Conway Polyhedron Notation - Geometric Coordinates of Derived Forms

Geometric Coordinates of Derived Forms

In general the seed polyhedron can be considered a tiling of a surface since the operators represent topological operations so the exact geometric positions of the vertices of the derived forms are not defined in general. A convex regular polyhedron seed can be considered a tiling on a sphere, and so the derived polyhedron can equally be assumed to be positioned on the surface of a sphere. Similar a regular tiling on a plane, such as a hexagonal tiling can be a seed tiling for derived tilings. Nonconvex polyhedra can become seeds if a related topological surface is defined to constrain the positions of the vertices. For example toroidal polyhedra can derive other polyhedra with point on the same torus surface.

Example: A dodecahedron seed as a spherical tiling

D
tD
aD
tdD
eD
teD
sD

dD
dteD
Example: An Euclidean hexagonal tiling seed (H)

H
tH
aH
tdH = H
eH
teH
sH

dH
dtH
daH
dtdH = dH
deH
dteH
dsH
Example: A transparent Tetrahedron seed (T)

T
tT
aT
tdT
eT
bT
sT

dT
dtT
jT
kT
oT
mT
gT
Example: A hyperbolic heptagonal tiling seed
{7,3}
"seed"		 truncate ambo
(rectify)		 bitruncate expand
(cantellate)		 bevel
(omnitruncate)		 snub
dual join kis
(vertex-bisect)		 ortho
(edge-bisect)		 meta
(full-bisect)		 gyro

Read more about this topic:  Conway Polyhedron Notation

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