LUGNET News : ~726/Diagonals


Gus Jansson / Diagonals

Here is the Diagonals spreadsheet in PDF form:

Diagonals PDF File

All the diagonal measurements are rounded to either the nearest quarter of a unit or half of a unit if the diagonal is greater then 5 units. For each measurement, I also show the error of the rounding in terms of the percentage of one Lego unit.

For example, lets say you want to mesh a 40 tooth gear with one of the new 12 tooth gears. When you look at the gear chart, you will see that the ideal distance between the axles is 3.25 Lego units. Now you can look at the diagonals chart and find a few close matches. With a vertical offset of 3.2 LU and a horizontal offset of 0.5 LU, there is an excellent fit with a diagonal whose length is only 1.12% of a LU off. Another close fit can be found at the vertical offset of 1.2 LU and horizontal offset 3 LU. This one has an error of 1.89%.

My testing shows that errors as great as 5% are usually no problem for gears. If the error is more then that, it may be too tight and jam or so loose that the gears risk skipping.

LEGO Unit terminology: In my terminology, one LEGO Unit (LU) is the width of a one stud block. It is also the distance between the studs on a block and the distance between axles holes on a beam. The height of a plate is .4 LU. Thus a standard LEGO brick, which is the same height as three plates, is 1.2 LU high. If you put two beams on top of each other without plates in between, the vertical distance between the holes of the two beams will be 1.2 LU apart. If you put two beams together with two plates in between them, then the vertical distance between the holes will by 2 LU, 1.2 from the height of the beams and another .8 from the two plates.

The reason that I have horizontal offsets in .5 increments is because it is possible to combine standard beams with beams that have the holes offset by half of an LU. Such as the 2 hole length 2 beam. Vertically, I have increments in .4 of a unit, the same as the height of a plate. I round to the nearest quarter of a unit when the diagonal is less then 5 because of the possible gear combinations that need them. Since the greatest distance between gears is 5 LU, I round to nearest half of LU above that.

For the new 12 and 20 tooth gears, there often doesn’t even exist a good vertical or horizontal way of laying out the gears. For example, lets say you want to mesh a 20 tooth with a 40 tooth gear. The ideal spacing is 3.75 LEGO Units. With a quick glance at the chart I found as many as 4 possible diagonal arrangements.

The chart is also designed to find acceptable diagonals that can be used for beams. Everyone knows about the 3:4:5 right triangle combinations, but did you know that you can also place a beam at the diagonal of a horizontal offset of 6 LEGO Units and a vertical offset of 3.6 LEGO Units? The diagonal will have a length of 7 with an error of -0.29% of one LEGO unit (that is .0029 LEGO Units!) Besides this one I found several more, especially when you can accept horizontal and diagonal lengths which include a half LEGO Unit.

Primary content in this document is © Gus Jansson. All other text, images, or trademarks in this document are the intellectual property of their respective owners.