# 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
W+mW+ 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 Z4Y5A - 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
K5 / 4(W+mW+) , 6Z4Z4(P4) can be combined into a single tunnel by connecting them with eight further copies of m. These now take up a cubic shape which I term [8W+12m]C. This again can be combined with a 6Z4Z4(P4) tunnel to make a remarkable toroid K5 /[8W+12m]C , 6Z4Z4(P4) which is six-fold threaded .

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

As the
[8W+12m]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 J914C 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

The copies of m in the majority of the above toroids in sections 2.1 and 2.2 can be replaced with copies of Z4, the exception being those containing 6Z4Z4(P4) as the Z4 links will interfere with at least one of the Z4(Z4(P4)Z4)Z4 rods. A new possibility arises as two copies of Z4 extending toward each other can then be joined by P4. This can be done up to six times to form K5 / [8W+6(Z4(P4)Z4)]C

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

The outer faces of the cubes can then be connected to K5 with six further copies of Z4 to form
K5 / [8W+6(Z4(P4)Z4)]C6Z4 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
J91J63-E can also be linked with cubes as above. Alternatively, the triangular faces can be linked with a further J91 which can be drilled with Z4. Only one such tunnel can be formed though as the J91/Z4 is almost central to the K5.No such substitutions of m or m* are possible with the Q5S5(m*)S5Q5 toroids in section 1.