Weil norm images

These are a few pictures I made while working on Weil height; I'm
archiving them here so I can dump what I've got and move on. These
were originally posted to Paul on Facebook so this is just an archival
backup.

===UNNORMALIZED TENNEY-WEIL UNIT BALLS, p=1, k varies===
/tuning-math/files/MikeBattaglia/TF_p_1-demo-different-balls-unnormalized.png

This is the interpolated "Tenney-Weil" norm, specifically the
"unnormalized" version. The hexagon in the center is the Weil norm,
the green diamond is the Tenney norm. This corresponds to log(n*d) +
k*|log(n/d)|. "TF" here stands for "Tenney-Farey," which is what I was
calling it before Farey became Weil.

===NORMALIZED TENNEY-WEIL UNIT BALLS, p=1, k varies===
/tuning-math/files/MikeBattaglia/TF_p_1-demo-different-balls-normalized.png:

These are all placed on top of each other, note that the top-right and
bottom-left of each of these unit spheres is exactly the same as
what's outlined in green. This corresponds to (log(n*d) +
k*|log(n/d)|)/(1+k).

k=0 is exactly log(n*d) and k=1 is exactly log(max(n,d)), with no
additional scaling required.

You can see that the red diamond is the Tenney norm, and the dark
green outer region (also given by the union of all of these) is the
Weil norm. The light green region in the bottom-left and upper-right
is all that remains of min(n,d). But you can see that this is how you
can transform a Tenney norm into a Weil norm; the top-left and
bottom-right regions of the diamond "pull" out to become the more
familiar looking Kees-esque hexagon.

Note also at the turquoise region given by k=0.5; this is right
between a Tenney and Weil norm. It's also one possible candidate to
finesse away the minimax problem and Carl Lumma's objection that it's
bad to rank 5/1 and 5/3 and 5/4 the same.

===NORMALIZED TENNEY-WEIL UNIT BALLS, p varies, k varies===
/tuning-math/files/MikeBattaglia/TF-Norm-Various-Demo-Normalized.png:

This lets us look at the influence of varying p as well, so we can
look at Weil-Euclidean norms and the like. Again, there are two
parameters, k and p; the former controls how much Tenneyness vs
Weilness is in the norm, and the latter controls how much L1ness vs
other Lpness is in the norm.

If our monzo is |a b c ...>, this correlates to ((|a|^p + |b|^p +
|c|^p + ... + k*|a+b+c|^p)/(1+k))^(1/p). Why did I put the 1+k in the
denominator INSIDE the 1/p? I dunno, because it worked better that
way.

We can look at the top left to see the usual Tp norm. The purple
diamond is the T1 or Tenney norm, the orange circle behind it is the
T2 or Tenney-Euclidean norm, and the green rectangle behind it is the
Tinf norm, whatever that means.

Look at the top-right to see the Weil norm. The blue hexagon is the
Weil norm, and the grey circle on the outside is its L2 equivalent.
You can see how, much like the T2 norm is an ellipse that mimics the
general shape and contour of the T1 norm, the W2 norm is an ellipse
that mimics the general shape and contour of the T2 norm. Then in the
background there's the Winf norm which is -exactly- the same as the W1
norm, but that's only for the 2D case (and not for higher dimensional
versions).

Note that k=0.5 on top is that intermediate norm again; note the L2
version still sort of follows the contour of it, but in the background
the Linf norm is doing weird things. I dunno what it means but it's
probably not important.

The bottom row is the same as the top row but with k set to the
negative version of what it was in the top row. We got some really
weird results for k=-0.5 and p=2. The bottom-right one was also very
strange - somehow k=~-1 and p=~Inf gives us something looking like the
ordinary Weil norm again. I dunno what to make of these cases but
they're pathological anyway.

-Mike