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Steiner Systems

Source Notebook

Sets of blocks where every possible pair or triple of symbols is in a unique block

Details

In the Fano plane there are 7 symbols in 7 blocks of 3 so that every set of 2 points is in exactly 1 block, also called Steiner triple system STS(7), a 2-(7,3,1) block design. These systems were studied by Pascal, Kirkman, Cayley, Plücker, Salmon, Steiner and many others.

Examples

Basic Examples (4) 

Some data for the Fano plane:

In[1]:=
ResourceData[\!\(\* TagBox["\"\<Steiner Systems\>\"", #& , BoxID -> "ResourceTag-Steiner Systems-Input", AutoDelete->True]\)]["STS(7)"]
Out[1]=

Complete graph K9 as a resolvable design using four colors:

In[2]:=
rst9g = UndirectedEdge @@@ Flatten[Subsets[#, {2}] & /@ #, 1] & /@ Partition[ResourceData[\!\(\* TagBox["\"\<Steiner Systems\>\"", #& , BoxID -> "ResourceTag-Steiner Systems-Input", AutoDelete->True]\)]["STS(9)"]["Blocks"], 3]; Graph[Join @@ rst9g, EdgeStyle -> Join[Thread[rst9g[[1]] -> Red], Thread[rst9g[[2]] -> Green], Thread[rst9g[[3]] -> Blue], Thread[rst9g[[4]] -> Brown]]]
Out[2]=

Sixteen golfers in foursomes for five days, each playing once with each other golfer:

In[3]:=
Grid[Partition[ResourceData[\!\(\* TagBox["\"\<Steiner Systems\>\"", #& , BoxID -> "ResourceTag-Steiner Systems-Input", AutoDelete->True]\)]["SQS(16)"]["Blocks"], 4]]
Out[3]=

Fifteen schoolgirls in triples for seven days, each meeting another girl only once per week:

In[4]:=
Grid[Partition[ResourceData[\!\(\* TagBox["\"\<Steiner Systems\>\"", #& , BoxID -> "ResourceTag-Steiner Systems-Input", AutoDelete->True]\)]["STS(15)"]["Blocks"], 5]]
Out[4]=

Wolfram Research, "Steiner Systems" from the Wolfram Data Repository (2021)  

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Publisher Information

  • Prepared for the Wolfram Data Repository By: Ed Pegg Jr
  • Publisher of Record: Wolfram Research