The first photo shows assorted collapsibles floating in a marine tank at the Gumbo Limbo Environmental Center at 1801 North
Ocean Blvd. in Boca Raton, Florida. In this particular tank there are several varieties of fish, a sea star, a few stingrays, and
a poisonous spiny looking creature that I didn't get a picture of (but plan to this summer 2003). The tank has a huge circumference, is fairly shallow,
with sand covering the bottom, probably because stingrays like sand. The middle photo shows a stingray with the collapsibles. The fish darted
from the structures, while the stingrays stayed close. The photo on the right shows a permutahedron for n=4 floating in a fountain
at Arizona State University. Its 24 vertices are the 24 permutations of the set {1,2,3,4} or any other set of 4 distinct elements. It happens that
the permutahedron for n=4 is either one and the same structure as the snub octahedron, or is isomorphic to the snub octahedron, one of the
Archimedean solids. The permutahedron for n=3 is a hexagon, whose vertices are the 6 permutations of {1,2,3}. The permutahedron for n=2 is
a line with endpoints being the 2 permutations of {1,2}, and the permutahedron for n=1 is a single point. Shelly Smith, a graduate student with
the Department of Mathematics and Statistics at ASU, has done research on the permutahedron, and helped me with wording/notation to ensure
accuracy/clarity of the information presented here.
