Post History
#3: Post edited
by
WheatWizard
·
2024-01-04T15:22:35Z (12 months ago)
Coxeter points out that for a self-dual configuration $(m_c,n_d)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $m_c$.- However I'm interested in the other direction of this implication, i.e. is there a configuration $(m_c,m_c)$ which is not self-dual? For $c=2$ there is none. All polygons are combinatorially self-dual.
- It's also clear that if the automorphism group of the configuration is flag-transitive, then it is of the form $_n\{p\}_n$ and must be self dual. (All polygons fall into this category.)
- However this still leaves a lot of open cases. In particular I am interested in 3-configurations.
- Is there a $(n_3)$ configuration which is not self-dual?
- ### References
- * Coxeter, (1950) *Self-dual configurations and regular graphs*.
- Coxeter points out that for a self-dual configuration $(m_c,n_d)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $(m_c)$.
- However I'm interested in the other direction of this implication, i.e. is there a configuration $(m_c,m_c)$ which is not self-dual? For $c=2$ there is none. All polygons are combinatorially self-dual.
- It's also clear that if the automorphism group of the configuration is flag-transitive, then it is of the form $_n\{p\}_n$ and must be self dual. (All polygons fall into this category.)
- However this still leaves a lot of open cases. In particular I am interested in 3-configurations.
- Is there a $(n_3)$ configuration which is not self-dual?
- ### References
- * Coxeter, (1950) *Self-dual configurations and regular graphs*.
#2: Post edited
by
WheatWizard
·
2024-01-04T04:26:42Z (12 months ago)
A little about the automorphism group.
- Coxeter points out that for a self-dual configuration $(m_c,n_d)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $m_c$.
- However I'm interested in the other direction of this implication, i.e. is there a configuration $(m_c,m_c)$ which is not self-dual? For $c=2$ there is none. All polygons are combinatorially self-dual.
However for 3-configurations it is not as apparent.- Is there a $(n_3)$ configuration which is not self-dual?
- ### References
- * Coxeter, (1950) *Self-dual configurations and regular graphs*.
- Coxeter points out that for a self-dual configuration $(m_c,n_d)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $m_c$.
- However I'm interested in the other direction of this implication, i.e. is there a configuration $(m_c,m_c)$ which is not self-dual? For $c=2$ there is none. All polygons are combinatorially self-dual.
- It's also clear that if the automorphism group of the configuration is flag-transitive, then it is of the form $_n\{p\}_n$ and must be self dual. (All polygons fall into this category.)
- However this still leaves a lot of open cases. In particular I am interested in 3-configurations.
- Is there a $(n_3)$ configuration which is not self-dual?
- ### References
- * Coxeter, (1950) *Self-dual configurations and regular graphs*.
#1: Initial revision
by
WheatWizard
·
2024-01-04T03:47:14Z (12 months ago)
Is there a $(n_3)$ configuration which is not self-dual?
Coxeter points out that for a self-dual configuration $(m_c,n_d)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $m_c$. However I'm interested in the other direction of this implication, i.e. is there a configuration $(m_c,m_c)$ which is not self-dual? For $c=2$ there is none. All polygons are combinatorially self-dual. However for 3-configurations it is not as apparent. Is there a $(n_3)$ configuration which is not self-dual? ### References * Coxeter, (1950) *Self-dual configurations and regular graphs*.