Relations between hypermap representations

The six most important hypermap representations (Walsh, James, Cori, Belyi, Chess, and Quad) are tightly interrelated by map operations. This diagram summarizes their relationship.



In short, Walsh and Chess are duals; Belyi and James are duals; Cori and Quad are duals; Belyi can be derived from Walsh by kis, James can be derived from Walsh by leapfrog; Cori can be derived from Walsh by medial; Quad can be derived from Walsh by radial.

Alternately, the derivations can begin at Chess, the dual of Walsh. James can be derived from Chess by truncate, Belyi can be derived from Chess by ko; Cori can be derived from Chess by medial; Quad can be derived from Chess by radial.

The map operation here called kis (following Conway) has sometimes been called P3, Su2, 2-dimensional subdivision, stellation, or omnicapping.

The map operation here called leapfrog (following most of the map operations literature) has sometimes been called tripling or dual √3 trisection.

The map operation here called ko is associated in the computer graphics literature with Kobbelt, and has sometimes been called primal √3 trisection. Note that Ki(M) = Ko(Du(M)). That corrects an error in my paper for ISAMA 2011.