In mathematics, Danzer's configuration is a self-dual
configuration
Configuration or configurations may refer to:
Computing
* Computer configuration or system configuration
* Configuration file, a software file used to configure the initial settings for a computer program
* Configurator, also known as choice board ...
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
Ludwig Danzer (15 November 1927 – 3 December 2011) was a German geometer working in discrete geometry. He was a student of Hanfried Lenz, starting his career in 1960 with a thesis about "Lagerungsprobleme".
Danzer's name is popularized in t ...
and was popularised by
Branko Grünbaum. The
Levi graph of the configuration is the
Kronecker cover of the
odd graph O
4, and is isomorphic to the
middle layer graph
Middle or The Middle may refer to:
* Centre (geometry), the point equally distant from the outer limits.
Places
* Middle (sheading), a subdivision of the Isle of Man
* Middle Bay (disambiguation)
* Middle Brook (disambiguation)
* Middle Creek ...
of the seven-dimensional
hypercube graph Q
7. 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.
Danzer's configuration DCD(4) is the fourth term of an infinite series of
configurations DCD(n), where DCD(1) is the trivial configuration (1
1), DCD(2) is the trilateral (3
2) and DCD(3) is the
Desargues configuration
In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues.
The Desargues configuration can be constructed in two dimensions f ...
(10
3). In configurations DCD(n) were further generalized to the unbalanced
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
Clifford configuration Clifford may refer to:
People
*Clifford (name), an English given name and surname, includes a list of people with that name
*William Kingdon Clifford
* Baron Clifford
*Baron Clifford of Chudleigh
* Baron de Clifford
*Clifford baronets
* Clifford f ...
. 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
In geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles.
Statement
The first theorem considers any four circles passing through a common poin ...
.
Example
See also
*
Miquel configuration
*
Reye configuration
References
Bibliography
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*.
*.
*.
Configurations (geometry)
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