The Mathematics Behind Shadow Art
By: Sean Jiang, Bryan Chan, Joonsuh Huh
Figure 1: The image shows the final product of shadow art of Sean
Figure 2: The image shows the final product of shadow art of Joonsuh
Introduction:
Shadows are a fascinating phenomena that reveal a lot about the world around us. They are the dark areas that occur when an object blocks light from a source, creating a two-dimensional silhouette on a surface. The shape and size of a shadow depend on the position and intensity of the light source, as well as the size, shape, and position of the object casting the shadow.
In the past Shadows have often been used to tell time. However, shadows are also greatly intertwined with math, as the shadow of three dimensional objects can hint at the contours and dimensions of the object.Overall, shadows have been used to allow us to better understand the fundamental nature of our visual world.
This year in the Concordia International School of Shanghai’s AP Precalculus class, our instructor DT gave us a challenge to use Art to show Transformations. The immediate idea of most students was to use reflective cylinders or plates to show polar to rectangular coordinates or vice versa. However, our group decided to take on the challenge of not only creating a wooden creation of our own faces so that we could shine a light through to show shadow art, but also make it where from two different perspectives two different faces could be seen.
Experiment:
Our team first took inspiration from famous online influencers that created projects to an exceptional degree, where when the light is shone on the object the details are still intricate. Figure 3, shows the inspiration behind the project. Where a man took scrap cardboard and turned it into art. We realized that if we wanted to make the project look good, we needed to use materials that were sturdy, and to also cut out the pieces accurately so that the shadow looks detailed.
Figure 3: Shadow art of Star Wars by Red Hong Yi.
Red Hong Yi is an artist who skillfully utilizes various objects and shapes to create art. She employs wires to align different shapes in a specific manner, allowing the resulting image to be reflected using light sources.
The first step we took was brainstorming and designing what we wanted to display in our artwork. The first idea was to create our instructor DT, but then it shifted towards creating portraits of two of our group mates. We then traced drawings of ourselves to serve as a basis and a template.
Experimental Drawings:
Figure 4: Digitally Drawn Image of Joonsuh
`Figure 5: Digitally Drawn Image of Sean
Our team started by creating prototypes from different materials. We originally started with cardboard, however we realized that we didn’t have the talent compared to Red Hong Yi. As a result, we took a mechanical approach to things.
Figure 6: The image shows an attempt to use cardboard to produce a shadow. However this caused issues as we weren’t able to precisely re-create figures 2&3.
Although cardboard was sturdy enough to stand up, the cardboard was hard to carve precisely, as seen in figure 6, especially when it came to carving out the specific features. As a result, this distorted the shadow that was supposed to re-create figure 3 as a shadow. Additionally, the cardboard was also not sturdy enough to hold other cardboard on each other, which didn’t allow us to make the 2 perspective shadow art.
This is why we moved onto our next approach 3D printing. 3D printing was overall successful in showing our faces, but the machine wasn’t able to connect the face effectively as our template was too thin. We also realized that with 3D printing it’s very easy to mess up cutting up the parts, as seen in figure 7, which will result in another few hour wait between each print. In addition to having fragile material, there were too many errors when it came to 3D printing, which meant that this process would take longer than expected.
Figure 7: 3D printed prototype that did not connect properly. You can also see the 3D model was too thin, which later could’ve resulted in the snapping of the image.
The last prototype which is also the one we used for our final product is using a wooden material paired with laser cutting. This prototype worked out especially well, since the laser cutting was more precise compared to both the cardboard and the 3d printing. The wooden material also allowed us to make a lot of changes so that the two faces could be seen in two different perspectives. We also used multi layer wood to show the complexity of the design.
Mathematical Analysis:
Geometric Transformations: Many shadow art techniques involve the use of geometric transformations such as scaling, rotation, and translation to manipulate the shape and size of the object casting the shadow. The changing shape of the shadow as the object or light source is moved demonstrates these transformations in action.
Perspective: Shadow art relies on the principles of perspective to create the desired illusion. As the position of the light source or the object changes, the shadow’s size, shape, and orientation transform accordingly, just as we observe with perspective in other visual art
Distance vs Shadow Relationship
If you were to put this into an experiment, assuming the distance of the image and the wall stays context, and only the distance between the image and the light source increases. We will notice that as distance increases, the image reflected on the wall decreases as well and vice versa.
Procedure (How to make your own)
To create Figures 1 and 2
- Sketch out desired image to be reflected using the concept of shadows, refer off of figure 4 and 5
- Transfer image to (app)
- With supervision and with a wooden board, begin using the laser cutter to cut out desired image
- Pop out the desired image, making sure you separate the outlined portions and the left overs.
5. Using the wooden outlined images and wire, from a distance, between placing the facial features, using the wires to elevate them as necessary
6. Do steps 1–4 (“For 1 Image Shadow Art”) for both images. However when placing the face, make sure they are perpendicular with each other. And not overlapping. Then place the facial features for both images making sure they don’t overlap.
Setting up the Experiment:
You are able to show this relationship through an experiment to prove that the length of the shadow increases as the distance between the image and light source decreases.
Set up ur variables:
- d1=distance between light source and image
- d2=distance between image and wall (constant)
- h1=height of the object (constant)
- h2=height of the shadow
- Starting from the end of the board, find the distance between the light source and the image. And the height of the shadow produced.
- Begin to slowly move the light source back by intervals of your choosing and record the new height of the shadow, and do so until you feel like you have enough data (at least 20 data points).
- Plot this on a graph, and you will notice as the distance between the light source and image increases, the height of the shadow decreases.
In our shadow art, the measured distance between the light source and the face was approximately 39.9 cm, the distance of the height of the face was 18.25, and the distance between the wall and the face was 34.23 cm. Using these values we could construct a model showing the height of the shadow. Using 18.25 and 39.9 we could set the initial slope as 18.25 / 39.9, and to show the change in distance between the face and light source we can put as 18.25 / (39.9 + x). Then, as we know that the distance between the wall and face was 34.23, we can add this to the change in distance between the light source, which gives us 74.13 + x, and since this is our new x-value, we can construct the model (18.25/(39.9+x))(74.13+x), which will give us the height of the shadow.
Figure 8: Relationship between distance and shadow graphed using desmos.
Figure 8 shows how the distance and the shadow have an inverse relationship, as previously mentioned. Also, we could see a vertical asymptote at -74.13 and a horizontal asymptote at 18.25, which makes sense as the distance decrease, meaning that the light source gets closer to the face, the angle will exponentially increase, making the height of the shadow increase exponentially too until it reaches the wall completely. The horizontal asymptote can be explained too as there is no certain distance that can give the exact height as the face since the face is elevated from the light source.
Conclusion
The overall impact of this project is to show transformation through our different works of art. Art that we thought was not related to math and transformations were actually surprisingly related. Overall, this project was challenging but also successful and fullfilling. For a better a look at the shadow art that we created, click on the link below to reach our video.
