**Augmenting the great snub icosidodecahedron**

The great snub icosidodecahedron, vertex figure (^{5}/_{3},3,^{5}/_{2},3,3,3)

This polyhedron offers a number of interesting
augmentations. The ^{5}/_{3} face can be excavated by a prism
or an antiprism. The ^{5}/_{2}
face can be excavated by a 5/3 cupola.
Combinations of these can occur. Prisms can be
augmented with pyramids or with 5/3
cupola . Cupolas can be augmented
with pyramids . The diagram below is
an attempt to summarise the augmentations found to date:

The notation above refers
first to
the polyhedron excavating the ^{5}/_{3} face (mandatory) and
after the comma, the polyhedron excavating the ^{5}/_{2} face
(optional). Where a polyhedron is subsequently augmented or excavated, a
second letter is added.

**P** refers to an excavation by a ^{5}/_{2}
prism, **
S** to an excavation by a

Q

Y

**PY _{4}** is a

PQ

Colour coding in all subsequent models is as follows:

^{ 5}/_{2}and^{5}/_{3}face of great snub icosidodecahedron: light green

3 face of great snub icosidodecahedron: blue

^{ 5}/_{2}face of^{5}/_{2}prism: dark green

4 face of^{5}/_{2}prism: orange

3 face of square pyramid augmented to^{5}/_{2}prism: orange4 face of

^{5}/_{3}cupola: yellow

3 face of^{5}/_{3}cupola: cyan

^{ 10}/_{3}face of^{5}/_{3}cupola: red

3 face of square pyramid augmented to^{5}/_{3}cupola: yellow

^{ 5}/_{2}face of^{5}/_{2}antiprism: dark green

3 face of^{5}/_{2}antiprism: pink

Left (P 69a): Each
retrograde pentagrammic face of the great snub icosidodecahedron can be
excavated with a pentagrammic prism completing a cycle
around the axis. This has the effect of uncrossing the original
vertex figure. The ^{5}/_{3} in the vertex figure is replaced by
(4,4) The original vertex figure becomes (__4,4__,3,^{5}/_{2},3,3,3)/2
with new vertices of (^{5}/_{2},4,4). All
vertices are on the exterior of the polyhedron.

Centre (PY_{4}
69aa): Each
square face of the pentagrammic pyramids in the above polyhedron (69a) be further augmented with a square
pyramid. Each 4 in the vertex figure is replaced by a (3,3). The original vertex figure becomes
(__3,3__,__3,3__,3,^{5}/_{2},3,3,3)/2. The (^{5}/_{2},4,4)
vertices in the first augmentation become (^{5}/_{2},__3,3__,__3,3__)
vertices. New vertices are of the form (3,3,3,3). The figure is
dominated by the square pyramids. The original vertices
remain just visible. All vertices are on the exterior of the
figure. Unfortunately I have not
yet been able to generate a VRML model of this polyhedron, but clicking on the
image above will show a larger image.

Right
(PQ 69ab): Alternatively, the cap of each of the pentagrammic
pyramids in the above polyhedron (69a) can be further augmented with a 5/3-cupola.
The cupolas themselves are not locally convex but the act of joining them to the
pentagrammic caps removes the retrograde polygon. The original vertices of (4,4,3,^{5}/_{2},3,3,3)/2
are unaffected. The ^{5}/_{2} in the (^{5}/_{2},4,4)
vertices is replaced by (4,3,4)/2 to give (__4,3,4__,4,4)/2
vertices. New vertices are added of the form (^{10}/_{3},3,4).
All
vertices are on the exterior of the polyhedron.

Left (S
69b): Another augmentation of the original great snub icosidodecahedron
can be
obtained by excavating each retrograde pentagrammic face with a pentagrammic
antiprism
which cycles the axis. This has the effect of uncrossing the original
vertex figure. The ^{5}/_{3} in the vertex figure is replaced by
(3,3,3) The original vertex figure becomes (__3,3,3__,3,^{5}/_{2},3,3,3)/2
with new vertices of (^{5}/_{2},3,3,3). All
vertices are on the exterior of the polyhedron.

