Finite Reductions = 3D > 2D > 1D..STOP( see 2nd law/principle of thermodynamics ) as it is simple not complex.

Following is a simple numerical reduction of a 3D polyhedron called a cubo-octahedron or Vector Equilibrium( VE )--- aka operating system of Universe ---that is based upon 12-around-1 sphericals.

http://www.rwgrayprojects.com/synergetics/s05/figs/f1603.html

http://www.rwgrayprojects.com/synergetics/s04/figs/f6008.html

..{ this link shows the VE as toy model called a jitterbug by R. B. Fuller }....

as it transforms in specific set of configurations that go from a 3Dimensions--- 12-around-1 to a 1Dimensional set of three points / nodal-vertexial events ergo a 1D triangle that has no angles but maintains a set of three numerical identifications with a triangles 3 nodal-vertexial events.

3D = 12-around-1, however if the central, 13th nuclear sphere is removed, then it becomes 12-around-0 and allows for the 3D VE/jitterbug, to transform into a few differrent 2D configurations.

http://www.rwgrayprojects.com/synergetics/s02/figs/f2230.html

In the link above we can see a 1/half octahedron--- square-based pyramid ---and in the VE there exists three sets of dipolar 1/half-octahedra i.e. they are likened to a di-polar dumb-bell / bar-bell or sand-clock configuration, that appears as only two 1/half octahedra sharing a common nodal vertexial event, tho it is really 4 nodal-vertexial events at at this shared connection location.

From a side view--- aka profile ---we see the aproximated dumbell or barbell-like shape as two triangles !><! where the each of the exclamation symbols is a side view of the square planes. With the VE/jbug these 2D squares collapse to be come 1D line of 4 nodal-vertexial events ex;

o......oo........o...is one of the 1D line of 4 nodal vertexial events

......oooo.............is the commonly shared 4 nodal-vertexial events

o......oo........o...is the other 1D line of 4 nodal-vertexial events.

So here above we have the total 12 nodal-vertexial events we started with as the 3D 12-around-1 then 12-around-0 polyhedron called the cubo-octahedron or VE/jitterbug and has been reduced to a 2D set of relationships.

So here above we the two outer sets--- above and below ---of 4 nodal-vertexial events ergo 8 and we are in 2D configuration both numerically and geometrically with the VE/jitterbug.

In the middle--- shared nuclear ---4 ergo my initial numerical set as and 8-to-4 ratio of outer sets to inner set.

Next we can numerically-- not geometrically --reduce this above set to a 1D line as;

oo.....oooooooo......oo..ergo we still have 12 nodal-vertexial events but have gone from 3D > 2D > 1D, however, we can numerically reduce this set from 8-4 ratio to a;

oo...oo....oo..ergo 4-2 ratio to a;

o..o...o..ergo 2-1 and this is the minimal, rationa,l numerical reduction we can do of the original 12 nodal-vertexial events.

This set of three is likened to 1D triangle i.e. we collapse the 2D triangle to a 1D because we maintain the its numerical three nodal-vertexial events. This is one way to turn a triangle inside-out is by contracting it to 1Dimension, the expanding back on to triangle diametrically opposite to the initial triangle.

The following shows both 2D triangle configurations and the a single 1D set as it turns itself inside-out.

........o........

o......o.......o

........o.......

There is simple analogy to this phenomena using a 3D tetrahedron, that turns itself inside-out, when its 4th nodal-vertexial event passes through its diametrically opposite opening we have momentarily gone from a seemingly 3D tetrahedron, to seemingly 2D subdivided triangle.

r6