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Stereographic Projections

What is a stereographic projection? What does it do?

So far, a lot of our math has had to do with surfaces we can see easily (for example, a ruled surface from last week). For this project, we're moving away from that and focusing on more complicated geometry. Instead of taking \(f:\mathbb{R} \mapsto \mathbb{R}\) and graphing points (\(x,f(x)\)) in the \(xy\)-plane like a more basic 2D function, we are going to graph \(f:\mathbb{R^2} \mapsto \mathbb{R^3}\). Notice that now we are working in 5 dimensions! Clearly, we live in a three-dimensional world, so how does that work? Let's dive into visualizing this type of function.

Consider the example below comparing 3D to 5D.
A function mapping points on a sphere to a plane is a stereographic projection. This projection is defined on the entire sphere except at one point denoted as "N". Stereographic projections allow us to visualize a sphere as a plane. Perhaps the best example of this is the world map; this is a flat image of a sphere (although with stereographic projections, Greenland looks huge).
(Source: www.custom-wallpaper-printing.co.uk)
For this project, we're going to decide on a shape to project onto the plane. Our model will be a sphere with the shape in many sizes cut out of it; however, when we shine a light through the sphere the image of these shapes will make them appear to be the same size!

Design (and challenges!)

The biggest challenge I had with this project was coding in OpenSCAD. I had more complex ideas of shapes to use, but I couldn't execute them (I've never written code before, and I'm not a big CS person!). I ultimately decided on a teardrop shape because I understood the hull function for this. My original idea was to make a crescent moon, but OpenSCAD is not my friend.

It's interesting to see that despite the teardrops appearing to be different sizes on the face of the sphere, they are going to project as the same size on the plane. You'll see this in class on Friday.

Why I chose this design

Other than needing to choose a design I could actually code, I chose a teardrop design for some fun reasons. Someone else in our class designed their center of mass object to be a teardrop (or blood) shape, so I thought this would be a way to continue exploring that object as a stereographic projection. Speaking of blood, it's Halloween! I've seen a lot of decorations that are fun lights projected onto people's houses, so this is my version of a spooky math Halloween decoration. I know that by the time I print this object Halloween will be over, but now I'll be prepared for next year.

Hopefully, you have now seen how interesting stereographic projections are! For designs that are extra unique, I recommend visiting https://mathemalchemy.org/author/henry-segerman/ to view Henry Segerman's work with 3D printing. This way, you can see some examples with shapes that aren't very standard.

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