Smallest number of $45^\circ-60^\circ-75^\circ$ triangles that tile a rectangle

Question

In this wonderful question we learned that a square can be divided into forty six $45^\circ-60^\circ-75^\circ$ triangles.

Now I am wondering what is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that can tile some rectangle? In other words, from all tilings of rectangles with such triangles, I am looking for the one with the smallest number of triangles.

Answer 1

Here's my 18-triangle solution.

From the coordinates, it seems the width/height ratio of the rectangle is 138/(96+6√₃) or (16-√₃)/11 which is about 1.297

How

Using my code from answering this question, I checked for 5-sided polygons that have 45-degree, 90-degree, 45-degree angles in sequence, with equal lengths for the (parallel) "45-degree" sides.

I was happy to find that a simple solution exists!

Coordinates

27+6√₃,48+3√₃
63-9√₃,33+15√₃
36+8√₃,36+8√₃
27+6√₃,69
27+6√₃,27+6√₃
0,63-9√₃
0,96+6√₃
0,0
84-12√₃,0
75+9√₃,63-9√₃
138,0
111-6√₃,48+3√₃
102-8√₃,60-2√₃
111-6√₃,27+6√₃
111-6√₃,69
138,33+15√₃
138,96+6√₃
54+12√₃,96+6√₃
{{0,1,2},{3,1,0},{4,0,2},{5,4,3},{6,3,5},{7,5,4},{7,8,2},{8,9,1},{10,9,8},{11,9,12},{13,9,11},{14,11,12},{15,14,13},{10,13,15},{16,15,14},{16,17,12},{17,1,9},{6,1,17}}

Larger solutions

20 triangles (48 by 30+6√₃)

22 triangles (772 by 402+54√₃)

22 triangles (222 by 228-30√₃)