As you stand in one place and look around, up, and down, there is a sphere's worth of directions you can look. One way to record what you see would be to construct a big sphere, with the image painted on the inside surface. To see the world as viewed from the one place, you would stand on a platform in the center of the sphere and look around. We will call this sphere the visual sphere. You can imagine a sphere, like a planetarium, with projectors projecting a seamless image. The image might be created by a robotic camera device, with video cameras pointing in enough directions to cover everything.
Question. What is the geometric relation of objects in space to their images on the visual sphere?
Unfortunately, you can't order spherical prints from most photographic shops. Instead, you have to settle for flat prints. Geometrically, you can understand the relation of a flat print to the `ideal' print on a spherical surface by constructing a plane tangent to the sphere at a point corresponding to the center of the photograph. You can project the surface of the sphere outward to the plane, by following straight lines from the center of the sphere to the surface of the sphere, and then outward to the plane. From this, you can see that given size objects on the visual sphere do not always come out the same size on a flat print. The further they are from the center of the photograph, the larger they are on the print.
Suppose we stand in one place, and take several photographs that overlap, so as to construct a panorama. If the camera is adjusted in exactly the same way for the various photographs, and the prints are made in exactly the same way, the photographs can be thought of as coming from rectangles tangent to a copy of the visual sphere, of some size. The exact radius of this sphere, the photograph sphere depends on the focal length of the camera lens, the size of prints, etc., but it should be the same sphere for all the different prints.
If we try to just overlap them on a table and glue them together, the images will not match up quite right: objects on the edge of a print are larger than objects in the middle of a print, so they can never be exactly aligned.
Instead, we should try to find the line where two prints would intersect if they were arranged to be tangent to the sphere. This line is equidistant from the centers of the two prints. You can find it by approximately aligning the two prints on a flat surface, draw the line between the centers of the prints, and constructing the perpendicular bisector. Cut along this line on one of the prints. Now find the corresponding line on the other print. These two lines should match pretty closely. This process can be repeated: now that the two prints have a better match, the line segment between their centers can be constructed more accurately, and the perpendicular bisector works better.
If you perform this operation for a whole collection of photographs, you can tape them together to form a polyhedron. The polyhedron should be circumscribed about a certain size sphere. It can give an excellent impression of a wide-angle view of the scene. If the photographs cover the full sphere, you can assemble them so that the prints are face-outwards. This makes a globe, analogous to a star globe. As you turn it around, you see the scene in different directions. If the photographs cover a fair bit less than a full sphere, you can assemble them face inwards. This gives a better wide-angle view.
One way to do this is just to take enough photographs that you cover a certain area of the visual sphere, match them up, cut them out, and tape them together. The polyhedron you get in this way will probably not be very regular.
By choosing carefully the directions in which you take photographs, you could make the photographic polyhedron have a regular, symmetric structure. Using an ordinary lens, a photograph is not wide enough to fill the face of any of the 5 regular polyhedra.
An Archimedean polyhedron is a polyhedron such that every face is a regular polygon (but not necessarily all the same), and every vertex is symmetric with every other vertex. For instance, the soccer ball polyhedron, or truncated icosahedron, is Archimedean.
Question. Show that every Archimedean polyhedron is inscribed in a sphere.
The dual Archimedean polyhedra are polyhedra which are dual to Archimedean polyhedra.