User:Gzaytman/Topologies

In addition to the bounded grids currently supported by LifeViewer/Golly, that there are other ways to glue sides together to obtain Klein bottles, cross-surfaces, or spheres.

Klein bottle 2

Joining adjacent sides as in Sphere but "twisting" both joins creates a surface that's topologically a Klein bottle.

x = 199, y = 100, rule = LifeHistory:T200+100,100 99.100D$74.2A9.2A12.75D2A9D2A12D$74.2A9.2A12.75D2A9D2A12D$99.100D$99. 75D2A9D2A12D$99.75D2A9D2A12D$99.100D$99.100D$32.2A12.2A2.2A12.2A33. 33D2A12D2A2D2A12D2A33D$31.A.A11.A.A2.A.A11.A.A32.32DADA11DADA2DADA11D ADA32D$30.A.A11.A.A4.2A12.2A32.31DADA11DADA4DADA11DADA31D$29.A.A11.A. A53.30DADA11DADA6D2A12D2A31D$2A27.2A12.2A48.2A.2A.D2AD2A23DADA11DADA 51D2AD$2A91.2A.2A.D2AD2A23D2A12D2A52D2AD$99.100D$99.100D$99.100D$99. 39D2C59D$99.37D2CD2C58D$99.37D4C59D$99.38D2C60D$99.100D$99.100D$2A91. 2A.2A.D2AD2A91D2AD$2A91.2A.2A.D2AD2A91D2AD$99.100D$99.100D$99.100D$ 99.100D$99.12D2A86D$10.2A4.3A80.11DADA3D3C80D$9.A.A6.A68.2A10.10DADA 6DC80D$8.A.A6.A69.A.A9.9DADA6DC70D2A9D$7.A.A78.A.A8.8DADA78DADA8D$7. 2A80.2A8.8D2A80D2A8D$99.100D$99.100D$99.100D$99.100D$99.100D$99.100D$ 99.100D$99.100D$99.12D2A86D$10.2A87.11DADA86D$9.A.A75.2A10.10DADA87D$ 8.A.A76.A.A9.9DADA77D2A9D$7.A.A78.A.A8.8DADA78DADA8D$7.2A80.2A8.8D2A 80D2A8D$99.100D$99.100D$7.2A80.2A8.8D2A80D2A8D$7.A.A78.A.A8.8DADA78DA DA8D$8.2A77.A.A9.9DADA76DADA9D$86.A.A10.10D2A75DADA10D$4.2A79.A.A11. 87D2A11D$4.A.A78.2A12.100D$4.A94.100D$80.A18.100D$79.3A17.100D$78.2A. A17.100D$78.3A18.100D$79.2A18.100D$99.100D$99.100D$7.2A80.2A8.8D2A80D 2A8D$7.A.A78.A.A8.8DADA78DADA8D$8.2A77.A.A9.9DADA76DADA9D$86.A.A10. 10D2A75DADA10D$85.A.A11.87D2A11D$85.2A12.100D$99.100D$99.100D$99.100D $99.100D$99.100D$99.100D$99.100D$99.100D$99.100D$67.2A30.68D2C30D$66. A.A30.67DCDC30D$68.A30.69DC30D$99.100D$99.100D$99.100D$54.2A12.2A29. 100D$53.A.A11.A.A29.54D2A12D2A30D$30.2A12.2A6.A.A11.A.A30.53DADA11DAD A30D$30.A.A11.A.A4.A.A11.A.A31.32D2A12D2A4DADA11DADA31D$31.A.A11.A.A 2.A.A11.A.A32.32DADA11DADA2DADA11DADA32D$32.2A12.2A2.2A12.2A33.33D2A 12D2A2D2A12D2A33D$99.43DC56D$99.44DC55D$74.2A9.2A12.42D3C55D$74.2A9. 2A12.100D$99.100D$74.2A9.2A12.75D2A9D2A12D$74.2A9.2A12.75D2A9D2A12D$ 99.100D! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 1000 HEIGHT 600 GPS 60 COLOR 3 0 0 0 COLOR 4 40 40 40 ]]

