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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI added a loose description of the dihedron, and commented that the 2-gon as a face should be possible (so as to have the $A_1$ case included, thinking of the [[ADE]] classification) <a href="https://ncatlab.org/nlab/revision/diff/dihedron/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/dihedron/2">v2</a>, <a href="https://ncatlab.org/nlab/show/dihedron">current</a>

   I added a loose description of the dihedron, and commented that the 2-gon as a face should be possible (so as to have the $A_1$ case included, thinking of the ADE classification)

   diff, v2, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. 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.)

   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.)

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWhy 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](https://en.wikipedia.org/wiki/Dihedral_group) 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](https://en.wikipedia.org/wiki/List_of_finite_spherical_symmetry_groups) helps. I have a feeling there's something else wrong with that table. [Wikipedia](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#Binary_polyhedral_groups) speaks of a 'binary cyclic group', which is what we should have presumably as a subgroup of $SU(2)$.

   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 $SU(2)$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I have a feeling there's something else wrong with that table. [Wikipedia](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#Binary_polyhedral_groups) 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?

   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?

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThe index in $D_{4n}$ issue.

   The index in $D_{4n}$ issue.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2018
   • (edited Apr 16th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI see now. Thanks. Let me see. [here](https://ncatlab.org/nlab/revision/Sandbox/1480) is a corresponding table in [Durfee 79](https://ncatlab.org/nlab/show/ADE+singularity#Durfee79) 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!

   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!

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexGah, 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).

   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).

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItex>> I have a feeling there's something else wrong with that table. [Wikipedia](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#Binary_polyhedral_groups) 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 $$ \left( \array{ e^{2\pi i / (2n + 1)} & 0 \\ 0 & e^{-2\pi i / (2n + 1)} } \right) $$

   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

   $$\left( \array{ e^{2\pi i / (2n + 1)} & 0 \\ 0 & e^{-2\pi i / (2n + 1)} } \right)$$
9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 16th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo the odd cyclics are subgroups of $SU(2)$ but are not binary polyhedral groups.

   So the odd cyclics are subgroups of $SU(2)$ but are not binary polyhedral groups.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes. In [Keenan 03, theorem 4](https://ncatlab.org/nlab/show/classification+of+finite+rotation+groups#Keenan03) it is phrased this way: > Every finite subgroup of $SU(2)$ is a cyclic, binary dihedral or binary polyhedral group.

    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.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 16th 2018
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexFor reference : <https://en.wikipedia.org/wiki/Dicyclic_group#Binary_dihedral_group>

    For reference : https://en.wikipedia.org/wiki/Dicyclic_group#Binary_dihedral_group

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexThat pointer should go to _[[binary dihedral group]]_. I have added it there.

    That pointer should go to binary dihedral group. I have added it there.

13. • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 17th 2018
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexThanks. I was looking up stuff on my phone while walking to the bus, not quite up to editing a lab page.

    Thanks. I was looking up stuff on my phone while walking to the bus, not quite up to editing a lab page.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexIn 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" <a href="https://ncatlab.org/nlab/revision/diff/dihedron/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/dihedron/4">v4</a>, <a href="https://ncatlab.org/nlab/show/dihedron">current</a>

    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”

    diff, v4, current