I’m bad at geometry and am hoping for some help. The path I’ve laid so far is 4 ft across on top left of the pic. I’ve made my turn and am about to connect to my deck. I plan to cut the edges of the path down to a width of 4ft across. My question is, how do I keep my path width 4ft and account for the turn at the same time?

Are there any websites that can generate printable nets for shapes given specific dimensions?

I am trying to make DEVO Energy Domes (hats) for Halloween. I have the dimensions required but it will be tricky (impossible) to make a net by hand for a truncated cone (aka tapered cylinder). I could just make straight up cylinders, but I’d like to get it right if I can.

Thanks!

Kobon Triangles - optimal arrangement for k=19 found!

Kobon Triangle Problem - optimal arrangement for k=19 found with 107 triangles! (previously unknown)

The Kobon Triangle Problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines.

Before this, the largest value k for which an optimal arrangement was known was k=17, with 85 triangles.

k=19 has an upper bound of 107 triangles, but the best known arrangement had 104 triangles. This arrangement I found has 107 triangles, and so has the maximum number of triangles possible!

I can only do one attachment, the image itself, so I can’t link my GitHub which has the code I used to find the arrangement. But here it is:

https://github.com/Bombardlos/Kobon_Triangle_Workspace

compile_mirror was used to find this arrangement in pure numerical form, then a separate program rendered 19_representation.png, and finally I made 19_final by hand. I also have compile_11, which is an algorithmic proof that k=11 CANNOT reach the current accepted upper bound of 33 triangles, and so the current best arrangement with 32 triangles is actually optimal. With the right equipment, it could ALSO find whether there is an arrangement for k=21 which meets the upper bound in a reasonable amount of time, but my laptop sucks and I don’t wanna cook it TOO badly lol.

I actually found the arrangement about a week ago, but it was with an algorithm that abstracts it really far away from the physical model. It took me awhile to turn a representation of the model into the model itself, and I had to do it largely by hand. I actually bought ribbon and wall tacks to be able to arrange part of it, since the first visual representation used VERY unstraight lines. I could move around the ribbons at certain points and restrict their movements with tacks, eventually sorting them into much straighter lines. Finally, I took a picture, opened a Google Slides file, uploaded the pic, turned the opacity down, and drew line objects overlayed on top of the pic. Did some more adjusting, and the final image is just a screenshot of the Google Slide 😂