Two copies of Klein bottle 2 combined into a shifted torus. (click above to open LifeViewer)

Sphere 2

"Rolling" two opposite sides as in Torus or Klein bottle but "gluing" each of the remaining two sides to itself creates a surface that's topologically a sphere. Note that each of the gluings can be shifted independently.

x = 199, y = 100, rule = LifeHistory:T200,100 99.100D$74.2A9.2A12.12D2A9D2A75D$74.2A9.2A12.12D2A9D2A75D$99.100D$99. 100D$99.100D$99.100D$99.100D$44.2A11.2A11.2A27.45D2A11D2A11D2A27D$44. A.A10.A.A10.A.A26.45DADA10DADA10DADA26D$45.A.A10.A.A10.A.A25.46DADA 10DADA10DADA25D$46.A.A10.A.A10.A.A24.47DADA10DADA10DADA24D$2A.2A.2A 39.2A11.2A11.2A18.2A.2A.D2AD2A42D2A11D2A11D2A15D2AD2AD2AD$2A.2A.2A85. 2A.2A.D2AD2A85D2AD2AD2AD$99.100D$99.100D$99.100D$99.59D2C39D$99.57D2C D2C38D$99.57D4C39D$99.58D2C40D$99.100D$99.100D$2A.2A.2A85.2A.2A.D2AD 2A85D2AD2AD2AD$2A.2A.2A85.2A.2A.D2AD2A85D2AD2AD2AD$99.100D$99.100D$ 89.2A8.90D2A8D$89.A.A7.90DADA7D$90.A.A6.91D2A7D$91.2A6.100D$99.100D$ 99.100D$99.100D$99.100D$99.100D$99.100D$99.18D2C80D$99.18D3C75DC3D$ 99.17DCD2C73DCDC3D$89.2A8.17D3C70D2A3D2C3D$89.A.A7.18DC71DADA7D$90.A. A6.91D2A7D$91.2A6.6D2A92D$6.2A91.6DADA91D$6.A.A90.7DADA90D$7.2A90.8D 2A90D$99.100D$99.100D$99.100D$99.100D$99.100D$99.100D$89.2A8.90D2A8D$ 89.A.A7.90DADA7D$90.A.A6.91D2A7D$91.2A6.6D2A92D$6.2A91.6DADA91D$6.A.A 71.A18.7DADA90D$2.2A3.2A70.3A17.8D2A90D$2.A.A73.2A.A17.100D$2.A75.3A 18.100D$79.2A18.100D$99.100D$99.100D$99.100D$99.100D$99.100D$99.100D$ 99.6D2A92D$6.2A91.6DADA91D$6.A.A90.7DADA90D$7.2A90.8D2A90D$99.100D$ 99.100D$2A.2A.2A85.2A.2A.D2AD2A85D2AD2AD2AD$2A.2A.2A85.2A.2A.D2AD2A 85D2AD2AD2AD$99.100D$99.100D$39.2A58.100D$38.4A57.100D$37.2A.2A57. 100D$38.2A59.100D$99.100D$99.100D$99.100D$2A.2A.2A85.2A.2A.D2AD2A85D 2AD2AD2AD$2A.2A.2A15.2A11.2A11.2A42.2A.2A.D2AD2A18D2A11D2A11D2A39D2AD 2AD2AD$23.A.A10.A.A10.A.A47.24DADA10DADA10DADA47D$24.A.A10.A.A10.A.A 46.25DADA10DADA10DADA46D$25.A.A10.A.A10.A.A45.26DADA10DADA10DADA45D$ 26.2A11.2A11.2A45.27D2A11D2A11D2A45D$99.100D$99.100D$99.100D$99.100D$ 99.100D$74.2A9.2A12.12D2A9D2A75D$74.2A9.2A12.12D2A9D2A75D$99.100D! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 1000 HEIGHT 600 GPS 60 COLOR 3 0 0 0 COLOR 4 40 40 40 ]]

