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Homework due October 30.

This assignment concerns the kth power of the standard 1-simplex, which we will call the simplicial k-cube and denote by C[k]. We will denote the standard k-cube by I[k].

1. Find a formula for a[k,i], the number of nondegenerate i-simplices in C[k].  In particular show that a[k,k]=k! and a[k,k-1]=k!(k+3)/2.

 
2. Find a formula for b[k,i], the number of  i-faces in I[k].

 
3. Describe the k! nondegenerate k-simplices of C[k] explicitly.

 
4. Show that the k! nondegenerate k-simplices intersect along the main diagonal of the k-cube and that each is the topological join of that diagonal with a nondegenerate (k-2)-simplex on the boundary of the k-cube.  Find the number of nondegenerate (k-2)-simplices on the boundary and show that all but k! of them touch the endpoints of the main diagonal.

 
5. For k=2, the k! nondegenerate (k-2)-simplices above are the vertices of the square not lying on the main diagonal.  For k=3 they are the 6 edges of the 3-cube not touching the endpoints of the main diagonal. Projecting these edges into the plane perpendicular to the diagonal gives a regular hexagon.  Give analogous descriptions for k=4 and k=5.  In the former case show that the image of the projection is a rhombic dodecahedron with each rhombic face bisected into two triangle as in C[2]. What is the 4-dimensional analog (in this sense) of the rhombic dodecahedron?

 
6. Describe the 2-fold product of the standard two simplex as explicitly as possible. Determine the number of nondegenerate i-simplices for each i.

This page was last revised on October 19, 2001, by Doug Ravenel .