Macdonald Polynomial Web page
Associahdra plane tiling
Permutahedron for n=5

The permutahedron for n=4

I drew some pictures of a sphere divided into fundamental chambers by hyperplanes.  This was drawn using MuPad and made into a sequence of pdf images that animate to rotate the sphere.  The final images have the midpoints of the fundamental chambers connected so that the permutahedron appears.

the sphere divided into hyperplanes and fundamental chambers
pdf format

More permutahedrons on the sphere.

pdf format

Permutahedrons can tile n-space.  Here is an attempt at drawing this phenomenon in 3-space.

permutahedrons tiling 3-space
pdf format
pdf with 4 permutathedra

These images were taken from a paper by Marcelo Aguiar and Frank Sottile: Structure of the Malvenuto-Reutenauer Hopf algebra of permutations.  I have modified them so that they can easily be printed, copied, and used for web applications.  The gif versions of these pictures have an invisible background and so can be placed over other images.  I needed it because I drew it several dozen times trying to see patterns in a problem related to this poset and all I wanted was a copy that I could color in.

Click on the following links for larger image
Color gif jpeg eps pdf
B&W gif jpeg

Click on the following links for a larger image
Color: gif jpeg eps pdf

Click on the following links for a larger image
B&W: jpeg gif eps pdf


Pentagons and hexagons vs. squares and hexagons