I learned in the process of creating my project.

To start I’ll explain what a fractal is. A fractal consists of patterns that build off of one another to create something more complex and beautiful. Below this is a picture of a pythagorean tree fractal. Here we see it starts with a square, then there are two more squares added, then to each of those two more, and this pattern continues on and on.

A musical fractal, as I explain in the video, is some sort of property about the music whether that be a rhythm, tension, pitch, or something else that is a part of a bigger picture and is building off of other dynamics within the music to create this musical fractal. It can also be described as a symmetry or pattern found within music. One composer who was a master at using musical fractals was ​Johann Sebastian Bach. Many of these musical fractals occur within his canons, which makes sense since canons themselves have many fractal-like properties of starting with a melody then creating different variations of that melody at different speeds, styles, instruments, or even different keys.

In my project I wanted to be able to show these fractals in a way that helped the listener be able to visually see the patterns as well as hear them. To do this I found a program called Music Animation Machine (MAM) created by Stephen Malinowski. This app took audio files and created visualizations of the music from abstract images to lines and dots scrolling across the page (this was the visualization I used in my video presentation). Each series of lines and dots represents a different instrument part and in my presentation I showed the example of Bach’s Brandenburg Concerto #4, movement 3. Each note is represented by a dot and the lines are there for fluidity, duration, and aesthetic purposes. As the video plays there are numerous patterns and symmetries we can see. We notice similar melodies and themes occur and bounce around from instrument to instrument.

Finally at the end of my video presentation I showed my own original composition of a musical fractal. I tried to incorporate the same melody in different octaves as well as note lengths. Instead of simply summarizing the video I wanted to explain what I learned during this process and really a key take away I got from this project. After I had spent countless days and hours trying to make my musical fractal composition at least not sound awful, I began to realize that to create symmetries and fractals in music is something that can not be done by anyone, but is a skill that takes expertise in many disciplines to be able to create something that actually sounds beautiful and aesthetically pleasing. Bach was successful in his compositions of musical fractals because he understood both mathematics as well as music and how to incorporate them both to create something amazing. Bach not only relied on the calculations and math to create his music (like I mainly focused on), but he also understood that audiences don’t simply want to hear something completely predictable. Music is a creative process because of its originality; the way Bach uses not just mathematics to create his music, but also the element of surprise and other musical qualities makes Bach’s music authentic and beautiful.

### Bibliography

DancingPhilosopher. ​Animation of an Imperfectly Self-Resembling Pythagoras Tree​. 16 Oct.

2019,en.wikipedia.org/wiki/File:Animated_self-resembling_Pythagoras_tree_(fractal).we bm. Accessed 3 June 2020.

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## Lucas, Meagan McCoy

Puzzle piece treestakingin sunlight through their leavesseem at home among bumblebees.Near bulbs with roots tangled in dirt sipping water from soil, blooms speak in codes of bright hue; cyphers of light refracted 'cross retinas transcribe the wavesfrom nature and numbersintovibrant color

For my math project, I decided to write a poem using the Lucas number sequence to determine the number of feet per line, where a foot is a pair of syllables. Once I reached seven feet I decided to repeat the sequence backward, reversing the form to create a horizontally symmetrical shape with two distinct halves.

Usually, poems will use feet that follow a particular rhythm of stressed and unstressed syllables. I did not pay particular attention to the rhythm of my poem, but it might have some kind depending on how people choose to read it. I attempted to use an iambic meter in the beginning, but the constraints brought by the form of the poem made it difficult to implement any other attributes.

For the content of the poem, I tried to draw on what we learned in class, and what my attempts at gardening have taught me. The first half focuses on flora, while the second half is on the concept of light. I’m not sure who scientifically accurate my description of light is, but it was inspired by our reading on rainbows. I thought it was interesting how light, as we know it, is a phenomenon that really only happens in our brains. Without someone to observe, the waves of light are not really color. This is interpreted in my poem as a form of communication, and translation. Without someone to observe, a flower is not really red or purple, and it is not pretty/aesthetically appealing either. Color then can be understood in two ways, mathematical or scientific analysis, shown by waves, or through experience through the senses, what we know as color. The path of the light waves from the flower to the eye in the second half of the poem means to synthesize these aspects. By understanding the miraculous nature of light and its interesting mathematical properties we can better appreciate its beauty.

The Lucas numbers sequence begins with two and one, and are added to produce the next number in the sequence, where the new number is always the summation of the last two numbers. For my project, I only went up to seven, but I explored the sequence before starting the poem. Up to seven, where this property seems to break down, you can take the values of each number in the sequence, make each individual number an area, and make a rectangle out of the Tetris-like pieces. I am not sure why it stops working at seven. It seems that it was just a coincidence, but I am sure that it works at least once more further down the sequence. The Lucas series is similar to the Fibonacci sequence in that it makes a spiral, but the Fibonacci sequence is more famous. The spirals produced look nearly the same, so I wonder if the Lucas series is found in nature as often. I am not sure I can spot much difference between the spirals visually, so there might be other factors that led to the reputation of the Fibonacci sequence. The pattern of summation between the two series is the same, but they just start with different numbers: one and one, and one and two, for Fibonacci and Lucas respectively. I have not looked into the timeline between when these sequences were discovered or rather invented, and if the Lucas numbers were more modern comparatively, which might explain the difference in their reputation.

