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#REDIRECT [[Block#Block arrays]]
{{Glossary}}
The '''block agar''' is an [[agar]] consisting of [[block]]s arranged periodically in rows and columns, with distance of 1 cell between any two adjacent rows or columns. The average field density of the block agar is 4/9.
 
Sometimes, more sparse arrangements of blocks are also described as block agars.{{refn|[[Lifeline Volume 3#Page 29]]}}
 
== Reactions in the block agar ==
{{about|a type of agar preturbation|the [[life-like cellular automaton]]|OCA:Virus}}
{| style="margin-left:16px; margin-bottom:8px; width:300px; float:right"
|{{EmbedViewer
|rle          = x = 20, y = 17, rule = B3/S23
2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob
2ob2ob2ob2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o$9bo$2ob2ob2o
b2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2o
b2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o!
|viewerconfig = #C [[ THUMBSIZE 2 ZOOM 16 HEIGHT 450 GPS 8 THEME Book AUTOSTART T 0 PAUSE 3 T 22 PAUSE 2 LOOP 23 ]]
|position    = center
|caption      = A pattern doomed by a virus (from [[Lifeline Volume 9]])
}}
|}
 
If an alive cell ("virus") is added to the block agar so that it touches the corners of four blocks, the agar self-repairs in two ticks. However, when a single cell is added so that it touches cells of two different blocks, it initiates an expanding disintegration of the agar.{{refn|Martin Gardner, ''[[Wheels, Life, and Other Mathematical Amusements]]'', The Game of Life, Part II}}{{refn|[[Lifeline Volume 9#Page 6]]}} Removing a single cell from any block will also destroy the agar.
 
[[Space clearer]]s for the block agar and for one quarter of the block agar were found by [[Hartmut Holzwart]];<ref name="post99952" /> these space clearers are reactions that expand through the infinite agar and leave empty space behind.<ref name="post160515" />
 
There are known p3 lightspeed "trimmers" on an array of blocks with zebra stripes boundary.<ref name="post104616" />
 
== Finite patches ==
Finite rectangular patches of the block agar (''n''-by-''m'' '''block arrays''') are [[pseudo still life]]s; the simplest example is the [[bi-block]], which is the {{times|1|2}} block array.
 
{| class="wikitable" style="margin-left:auto; margin-right:auto; clear:both; text-align:center"
! x\y
! 1
! 2
! 3
! ...
|-
! 1
! [[File:1×1 blocks.png]]
! [[File:2×1 blocks.png]]
! [[File:3×1 blocks.png]]
! rowspan="3" | →
|-
! 2
! [[File:1×2 blocks.png]]
! [[File:2×2 blocks.png]]
! [[File:3×2 blocks.png]]
|-
! 3
! [[File:1×3 blocks.png]]
! [[File:2×3 blocks.png]]
! [[File:3×3 blocks.png]]
|-
! ...
! colspan="3" | ↓
! ↘
|}
 
=== Glider synthesis ===
 
{|style="margin-left:16px;margin-bottom:8px;width:300px;float:right;"
|{{EmbedViewer
|rle          = x = 39, y = 45, rule = B3/S23
bo$2bo$3o2$28bo$28bobo$6bobo19b2o$7b2o$7bo5$38bo$18bo5bo11b2o$17bobo3b
obo11b2o$17bo2bobo2bo$18b2o3b2o3$7b2ob2ob2ob2o$7b2ob2ob2ob2o3b2o$20bo
2bo$7b2ob2ob2ob2o3b2o$7b2ob2ob2ob2o3$18b2o3b2o8b3o$17bo2bobo2bo7bo$17b
obo3bobo8bo$18bo5bo6$7bo$7b2o$6bobo19b2o$28bobo$28bo2$3o$2bo$bo!
|viewerconfig = #C [[ THUMBSIZE 2 ZOOM 10 HEIGHT 500 GPS 8 THEME Book AUTOSTART T 0 PAUSE 1 T 20 PAUSE 3 T 68 PAUSE 2 LOOP 69 ]]
|position    = center
|caption      = A way to extend a {{times|2|''n''}} block array<ref name="lifenews_2005_04" />
}}
|}
 
In {{year|2005}}, Jason Summers found a way to extend any {{times|2|''n''}} block array. The method involves building a constellation of beehive and four loaves and hitting it with 8 gliders.<ref name="lifenews_2005_04" />
 
On December 3, {{year|2015}}, [[Martin Grant]] found a method of constructing a {{times|3|''n''}} block array via [[glider synthesis]].<ref name="post25637" />
 
On May 6, {{year|2020}}, [[Goldtiger997]] found a method of constructing a {{times|4|''n''}} block array.<ref name="post96210" />
 
The [[problem]] of constructing an arbitrary ''m''-by-''n'' block array is currently still unsolved.<ref>https://conwaylife.com/forums/viewtopic.php?p=140125#p140125</ref><ref>https://conwaylife.com/forums/viewtopic.php?p=146722#p146722</ref>
 
== See also ==
Objects in which a temporary {{times|2|2}} block array is formed for several consecutive phases:
* [[62P83]], a C2_1 oscillator
* A C4_1 [[queen bee shuttle]] [[tetramer]] ({{LinkCatagolue|xp30_yh33zye8kqrqk8zzyvg8gzccx8s2tuy6rr0rry6fn872x66zy3121zzyg25brb52zyioo|style=brief}})
<!--Other related articles:-->
 