Two copies of sphere 2 combined into a torus. (click above to open LifeViewer)

Sphere 3

"Gluing" each of the four sides to itself creates another surface that's topologically a sphere. Note that a 2m × n sphere 3 is equivalent to a m × 2n sphere 2.

x = 199, y = 100, rule = LifeHistory:T200+100,100 99.100D$11.2A9.2A50.2A9.2A12.12D2A9D2A50D2A9D2A12D$11.2A9.2A50.2A9.2A 12.12D2A9D2A50D2A9D2A12D$99.100D$99.12D2A9D2A50D2A9D2A12D$99.12D2A9D 2A50D2A9D2A12D$99.100D$43.2A11.2A11.2A28.100D$43.A.A10.A.A10.A.A27. 45D2A11D2A11D2A27D$44.A.A10.A.A10.A.A26.45DADA10DADA10DADA26D$45.A.A 10.A.A10.A.A25.46DADA10DADA10DADA25D$46.A.A10.A.A10.A.A24.47DADA10DAD A10DADA24D$2A.2A.2A39.2A11.2A11.2A12.2A.2A.2A.2A.D2AD2AD2AD2A36D2A11D 2A11D2A15D2AD2AD2AD$2A.2A.2A79.2A.2A.2A.2A.D2AD2AD2AD2A79D2AD2AD2AD$ 99.100D$99.100D$99.100D$99.59D2C39D$99.57D2CD2C38D$99.57D4C39D$99.58D 2C40D$99.100D$99.100D$2A.2A.2A79.2A.2A.2A.2A.D2AD2AD2AD2A79D2AD2AD2AD $2A.2A.2A79.2A.2A.2A.2A.D2AD2AD2AD2A79D2AD2AD2AD$99.100D$99.100D$89. 2A8.90D2A8D$89.A.A7.90DADA7D$90.A.A6.91D2A7D$91.2A6.100D$99.100D$99. 100D$99.100D$99.100D$99.100D$99.100D$99.18D2C80D$99.18D3C75DC3D$96.A 2.17DCD2C73DCDC3D$89.2A3.A.A2.17D3C70D2A3D2C3D$89.A.A3.2A2.18DC71DADA 7D$90.A.A6.91D2A7D$91.2A6.6D2A92D$6.2A91.6DADA91D$6.A.A90.7DADA90D$7. 2A90.8D2A90D$99.100D$99.100D$99.100D$99.100D$99.100D$99.100D$89.2A8. 90D2A8D$89.A.A7.90DADA7D$90.A.A6.91D2A7D$91.2A6.6D2A92D$6.2A91.6DADA 91D$6.A.A71.A18.2D2C3DADA90D$2.2A3.2A70.3A17.2DCDC3D2A90D$2.A.A73.2A. A17.2DC97D$2.A75.3A18.100D$79.2A18.100D$99.100D$99.100D$99.100D$99. 100D$99.100D$99.100D$99.6D2A92D$6.2A91.6DADA91D$6.A.A90.7DADA90D$7.2A 90.8D2A90D$99.100D$99.100D$2A.2A.2A79.2A.2A.2A.2A.D2AD2AD2AD2A79D2AD 2AD2AD$2A.2A.2A79.2A.2A.2A.2A.D2AD2AD2AD2A79D2AD2AD2AD$99.100D$99. 100D$39.2A58.100D$38.4A57.100D$37.2A.2A57.100D$38.2A59.100D$99.100D$ 99.100D$99.100D$2A.2A.2A79.2A.2A.2A.2A.D2AD2AD2AD2A79D2AD2AD2AD$2A.2A .2A15.2A11.2A11.2A36.2A.2A.2A.2A.D2AD2AD2AD2A12D2A11D2A11D2A39D2AD2AD 2AD$23.A.A10.A.A10.A.A47.24DADA10DADA10DADA47D$24.A.A10.A.A10.A.A46. 25DADA10DADA10DADA46D$25.A.A10.A.A10.A.A45.26DADA10DADA10DADA45D$26. 2A11.2A11.2A45.27DADA10DADA10DADA44D$99.28D2A11D2A11D2A44D$99.100D$ 11.2A9.2A50.2A9.2A12.100D$11.2A9.2A50.2A9.2A12.100D$99.100D$11.2A9.2A 50.2A9.2A12.12D2A9D2A50D2A9D2A12D$11.2A9.2A50.2A9.2A12.12D2A9D2A50D2A 9D2A12D$99.100D! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 1000 HEIGHT 600 GPS 60 COLOR 3 0 0 0 COLOR 4 40 40 40 ]]

