The E-module in Tetravolumes

by D. Koski & K. Urner, June 2018

The E-module has long been a standard module, starting with Synergetics and persisting through the writings and research of several students of polyhedral dissections.

The E is 1/120th of the Rhombic Triacontahedron with:

  • radius = 1 (from center to surface face center)
  • surface face diagonals of 1/Ø and (1/Ø)(1/Ø)

In Synergetics, our unit of volume is the regular tetrahedron of edges D, where D is the diameter of a CCP sphere in densest packing, with twelve-around-one at any interior vertex or node. D may be set at 1 or 2, in which case R (radius) will be 0.5 or 1 respectively.

Fig 1. RT in Synergetics (Fig. 986.419)

The E-mod's volume, in tetra-volumes, is: (√2/8)/ØØØ

The E-mod's volume is a little bigger than 1/24, the T-mod's volume i.e. 1/120th of a RT of volume 5 exactly. Ts and Es come in left and right handed flavors.

The E-mods RT, scaled up by Ø in all linear dimensions, is the "child" of the Concentric Hierarchy Icosahedron of volume ~18.51 and its dual, the Pentagonal Dodecahedron. When their edges cross, the thirty diamond-face mid-points are established, along with the corresponding vertexes.

When the RT is scaled up by Ø in all linear dimensions, volume goes up as a factor of Ø to the third power (we do not feel compelled to say "cubed" given the shape in question). David's notation for the larger Emod shapes is E3, whereas if we shrink Emod down by Ø in all linear dimensions, we call it e3 (lowercase means "smaller than"), e6 etc. Ditto for the Smod.

For more on how the Platonics "give birth" to our Concentric Hierarchy players, check out Genesis Story on Youtube.

In [1]:
import tetravolume
Ø = (1 + 5**0.5)/2
from math import sqrt
Emod = sqrt(2)/8 * Ø**-3


Fig 2. Emod by D. Koski

The E mod tetravolume is (√2/8)/ØØØ or .041731

In the figure above the E mod is further subdivided by additional intersecting planes creating what we might call Phe, Phi, Pho, Phum submodules. This subdivision of the E is discussed in more detail elsewhere. The planes that cut the E3 per LCD trianges (derived from the 31 great circles of the spinning icosahedron), define a way to cut an E lying sideways on the E3's external face.

The E module proper, as depicted, has three blue orthogonal lengths:

  • short_face_diag = 1/(Ø*Ø) = 0.381966
  • long_face_diag = 1/Ø = 0.618033
  • radius = 1.000000

It then has 2 red lengths, serving as hypotenuses:

  • long_hypot = √(5-√5)/2 = 1.175570
  • short_hypot = √(5-2√5) = .726542

... and finally, a yellow length, also an hypotenuse:

  • yellow = (Ø^-1)(√3) = 1.070466

In [2]:
short_face_diag = 1/(Ø*Ø)
long_face_diag = 1/Ø
radius = 1.000000
long_hypot = sqrt((5-sqrt(5))/2)
short_hypot = sqrt(5-2*sqrt(5))
yellow = sqrt(3)/Ø

Below, we convert these to Python and feed them to our volume computer, which sets D=1 i.e. our dimensions are twice as big (assuming R=1). So the final step involves taking 1/8th the computed volume.

In [3]:
print("   short_face_diag:", short_face_diag)
print("    long_face_diag:", long_face_diag)
print("            radius:", radius)
print("        long_hypot:", long_hypot)
print("       short_hypot:", short_hypot)
print("            yellow:", yellow)

   short_face_diag: 0.38196601125010515
    long_face_diag: 0.6180339887498948
            radius: 1.0
        long_hypot: 1.1755705045849463
       short_hypot: 0.7265425280053608

            yellow: 1.0704662693192697

In [4]:
ivm_vol = tetravolume.Tetrahedron(radius, long_face_diag, short_face_diag,
                                 long_hypot, short_hypot, yellow).ivm_volume()


Fig 3. Same Volume as Emod, by D. Koski

We may follow essentially the same workflow with this alternative tetrahedron which turns out to have the same volume as the E module. Note the division by 8 in the final step.

  • Short Blue = 1/Ø
  • Long Blue = 2/Ø
  • Yellow = √3/Ø
  • Purple = √3/Ø
  • Green = √2/Ø
  • Red = √((25-11√5)/2)

In [5]:
purple = yellow = (3**0.5)/Ø
short_blue = 1/Ø
long_blue = 2/Ø
green = sqrt(2)/Ø
red = sqrt((25-11*sqrt(5))/2)

In [6]:
print("phi", Ø)
print("Purple, Yellow:", purple)
print("    Short Blue:", short_blue)
print("     Long Blue:", long_blue)
print("         Green:", green)
print("           Red:", red)

phi 1.618033988749895
Purple, Yellow: 1.0704662693192697
    Short Blue: 0.6180339887498948
     Long Blue: 1.2360679774997896
         Green: 0.8740320488976422
           Red: 0.4490279765795847

In [7]:
ivm_vol = tetravolume.Tetrahedron(red, yellow, green, long_blue, short_blue, purple).ivm_volume()

In [8]:


So far, we've depending on the built-in float type for our computations, but lets remember our 3rd party assets, in addition to decimal.Decimal in the Standard Library.

The computations below run through some of the same volumes already talked about, such as the E-module.

In [9]:
import gmpy2

Ø = (1 + gmpy2.root(5, 2))/2
vol_scale_factor = 1.5
rad_scale_factor = gmpy2.root(vol_scale_factor, 3)

h = (Ø / gmpy2.root(2,2)) * (1/rad_scale_factor)
print("RT5 radius: {:42.40f}".format(h))
S3 = gmpy2.root(9/8, 2)
SuperRT_vol = S3 * 20
print("SuperRT:   {:42.40f}".format( SuperRT_vol))
Emod_RT_vol = SuperRT_vol * (1/Ø)**3
print("5+ RT:      {:42.40f}".format( Emod_RT_vol ))
Tmod_RT_vol = gmpy2.mpfr(float(5.0))
print("5  RT:      {:42.40f}".format( Tmod_RT_vol ))
print("Emod:       {:42.40f}".format( Emod_RT_vol/120 ))
print("Tmod:       {:42.40f}".format( Tmod_RT_vol/120 ))

RT5 radius: 0.9994833322623434400464260276814215953818
SuperRT:   21.2132034355964257320253308631454711785451
5+ RT:      5.0077580313328385159327414401998356487032
5  RT:      5.0000000000000000000000000000000000000000
Emod:       0.0417313169277736542994395120016652970725
Tmod:       0.0416666666666666666666666666666666666667