the ari / gra tree, one dimension up
the gra arrangement is not planar: chokhmah, binah, chesed, gevurah and tiferet form a k₅. so the plane is the wrong surface for it. three places where it does fit without any crossing, in order of increasing dimension.
it lives on a torus
a graph with crossing number 1 embeds on a torus: punch a handle where the crossing was and send one edge through it. the gra tree's crossing number is exactly 1, so its genus is exactly 1. the donut is its natural home the way the sphere is the kircher tree's.
left, the flat version: a rectangle whose top is glued to its bottom and left to right. red edges leave one side and come back in the matching place on the other. there are no crossings anywhere, including across the seams. right, the same coordinates wrapped onto the actual surface.
euler characteristic of the torus is 0, so a cellular embedding has f = e − v = 12 faces.
a tetrahedron with a point inside
k₅ is impossible in the plane but trivial in space: four vertices as a tetrahedron, the fifth inside it, and the ten edges are the six tetrahedron edges plus four spokes. here chokhmah, binah, chesed, gevurah are the tetrahedron and tiferet is the interior point. keter hangs off the chokhmah–binah edge; the lower triad and malkhut hang off the chesed–gevurah edge. every edge is a straight segment and none of them meet.
what you cannot do is make this a convex solid: steinitz needs 3-connectivity and malkhut is a pendant vertex. the tree sits in space, but it is not the skeleton of anything.
the k₅ is a regular 4-simplex
in ℝ⁴ five points can all be the same distance apart. that figure is the 4-simplex (pentachoron), and its skeleton is exactly k₅. so the upper gra tree, {2,3,4,5,6}, is the vertex-and-edge set of the simplest regular 4-polytope, with keter and the lower sefirot attached. the canvas rotates the whole thing in two independent planes of ℝ⁴ and projects it down; the k₅ is green.
this is also what the clique complex says. fill in every clique as a simplex and you get:
| kircher | gra | |
|---|---|---|
| f-vector | (10, 22, 16, 3) | (10, 22, 19, 7, 1) |
| dimension | 3 | 4 |
| euler char. | 1 | 1 |
| homology | contractible | contractible |
both complexes are contractible, all the triangles fill in all the holes, but the kircher tree is three tetrahedra glued in a line and the gra tree contains a solid 4-simplex. the two edges really do add a dimension.