Stewart notation
The Stewart notation is used to describe Stewart toroids. It was developed by Bonnie Stewart in the book Adventures among the Toroids^{[1]}. It abbreviates certain polyhedra each to a letter and number, and shows how groups of them are connected to or removed from one another.
Convex polyhedral units[edit | edit source]
The series of polygonal prisms, polygonal antiprisms, and polygonal pyramids each have a letter associated with them. The polygon base of a polyhedron determines its number.
n -gonal pyramid | Y_{n } |
n -gonal prism | P_{n } |
n -gonal antiprism | S_{n } |
Three Platonic solids can be represented as members of these series. The two that can't have new letters set aside for them.
Tetrahedron | Y_{3} |
Cube | P_{4} |
Octahedron | S_{3} |
Dodecahedron | D_{5} |
Icosahedron | I_{5} |
The letters used for Archimedean solids loosely follow some rules: truncation of a polyhedron with 3 edges at a vertex = T, rectification = B, cantellation/expansion = E, omnitruncation = K. The numbers used are: 3 for polyhedra with tetrahedral symmetry, 4 for cubic symmetry, and 5 for dodecahedral symmetry.
These letters act like a weaker version of the Conway polyhedron notation's operators. Without an operator for bitruncation, Stewart notation takes advantage of the fact that the truncated octahedron is the omnitruncated tetrahedron, and sets aside another new letter for the truncated icosahedron.
The two snub Archimedean solids are barely mentioned in Adventures.
Truncated tetrahedron | T_{3} |
Truncated cube | T_{4} |
Truncated dodecahedron | T_{5} |
Truncated icosahedron | C_{5} |
Cuboctahedron | B_{4} |
Icosidodecahedron | B_{5} |
Small rhombicuboctahedron (Rhombicuboctahedron) |
E_{4} |
Small rhombicosidodecahedron (Rhombicosidodecahedron) |
E_{5} |
Truncated octahedron | K_{3} |
Great rhombicuboctahedron (Truncated cuboctahedron) |
K_{4} |
Great rhombicosidodecahedron (Truncated icosidodecahedron) |
K_{5} |
Snub cube (Snub cuboctahedron) |
sB_{4} |
Snub dodecahedron (Snub icosidodecahedron) |
sB_{5} |
Johnson solids are mostly notated by a "J" followed by their index in Norman Johnson's enumeration. However, the simpler dome-like ones that cannot be decomposed into other Johnson solids - the polygonal cupolae and the pentagonal rotunda, as well as the aforementioned pyramids - get their own letters.
Many Johnson solids are unused. Among the ones that are used, if they can be built with augmentations and elongations, they are usually referred to as a stack of smaller units. For example, the elongated pentagonal orthobirotunda J_{42} is usually referred to as R_{5}P_{10}R_{5}. This informs us of its structure, and keeps us from having to memorize or look up the indices of all 92 Johnson solids.
n -gonal cupola | Q_{n } |
pentagonal rotunda | R_{5} |
Some relatively common "J" ones | |
---|---|
Tridiminished icosahedron | J_{63} |
Bilunabirotunda | J_{91} |
Triangular hebesphenorotunda | J_{92} |
Other polyhedral units[edit | edit source]
Since Stewart toroids are by definition nonconvex, there is no reason to impose a restriction of convexity upon the polyhedra they're made of (except for the excavated-from polyhedron in a quasi-convex toroid). Some nonconvex polyhedra occurring relatively frequently in Stewart's investigations were given names in order to more easily refer to them.
Stewart notation | Notes | |
---|---|---|
G_{3} | Involved in a dissection of the small rhombicosidodecahedron | |
Z_{4} | Has a square-square height of | |
A_{5} A_{5}' A_{5}'' |
Have pentagon-pentagon heights of .
This is the height of two pentagonal antiprisms outer-blended together at a pentagon. | |
m m* |
They can modify some of the A_{5} polyhedra. | |
X | X is an outer-blend of m and J_{63} | |
W W' W'' |
Modifications of the triangular hebesphenorotunda | |
J_{91}' T T' |
J_{91}' is half of a bilunabirotunda, cut along one of its irregular-hexagonal pseudofaces.
