Bounded grids: Difference between revisions

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A bounded grid is a finite grid. In this page, it shows where diagonal-moving objects (represented as [[gliders]]) and orthogonal-moving objects (represented as [[lightweight spaceships]]) will go after reaching the edge. On a sphere, an object too close to the corners will have affected evolution.
{{Glossary}}
'''Bounded grids''' are a special class of [[universe]]s (or surfaces) on which a [[cellular automaton]] can operate. A surface – e.g. the grid used to play the [[Game of Life]] – may be infinite, like a plane or a cylinder, or finite, like a sphere or a torus. A finite surface has to be curved to form a cylinder (infinite in only one direction), torus or Klein bottle. The image shows how to "roll" and connect the sides in each case:


This page uses the following '''gates''':
[[Image:topol.png|center|frame|Topologies]]


For orthogonal:
* Infinite plane: default.
* [[Finite plane]]: cells outside of the plane are always considered to be dead.
* [[Cylinder]]: "rolling" the plane and connecting the opposite sides marked "1".
* [[Torus]]: "rolling" the cylinder and connecting the opposite circles marked "2".
* [[Klein bottle]]: "rolling" the cylinder, "twisting" it in the fourth dimension and connecting the opposite circles marked "2" and "5"; note that the "5" becomes a "2" after twisting.
* [[Cross-surface]]: like the Klein bottle, but "twisting" the opposite sides while creating the cylinder and then "twisting" the opposite circles when creating the cross-surface.
* [[Sphere]]: joining adjacent sides, rather than opposite sides as is done for the torus.


* The gates with 2 singular [[block]]s correspond to each other.
Note that there are other ways to glue sides together to obtain Klein bottles, cross-surfaces, or spheres.
* The gates with 2 [[bi-block]]s correspond to each other.


For diagonal:
Formal discussion on bounded grids uses knowledge about topology. '''Topology''' is a branch of mathematics dealing with the properties of space - whether it is contiguous, contains holes, and so on. As cellular automata can be characterized in entirely topological terms, thus they can be studied within the scope of topology; see [[Cellular automaton#Generalizations and topological characterization]].


* The methyl gates ([[ship]]s) correspond to each other.
==Travelling spaceships==
* The ethyl gates ([[long ship]]s) correspond to each other.
* The propyl gates ([[long long ship]]s) correspond to each other.
* The butyl gates ([[long long long ship]]s) correspond to each other.


For [[finite plane]] and [[cylinder]], please see the links mentioned.
To demonstrate behaviours of some bounded grids, this section shows where diagonal-moving objects (represented as [[glider]]s) and orthogonal-moving objects (represented as [[lightweight spaceship]]s) will go after reaching the edge. Several "gates" are used to note correspondance:


Below are 3 ways all opposite edges can connect. In brackets, the x represents that this direction's connection is flipped.
* For orthogonal:
** The gates with two singular [[block]]s correspond to each other.
** The gates with two [[bi-block]]s correspond to each other.
** The gates with two [[tri-block]]s correspond to each other.
*For diagonal:
** The gates with [[ship]]s correspond to each other.
** The gates with [[long ship]]s correspond to each other.
** The gates with [[very long ship]]s correspond to each other.
** The gates with [[long^3 ship]]s correspond to each other.


Torus: (XY)
There are two pairs of opposite edges for a plane, one horizontal and another vertical. When connecting edges in pairs, each pair can either be flipped or not, giving three distinct ways as shown below. In brackets, the x represents that this direction's connection is flipped.
 
[[Torus]]: (XY)
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Klein bottle: (XxY)
[[Klein bottle]]: (XxY)
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Cross surface: (XxYx)
[[Cross-surface]]: (XxYx)
The corners have themselves as neighbors, so an object near a corner will evolve as a pattern with [[C2_4]] symmetry.
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The sphere connects the north with west edge, and south with east. The corners have themselves as neighbors.
Besides these, other connectivity is also possible. Demonstrations for [[finite plane]] and [[cylinder]] are shown on the respective pages. Finally there is [[sphere]], which connects the north with west edge, and south with east. The corners have themselves as neighbors, so an object near the upper right or lower left corner will evolve as a pattern with [[C4_4]] symmetry and one near the upper left or lower right will evolve as a pattern with [[C2_4]] symmetry.
 
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|viewerconfig = #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 600 HEIGHT 600 GPS 60 LOOP 801 ]]
|viewerconfig = #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 600 HEIGHT 600 GPS 60 LOOP 801 ]]
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==See also==
* [[Gutter]]
==External links==
* {{LinkWikipedia|Topology}}
* {{LinkGollyHelp|bounded.html|title=Bounded grids}}
[[Category:Topologies| ]]

Latest revision as of 09:26, 22 August 2025

Bounded grids are a special class of universes (or surfaces) on which a cellular automaton can operate. A surface – e.g. the grid used to play the Game of Life – may be infinite, like a plane or a cylinder, or finite, like a sphere or a torus. A finite surface has to be curved to form a cylinder (infinite in only one direction), torus or Klein bottle. The image shows how to "roll" and connect the sides in each case:

Topologies
  • Infinite plane: default.
  • Finite plane: cells outside of the plane are always considered to be dead.
  • Cylinder: "rolling" the plane and connecting the opposite sides marked "1".
  • Torus: "rolling" the cylinder and connecting the opposite circles marked "2".
  • Klein bottle: "rolling" the cylinder, "twisting" it in the fourth dimension and connecting the opposite circles marked "2" and "5"; note that the "5" becomes a "2" after twisting.
  • Cross-surface: like the Klein bottle, but "twisting" the opposite sides while creating the cylinder and then "twisting" the opposite circles when creating the cross-surface.
  • Sphere: joining adjacent sides, rather than opposite sides as is done for the torus.

