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Thanks. I made Platonic solid a hyperlink.
(Let’s remember to hyperlink at least the key technical terms in an entry. That’s what make a wiki be more useful than a book.)
Why is the index $4 n$ for the dihedral group $D_{4n}$? I have a vague recollection of a difference in terminology about say $D_5$ or $D_10$ for symmetries of the pentagon. Ah yes, wikipedia mentions this. But this concerns $n$ or $2n$, not $4n$.
Another point, we claim that the ADE classification concerns Platonic solids, and yet don’t associate anything with the $A$ series in the table. Is there a way of associating degenerate solids to both $A$ and $D$? Perhaps this page helps.
I have a feeling there’s something else wrong with that table. Wikipedia speaks of a ’binary cyclic group’, which is what we should have presumably as a subgroup of $SU(2)$.
I have a feeling there’s something else wrong with that table. Wikipedia speaks of a ’binary cyclic group’, which is what we should have presumably as a subgroup of $SU(2)$.
Thanks for catching that. I created binary cyclic group and fixed the ADE – table.
But why “something else”? What else is wrong?
The index in $D_{4n}$ issue.
I see now. Thanks.
Let me see. here is a corresponding table in Durfee 79
It seems to say that
Dynkin | fin group | symbol | order |
---|---|---|---|
$D_k$ | binary dihedral | $D_k$ | 4(k-2) |
for $k \geq 4$.
I would like us to start counting at 0. That should give
Dynkin | fin group | symbol | order |
---|---|---|---|
$D_{n+4}$ | binary dihedral | $D_{n+4}$ | 2(2n+4) |
for $n \in \mathbb{N}$
This seems to fit with neither of the two conventions that Wikipedia offers, even if one accounts for the binary version.
But we get from it that the non-binary dihedral group corresponding to the Dynkin diagram $D_{n+4}$ has order $2(n+2)$. If we follow Wikipedia, then this should be called either $D_{n+2}$ or $D_{2n+4}$.
What a mess!
Gah, is it too much to ask for Wikipedia to give the collection of unit quaternions corresponding to the binary cyclic group? Cf https://groupprops.subwiki.org/wiki/Dicyclic_group (which I will add later if no one beats me to it).
I have a feeling there’s something else wrong with that table. Wikipedia speaks of a ’binary cyclic group’, which is what we should have presumably as a subgroup of $SU(2)$.
Thanks for catching that. I created binary cyclic group and fixed the ADE – table.
Sorry, that was wrong. I changed it back. The non-binary $\mathbb{Z}_{2n+1}$ are still finite subgroups of $SU(2)$, of course: The generator is
$\left( \array{ e^{2\pi i / (2n + 1)} & 0 \\ 0 & e^{-2\pi i / (2n + 1)} } \right)$So the odd cyclics are subgroups of $SU(2)$ but are not binary polyhedral groups.
Yes.
In Keenan 03, theorem 4 it is phrased this way:
Every finite subgroup of $SU(2)$ is a cyclic, binary dihedral or binary polyhedral group.
That pointer should go to binary dihedral group. I have added it there.
Thanks. I was looking up stuff on my phone while walking to the bus, not quite up to editing a lab page.
In the sentence
A dihedron is a degenerate Platonic solid with only two (identical) faces, which may be any polygon (including possibly the degenerate
I have replaced “polygon” by “regular polygon”
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