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

2. Toroids based on K₅ with W⁺mW⁺ tunnels.

2.1 Five Independent Tunnels	2.2 K5 with ringed tunnels and with one tunnel passing through another
2.3 Variations on m

2.1 Five independent tunnels

On page 168 Stewart describes W (link), a non-convex polyhedron related to J₉₂ which can be formed by augmenting J₉₂ with three pentagonal pyramids, then extracting three pentagonal pyramids adjacent to the hexagonal face. The result is a polyhedron with coplanar faces, this coplanarity can be resolved by excavating tetrahedra. In order to maximise flexibility for subsequent excavations, I have chosen to excavate one tetrahedron from the centre of each complex of three coplanar triangles and three tetrahedra from the corner triangles of the large coplanar triangular face to create what I shall term W⁺ (alternatively WY₃^-6)

W+

Two copies of W⁺ can be connected by m to form the complex W⁺mW⁺. The reason for the excavation of three tetrahedra from the large triangular face rather than just the central one is to prevent coplanarity between W⁺ and m. Alternatively m* could have been used such that the excavations would have been W"m*W". In this section I have chosen to use W⁺ and m. Note: although the same connecting polyhedrons m or m* can used in both this section and the previous one, they are not the same set of polyhedra as they are in different positions and different orientations. It also does not appear that they can be directly connected in any single tunnel.

W+mW+

The position of the hexagonal faces in W+ is such that

can be excavated from K₅ to form a tunnel. The proof of this is similar to the situation in section 1. As W⁺ has the same height from the hexagon to the opposing triangle as J₉₂ then W⁺ will connect K₅ to a central E₅. The internal vertices of the pentagonal faces of two adjacent copies of W⁺ can be shown to be one edge length apart and can be connected by a square face (to do this augment adjacent W⁺ with Z₄Y₅^A - this will introduce faces from S₅, the relevant vertices will be connected by edges of S₅). 5-4-3 vertices can then be formed using these square faces, the triangular face from the W⁺ and a diagonal across the pentagonal face of E₅ to show the same dihedral angle ABC as in section 1

The angle ABC between two adjacent copies of W+ appended to E5.

Unfortunately while W⁺ spreads less than J₉₂ it is still not possible to excavate two adjacent hexagonal faces of K₅, so a maximum of four such tunnels can be excavated simultaneously to form the toroid K₅ / 4(W⁺mW⁺)

Four independent tunnels: K5 / 4(W+mW+)   Genus p=4.

If the excavations are m rather than m* the tunnels do not penetrate the central E₅. This allows a fifth independent tunnel with six entrances of the form 6Z₄Z₄(P₄) to also be drilled through this toroid.

Five independent tunnels K5 / 4(W+mW+) , 6Z4Z4(P4)  Genus p=9.

 

2.2 K₅ with ringed tunnels and with one tunnel passing through another

Rings are also possible. The holes in these rings are again large enough to admit threading by a further tunnel from either decagonal-to-decagonal face or from square to square. The example shown below shows an S* containing 3 copies of R₅ threading the [W⁺m]^5C ring.

Threaded tunnels K5 / [W+m]5C, S*; Genus p=6.

The rod Q₅E₅S₅²Q₅ will also pass through this ring meaning the toroids generated in section 1.3 can also be generated with this ring tunnel.

Threaded tunnels K5 / [W+m]5C, Q5E5S52Q5; Genus p=6.

The W⁺mW⁺ tunnels in

which is six-fold threaded .

K5 / [8W+12m]C, 6Z4Z4(P4); Genus p=17.

.

As the

tunnel would not penetrate a central E₅, there is room in the centre of the polyhedron to replace the central 6Z₄(P₄) with 6J₉₁(P₄). However, instead of inserting the whole complex, instead form a toroid J₉₁^4C with a central square hole from four copies of J₉₁ (alternatively this could be described as E₅ / 8G₃2J₉₁(P₄) ). A toroid can now be formed with one tunnel, which is now two storied, threading another four times.

K5 / [8W+12m]C, 4Z4J914C; Genus p=16.

2.3 Variations on m

K5 / [8W+6(Z4(P4)Z4)]C ; Genus p=18.

6Z₄ with genus p=24

K5 / [8W+6(Z4(P4)Z4)]C6Z4; Genus p=24.

It is also tempting to replace m with J₉₁. However, this does not result in a valid toroid as one vertex of J₉₁ lies on the decagonal face of K₅. Instead excavate J₆₃ from J₉₁ to remove the problematic vertex and form toroids containing the combination W⁺(J₉₁J₆₃^-E)W⁺. Again, most of the above toroids in sections 2.1 and 2.2 can be formed with this combination. The square faces of