# Pascal's pyramid

Here's the image you would provide your students:

The point at the apex of the pyramid is given the value of 1. Each point below the apex is to be labelled with the sum of its nearest neighbors in the layer directly above. The apex is itself a layer. Every point in the second layer, the layer directly below the apex, is labelled 1, because each point has exactly one nearest neighbor, which is the apex. Students will likely notice that every point on an edge of the pyramid will have exactly one nearest neighbor, on the same edge, and so all of these points should be labelled one. Further, each point on a face of the tetrahedron has only nearest neighbors on the same face, and thus, each face of the tetrahedron is a copy of Pascal's triangle. Another observation students might make, is that no point can have more than 3 nearest neighbors in the layer above. And yet another observation, each layer has degree 3 rotational symmetry, because the lattice of points itself has degree 3 rotational symmetry about its vertical axis. Once students have completed the pyramid, it should look like this: