infrapinklizzard
Angles demonstrated
This was an exercise that was inspired by Gregory Brick's picture of various angles, and showing one (C) that is not a 3-4-5 triangle but is used in an official set.
On each of these constructions, the 1x1 round plates are directly over a connection to a stud below.
A is an extension of tilting a 2x4 so that one diagonal from extreme studs takes the place of the other diagonal. It's a 3-4-5 triangle. The next stud that meshes on the right side is the sixth (five from the start). It meshes three over and four up.
B is an extension of taking the 2x4 and tilting it so that the diagonal from extreme corners takes the place of the other diagonal. And it is also a 3-4-5 triangle. The next connection on the right side will be five studs along. That will be four studs sideways and three up. (This one was actually easier to demonstrate on the left side - but the principal remains.)
C is complicated. It is an extension of going from one corner to the opposite extreme stud. That is approximately 45°. (It's actually 46.4° from the vertical in his diagram and 43.6° from the horizontal.)
Since that is so close to 45°, it means that you can make connections over the short term of every 5 studs/5 studs with a run of 7 studs. (This is actually off by ~0.07 studs in length since we're forcing it to be a true 45°.)
Angles demonstrated
This was an exercise that was inspired by Gregory Brick's picture of various angles, and showing one (C) that is not a 3-4-5 triangle but is used in an official set.
On each of these constructions, the 1x1 round plates are directly over a connection to a stud below.
A is an extension of tilting a 2x4 so that one diagonal from extreme studs takes the place of the other diagonal. It's a 3-4-5 triangle. The next stud that meshes on the right side is the sixth (five from the start). It meshes three over and four up.
B is an extension of taking the 2x4 and tilting it so that the diagonal from extreme corners takes the place of the other diagonal. And it is also a 3-4-5 triangle. The next connection on the right side will be five studs along. That will be four studs sideways and three up. (This one was actually easier to demonstrate on the left side - but the principal remains.)
C is complicated. It is an extension of going from one corner to the opposite extreme stud. That is approximately 45°. (It's actually 46.4° from the vertical in his diagram and 43.6° from the horizontal.)
Since that is so close to 45°, it means that you can make connections over the short term of every 5 studs/5 studs with a run of 7 studs. (This is actually off by ~0.07 studs in length since we're forcing it to be a true 45°.)