Programs that generate map tiles default to generating tiles for abounding box whose dimensions are fixed up and down the zoom stack. Butthe overarchingly common use case calls this default behavior intoquestion. If the ultimate goal of a map is to lock down the display of afeature at a high zoom level, then any tile outside of the invertedpyramid whose truncated top bounds the feature at the desired zoom levelis extraneous, unnecessary.Inspired by a game of marbles that uses a similar shape in its playing,I call this truncated, inverted pyramid the "Tower of Prince Henry,Reversed"[1], and abbreviate it TOPHR.This presentation describes modifications to TileMill, the same strategyimplemented directly through Mapnik XML, the use of the flexible mbtileformat to store the generated tiles, and presents several measures ofthe resulting savings (tile generation time, number of tiles, diskspace). I'll also describe a plug in for Leaflet and an approach forOpenLayers that ensures that map users cannot stray outside the boundsof TOPHR.1. It's also reminiscent of the name of a real album by The Fall or an unreal tarot card.