Surface Minimizing Bubbles🫧
Mathus taught me how to use soap bubbles to find the minimal surface area of a shape I create. I created these shapes using Zometools.
Mathus gave me some guidance on what kind of shapes would be cool to make for this project, but left me with a lot of creativity. Mathus let me know that starting out simple with the blue Zometools would be easier than working with the green ones. So, initially I started out with a cube:
![](https://mathandmaking.com/wp-content/uploads/2024/02/Screenshot-2024-02-18-at-4.38.46 PM.png)
I saw that the bubble edge lines came together in the center to create a square. It looked almost like a 4D hypercube!
I then got rid of 6 edges to create a sort-of “half cube,” and saw a really cool bubble surface from the side-view that looked like a curve:
![](https://mathandmaking.com/wp-content/uploads/2024/02/Screenshot-2024-02-18-at-4.38.50 PM-894x1024.png)
Lastly, I tried creating two of these “half cubes” and linking their corresponding vertices together with the green Zometools. This bubble surface created like a diagonal hexagon with bubble edges coming from the vertices:
![](https://mathandmaking.com/wp-content/uploads/2024/02/Screenshot-2024-02-18-at-4.38.56 PM-1010x1024.png)
I thought this project about minimal surfaces was really interesting! I saw a blog online that solved this Plateau problem using variational calculus. I thought it was really neat how the solution that they solved for looked almost identical to the actual soap bubble shape made with the rings!
https://mathematicalgarden.wordpress.com/2014/09/06/soap-film-and-minimal-surface/: Surface Minimizing Bubbles and PenRose TilingsLastly, I thought that this video of a transformable hypercube soap bubble was super cool! The motion of the hypercube adds a cool dimension to the model:
PenRose Tilings!
Ashley provided me with an in-progress version of the final Penrose tiling, along with materials of leaves and flower petals, and a wooden framing device to make the cut shapes more consistent. She also provided me with a reference diagram of the fractal:
![](https://mathandmaking.com/wp-content/uploads/2024/02/penrose_tiling.png)
https://en.wikipedia.org/wiki/Penrose_tiling
First, I tried making more units. I used a stencil to cut shapes from leaves and petals. I used the provided wooden framing block to make the shapes consistent. It was a bit hard to reach the corners at first, but I eventually got the hang of it. Below is a photo of the cut out plant material along with the wooden frame and the knife.
![](https://mathandmaking.com/wp-content/uploads/2024/02/IMG_7267-768x1024.jpg)
Afterwards, I put glue on one side of each of the shapes and attached it at the relevant location to the rest of the tiling. Below is the in progress photo.
![](https://mathandmaking.com/wp-content/uploads/2024/02/IMG_7268-768x1024.jpg)
Some challenges included dealing with slightly out of line tilings and having to overlap shapes to ensure everything still fit. The flower petals were also pretty fragile, and I broke one when trying to glue it down.
![](https://mathandmaking.com/wp-content/uploads/2024/02/IMG_7269-768x1024.jpg)
Here is the final (for now) product!
![](https://mathandmaking.com/wp-content/uploads/2024/02/IMG_7270-768x1024.jpg)
Finally, there are variations of Penrose tilings that could be cool to explore or generate, such as this one that involves truncation of the polygons:
![](https://mathandmaking.com/wp-content/uploads/2024/02/penrose_tiling_2-1024x683.png)
https://www.iucr.org/news/newsletter/volume-27/number-3/deformed-penrose-tiling-and-quasicrystals
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