four more ways to draw the tree of life
the same graph every time: ten sefirot, twenty-two paths. each drawing is chosen to make a different theorem visible. the three rungs that cause all the trouble in the traditional picture stay green throughout.
it is a convex polyhedron
the graph is planar and 3-connected, so by steinitz's theorem it is the skeleton of a convex solid. this one has 12 triangular faces and 2 quadrilaterals, and the two quads meet along chesed–gevurah.
read it as an octagonal pyramid with tiferet at the apex whose base has been creased along the three rungs and capped with malkhut. drag to turn it.
straight lines only, no crossings
every planar graph has a crossing-free drawing with straight edges. here the outer face is a rectangle with chesed, netzach, hod and gevurah at the corners; the upper and lower pillars fold inward toward tiferet so the rungs can pass outside them.
mirror symmetry survives. this is the drawing where "tiferet is the center" is literally true.
two pages suffice
put the sefirot on a line and draw each path as a half-circle above or below. a planar graph with a hamiltonian cycle can always be done on two pages without crossings: here the cycle 1-2-4-7-10-9-8-5-3-6 is the spine.
the traditional lightning-flash order 1 → 10 is a hamiltonian path, not a cycle, and the best two-page drawing along it has five crossings. the same number as the traditional picture, by coincidence.
the graph draws itself
x is the fiedler vector (second laplacian eigenvector, λ₂ ≈ 1.28), y is the first eigenvector that is odd under the mirror (λ ≈ 3.75). nothing about kabbalah went in; the coordinates come from the adjacency alone.
and yet the fiedler vector sorts the sefirot exactly in emanation order, keter to malkhut, and the odd eigenvector separates the pillars of mercy and severity with the middle pillar pinned at zero. the cut it suggests is between tiferet and netzach–hod.