Note that there are other ways to glue sides together to obtain Klein bottles, cross-surfaces, or spheres.

Formal discussion on bounded grids uses knowledge about topology. Topology is a branch of mathematics dealing with the properties of space - whether it is contiguous, contains holes, and so on. As cellular automata can be characterized in entirely topological terms, thus they can be studied within the scope of topology; see Cellular automaton#Generalizations and topological characterization.

Travelling spaceships

To demonstrate behaviours of some bounded grids, this section shows where diagonal-moving objects (represented as gliders) and orthogonal-moving objects (represented as lightweight spaceships) will go after reaching the edge. Several "gates" are used to note correspondance:

  • For orthogonal:
    • The gates with two singular blocks correspond to each other.
    • The gates with two bi-blocks correspond to each other.
    • The gates with two tri-blocks correspond to each other.
  • For diagonal:

There are two pairs of opposite edges for a plane, one horizontal and another vertical. When connecting edges in pairs, each pair can either be flipped or not, giving three distinct ways as shown below. In brackets, the x represents that this direction's connection is flipped.

Torus: (XY)

x = 98, y = 98, rule = B3/S23:T100,100 10b2o6bo4b2o8b2o9b2o$10bobo4b2o4bobo7b2o9b2o$11b2o4bobo4b2o50$38b3o$ 38bo2bo$38bo$38bo$39bobo15$94b2o$94bobo$95bobo$96b2o3$2o$obo$bobo$2b2o 4$94b2o$94bobo$95bobo$96b2o3$2o$obo$bobo$2b2o2$5b2o11b2o$5bobo10bobo 12b2o9b2o$6b2o11b2o12b2o9b2o! #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

Klein bottle: (XxY)

x = 98, y = 98, rule = B3/S23:K100,100* 73b2o9b2o$73b2o9b2o5$38b2o11b2ob2o11b2o$37bobo10bobobobo10bobo$36bobo 10bobo3bobo10bobo$35bobo10bobo5bobo10bobo$34bobo10bobo7b2o11b2o$34b2o 11b2o5$2ob2o85b2ob2ob2o$2ob2o85b2ob2ob2o10$2ob2o85b2ob2ob2o$2ob2o85b2o b2ob2o3$13b2o77b2o$12bobo77bobo$11bobo79bobo$11b2o81b2o10$13b2o77b2o$ 12bobo77bobo$11bobo79bobo$11b2o81b2o3$11b2o81b2o$11bobo65bo13bobo$12b 2o64b3o12b2o$77b2obo$77b3o$78b2o8$11b2o81b2o$11bobo79bobo$12b2o79b2o4$ 2ob2ob2o85b2ob2o$2ob2ob2o85b2ob2o4$31b2o$29b2ob2o$29b4o$30b2o3$2ob2ob 2o85b2ob2o$2ob2ob2o85b2ob2o2$37b3o$37bo$38bo$58b2o11b2o$35b2o11b2o7bob o10bobo$35bobo10bobo5bobo10bobo$36bobo10bobo3bobo10bobo$37bobo10bobobo bo10bobo$38b2o11b2ob2o11b2o5$73b2o9b2o$73b2o9b2o! #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

Cross-surface: (XxYx) The corners have themselves as neighbors, so an object near a corner will evolve as a pattern with C2_4 symmetry.

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

Besides these, other connectivity is also possible. Demonstrations for finite plane and cylinder are shown on the respective pages. Finally there is sphere, which connects the north with west edge, and south with east. The corners have themselves as neighbors, so an object near the upper right or lower left corner will evolve as a pattern with C4_4 symmetry and one near the upper left or lower right will evolve as a pattern with C2_4 symmetry.

x = 98, y = 98, rule = B3/S23:S100 11b2o9b2o$11b2o9b2o5$29b2o11b2ob2o11b2o$28bobo10bobobobo10bobo$27bobo 10bobo3b2o11b2o$26bobo10bobo$26b2o11b2o$2o91b2ob2o$2o91b2ob2o10$2o91b 2ob2o$2o91b2ob2o5$7b2o$6bobo$6b2o4$86b2o$86bobo$87bobo$88bobo$89bobo$ 90b2o2$7b2o$6bobo$6b2o2$6b2o$6bobo$7bobo76b2o$8bobo75bobo$9b2o76bobo$ 88bobo$89bobo$90b2o2$90b2o$89bobo$88bobo$17bo70b2o$6b2o8b3o$6bobo7bob 2o$7bobo7b3o$8bobo6b2o$9b2o5$90b2o$89bobo$88bobo$88b2o9$34b2o$34bobo$ 34bo5$58b2o11b2o$57bobo10bobo$36b2o11b2o5bobo10bobo$36bobo10bobo3bobo 10bobo$37bobo10bobobobo10bobo$38b2o11b2ob2o11b2o2$11b2o9b2o$11b2o9b2o 2$11b2o9b2o$11b2o9b2o! #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

See also

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