Talk:The Seven Limit Symmetrical Lattices: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Wikispaces>FREEZE
No edit summary
 
Sintel (talk | contribs)
No edit summary
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
= ARCHIVED WIKISPACES DISCUSSION BELOW =
{{WSArchiveLink}}
'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''
----


== Question about [a b c] notation ==
== Some notes ==
I'm confused by this sentence:


"The 4:5:6:7 major tetrad consists of the notes |* 0 0 0>, |* 1 0 0>, |* 0 1 0>, and |* 0 0 1>; the centroid of this is |* 1/2 1/2 1/2>; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2>. If we shift the origin to |* 1/2 1/2 1/2>, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd"
Interesting page, actually. I'm trying to work through it bit by bit to make sure I understand it.
The page doesn't really explain what is "symmetric" about these lattices. The original lattice is of course already full of symmetries since it's just the usual cubical lattice.


This [a b c] notation wasn't defined anywhere on the page. What does it mean, and how does it define a tetrad, rather than an interval?
For the first part, there is some bit about coming up with various norms and then comes to the conclusion that you can make a cuboctahedron out of the 7-odd limit.
I feel like you can explain this quite simply by considering the transformation:


- '''mbattaglia1''' September 15, 2011, 12:29:35 AM UTC-0700
:<math>
----
A = \begin{bmatrix}
I'm going to rewrite this old posting.
0 & -1 & -1 \\
-1 &  0 & -1 \\
-1 & -1 &  0 \\
\end{bmatrix}
</math>


- '''genewardsmith''' September 15, 2011, 09:16:55 AM UTC-0700
Computing explicitly:
----
 
:<math>
\begin{aligned}
A[1,0,0]^\mathsf{T} &= [0, -1, -1]^\mathsf{T} \\
A[0,1,0]^\mathsf{T} &= [-1, 0, -1]^\mathsf{T} \\
A[0,0,1]^\mathsf{T} &= [-1, -1, 0]^\mathsf{T} \\
A[1,-1,0]^\mathsf{T} &= [1,-1,0]^\mathsf{T} \\
A[1,0,-1]^\mathsf{T} &= [1,0,-1]^\mathsf{T} \\
A[0,1,-1]^\mathsf{T} &= [0,1,-1]^\mathsf{T} \\
\end{aligned}
</math>
 
Which are the vertices of a cuboctahedron, as we wanted.
 
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 22:00, 26 April 2025 (UTC)

Latest revision as of 22:00, 26 April 2025

This page also contains archived Wikispaces discussion.

Some notes

Interesting page, actually. I'm trying to work through it bit by bit to make sure I understand it. The page doesn't really explain what is "symmetric" about these lattices. The original lattice is of course already full of symmetries since it's just the usual cubical lattice.

For the first part, there is some bit about coming up with various norms and then comes to the conclusion that you can make a cuboctahedron out of the 7-odd limit. I feel like you can explain this quite simply by considering the transformation:

[math]\displaystyle{ A = \begin{bmatrix} 0 & -1 & -1 \\ -1 & 0 & -1 \\ -1 & -1 & 0 \\ \end{bmatrix} }[/math]

Computing explicitly:

[math]\displaystyle{ \begin{aligned} A[1,0,0]^\mathsf{T} &= [0, -1, -1]^\mathsf{T} \\ A[0,1,0]^\mathsf{T} &= [-1, 0, -1]^\mathsf{T} \\ A[0,0,1]^\mathsf{T} &= [-1, -1, 0]^\mathsf{T} \\ A[1,-1,0]^\mathsf{T} &= [1,-1,0]^\mathsf{T} \\ A[1,0,-1]^\mathsf{T} &= [1,0,-1]^\mathsf{T} \\ A[0,1,-1]^\mathsf{T} &= [0,1,-1]^\mathsf{T} \\ \end{aligned} }[/math]

Which are the vertices of a cuboctahedron, as we wanted.

Sintel🎏 (talk) 22:00, 26 April 2025 (UTC)