Overall, I think I was successful in producing an art piece that really engages with the material. I used mathematical content and form, which we looked at in our reading, and I thought about aesthetics in a meaningful way. I have learned quite a bit from this project, and I feel that the form of the poem both enhances its content, and forced me to write in more creative ways, really appreciating every syllable I used.

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## Warped Perspective, Amelia Hamerlynck

### Warped Perspective:

#### a Mathematical and Aesthetic Journey with the Fisheye Lens

In 1906, optical physicist Robert W. Wood wanted to invent a lens that would show the world from the perspective of a fish, “whose view from underwater compresses the entire horizon” (Vox). His paper discussed an experiment involving “a camera in a water-filled pail starting with a photographic plate at the bottom, a short focus lens with a pinhole diaphragm located approximately halfway up the pail, and a sheet of glass at the rim to suppress ripples in the water” (Vox). The resulting fisheye lens might have remained a novel item of early photography consigned to oblivion, but instead it took on a life of its own as fisheye lenses became commercially available in the 1960s. The fisheye’s multipurpose style has proven successful and popular from the lens’s first prototypes to modern smartphone apps.

While viewing the world from the eyes of a fish may seem a peculiar goal for a physicist, Wood’s work was both a natural outgrowth of early experimentation with a relatively recent invention and highly important in furthering our understanding of light. His other inventions, from a “disk whose microscopic experiments helped determine the age of stars” to infrared photography often used in nature documentation and thermodynamics, have occupied realms of physics without influencing the art world as strongly (Vox). The fisheye lens, then, is that unique instance of math meeting pop culture. I attempted to showcase this phenomenon in my project by explaining how each image represents a unique ability of the fisheye lens to interpret the world around us, interspersed with math, physics, and scientific history lessons on how and why the fisheye lens works.

The mathematical qualities that create the fisheye mainly involve manipulation of angles, distortion, and perspective. Rectilineal lenses represent straight features as straight lines. Distortions, therefore are deviations from rectilineal projection (Wikipedia). Fisheye lenses are extremely wide, massively distorting their subject matter and creating an illusion of convexity or concavity. The result is either hemispherical, meaning all the points in the image appear to be of

equal distance from the center; or panoramic, conveying a massive scale using wide angles (Wikipedia). In other words, widest highest quality fisheye lenses will capture absolutely everything from objects directly to their left to directly to the right of the lens; some are so wide one can take a photograph of a ceiling and capture the floor (Brownlee). “While wide-angle rectilinear lenses can capture angles of view approaching 100 degrees, fisheye lenses can stretch that to 180 degrees — impossible to do without the light bending science they employ” (Cunningham). In principle, this occurrence is simple refraction, the bending of light as it changes medium, just as one sees upon placing a pencil in a glass of water, hence the connection to a fish’s worldview and the usage of water in Wood’s experiment (Hashem). “The tradeoff is distinct: Straight lines anywhere but dead center in the fisheye image appear to curve. The farther they are from center, the greater the curved distortion” (Cunningham). So, whether one reenacts Wood’s experiment with light and water; utilizes an actual fisheye lens, which functions because of its physical shape; or, like me, takes photos using an app created by computer code; the origins of the fisheye’s instantly recognizable aesthetic is deeply rooted in mathematics.

Many of its applications are scientific. In nature, landscape, astronomical, and meteorological photography, the fisheye lens is uniquely qualified to capture both tiny spaces and huge spaces. Astronomers have therefore used it to photograph both the surface of mars and the movement of stars, as well as the confined quarters of a space vessel (Vox). I unfortunately did not have access to this subject matter, so instead I juxtaposed my use of the fisheye to capture me in my bathtub to my feet dangling over a cliff at a lookout point. My photographs of trees are meant to mimic the fisheye’s portrayal of the sky as a dome rather than a vast expanse.

Of course, many common usages of the fisheye are probably completely unaware of the lens’s mathematical origins, but they nonetheless make unique use of mathematical phenomena. Countless album covers from jazz to hip hop have utilized the lens to make artists appear larger-than-life. It helped that the release of the first consumer-grade fisheye lens in 1962 roughly coincided with the advent of rock ‘n’ roll and wider youth culture (Vox). The fisheye’s ability to distort made it instantly popular among the psychedelic rock stars of the 1960s including the Birds and Jimi Hendrix, who worshipped everything surrealistic and trippy. 1990s skaters filmed countless videos with the lens to properly capture the curves and swerves of a skatepark, one invention of optical physics capturing another invention of motion physics. I attempted to pay tribute to both art movements with my fake album cover photographs. And, of course, harkening back to Wood’s original intent, fisheye lens photos of my dog’s face represent the last thing a fish sees before being eaten.

### Sources

Brownlee, Marques. “What is a Fisheye Lens?” YouTube. November 30, 2011.

Cunningham, Matt. “What is fisheye lens photography?” HowStuffWorks.

https://electronics.howstuffworks.com/cameras-photography/tips/fisheye-lens-photography.htm

“Fisheye lens.” Wikipedia, the Free Encyclopedia. https://en.wikipedia.org/wiki/Fisheye_lens

Hashem, Amin. “What Is Fisheye Lens – The Full Guide.” Ehab Photography. May 11, 2018.

“How the fisheye lens took over music.” Vox. YouTube. Film. December 17, 2019.

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