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

1. Toroids based on K5 with Q5S5(m*)S5Q5 tunnels.

1.1 Six Independent Tunnels

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

1.3 Rods including E5

1.4 Very long tunnels and loops

1.5 Two interlinked tunnels

1.1 Six independent tunnels

A number of toroids can be generated from a truncated icosidodecahedron K5 by drilling the decagonal faces with a pentagonal cupola (Q5) then an antiprism (S5). The inner pentagonal faces of the antiprisms could then be drilled by a central rhombicosidodecahedron (E5) to generate toroids of genus up to 11. It is this approach that led to Stewart's four tunnelled model where he further drilled these faces in sets of three with pentagonal pyramids (Y5) and a non convex polyhedron he terms G3 (see page 129).

Rather than drilling the inner faces of the antiprisms, consider the lateral faces edge adjacent to the inner pentagons. On page 157 Stewart describes a non convex polyhedron he terms m* (link). This fits perfectly between the lateral faces of two adjacent antiprisms to form Q5S5(m*)S5Q5.
 

Q5S5(m*)S5Q5

To show this, consider the toroid K5 / 30gZ4(E5) (link). Z4 has the same square to square height as J91 meaning the dihedral angle ABC of J91 must be complementary to the angle A'B'C' formed by two adjacent copies of S5 in the toroid K5 / nQ5S5(E5) (link). Now consider excavating J63 from J91 to get J91J63-E (link) which contains a 5-4-3 vertex and still contains the angle ABC. The dihedral angle between the square and triangular faces adjacent to this vertex must be the same as the equivalent dihedral angle of m*. Therefore the dihedral angle A"B"C" in m* (and in m) is the same as ABC and A'B'C' (see below).

Equivalent dihedral angles (a) A'B'C' in K5 / nQ5S5(E5),
(b) ABC in J91, (c) ABC in J91J63-E ,(d) A"B"C" in m*

With the 12 Q5S5 excavations of K5, there are 30 potential such connections between copies of S5 but unfortunately it is not possible to connect all 30 at once without the m* excavations interfering. The maximum number I have connected without interference is 20. There is a simpler polyhedron 'm' (link) which would also connect two antiprisms but this results in coplanar faces. This can be resolved by excavating two tetrahedra from the antiprisms. However, in this section I have chosen to use m*.

Six pairs of faces can be selected in a number of ways, each of which results in a toroid with six independent tunnels, each one being of the form Q5S5(m*)S5Q5

Toroid with six independent tunnels
K5 / 6(Q5S5(m*)S5Q5). Genus p=6


On page 138 Stewart connects opposite pentagonal faces of the E5 with a combination of an antiprism, two cupolas and a dodecahedron S5(gQ52)D5. This means two opposite faces of K5 can be connected with the combination Q5S5(S5(gQ52)D5)S5Q5 (for brevity I term this a "sceptre" or S*). The central section can also be of the form "R5Q5D5" where R5 is the pentagonal rotunda. The Q5S5 ends of the sceptre can also be replaced by pentagonal rotundas (R5) meaning variations on S* can be formed with 0, 1, 2 or 3 copies of R5. Some variation in the sequence of the elements is also possible. Also, given the S5S5 combination in some versions of S*, chapter XIII 'Accordion Polyhedra' opens up many further possibilities including using the Stewart A5 (link) or A5'' (link). My notation S* will normally refer to any of these possible variations. An alternative 6 holed toroid can be generated by excavating K5 once with S* and then five times with the Q5S5(m*)S5Q5 combination. Again options exist for the selection of the pairs.

The 'sceptre' or S*, in this instance: Q5S5(S5(gQ52)D5)S5Q5.

Toroid with six independent tunnels K5/5(Q5S5(m*)S5Q5), S*; Genus p=6.


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


Note: I introduce the term 'threaded' to refer to a set of tunnels where one passes through another. "N-fold threaded" can refer either to one tunnel passing through n others or one tunnel passing through another n times or a combination of both.

The tunnelled decagons can be grouped in rings of 3. Stewart does not have any specific notation for cyclic polyhedral, so I introduce the terminology [X]nC to denote a cycle of n copies of polyhedron X as in [Q5S5m*]3C below.

[Q5S5m*]3C

Four copies of [Q5S5m*]3C can be excavated from K5 to form a toroid with tetrahedral symmetry and four independent tunnels.
 

K5/4[Q5S5m*]3C
Four independent tunnels.  Gensus p=12.

Rings of 4 are also possible

[Q5S5m*]4C
Four independent tunnels.  Gensus p=12.

Three copies of [Q5S5m*]4C can be excavated from K5 to form a toroid with chiral 3-fold dihedral symmetry and three independent tunnels.

K5/3[Q5S5m*]4C
Three independent tunnels.  Gensus p=12.


There is also a ring of five such excavations [Q5S5m*]5C


Ring formed by [Q5S5m*]5C

This ring has a hole large enough to accommodate S* (of any form) without interference.

Threaded tunnels K5/[Q5S5m*]5C , S*; Genus p=6.


A second ring can also be excavated from the five remaining decagons such that S* threads both rings.

