For our Family Math project today we played with the idea of a spheres intersecting objects other than a plane. The idea was to explore how a 2d being on a cone, say, would “see” and describe a sphere passing through the cone. That project is here:
Tonight I revisited the project just to see the different images of a 4d sphere intersecting a 3d cone. I wanted to see the analogous and presumably not really spherical shapes.
Here are some pictures.
(1) A 3d sphere intersecting a 2d cone as it descends down the middle of the cone:
(2a) The analogous picture of a 4d sphere intersecting a 3d cone:
(2b) Note the intersection does have a hole in the middle:
(3) Now I shift the sphere so that it is descending down the cone slightly off center. For the 3d sphere intersecting the 2d cone you get this picture:
(4) And here’s the same situation with a 4d sphere intersecting a 3d cone:
It is fun to see and try to imagine these images. The usual way of thinking about constructing a sphere (either 3d or 4d) makes some sense when you think about stacking up the images of the intersection with a 2d plane or with 3d Euclidean space. Trying to “stack” the images if the sphere intersecting the cone to make either a 3d or 4d sphere is really not intuitive for me at all!