Spotlight on Stereographic Projections

Watch Henry Segerman’s video chronicle

Read the video transcript

About Mathemalchemy

Hi I’m Henry Segerman,

For the past year and a bit, I have been working with a team of 24 other mathematicians and mathematical artists to create a large multimedia art installation celebrating the creativity and beauty of mathematics. The project is headed by Ingrid Daubechies, a mathematician at Duke University, and Dominique Ehrmann, a fiber artist based in Canada.

This all started when Ingrid saw Dominique’s installation “Time to Break Free“, in which a “machine” of some sort converts flat figures on a quilt into fully three-dimensional characters.

This inspired Ingrid to wonder if something like this could similarly bring the beauty and creativity of mathematical ideas to life.

Ingrid and Dominique presented at the Joint Mathematics Meetings in January 2020, and a group formed around the idea, called “Mathemalchemy”, which now has 24 mathematicians and artists involved.

Mathemalchemy’s final maquette

This is the current 1/4 scale maquette of the piece – a magical island filled with whimsical scenes, made from :

  • crochet,
  • knitting,
  • quilting,
  • sewing,
  • weaving,
  • beadwork,
  • needle felted wool,
  • embroidery,
  • woodworking,
  • ceramics,
  • polymer clay,
  • 3D printing,
  • painting,
  • stained glass,
  • steel welding,
  • acrylic,
  • wire bending,
  • origami,
  • temari,
  • leather,
  • light.

Wherever a mathematical idea or reference can fit in, we have been fitting it in.

The permanent home of the Mathemalchemy installation will be at Duke University, but it will also go on tour, with stops planned at the National Academy of Sciences in Washington DC, as well as other international venues (all pending possible disruption due to the pandemic).

Stereographic projection

One of the mathematical scenes that I’ve been involved with is the lighthouse. The top of the lighthouse will have two lights – one projecting horizontally from within a stained glass dodecahedron made by Bronna Butler, and the other projecting up on the the ceiling.

This will repurpose an old project of mine illustrating something called stereographic projection. Stereographic projection is a map from the sphere to the plane. So just like the Mercator map is a way of getting the continents of the globe on a flat piece of paper, stereographic projection is another way to do that.

Stereographic projection

In the model, what’s happening is the light ray start at the north pole of the sphere, it goes inside the sphere, hits the sphere from the inside somewhere and then continues on to hit the plane somewhere.

And that’s the map, the light rays do it, where does the light will hit the sphere, where does it hit the plane, it maps one to the other.

And you can see what stereographic projection does just by looking at the shadow. For example, the triangles in the middle are about the same size as the triangles on the sphere itself but then as you get further out, they start getting much much bigger. And there are lots of other nice features of stereographic projection to see here, for example: angles are preserved, all of the ninety degree and sixty degree angles and so on the sphere come out as ninety degree and sixty degree on the shadow. Also, circles map to circles there are all of these great circles on the sphere that go to circles on the plane.

The shadow is very sensitive to where the light is relative to the sphere, but it’s not too sensitive to where the table is. So, if I move the sphere and the light up away from the table, all that happens is that the shadow scales up, it doesn’t otherwise change the image. This is good for us because on the lighthouse, this will be pointed upwards casting the shadow on the ceiling and since installation will be in many different locations there is no way to know how high the ceiling is going to be.

We have a second version planned in case it’s necessary. If the ceiling is too high in some venue, then the pattern might get scaled up too much and most of it get projected onto the walls. The pattern would still be visible but the shadow on the walls will not have all the nice properties of stereographic projection. Our plan in this case is to use a different repurposed shadow artwork which projects the Poincaré disk model of the hyperbolic plane onto a disk. This pattern only uses a 90 degrees cone of light which will project on a smaller patch rather than the 180 degrees which is used in the stereographic projection full sphere.

Poincaré disk model

Alright, that’s the sphere on the top of the lighthouse. The plan is that assembly of all of the independently constructed parts of the installation will happen in July of this year – so hopefully, more to follow!

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