Two copies of sphere 3 combined into a shifted torus. (click above to open LifeViewer)

Cross surface 2

"Gluing" each of two adjacent sides to itself joining the other two adjacent sides with a twist creates a topological cross surface.

x = 399, y = 100, rule = LifeHistory:T400+100,100 99.300D$11.2A9.2A50.2A9.2A12.12D2A9D2A50D2A9D2A24D2A9D2A50D2A9D2A24D 2A9D2A50D2A9D2A12D$11.2A9.2A50.2A9.2A12.12D2A9D2A50D2A9D2A24D2A9D2A 50D2A9D2A24D2A9D2A50D2A9D2A12D$99.159D3C138D$99.12D2A9D2A50D2A9D2A24D 2A9D2A34DC15D2A9D2A24D2A9D2A50D2A9D2A12D$99.12D2A9D2A17D2A11D2A11D2A 5D2A9D2A24D2A9D2A35DC14D2A9D2A24D2A9D2A17D2A11D2A11D2A5D2A9D2A12D$99. 42DADA10DADA10DADA171DADA10DADA10DADA29D$43.2A11.2A11.2A28.12D2A9D2A 18DADA10DADA10DADA3D2A9D2A56D2A11D2A11D2A40D2A9D2A18DADA10DADA10DADA 3D2A9D2A12D$43.A.A10.A.A10.A.A27.12D2A9D2A19DADA10DADA10DADA2D2A9D2A 56DADA10DADA10DADA39D2A9D2A19DADA10DADA10DADA2D2A9D2A12D$44.A.A10.A.A 10.A.A26.45D2A11D2A11D2A72D2A11D2A11D2A72D2A11D2A11D2A27D$45.2A11.2A 11.2A26.75D2A9D2A124D2A9D2A75D$99.75D2A9D2A124D2A9D2A75D$2A.2A82.2A. 2A.2A.2A.D2AD2AD2AD2A85D2A2D2A88D2AD2AD2A2D2AD2AD2A85D2AD2AD$2A.2A82. 2A.2A.2A.2A.D2AD2AD2AD2A85D2A2D2A88D2AD2AD2A2D2AD2AD2A85D2AD2AD$99. 300D$99.300D$99.300D$99.39D2C259D$99.37D2CD2C258D$99.37D4C259D$99.38D 2C260D$99.300D$99.300D$2A.2A82.2A.2A.2A.2A.D2AD2AD2AD2A85D2A2D2A88D2A D2AD2A2D2AD2AD2A85D2AD2AD$2A.2A82.2A.2A.2A.2A.D2AD2AD2AD2A85D2A2D2A 88D2AD2AD2A2D2AD2AD2A85D2AD2AD$99.300D$99.89D2A209D$89.2A8.89DADA98D 2A98D2A8D$89.A.A7.90DADA97DADA97DADA7D$90.A.A6.91D2A98DADA97D2A7D$91. A.A5.192DADA105D$92.2A5.193D2A105D$99.300D$99.300D$99.300D$99.300D$ 99.300D$99.218D2C80D$99.218D3C75DC3D$99.89D2A126DCD2C73DCDC3D$89.2A8. 89DADA98D2A25D3C70D2A3D2C3D$89.A.A7.90DADA97DADA25DC71DADA7D$90.A.A6. 5D2A84D2A98DADA11D2A84D2A7D$91.A.A5.5DADA184DADA10DADA92D$6.2A84.2A5. 