Flying
Further
into
the K5 Galaxy
An
extension
to
Chapter
XVI
of
'Adventures
Among
The
Toroids'
by
B.M.Stewart
3. Threaded examples involving
dodecahedra
Copies of J92
excavated from K5 can be linked with dodecahedra (D5).
One
vertex
of
this
D5 lies on a hexagonal face of K5.
This vertex can
be removed by excavating G3 (link).
Up
to three independent tunnels
can be
formed by excavating J92(D5G3-E)J92
from K5. A ring can be formed of
five dodecahedral links. Neighbouring copies of D5 now share
a common
edge, this is resolved by adding J63. The toroid K5
/ [J92D5G3-EJ63]5C
can then be generated
K5 /
[J92D5G3-EJ63]5C
Genus
p=5
The hole in this ring can be enlarged
by excavating a further copy of
G3 from each D5, this hole is then large enough
to allow a specifically
sequenced S* to pass through to form the threaded toroid K5 / [J92D5G3-2EJ63]5C, S*.
K5 /
[J92D5G3-2EJ63]5C, S* Genus p=6
.
The connection between D5
and J63 means that the J92 excavations can be
omitted. In this case each J63 can be connected to the K5
with a Q5 to
form the threaded toroid K5 / [Q5J63D5G3-2E]5C,
S*
K5 /
[Q5J63D5G3-2E]5C,
S*
Genus
p=6
It is this toroid which allows the
proof of the validity of this
family. The triangular faces of J63 are coincident with the
faces of S5
in section 2.1. Two facing pentagonal faces have
their nearest vertices
1 edge length apart (as they are neighbouring S5 vertices),
the two
vertices shared with the Q5 are Φ units apart (as they are
also
vertices of J91). On page 187 Stewart dissects E5 into 6J91
• (P4) •
12Y5 • 8G3. The top vertices of our copies of J63
are the top vertices
of Y5 in this dissection, they are thus also separated by Φ
units.
Thus, pentagons can be inserted between the two copies of J63
to
generate 5-5-5 vertices which show the pentagons are faces of a
dodecahedron.
Separation of J63 showing validity of
connection by
pentagonal faces.
