Comparing triangles using Java 8 streams
Since (as mentioned in my comment) the question is very tightly scoped, this review will be mostly style-focused:
Names
Some of your names are good. Some of your names are bad. I didn't see anything really ugly though, so you got that going for you ;)
Some examples of the good names include:

original, permutation, bestDeviation, trial


These names are speaking for themselves, it's almost instantly clear what they  mean.
On the other hand there's things like:

t1, t2, v, v2


These names are completely meaningless. I can't tell what they stand for, I have no idea (not in the slightest) what is in the variables with these names. It would tremendously help to understand your code's purpose, if I (or Mr. Maintainer) knew what the smallest units the code works with are.
I'd guesstimate you're operating on Vectors in a graphical representation of bodies, but I simply cannot be sure.
Anyways, from naming to:
Spacing and Linebreaks:
Thge code could use some additional linebreaks here and there, if only just to reduce the &quot;verical complexity&quot; of your code.
Normally I try to portion my code so you can read it one line at a time, and almost instantly understand what happens on it.
That's extremely hard when I need to scroll and there's multiple (low-level) transformations and operations. Try to write code, which doesn't take long to skim over and get the gist of.

points1.stream().map(v -&gt; v.extend(1)).map(trial::multiply).forEach(v -&gt; sum[0] += points2.stream().mapToDouble(v2 -&gt; distanceCheated(v, v2)).min().orElse(0));


then becomes:
points1.stream().map(v -&gt; v.extend(1))
        .map(trial::multiply)
        .forEach(v -&gt; sum[0] += points2.stream()
                .mapToDouble(v2 -&gt; distanceCheated(v, v2))
                .min().orElse(0)
        );

On the other hand there's one place where I feel you left too much space:

t2.forEach(transformed -&gt;
{

    for (int[] permutation : permutations) {


I'd write it like this:
t2.forEach(transformed -&gt; {
    for (int[] permutation : permutations) {

Other than this, there's no stylistic points to make here, and as mentioned I don't want to review the functionality without more information and context.