== References ==
<references>
<ref name="lifenews_2005_04">{{cite web
|url    = https://web.archive.org/web/20160409092832/http://pentadecathlon.com/lifeNews/2005/04/block_array_constructions.html
|title  = Block Array Constructions
|author = Heinrich Koenig
|date  = April 20, 2005
|work  = Game of Life News
}}</ref>
<ref name="post25637">{{LinkForumThread
|format = ref
|p      = 25637
|title  = Re: Synthesising Oscillators
|author = Martin Grant
|date  = December 3, 2015
}}</ref>
<ref name="post96210">{{LinkForumThread
|format = ref
|p      = 96210
|title  = Re: Synthesising Oscillators
|author = Goldtiger997
|date  = May 6, 2020
}}</ref>
<ref name="post99952">{{LinkForumThread
|format = ref
|p      = 99952
|title  = Re: Agar crawlers
|author = Hartmut Holzwart
|date  = July 2, 2020
}}</ref>
<ref name="post104616">{{LinkForumThread
|format = ref
|p      = 104616
|title  = Re: 'Skimmers'
|author = wildmyron
|date  = September 21, 2020
}}</ref>
<ref name="post160515">{{LinkForumThread
|format = ref
|p      = 160515
|title  = Re: Thread for basic questions
|author = Dave Greene
|date  = April 15, 2023
}}</ref>
</references>
 
[[Category:Stable agars]]
[[Category:Agars]]
[[Category:Pseudo still lifes]]
[[Category:Problems]]
[[Category:Blockic constellations]]
__NOTOC__

Latest revision as of 15:41, 15 January 2026

The block agar is an agar consisting of blocks arranged periodically in rows and columns, with distance of 1 cell between any two adjacent rows or columns. The average field density of the block agar is 4/9.

Sometimes, more sparse arrangements of blocks are also described as block agars.[1]

Reactions in the block agar

This article is about a type of agar preturbation. For the life-like cellular automaton, see OCA:Virus.
x = 20, y = 17, rule = B3/S23 2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob 2ob2ob2ob2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o$9bo$2ob2ob2o b2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2o b2ob2o2$2ob2ob2ob2ob2ob2ob2o$2ob2ob2ob2ob2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 16 HEIGHT 450 GPS 8 THEME Book AUTOSTART T 0 PAUSE 3 T 22 PAUSE 2 LOOP 23 ]]
A pattern doomed by a virus (from Lifeline Volume 9)
(click above to open LifeViewer)

If an alive cell ("virus") is added to the block agar so that it touches the corners of four blocks, the agar self-repairs in two ticks. However, when a single cell is added so that it touches cells of two different blocks, it initiates an expanding disintegration of the agar.[2][3] Removing a single cell from any block will also destroy the agar.

Space clearers for the block agar and for one quarter of the block agar were found by Hartmut Holzwart;[4] these space clearers are reactions that expand through the infinite agar and leave empty space behind.[5]

There are known p3 lightspeed "trimmers" on an array of blocks with zebra stripes boundary.[6]

Finite patches

Finite rectangular patches of the block agar (n-by-m block arrays) are pseudo still lifes; the simplest example is the bi-block, which is the 1 × 2 block array.

x\y 1 2 3 ...
1 →
2
3
... ↓ ↘

Glider synthesis

x = 39, y = 45, rule = B3/S23 bo$2bo$3o2$28bo$28bobo$6bobo19b2o$7b2o$7bo5$38bo$18bo5bo11b2o$17bobo3b obo11b2o$17bo2bobo2bo$18b2o3b2o3$7b2ob2ob2ob2o$7b2ob2ob2ob2o3b2o$20bo 2bo$7b2ob2ob2ob2o3b2o$7b2ob2ob2ob2o3$18b2o3b2o8b3o$17bo2bobo2bo7bo$17b obo3bobo8bo$18bo5bo6$7bo$7b2o$6bobo19b2o$28bobo$28bo2$3o$2bo$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 10 HEIGHT 500 GPS 8 THEME Book AUTOSTART T 0 PAUSE 1 T 20 PAUSE 3 T 68 PAUSE 2 LOOP 69 ]]
A way to extend a 2 × n block array[7]
(click above to open LifeViewer)

In 2005, Jason Summers found a way to extend any 2 × n block array. The method involves building a constellation of beehive and four loaves and hitting it with 8 gliders.[7]

On December 3, 2015, Martin Grant found a method of constructing a 3 × n block array via glider synthesis.[8]

On May 6, 2020, Goldtiger997 found a method of constructing a 4 × n block array.[9]

The problem of constructing an arbitrary m-by-n block array is currently still unsolved.[10][11]

See also

Objects in which a temporary 2 × 2 block array is formed for several consecutive phases:

References

  1. ↑ Lifeline Volume 3#Page 29
  2. ↑ Martin Gardner, Wheels, Life, and Other Mathematical Amusements, The Game of Life, Part II
  3. ↑ Lifeline Volume 9#Page 6
  4. ↑ Hartmut Holzwart (July 2, 2020). Re: Agar crawlers (discussion thread) at the ConwayLife.com forums
  5. ↑ Dave Greene (April 15, 2023). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
  6. ↑ wildmyron (September 21, 2020). Re: 'Skimmers' (discussion thread) at the ConwayLife.com forums
  7. ↑ 7.0 7.1 Heinrich Koenig (April 20, 2005). "Block Array Constructions". Game of Life News.
  8. ↑ Martin Grant (December 3, 2015). Re: Synthesising Oscillators (discussion thread) at the ConwayLife.com forums
  9. ↑ Goldtiger997 (May 6, 2020). Re: Synthesising Oscillators (discussion thread) at the ConwayLife.com forums
  10. ↑ https://conwaylife.com/forums/viewtopic.php?p=140125#p140125
  11. ↑ https://conwaylife.com/forums/viewtopic.php?p=146722#p146722
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