The T polyhedra are outer-blends of J_{91}' and J_{63} | |
M | M is a simple quasi-convex Stewart toroid. It also has the longer symbol Q_{3}Q_{3}/S_{3}S_{3}. |
Stacking polyhedra[edit | edit source]
A common feature in Stewart toroids is the use of many polyhedral units outer-blended together. When these occur in a linear fashion (i.e. not forming branches or loops), the Stewart notation represents them by writing in a sequence the abbreviations of the involved polyhedra. A simple example of this, as stated before, is how the elongated pentagonal orthobirotunda can be referred to as R_{5}P_{10}R_{5} which more clearly describes its structure. But a stack of polyhedra does not need to be convex, and has no constraints on its length.
Especially repetitive linear stacks may be denoted with a numerical exponent after the repeated unit, like P_{3}^{4}. If these are made of prisms, they will contain many coplanar adjacent faces, so further modifications may be made to satisfy Stewart's (A) condition. These can be as short as Q_{5}^{2} or the pentagonal orthobicupola.
If a stack of polyhedra loops back on itself, a description of it as a sequence would fall short. The Stewart notation represents such loops of polyhedra as a list of counts of each unit in the loop. For example, a ring of heptagonal prisms, with (A) met by inserting hexagonal prisms in between (each one rotated 90° in the plane of its blended-away squares), is referred to as 14P_{7}14P_{6}.
If there is a polyhedron that is in the center of a stack, it may be surrounded by parentheses, like in Z_{4}Z_{4}(P_{4})Z_{4}Z_{4}.
If a polyhedron is in the center of an arrangement that many congruent stacks branch off of, write the number of those stacks, then write the stack, then write the central polyhedron in parentheses. An example of this is 6Z_{4}(P_{4}).
Tunnels: genus-changing[edit | edit source]
The operation of excavating one polyhedron from another larger one in a way that changes the genus of the latter (i.e. the excavation goes all the way through the larger polyhedron) is represented by a forward-slash or division sign . On its left side is written the larger polyhedron that has a tunnel dug through it, and on its right side is written the shape of the polyhedral tunnel that is being dug out of the larger polyhedron.
Stewart toroids with higher symmetry and higher genus can have many copies of the same tunnel going from the outside to an inner, central polyhedron. In this case, the syntax for central polyhedra with many congruent stacks branching off is used on the right side of the .
Augmentations and excavations: non-genus-changing[edit | edit source]
For augmentations and excavations of polyhedra to a "base" polyhedron that don't change the genus of the base polyhedron, the name of the augmented or excavated polyhedron is written after the base polyhedron, along with an exponent that contains the number of instances of it on the base polyhedron and either the letter "A" (for augmentation) or "E" (for excavation). It is also customary to precede the number of an excavation with "-".
For example, a triangular prism of edge length 4 made out of 64 unit-edge-length triangular prisms can be tunnelled along its axis by four triangular prisms. This can be referred to as 64P_{3}/P_{3}^{4}, but both the original polyhedron and the tunnel will have coplanar adjacent faces. Eight square pyramids can be excavated from each "square face" of the large triangular prism, and six tetrahedra from each "triangular face." These inward-facing tetrahedra surround the ends of the tunnel closely, so we can't augment every other triangular prism in the tunnel, but we can augment every face of the middle two. This gives us 64P_{3}Y_{4}^{-24E}Y_{3}^{-12E}/P_{3}^{4}Y_{4}^{6A}.
Also, Stewart's T polyhedron is J_{91}'J_{63}^{A}, W is W'Y_{3}^{-3E}, and W" is W'Y_{3}^{-4E}.
The fact that this part of the notation doesn't specify exactly which faces or orientations the augmentations or excavations occur on could be called a weakness. However, this probably wasn't a big concern for Stewart, as this part of the notation was mostly only used to keep adjacent faces from being coplanar. As long as one saw that augmentations or excavations were demanded by a name or problem, they could look into how to lay out those modifications in a way that kept adjacent faces from being coplanar (and that avoided self-intersections), and any valid solution they arrived at would be fine by Stewart.
External links[edit | edit source]
- Doskey, Alex. "Chapter 2 - Polyhedra"
- Doskey, Alex. "Chapter 8 - Limiting Properties"