As you are here, I suppose that you might be interested in the TilingBot living on twitter. Each day it posts the image of a new tiling. Have a look at https://twitter.com/TilingBot and become its follower. I have fun to recognize the symmetries of these tilings. It is surprising to see the effect of different projections. So I upgraded my app that generates kaleidoscopic images. It now offers different projections of a tiling and you can find it at http://geometricolor.ch/sphericalKaleidoscopeApp.html.
Today I present different projections of an image with elliptic geometry. It uses a kaleidoscope of three mirrors with corner angles that make four-, three- and two-fold rotational symmetries. As such it generates the symmetries of a octahedron or cube. An additional mirror cuts of the corners as discussed in my earlier post “Decorations of semi-regular tessellations” and results in a decoration of a cuboctahedron projected on a sphere.
The most common view is the orthographic projection. It simply shows one hemisphere as seen from far away. This is a typical result:
The mirrors of the kaleidoscope are planes that go through the center of the sphere. Thus, on the surface of the sphere they appear as great circles being mirror lines. In this projection the circles become ellipses, depending on the inclination of their planes. Great circles passing through the center become straight lines.
Going close to the sphere we get the stereographic projection. Imagine that the sphere is decorated with transparent color and that you have a source of light at its north pole. Then you look at the projected image on a tangent plane at the south pole. What you see is a stereographic projection of the sphere. The decoration shown above in orthographic projection now gives us this image:
Here, great circles appear as true circles that invert the image. Circles going through the center have an infinite radius and thus appear as straight lines. Looking at small regions, the mapping preserves shapes and changes only their size because it is conformal. For programming the kaleidoscope I am using this view, as you can see in my post “elliptic kaleidoscopes” and in my paper for the Bridges 2008 conference.
Putting the projecting light source at the center of the sphere, we get the gonomic projection. It only shows a hemisphere and looks like this:
The great circles now appear as straight lines because the projecting light is at their center. Look out for local mirror lines in this image to find the great circles. The strong distortions make a dramatic effect. Globe lamps might project similar images on the ceiling if the source of light is small enough.
Finally, projections on cylinders are interesting too, in particular the conformal Mercator map:
This projection is conformal and preserves the shape of small regions similarly to the stereographic projection. But now we get a repeating image in the horizontal direction. The great circles appear as sinusoidal lines.
Obviously, there exist many other projections and I can’t do them all. Tell me if you are missing your favorite one.