We have a colloquialism, “the deficit,” which means one in particular, namely some number representing what the Treasury has to borrow, to stay in the green.
The annual deficit feeds “the debt” which is the accumulated principal plus interest against which the Treasury must pay. We call this “central banking” for some reason, or maybe not. I leave it to economists to duke it out.
The deficit I’m talking about is different, but we’re free to borrow the same lingo, and even talk about the Debt, which is: 720 degrees or two circles worth of angle.
One needs to go around twice to carve out enough turn, to undo a System, yet another jargon term, or Tetrahedron, to distill it further.
This 720 degrees has been christened Descartes’ Deficit, and that’s what I’m focused upon in this Medium story.
To read further, with comprehension, you need not familiarize yourself with “the deficit” I’ve mentioned above, as it belongs to another namespace, frequented by bankers and Treasury bond dealers.
Consider the Platonics, as a “subclass” of Polyhedron.
In ancient Greek, we say “subclass” or “subset” whereas in Anthropology we speak of “subcultures” or simply “cults” for short. Cultures, more than the sums of their parts, consist of partially overlapping subcultures, or cults, such as we study with Venn Diagrams.
The Platonics comprise an elite cult (subset) indeed, of those Polyhedrons with congruent faces and vertexes both. We have only five of them.
The Rhombic Dodecahedron (RD), for example, is not a Platonic? Why?
Yes, every face is an identical diamond. However, those diamonds meet up in two ways: the sharper corners cram four around a vertex, whereas the more spread out corners gather in threes.
The vertexes are therefore not identical. A Platonic the RD is not.
Consider the Cube, then, likewise a rhombohedron and zonohedron.
Each face is a Rhombus.
A subclass of Rhombus is Square.
Square is a subclass of rectangle, in turn of a subclass of “parallelogram,” in turn of “trapezoid” in American English.
Lets get on with Descartes’ Deficit, which pops up in American literature within Synergetics, the magnum opus of one of the Transcendentalist writers, a poet with the initials RBF.
The deficit gets spelled out in prose, with illustrations, in order to set the stage for an “omnidirectional halo” concept, further refined in No More Secondhand God by the same writer.
A System connects around, in all circumferential directions, as a polyhedron does, as a closed circuit graph. The Platonics all do this, and around each vertex: the same number of degrees.
Consider the Cube: eight vertexes each defined by three squares, mutually perpendicular, of “three dimensional” fame (x, y and z dimensions). That’s three right angles for a total of 270 degrees — per vertex.
How different is 270 from “flat”?
Perfectly flat is 360 and 360–270 is 90.
So a cube is “missing” (absent) 90 times 8 degrees, of 720.
That 720 total is the “sphere tax” levied cumulatively on every vertex, to give it some curvature.
In other words, the amount of angle withdrawn from a vertex (taxpayer) to ensure a system’s systemhood, is perhaps minuscule. Yet no matter how minuscule, it’s still discrete, prompting a meditation on limits.
As the number of total triangles increases, on a spherical network of increasing frequency, the number of degrees contributed by each vertex, to 720, approaches zero. But it never gets there, thanks to finite definite curvature overall.
That same total amount, 720, holds, no matter what Platonic we’re talking about, and this rule further generalizes.
We’ve found something true about systemhood to a very high order.
In a mathematics, Descartes’ Deficit may go by as another interesting, provable truth, but will probably not be the root concept of some entire cosmology or Universe.
In Synergetics, on the other hand, a System has both Hegelian connotations of “any thinkability” (as in “belief system”), and geometric denotations.
The System is self booting thanks to a “missing tetrahedron” (a deficit), which is what creates a firmament in the first place.
The Negative Tetrahedron is fundamental to the positive one.
They beget one another.
Explorations in the geometry of thinking: Synergetics isn’t called that for nothing.
 in American English, a trapezoid is a quadrilateral with at least one pair of edges square.