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"	ambo (rectify)	bitruncate	expand (cantellate)	bevel (omnitruncate)	snub

dual	join	kis (vertex-bisect)	ortho (edge-bisect)	meta (full-bisect)	gyro

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