10
$\begingroup$

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.

$\endgroup$
6
  • $\begingroup$ 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? $\endgroup$
    – RavenclawPrefect
    Mar 2, 2021 at 21:30
  • $\begingroup$ No you misunderstood. For all possible tilings of rectangles I am looking for the one with the smallest number of triangles. $\endgroup$
    – Dmitry Kamenetsky
    Mar 2, 2021 at 23:17
  • 1
    $\begingroup$ Ah, I see. Thanks for the clarification! $\endgroup$
    – RavenclawPrefect
    Mar 2, 2021 at 23:20
  • $\begingroup$ Clearly all rectangles with rational ratio between the side has a solution - not sure that this is the minimal. $\endgroup$
    – Moti
    Mar 6, 2021 at 6:33
  • 4
    $\begingroup$ 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. $\endgroup$
    – Tom Sirgedas
    Mar 7, 2021 at 20:24

1 Answer 1

Reset to default
12
$\begingroup$

enter image description here

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

enter image description here

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

enter image description here

20 triangles (48 by 30+6√₃)

enter image description here

22 triangles (772 by 402+54√₃)

enter image description here

22 triangles (222 by 228-30√₃)

$\endgroup$
10
  • $\begingroup$ Nice! This may not be minimal though. $\endgroup$
    – markvs
    Nov 19, 2021 at 2:09
  • $\begingroup$ 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. $\endgroup$
    – Tom Sirgedas
    Nov 19, 2021 at 3:19
  • $\begingroup$ Perhaps. Is it clear that the number should be even? $\endgroup$
    – markvs
    Nov 19, 2021 at 3:24
  • 1
    $\begingroup$ 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. $\endgroup$
    – markvs
    Nov 19, 2021 at 4:04
  • 1
    $\begingroup$ This is awesome, thank you Tom! $\endgroup$
    – Dmitry Kamenetsky
    Nov 19, 2021 at 6:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .