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

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.

• I do not think this is even possible for an arbitrary rectangle. Do you have a reason to suspect that such a tiling exists in the case of a rectangle whose sides have an irrational ratio? Mar 2, 2021 at 21:30
• No you misunderstood. For all possible tilings of rectangles I am looking for the one with the smallest number of triangles. Mar 2, 2021 at 23:17
• Ah, I see. Thanks for the clarification! Mar 2, 2021 at 23:20
• Clearly all rectangles with rational ratio between the side has a solution - not sure that this is the minimal.
– Moti
Mar 6, 2021 at 6:33
• Is there a nice app for testing out ideas? I tried GeoGebra, but adding a new triangle takes quite a few steps (Angle with given sizex2+Rayx2+Intersection). Doing the calculations by hand is very tedious. Mar 7, 2021 at 20:24 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)

• Nice! This may not be minimal though. Nov 19, 2021 at 2:09
• Thanks! It's clear that at least 12 triangles are needed just to complete the perimeter, so proving a minimal solution might be within reach. Nov 19, 2021 at 3:19
• Perhaps. Is it clear that the number should be even? Nov 19, 2021 at 3:24
• I understood where $12$ came from. It is probably clear that the number is at least $14$. But the cases $15,16,17$ are not clear at all. Nov 19, 2021 at 4:04
• This is awesome, thank you Tom! Nov 19, 2021 at 6:37