Right (S,Q 69d) : Each
retrograde pentagrammic face of the great snub icosidodecahedron can be
excavated with a pentagrammic anti-prism completing a cycle
around the axis. Each prograde pentagrammic face of the great snub icosidodecahedron
is then excavated
with a 5/3-cupola.
The cupolas themselves are not locally convex but the act of joining them to the
pentagrammic caps removes the retrograde polygon. The overall effect is to
uncross the original
vertex figure. The ^{5}/_{3} in the vertex figure is replaced by
(3,3,3) and the ^{5}/_{2} is replaced by (4,3,4)/2 The original vertex figure becomes (__3,3,3__,3,__4,3,4__,3,3,3)/3
with new vertices of (^{5}/_{2},3,3,3) from the antiprisms and (^{10}/_{3},3,4)
from the cupolas. All
vertices are on the exterior of the polyhedron.

Left (P,Q 69c) : Each
retrograde pentagrammic face of the great snub icosidodecahedron can be
excavated with a pentagrammic prism completing a cycle
around the axis. Each prograde pentagrammic face of the great snub icosidodecahedron
is then excavated with a 5/3-cupola.
The cupolas themselves are not locally convex but the act of joining them to the
pentagrammic caps removes the retrograde polygon. The overall effect is to
uncross the original
vertex figure. The ^{5}/_{3} in the vertex figure is replaced by
(4,4) and the ^{5}/_{2} is replaced by (4,3,4)/2 The original vertex figure becomes (__4,4__,3,__4,3,4__,3,3,3)/3
with new vertices of (^{5}/_{2},4,4) from the prisms and (^{10}/_{3},3,4)
from the cupolas. All
vertices are on the exterior of the polyhedron.

Right (PQ,Q 69ca) :
A second
order augmentation can again be formed from the previous polyhedron (69c) by excavating the pentagrammic prismatic caps
with 5/3-cupolas. The original vertices of
(4,4,3,4,3,4,3,3,3)/3
are unaffected. The ^{5}/_{2} in the (^{5}/_{2},4,4)
vertices is replaced by (4,3,4)/2 to give (__4,3,4__,4,4)/2
vertices. A second set of (^{10}/_{3},3,4)
vertices are added. All
vertices are on the exterior of the polyhedron.

Left (P,QY_{4}
69cb) : Another second
order augmentation can again be formed from the previous polyhedron (69c) by
augmenting the square faces of the 5/3-cupolas
with square pyramids. The 4's
originating from the cupolas in the original vertices of
(4,4,3,4,3,4,3,3,3)/3
are replaced by (3,3) to give vertices of (4,4,3,__3,3__,3,__3,3__,3,3,3)/3. The
(^{5}/_{2},4,4)
vertices are unaffected. The 4 in the (^{10}/_{3},3,4)
vertices is again replaced by (3,3) to give vertices of the form (^{10}/_{3},3,__3,3__).
New vertices are added of the form (3,3,3,3). All
vertices are on the exterior of the polyhedron.

Right (PQ,QY_{4}
69cc) : In a combination of the previous two
operations, the square faces of the 5/3-cupolas
can be excavated with square pyramids and
the pentagrammic prismatic caps
with 5/3-cupolas. The 4's originating
from the cupolas in the original vertices of
(4,4,3,4,3,4,3,3,3)/3
are replaced by (3,3) to give vertices of (4,4,3,__3,3__,3,__3,3__,3,3,3)/3.
The ^{5}/_{2} in the (^{5}/_{2},4,4)
vertices is replaced by (4,3,4)/2 to give (__4,3,4__,4,4)/2
vertices. The 4 in the (^{10}/_{3},3,4)
vertices from the original set of 5/3 cupolas is replaced by (3,3) to give
vertices of the form (^{10}/_{3},3,__3,3__). New
vertices are added of the form (3,3,3,3). A new set of (^{10}/_{3},3,4)
vertices is added. All
vertices are on the exterior of the polyhedron.

Note: the following further combinations did not lead to
valid augmentations:

PY_{4}Q PY_{4},Q P,QY_{3}
SQ S,QY_{4} S,QY_{3}