Danzer's configuration

Image:danzer_graph.svg of Danzer's configuration as unit distance graph.]]

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.{{sfnp|Grünbaum|2008}} The Levi graph of the configuration is the Kronecker cover of the odd graph O₄,{{sfnp|Boben|Gévay|Pisanski|2015}} 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 Q_2n+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.{{sfnp|Mütze|2016}}

Danzer's configuration DCD(4) is the fourth term of an infinite series of $(\tbinom {2n-1}{n}_n)$ 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 {{sfnp|Gévay|2018}} configurations DCD(n) were further generalized to the unbalanced $(\tbinom {n}{d}_d, \tbinom {n}{d-1}_{n-d+1})$ 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 $(2^{2n}_{2n+1})$ 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

Image:Hasse diagram of powerset of 3.svg of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Distinct sets on the same horizontal layer are incomparable with each other. Two consecutive layers form a Levi graph of a suitable DCD-configuration.]]

References

Bibliography

{{citation

| last1 = Boben | first1 = Marko

| last2 = Gévay | first2 = Gábor

| last3 = Pisanski | first3 = T. | author3-link = Tomaž Pisanski

| doi = 10.1515/advgeom-2015-0019

| issue = 4

| journal = Advances in Geometry

| mr = 3406469

| pages = 393–408

| title = Danzer's configuration revisited

| volume = 15

| year = 2015| arxiv = 1301.1067

| s2cid = 117048451

}}.

{{citation

| last1 = Gévay | first1 = Gábor

| editor1-last = Conder | editor1-first = Marston D. E.

| editor2-last = Deza | editor2-first = Antoine

| editor3-last = Weiss | editor3-first = Asia Ivić

| mr = 3816877

| series = Springer Proceedings in Mathematics & Statistics

| pages = 181–199

| chapter = Pascal's triangle of configurations

| title = Discrete Geometry and Symmetry

| volume = 234

| doi = 10.1007/978-3-319-78434-2_10

| year = 2018| isbn = 978-3-319-78433-5

}}.

{{citation

| last = Grünbaum | first = Branko | author-link = Branko Grünbaum

| doi = 10.1016/j.ejc.2008.01.004

| journal = European Journal of Combinatorics

| mr = 2463166

| page = 1910-1918

| title = Musing on an example of Danzer's

| year = 2008

| volume = 29| issue = 8 | doi-access =

}}.

{{citation

| last = Mütze | first = Torsten

| journal = Proc. Lond. Math. Soc.

| mr = 3483129

| page = 677-713

| title = Proof of the middle levels conjecture

| year = 2016

| volume = 112| issue = 4

| doi = 10.1112/plms/pdw004

| s2cid = 119119260

| url = http://wrap.warwick.ac.uk/118871/2/WRAP-proof-middle-levels-conjecture-Mutze-2016.pdf

}}.

Category:Configurations (geometry)

Danzer's configuration

Example

See also

References

Bibliography