2-fold threaded tunnels K5/2[Q5S5m*]5C , S*; Genus p=11.

Terming the two holes as the 'ring' and the 'rod', then assuming all decagons are excavated and by selectively including connecting copies of m*, alternatives exist where ring can utilise any number between 5 and 10 decagons with the rod utilising the remaining 7 to 2 decagons. The number of holes can also be arbitrarily reduced by replacing the Q5 such that the rod or ring remains connected by the underlying S5.

 In the two images below (a) The ten decagons in the two rings above are instead connected by copies of m* in a zig-zag fashion to generate one alternating ring which S* threads; and (b) S* is connected by 5 copies of m* to one ring generating a toroid where a hole with 7 entrances threads a hole with 5 entrances.

Threaded tunnels K5/[Q5S5m*]10C , S*; Genus p=11.

Threaded tunnels K5/[Q5S5m*]5C , [Q5S5m*]5C(S*5m*); Genus p=11.

A ring of six excavations can also be excavated leaving a cap of three decagons at each end.


Ring formed by  K5/[Q5S5m*]6C

A hole can be excavated through these caps of K5 from either the decagonal, square or hexagonal faces without interfering with this ring.

Decagonal case: S* (of any form) can then be excavated between these caps even though it is skew to the ring




Square case: Alternatively, and somewhat closer to Stewart's original attempts, a hole of the format Z4(Z4(P4)Z4)Z4 can also be excavated in a skew fashion through this ring (P4 is a cube, see Stewart page 132 for Z4 (link)). The two internal copies of Z4 can also be replaced by J91.


Threaded tunnels K5/[Q5S5m*]6C , Z4(Z4(P4)Z4)Z4; Genus p=7.

Hexagonal case: Stewart does not give an explicit method of connecting opposite hexagonal faces of K5, although one can be implied. E5 can be drilled between triangular faces with the complex X(P4)X (X (link) is derived from J91 - see Stewart page 134). Then J92 can be used to connect a hexagonal face of K5 to a triangular face of E5 (see Stewart page 186). Thus a complex of J92(X(P4)X)J92 can be excavated from K5.


  K5/J92(X(P4)X)J92; Units coloured separately for clarity. Genus p=1.

This can be done in conjunction with the above ring of six to give further examples of threaded tunnels. (J91 cannot be used in place of X as this causes the holes to have a common vertex).




A further possibility exists with the ring of six excavations. In this case for clarity, only two of the Q5 will be excavated making the ring Q5[S5m*]6CQ5. Append two further copies of m* on opposite sides of the ring and join them with the central section of S*. The resulting toroid has one tunnel which is self-threading K5/Q5[S5m*]6CQ5S52Q52D5S5


A self-threading tunnel. K5/Q5[S5m*]6CQ5S52Q52D5S5 ;Genus p=3.
The 'ring' is shown in yellow.
The connecting M* in blue and the S* section in red


1.3 Rods including E5


Many of the toroids above are based on K5 / nQ5S5(E5) - Stewart discusses variations on this toroid on page 188. In the case with just two opposing cupolas, we can form K5 / Q5S5(E5)S5Q5. The polyhedra in the tunnel can be re-sequenced to form K5 / Q5E5S5S5Q5 with an off centre E5


K5 / Q5E5S5S5Q5.  Genus p=1

This can be used in conjunction with the ring of five Q5S5m* excavations [Q5S5m*]5C to form the threaded toroid K5 / [Q5S5m*]5C, Q5E5S5S5Q5 with a clear separation between the tunnels.


K5 / [Q5S5m*]5C, Q5E5S5S5Q5.  Genus p=6.


E5 can itself be tunnelled in a number of ways (Stewart devotes Chapter XII 'Opening the E5-Realm' to such toroids). Examples being E5 / 6X(P4) (link) and E5 / 6J91(P4) (link) - which is now valid as it is not the exterior of the toroid. These can both be used to generate threaded toroids where the rod has genus p=5


Threaded toroid with rod of genus p=5.
K5 / [Q5S5m*]5C , Q5(E5 / 6X (P4))S5S5Q5.
E5 translucent to show internal structure; Genus p=11.


Threaded toroid with rod of genus p=5.
K5 / [Q5S5m*]5C , Q5(E5 / 6J91 (P4))S5S5Q5.
E5 translucent to show internal structure; Genus p=11.

Stewart also describes an E5 based toroid with two independent tunnels which I shall denote as E5 / 2(Z4I5Y5-E) (link) (Stewart page 142) and a two storied toroid I shall denote as E5 / I5(J91 / Z4)I5 (link) (Stewart page 142). We can drill our E5 excavation in either fashion to form (i) a threaded toroid K5 / [Q5S5m*]5C , Q5(E5 / 2(Z4I5Y5-E))S5S5Q5 where the rod has two independent holes and (ii) a three storied threaded toroid K5 / [Q5S5m*]5C,Q5(E5 / I5(J91 / Z4)I5)S5S5Q5 where the rod is two storied.


Threaded toroid where rod has two independent tunnels.
K5 / [Q5S5m*]5C , Q5(E5 / 2(Z4I5Y5-E))S5S5Q5.
E5 translucent to show internal structure; Genus p=8.


