ALGORITHM FOR FINDING ALL SAND HEAPS HAVING A CONVEX POLYGONAL BASE

REPRESENTATION OF A SAND HEAP BY ITS DUAL

The sides of a sand heap are numbered, exept the base.
Each side is represented by a point.
A polygon is obtained, called DUAL of the sand heap.
The edges of the dual correspond to the ridges of the sides of the sand heap wich strated from the base.
The diagonals of the dual correspond to the ridges of the heaps.
The edges of the base of the sand heap are no longer represented on the dual.

Example:

On the sand heap, the ridge IJ links the faces 2 and 4.
On the dual, the diagonal IJ links the points 2 and 4.

On the sand heap, the ridge IJ links the faces 1 and 3.
On the dual, the diagonal IJ links the points 1 and 3.

HOW TO FIND THE FORMULA OF A SAND HEAP FROM IS DUAL ?

For the side number i of the sand heap, the number of its edges is equal to :
riges starting from the base + ridges + edges of the base

This gives for the top number i of the dual : 2 + number of diagonals starting frompoint i + 1

HOW TO FIND ALL THE SAND HEAPS FOR A GIVEN BASE ?

One utilises the dual .
One starts from the configuration where all diagonals strat from point 1

This corresponds to the formula : n 3 4 .... 4 3
Then, in a lexigological order, one removes the diagonals of point 1 from putting them on the places left empty
One shall observe that two diagonals can not cross.

The removed diagonals are in dotted lines.