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`3,6-duoprism:`

- First ring can be augmented with triangle||6-prism. Adjacent augments

allowed.

- 2 augments can be ortho (o) or gyro (g)

- 3 augments can be ortho-ortho (oo) or ortho-gyro (og)

- Second ring augmentable with 3-prism pyramid.

- Enumeration:

- 1 augment

- 2o augments

- 2g augments

- 3oo augments

- 3og augments

- Total: 5+4=9 augmentations (4 augmentations from 3-prism pyramids in the

second ring)

4,6-duoprism:

- First ring augmentable with triangle||6-prism. Adjacent augments NOT allowed.

- So only up to 2 augments, ortho (o) or gyro (g).

- Second ring augmentable with 4-prism pyramid.

- Cross-ring angle sum ~= 102.69°, so only one ring can be augmented at a time.

- Enumeration:

- 1 augment

- 2o augments

- 2g augments

- Total: 3+4=7 augmentations (4 augmentations with 4-prism pyramid).

5,6-duoprism:

- First ring augmentable with triangle||6-prism. Adjacent augments NOT allowed.

- Only up to 2 augments, meta (m) or para (p). Both cases can be either

ortho (o) or gyro (g).

- Second ring augmentable with 5-prism pyramid.

- Cross-ring angle sums:

- Vertex-aligned: 70.97°

- Edge-aligned: 79.19°

- So both v-aligned and e-aligned cross-ring augments are allowed.

- Cross-ring constraints:

- When the first ring has no augments, the second ring has the set of 12

augmentations by 5-prism pyramids.

- When the first ring has 1 augment, the second ring becomes oriented, and

so will have > 12 augmentations.

- When the first ring has 2 ortho augments, the second ring also becomes

oriented, and has the same number of 5-prism pyramid augmentations as

the 1-augment case.

- When the first ring has 2 gyro augments, the split orientations become

equivalent again, so it has the set of 12 augmentations as in the

no-augment case.

- Enumeration:

- No augment in 1st ring:

- .....*

- ....**

- ...*.*

- ...***

- ..*..*

- ..*.**

- ..****

- .*.*.*

- .*.***

- .**.**

- .*****

- ******

- Subtotal: 12 augmentations

- 1 augment in 1st ring (imposes alternating configuration on 2nd ring):

(* = vertex-aligned, # = edge-aligned)

- .....*

- ....#.

- ....#*

- ...*#.

- ...*.*

- ..#.#.

- ...*#*

- ..#*#.

- ..#..*

- ..#.#*

- .*.*#.

- ..#*#*

- .*.*.*

- #.#.#.

- .*.*#*

- #.#*#.

- .*#.#*

- .*#*#*

- #*#*#.

- #*#*#*

- Subtotal: 20 augmentations

- 2 ortho augments in 1st ring:

- Same as 1 augment case

- Subtotal: 20 augmentations

- 2 gyro augments in 1st ring:

- Same as 0 augment case

- Subtotal: 12 augmentations

Total: 12+20+20+12 = 64 augmentations

6,6-duoprism:

- Only non-adjacent augments allowed, can be meta (m) or para (p). No

cross-ring augments allowed.

- Enumeration:

- 1 augment

- 2m augments

- 2p augments

- 3 augments (can only be meta)

- Total: 4 augmentations.

As you can see, the number of augmentations is quite small (<10) except for the 5,6-duoprism where the count jumps to 64 suddenly, and then drops back down quickly.