6DADA98D2A84D2A11DADA91D$6.A.A90.7DADA97DADA97DADA90D$7.2A90.8D2A98DA DA97D2A90D$99.109D2A189D$99.300D$99.300D$99.300D$99.300D$99.89D2A209D $89.2A8.89DADA98D2A98D2A8D$89.A.A7.90DADA97DADA97DADA7D$90.A.A6.5D2A 84D2A98DADA11D2A84D2A7D$91.A.A5.5DADA184DADA10DADA92D$6.2A84.2A5.6DAD A98D2A84D2A11DADA91D$6.A.A71.A18.7DADA97DADA97DADA90D$2.2A3.2A70.3A 17.8D2A98DADA97D2A90D$2.A.A73.2A.A17.109D2A189D$2.A75.3A18.300D$79.2A 18.300D$99.300D$99.300D$99.300D$99.300D$99.300D$99.5D2A198D2A93D$99. 5DADA197DADA92D$6.2A91.6DADA98D2A97DADA91D$6.A.A90.7DADA97DADA97DADA 90D$7.2A90.8D2A98DADA97D2A90D$99.109D2A189D$99.300D$2A.2A85.2A.2A.2A. D2AD2AD2A88D2A2D2A85D2AD2AD2AD2A2D2AD2AD2AD2A82D2AD2AD$2A.2A85.2A.2A. 2A.D2AD2AD2A88D2A2D2A85D2AD2AD2AD2A2D2AD2AD2AD2A82D2AD2AD$99.300D$99. 300D$99.160D2C138D$99.159D4C137D$99.158D2CD2C137D$99.159D2C139D$99. 300D$99.300D$99.300D$2A.2A85.2A.2A.2A.D2AD2AD2A88D2A2D2A85D2AD2AD2AD 2A2D2AD2AD2AD2A82D2AD2AD$2A.2A85.2A.2A.2A.D2AD2AD2A88D2A2D2A85D2AD2AD 2AD2A2D2AD2AD2AD2A82D2AD2AD$74.2A9.2A12.112D2A9D2A175D$74.2A9.2A12. 112D2A9D2A101D2A11D2A11D2A46D$26.2A11.2A11.2A45.27D2A11D2A11D2A72D2A 11D2A11D2A71DADA10DADA10DADA45D$11.2A9.2A2.A.A10.A.A10.A.A19.2A9.2A 12.27DADA10DADA10DADA56D2A9D2A2DADA10DADA10DADA19D2A9D2A39DADA10DADA 10DADA44D$11.2A9.2A3.A.A10.A.A10.A.A18.2A9.2A12.28D2A11D2A11D2A56D2A 9D2A3DADA10DADA10DADA18D2A9D2A40D2A11D2A11D2A44D$28.A.A10.A.A10.A.A 42.129DADA10DADA10DADA142D$11.2A9.2A5.2A11.2A11.2A17.2A9.2A12.12D2A9D 2A14DC35D2A9D2A24D2A9D2A5D2A11D2A11D2A17D2A9D2A112D$11.2A9.2A50.2A9. 2A12.12D2A9D2A15DC34D2A9D2A24D2A9D2A50D2A9D2A112D$99.38D3C259D$11.2A 9.2A50.2A9.2A12.12D2A9D2A50D2A9D2A24D2A9D2A50D2A9D2A24D2A9D2A50D2A9D 2A12D$11.2A9.2A50.2A9.2A12.12D2A9D2A50D2A9D2A24D2A9D2A50D2A9D2A24D2A 9D2A50D2A9D2A12D$99.300D! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART THUMBSIZE 2 ZOOM 4 WIDTH 1800 HEIGHT 600 GPS 60 COLOR 3 0 0 0 COLOR 4 40 40 40 ]]

Four copies of cross surface 2 combined into a shifted torus. (click above to open LifeViewer)