What are...Steiner systems?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What are...Steiner systems? Or: Finite geometry and puzzles.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
www.dtubbenhaue...
Thumbnail picture.
commons.wikime...
Material used.
www.maths.qmul....
math.ucdenver.e...
cameroncounts....
Steiner systems & block designs.
en.wikipedia.o...
mathworld.wolf...
mathworld.wolf...
mathworld.wolf...
ncatlab.org/nl...
en.wikipedia.o...
A bit of history.
www.maa.org/si...
Finite projective planes and finite geometry.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Kirkman schoolgirl problem.
en.wikipedia.o...
math.stackexch...
Witt graph.
mathworld.wolf...
www.distancere...
Applications.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...)
en.wikipedia.o...)
Mathematica.
demonstrations...
demonstrations...
demonstrations...

[@弗雷克]

What if I add another constraint that every element should appear exactly m times?

[@VisualMath]

Hmm, I have never seen discussed that anywhere...🧐 But Steiner systems are already tricky to find, so I am not sure what one could say.

[@leewilliam3417]

Mmmmm😊

[@VisualMath]

Yes, its Steiner systems are tasty ☺

[@coconutart6874]

First of all thanks for the quick and detailed response I will utilize your kindness and knowledge for further question. Suppose I want to construct an array (6,6,36) S
As you explained in the video, a form of determining the number of lines and points is pp have n ^ 2 + n + 1 point and line
On each line is a number of points N + 1
This means that in the case of (6,6,36) S
point and line = 6^6 + 6 + 1
Number of dots placed on each line is 7?
Thank you very much for your heart junction!

[@VisualMath]

Maybe that was not clear, sorry for that! But only for certain values of t,k,n for the Steiner system S(t,k,n) you get some associated finite geometry. For t=k=6 and n=36 the solution is to simply take all 6-element subsets as your blocks, and I would not try to use any points and line picture.

[@coconutart6874]

i thanks for the great explanation!
I would love to get answered some questions to better understanding
Suppose I want to build a Steiner system for (3,3,9) s
For the initial 9 static of several points
For 3 mid length subgroup
For the first 3 on the left I did not understand the meaning of this entry can I get an explanation?
For the number of points (3,3,9) s to understand the number of points one has to perform for N ^ 2 + N + 1 I mean for this value N = 3 which means that there are 13 lines and points?
N + 1 means for this case there are 4 points in each line?
Hope I did not waste your time
Lots of congratulations
moshe

[@VisualMath]

Ok, let me try a different formulation.
Let me modify your example to S(2,3,9), which is a bit easier to explain.
Suppose you are looking for S(t,k,n). Here n is the size of the underlying set X, in your example we take X={1,…,9}. Now take a set Y, called blocks, of subsets of X, all having size k. So in your example I could take Y consisting of 123, 456, 789, 147, 258, 369, 159, 267, 348, 168, 249 and 357. Now comes the condition given by t: every collection of t elements from X has to be contained in precisely one block. For example, if I take 37 from X, then it is contained in 357 and in no other block.
I hope that this makes it clear that there is no reason for arbitrary S(t,k,n) to exist: the condition given by t is killing most triples (t,k,n). A Steiner system S(2,3,9) for example does exist, but that is not obvious. One needs an explicit realization, e.g. via the affine plane of order 3 to see that. A Steiner system S(3,3,9) does exist, but it is rather boring: Take Y to be all 3-element subsets of X. Since t=3 this works out. In general S(t,t,n) is always rather boring.
And be careful with the numbers: if t=2, k=q+1 and n=q^2+q+1 (for example S(2,3,7), which is q=2), then the Steiner system comes from a finite projective plane with q^2+q+1 points, and each line passes through q+1 points, and each pair of distinct points lies on exactly one line. So in your example, q=3 and we get S(2,4,13) which is a finite projective plane with 13 points and each line passes through 4 points. And indeed, such a projective plane exists: en.wikipedia.org/wiki/Projective_plane#A_finite_example
I hope that helps!

[@luketaylor1257]

I've been looking to find the actual blocks of both S(5,6,12) and S(5,8,24). I've found a lot of proofs of uniqueness and stuff like that, but I want to get a hold of the actual "Block 1: (A,B,C,D,E,F) Block 2: (A, F, G...) or whatever. Anyone know where I could find that?

[@VisualMath]

They are rather complicated, I hope someone has some knowledge to share. I would be happy to know this as well!
What I would try is (potentially useless): I think you need to look at the explicit constructions (involving some Mathieu group maybe), which you might be able to do via SageMath. This side linear.ups.edu/eagts/section-24.html looks promising; maybe one can take it from there.

[@hamzaaallouche9452]

hey . hope u're doing great
please i have struggles to build systems S(6,6,36) for long term support succesfully
how can deal with it .
like the instructions on how ti build it .
thank u for u time

[@VisualMath]

Not sure what you want. Steiner systems S(t,k,n) with t=k are trivial: just take all k-element subsets as the blocks. Is that what you are looking for?

[@devrimturker]

Conway felt great pressure on him to find a new thing until he discovered Mathieu M12 Leech Lattice. Then he felt more freedom on his researches.

[@VisualMath]

Well, I certainly would be happy if I would discover 1% of what Conway did.

[@kamnovak5268]

thank you for your explanation

[@VisualMath]

You are very welcome! I hope you enjoyed it.