Threaded toroid with two storied rod.
K5 / [Q5S5m*]5C,Q5(E5 / I5(J91 / Z4)I5)S5S5Q5.
E5 translucent to show internal structure; Genus p=8.

1.4 Very long tunnels and loops


Consider excavating all 12 decagonal faces of E5 with Q5S5 and connecting them in a spiral chain with m*, then re-instating ten of the copies of Q5 leaving just the two decagonal faces at the ends of the chain filled. This results in the toroid K5 / Q5[S5m*]11S5Q5 with just a single tunnel. Assuming unit edge lengths, the shortest distance through this tunnel (connecting to the closest points on the surface of the enclosing K5) can be shown to be > 22.90, this is over 3.328 times the decagon-to-decagon height of K5 (6.882…). Note there is still room through this figure to excavate between the two opposing copies of S5 with the remaining parts of S* (figure 2.4.2) to create K5 / K5 / Q5[[S5m*]11S5D5gQ52S5]CQ5 with a looped tunnel. The shortest distance around of this loop can be shown to be >26.01, more than 3.78 times the decagon-to-decagon height of K5. Note that these configurations were chosen partly for aesthetic reasons and there may well be similar configurations with longer tunnels and loops.


K5 / Q5[S5m*]11S5Q5 with a tunnel over 22.9 edge lengths long;
Genus p=1.



K5 / Q5[[S5m*]11S5D5gQ52S5]CQ5.
The previous toroid with the insertion of the remaining parts of S*
creating a loop over 26 edge lengths long; Genus p=2.

1.5 Two interlinked tunnels


In Ex 153 on page 193 Stewart muses whether there exists a toroid with two interlinked tunnels but did not find one. Here I show how it is possible to place two interlinked rings within K5. The generation of this toroid is however not straightforward. In essence the solution is as follows:

(i) insert two orthogonal polar tunnels through E5 avoiding the centre
(ii) connect these tunnels to opposite edges of K5 using Z4
(iii) use the S5m*chains from the previous section to connect the opposing copies of Z4 around the outside of E5 and hence around the outside of the other tunnel.

The solution to part (i) is based on E5 / 6J91(P4). Consider just three copies of J91 drilling E5 around the central P4 and replace the central J91 with m* to form the toroid E5 / J91(m*)J91. Now excavate J63 from each remaining J91. A remaining face of the central P4 can be removed by excavating m. However, as the J91 is not wide enough to accommodate m, it must first have the pentagons of the J91J63-E combinations augmented with pyramids. The resulting complex can be written in full as 2J91J63-EY52Am-E(m*), which for brevity I shall term L4


L4 - shown here inside E5

Two orthogonal copies of L4 can be drilled through E5 without interference to form E5 / 2L4. The resulting toroid has some issues: one copy of L4 extends beyond E5 and L4 itself also has coplanar faces.


E5 / 2L4. Not a Stewart Toroid.

The former is not an issue as the enclosing E5 will not be present in the final toroid, and the coplanarity issues can be dealt with by augmenting two triangles and excavating two triangles with tetrahedra to form L4* (or L4Y32AY3-2E)


L4* - coplanarity issues resolved.


With a Z4 attached at each end, the combination gZ4L4*gZ4 will drill K5 and the toroid K5 / 2gZ4L4*gZ4 can be formed (see here for more on gZ4)


K5 / 2gZ4L4*gZ4;Genus p=2.


The next step is to use the lateral faces of Z4 and connect them around the outside of where the E5 would have been with S5m* chains. Unfortunately, the same method cannot be used for both tunnels.

For the first tunnel, the connecting chain is 2S5Y3-Em*m*(S5) (The tetrahedral excavations are to avoid coplanarity).


First tunnel K5 / gZ4L4*gZ4[2S5Y3-Em*m*(S5)]C

For one tunnel this chain will fit inside the enclosing K5. However, for the other tunnel, orthogonal to this one, the chain does not sit in the same position relative to the faces of K5 and the outer points of the m*m* combinations extrude beyond the decagonal faces. An alternative chain for the second tunnel involves connecting S5 at right angles to the intended direction of the loop and then completing the connection with a skew S5m* chain. In full the connection is 2S5Y3-Em*S5(m*). Attempting to use this form of the connection for both tunnels results in them interfering.


Second tunnel: K5 / gZ4L4*gZ4[2S5Y3-Em*S5(m*)]C

With one of each form of the chain above, the resulting toroid has two interlinked tunnels which do not interfere and can be written as K5 / gZ4L4*gZ4[2S5Y3-Em*m*(S5)]C, gZ4L4*gZ4[2S5Y3-Em*S5(m*)]C.

Two interlinked tunnels:
K5 / gZ4L4*gZ4[2S5Y3-Em*m*(S5)]C, gZ4L4*gZ4[2S5Y3-Em*S5(m*)]C; Genus p=4
 

It is tempting to try to connect the two interlinked tunnels to form a toroid with a single knotted tunnel. My attempts to do so have though not been successful to date.

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