*Note that Augusta actually was aware of this cube and mentioned it in a 1985 article on the Reinin dichotomies. However, the idea that the relationships are transformations of the cube was not mentioned.*

Model A is generally presented as a linear, 1-dimensional model, or as a 2-dimensional grid. You have functions 1, 2, 3, 4, 5, 6, 7, 8, also known as leading, creative, role, etc., which can be arranged in two different "loops".

But in reality, Model A is neither a line nor a grid. It's a cube. It is not 1-dimensional or 2-dimensional, but 3-dimensional.

That is to say, you can arrange the functions of Model A like so:

Let's say that this placement of the cube represents LII, so that the closest bottom edge represents the ego block, with the leading function on the left. Then

-the duality relation is the vertical reflection (switching TiNe with FeSi)

-supervision is a 90 degree rotation about the vertical axis

-mirror is a reflection across the yz-plane

-comparative is a reflection across a diagonal plane

-extinguishment is the antipodal map / complete reflection which sends each point to the point opposite from it

Supervision can be replaced with benefit as the rotation to produce a "Model G cube" as it were, making the benefit rings apparent instead of the supervision rings.

Duality can also be replaced with extinguishment as the vertical reflection.

And comparative can be replaced by any of the "odd" relations - the ones that reverse the Process/Result dichotomy.

These three choices determine the model, so that there are 2*2*8 = 32 versions. The one above is the "dsk" (dual/supervision/comparative) cube. In all cases, the possible transformations are exactly the ones that preserve the cube itself

**and**preserve the vertical axis. Half (eight) of them reverse the cube's orientation and half of them preserve it. In the dsk cube the orientation corresponds to the Negativist/Positivist dichotomy.

The diagonal reflection (which is the comparative relation in Model A) fixes half of the functions. We can expand the cube to a 4D hypercube, and Model A to a 16-function model, so that no relation fixes any functions:

In this case we can actually identify the types with the IM elements. I interpret this to mean that each type has an overarching goal in life, which is the leading function. Typically these are written as "signed" versions of the regular IM elements:

+Ne | ILE |

-Si | SEI |

+Fe | ESE |

-Ti | LII |

-Fe | EIE |

+Ti | LSI |

-Se | SLE |

+Ni | IEI |

+Se | SEE |

-Ni | ILI |

+Te | LIE |

-Fi | ESI |

-Te | LSE |

+Fi | EII |

-Ne | IEE |

+Si | SLI |

The elements of Model A (and any other cube model) are coarser versions of these elements, e.g. Ti is just +Ti and -Ti (LSI and LII) grouped together.*

Because types are identified with IM elements, in the 4D model we can simply choose which relations are adjacent to the type:

-(beneficiary and benefactor) or (supervisor and supervisee)

-one of the odd relations

-extinguisher or dual

There is much more to say about the cube, but this will do for now. This is the perfect logical system underlying socionics.

* Notice also that although the look-a-like relation is also a diagonal reflection in the Model A cube, it technically defines different elements than the comparative relation: Reasonable EJ, Reasonable IJ, Merry EP, Merry IP, etc.