Danzer's configuration
In mathematics, Danzer's configuration is a self-dual configuration of 35 lines and 35 points, having 4 points on each line and 4 lines through each point. It is named after the German geometer Ludwig Danzer and was popularised by Branko Grünbaum.[1] The Levi graph of the configuration is the Kronecker cover of the odd graph O4,[2] and is isomorphic to the middle layer graph of the seven-dimensional hypercube graph Q7. The middle layer graph of an odd-dimensional hypercube graph Q2n+1(n,n+1) is a subgraph whose vertex set consists of all binary strings of length 2n + 1 that have exactly n or n + 1 entries equal to 1, with an edge between any two vertices for which the corresponding binary strings differ in exactly one bit. Every middle layer graph is Hamiltonian.[3]
Danzer's configuration DCD(4) is the fourth term of an infinite series of [math]\displaystyle{ (\tbinom {2n-1}{n}_n) }[/math] configurations DCD(n), where DCD(1) is the trivial configuration (11), DCD(2) is the trilateral (32) and DCD(3) is the Desargues configuration (103). In [4] configurations DCD(n) were further generalized to the unbalanced [math]\displaystyle{ (\tbinom {n}{d}_d, \tbinom {n}{d-1}_{n-d+1}) }[/math] configuration DCD(n,d) by introducing parameter d with connection DCD(n) = DCD(2n-1,n). DCD stands for Desargues-Cayley-Danzer. Each DCD(2n,d) configuration is a subconfiguration of the [math]\displaystyle{ (2^{2n}_{2n+1}) }[/math] Clifford configuration. While each DCD(n,d) admits a realisation as a geometric point-line configuration, the Clifford configuration can only be realised as a point-circle configuration and depicts the Clifford's circle theorems.
Example
![](/wiki/images/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/250px-Hasse_diagram_of_powerset_of_3.svg.png)
See also
References
Bibliography
- Boben, Marko; Gévay, Gábor (2015), "Danzer's configuration revisited", Advances in Geometry 15 (4): 393–408, doi:10.1515/advgeom-2015-0019.
- Gévay, Gábor (2018), "Pascal's triangle of configurations", in Conder, Marston D. E.; Deza, Antoine; Weiss, Asia Ivić, Discrete Geometry and Symmetry, Springer Proceedings in Mathematics & Statistics, 234, pp. 181–199, doi:10.1007/978-3-319-78434-2_10, ISBN 978-3-319-78433-5.
- "Musing on an example of Danzer's", European Journal of Combinatorics 29 (8): 1910-1918, 2008, doi:10.1016/j.ejc.2008.01.004.
- Mütze, Torsten (2016), "Proof of the middle levels conjecture", Proc. Lond. Math. Soc. 112 (4): 677-713, doi:10.1112/plms/pdw004, http://wrap.warwick.ac.uk/118871/2/WRAP-proof-middle-levels-conjecture-Mutze-2016.pdf.
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