Flying Further into the K5 Galaxy
An extension to Chapter XVI of 'Adventures Among The Toroids' by B.M.Stewart 



4. Toroids based on K5 with Z4Y5Z4 tunnels

4.1  Six Independent Tunnels


Two adjacent copies of Z4 can be excavated from K5 and if suitably oriented can be connected with a pentagonal prism (Y5).  See the bottom of the page for a note on orientation.  The validity of this construction is as follows:  The height of Z4 is the same as that of J91 which has a width of Φ.  The relevant vertices of Z4 in the required orientation ("gZ4") are both also vertices of J91.  They therefore also lie on the vertices of an S5 in the Q5S5 excavation of K5 and are 1 unit apart.  The Y5 connector can be any one of a number of facets of the icosahedron, but most then require additional augmentation to avoid coplanarity issues.  This particular orientation of Z4 and Y5 appears unique in avoiding such issues.

gZ4Y5gZ4

Six such excavations can be made from E5 resulting in the toroid K5 / 6gZ4Y5gZ4.  Choices exist as to which excavations to make.


K5 / 6gZ4Y5gZ4  Genus p=6
.

4.2  K5 with ringed and threaded tunnels

A ring of 3 copies of Z4 can be linked with Y5. If suitably oriented then co-planarity can be avoided.  Four such rings can be excavated from E5 to form a toroid K5 / 4[gZ4Y5]3C with tetrahedral symmetry.


K5 / 4[gZ4Y5]3C  Genus p=12
 
A ring of 10 copies of Z4 can also be excavated around the equator of E5.  If suitably oriented, connections of Y5 can again be made, but 5 of these connections will require an additional tetrahedron (Y3) to be added to avoid coplanarity.  The resulting toroid can be termed K5 / [gZ4Y5gZ4Y5Y3A]5C

K5 / [gZ4Y5gZ4Y5Y3A]5C   Genus p=10.
Y5 in yellow,  Y5Y3A in green.


A sceptre (S*) see section1, can be threaded through the ring to give K5 / [gZ4Y5gZ4Y5Y3A]5C , S*


K5 / [gZ4Y5gZ4Y5Y3A]5C , S*.  Genus p=11.
Here S* contains a Stewart A5'' (link)

Returning to Z4:  5 copies of Z4 can be excavated around one of the decagons of E5, and if suitably oriented can be connected with a central S5.  Again coplanarity issues arise which can be resolved with Y3 augmentations.

5gZ4(S5Y35A)


Two copies of 5gZ4(S55Y3A) can be excavated from opposite ends of K5.  As the faces at the base of Z4 are on the vertices of a central E5, one pair of opposing faces can be connected by a Z4(P4)Z4 rod.


K5 / 2[5gZ4(S5Y35A)](Z4(P4)Z4).  Genus p=9.

If care is taken as to the orientation of the copies of Z4 in the rod, the ring [gZ4Y5gZ4Y5Y3A]5C can also be excavated from E5 leaving a threaded toroid of genus p=19 where the ring has genus p=10 and the rod genus p=9.


K5 / 2[5gZ4(S5Y35A)](Z4(P4)Z4) , [gZ4Y5gZ4Y5Y3A]5C.  Genus p=19.


4.3  A genus p=53 Stewart Toroid

Returning to the 5gZ4(S5Y35A) above.  All 30 square faces of E5 can be excavated with gZ4 and to connect them with 20 copies of S5.  Coplanarity issues can again be resolved with augmentation by Y3.  The resulting toroid can be termed K5 / [30gZ420(S5Y35A)]C (strictly speaking, not all 100 Y3 are actually necessary as occasionally the gZ4s will be aligned such that there is no coplanarity).  This toroid has p=48.

K5 / [30gZ420(S5Y35A)]C   Genus p=48.
Z4 in red, S5 in yellow, Y3A in green

The above excavations of K5 leave the square and pentagonal faces of the notional central E5 in place.  Thee can be further excavated with the complex 6Z4(P4) to generate the toroid K5 / [30gZ420(S5Y35A)]C(6Z4(P4)) with genus p=53.
  


K5 / [30gZ420(S5Y35A)]C(6Z4(P4))   Genus p=53.
Colours as above plus 6Z4(P4) in orange.

Note: Z4 and gZ4

On page 135, Stewart describes a toroid of genus p=46, his 'Holey Monster', K5 / 12Q5S5(E5/6X(P4))30Z4.  This was the highest genus he could obtain.  It is interesting that in Stewart's examples he orients each Z4 such that it's symmetry axis is parallel to the square-hexagon edges of E5.  This then allows the decagonal faces to also be excavated with Q5 but does not allow the S5 linkage between the Z4s.  In the examples on this page above each Z4 is oriented it's symmetry axis is parallel to the square-decagon edges of E5.  This allows the S5 linkage but now does not allow the degagonal faces to be excavated as the triangular edges of Q5 and Z4 now co-incide.  I have left Stewart's square-hexagon orientation as Z4 but term the square =decagon orientation as gZ4.

Next: A Solution to Stewart Exercise 153.
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