Oregon Curriculum Network
School of Tomorrow
These three quants (quantim modules) are not in Synergetics as published, but do extend the same ideas. David Koski is behind this naming schema, with the U an allusion to Tell Anderson's U.
V6 made of (3U+2V+1W)
from IPython.display import YouTubeVideo
YouTubeVideo("EbDKiOCcfBY")
Synergetics as published focused on the A, B, T (1/24), E and S modules.
To review, A and B modules together build the regular tetrahedron and octahedron, the complementary space-fillers of the IVM. 24 As (12 left, 12 right) build the regular tetrahedron.
S module: 24 of them (12 left, 12 right) nestle in the voids twixt the octa of volume 4, and a faces-flush icosahedron of volume approximately 2.91.
The E module: 120 of them (60 left, 60 right) build the thirty diamond-faced rhombic triacontahedron (RT). The RT made of Es "shrink wraps" an IVM ball.
The RT made of 120 Ts have a volume of precisely 5. The E and T share the same angles (have the same shape) and differ in size only minutely.
U3 means all edges phi-up (φ scaled) so volume goes up as a factor of φ to the 3rd.
u3 means all edges phi-down (1/φ scaled) so volume decreases as a factor of (1/φ) to the 3rd.
Ditto for other letters (E3, S3, v3, w3 and so on).
from math import sqrt as rt2
φ = (rt2(5)+1)/2 # golden ratio
Syn3 = rt2(9/8) # not to be confused with Smod
S = (φ**-5) / 2 # home base Smod
Cubocta = 20
SuperRT = Syn3 * Cubocta
E3 = (rt2(2) / 8)
E = E3 * (φ**-3) # home base Emod
e3 = E * (φ**-3)
e6 = e3 * (φ**-3)
S_factor = S / E # 2*sqrt(7-3*sqrt(5))
For computing tetrahedron volumes from six edges, we have a choice of formulae. I'm using Gerald de Jong's (GdJ). David works with Piero della Francesca's.
from tetravolume import Tetrahedron as T
Expected volume: E3 + E or $(\sqrt{2}/4)\phi^{-1}$
W = E3 + E
W
0.21850801222441055
(rt2(2)/4)*φ**-1
0.21850801222441052
# in R = 1 units
a = rt2(φ**-2 + 1)
b = rt2(φ**2 + 1 )
c = rt2(3)
d = rt2(3)
e = rt2(3*φ**-2)
f = rt2(φ**-2 + 1)
a, b, c, d, e, f
(1.1755705045849463, 1.902113032590307, 1.7320508075688772, 1.7320508075688772, 1.07046626931927, 1.1755705045849463)
T(a/2, c/2, e/2, b/2, f/2, d/2).ivm_volume() # D = 1
Expected volume: 2E3 or $(\sqrt{2}/4)$
U3 = 2 * E3
U3
0.3535533905932738
rt2(2)/4
0.3535533905932738
# in R = 1 units
a = rt2(φ**-2 + 1)
b = rt2(φ**2 + 1)
c = rt2(3)
d = rt2(3)
e = rt2(φ**2 + 1)
f = rt2(φ**-2 + 1)
a, b, c, d, e, f
(1.1755705045849463, 1.902113032590307, 1.7320508075688772, 1.7320508075688772, 1.902113032590307, 1.1755705045849463)
T(a/2, c/2, e/2, b/2, f/2, d/2).ivm_volume() # D = 1
Expected volume: V3 = 3E3 + E or or $(\sqrt{2}/4)\phi$
V3 = 3 * E3 + E
V3
0.5720614028176844
(rt2(2)/4)*φ
0.5720614028176844
a = rt2(φ**-2 + 1)
b = rt2(φ**2 + 1)
c = rt2(3)
d = rt2(3)
e = rt2(φ**2 + 1)
f = rt2(3*φ**2)
a, b, c, d, e, f
(1.1755705045849463, 1.902113032590307, 1.7320508075688772, 1.7320508075688772, 1.902113032590307, 2.8025170768881473)
V3 = T(a/2, c/2, e/2, b/2, f/2, d/2).ivm_volume() # D = 1
V3
V3/U3
U3/W
1.618033988749895
U = U3 * (φ**-3)
V = V3 * (φ**-3)
U + V
W
0.21850801222441055
U/V
V/W
V + W
U3
0.3535533905932738