Cross-surface

A cross-surface (or real projective plane) is a possible finite Life universe (i.e. one of the bounded grids) in the form of a real projective plane. The simplest way to do this is to use an m × n rectangle with the top edge twisted and joined to the bottom edge and the left edge twisted and joined to the right edge. It is similar to the Klein bottle except it has both pairs of sides twisted instead of just one.

The following pattern demonstrates how diagonal and orthogonal spaceships travel in a cross-surface universe.

x = 86, y = 98, rule = B3/S23:C100,100 6b2o9b2o48b2o9b2o$6b2o9b2o48b2o9b2o2$67b2o9b2o$67b2o9b2o2$23b2o11b2ob 2o11b2o$22bobo10bobobobo10bobo$21bobo10bobo3b2o11b2o$21b2o11b2o16$4b2o $3bobo$2bobo$bobo$obo$2o4$80b2o$80bobo$81bobo$82bobo$4b2o77bobo$3bobo 78b2o$2bobo$bobo$obo$2o2$2o$obo$bobo76b2o$2bobo75bobo$3b2o76bobo$82bob o$83bobo$84b2o2$84b2o$83bobo$82bobo$81bobo$2o79b2o$obo$bobo$2bobo$3b2o 5$84b2o$83bobo$82bobo$81bobo$81b2o4$11b3o$11bo2bo$11bo$11bo$12bobo9$ 30b2o6bo4b2o$30bobo4b2o4bobo3b2o11b2o$31bobo3bobo4bobobobo10bobo$32b2o 11b2ob2o11b2o2$6b2o9b2o$6b2o9b2o$67b2o9b2o$6b2o9b2o48b2o9b2o$6b2o9b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 600 HEIGHT 600 GPS 60 LOOP 801 ]]

(click above to open LifeViewer) RLE: here Plaintext: here

Following from topology, any pattern on an m × n cross-surface bounded grid is equivalent to a pattern on a 2m × 2n toroidal bounded grid, where the upper left quadrant of the toroidal bounded grid contains the pattern on the cross-surface bounded grid, the upper right quadrant of the toroidal bounded grid contains the pattern on the cross-surface bounded grid mirrored vertically, the lower left quadrant is mirrored horizontally, and the lower right quadrant is mirrored both ways.

External links

• Real projective plane at Wikipedia
• Bounded grids at Golly's online help