Part 2: Geometrical Models in the Sepher Yetzirah and Bahir

This post follows on from Part 1.

Second Model

An alternative way of representing the same geometry is to use a model constructed from spheres (eg. table-tennis balls) instead of lines and corners.  To do this we start by making a 5-layer tetrahedron, and then add a group of four balls to each face, thus making two intersecting tetrahedra – a ‘star’ shape.


5-layer tetrahedron



Adding to each face to make the star shape.

Looking at the star shape from a corner shows the intersecting tetrahedra – one ‘pointing up’ towards the viewer and the other ‘pointing down’. Looking at the star shape from the side shows the cube outline:


Although less obvious, the octahedron is still embedded within the star shape, with the six directions at the intersections of the edges of the tetrahedra (the centers of the cubic faces). For example, the ball marked ‘E’ in the diagram above represents one of the six directions. By cutting off the 8 corners of the tetrahedra, the octahedral shape is exposed:


Also concealed within the star shape (and indeed within the octahedron) is the dymaxion or cuboctahedron shape. This shape, where one central ball is surrounded by 12 others, is an aspect not seen in the ‘line-and-corner’ model.


The dark coloured balls make the dymaxion



One way of thinking about the star shape is that there is:

  • One ball at the very centre
  • Twelve balls that surround it and make the dymaxion (13 balls)
  • Add a ball for each of the six directions to make the octahedron (19 balls)
  • Add four groups of 4 balls to make a 5-layer tetrahedron (35 balls)
  • Add four groups of 4 balls to make the star shape (51 balls).


The introductions to the Yetzirah and Bahir include other numerical interests in the star shape:

  • The star shape has seven levels.
  • The surface contains 50 balls (all but the central ball is on the surface). The letter Nun equals 50 in numerical value.
  • Each tetrahedron contains six edges (Vav) and the number of spheres along these edges adds up to 22.
  • When it is on its base, the number of visible spheres in a tetrahedron is thirty-one (all except the one in the middle and the three on the inner part of the bottom face). The total of all spheres in the tetrahedron is 35. when it is remembered that it was customary to include in the numbering the whole that is formed, we have the combination of the 32 and the 36 upon which much of the Bahir’s explanation is built.

One of the comments in the introduction to the Yetzirah is a little unclear: “Also each of the subsidiary tetrahedra contain 10 spheres which answer to the four worlds of logos, creation, fashioning and making.” A three-level tetrahedron contains 10 spheres, and there are several ways that three-level tetrahedra can be seen in the star shape. Precisely which four of these are meant to correspond to the four worlds is not clear.


A PDF version of Part 1 and Part 2 is available here.

About singinghead

druid, mathematician, blogger, gardener...
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