In case my previous musings on lego triangles weren’t geeky enough for you, here’s some more.
60 degree 3-7-8 / 5-7-8 math
In my prior note I pointed out that you could construct an exact 60 degree angle with a 3-7-8 or 5-7-8 construction, such as this one:
The blue technic beam is the “7” unit leg. Building this within an 8-8-8 equilateral triangle helps convince us that the angle opposite the 7 unit beam is a 60 degree angle.
In the prior post I wrote “This construction is probably demonstrating some deep mathematical relationship between the 3-7-8 60 degree construction and the 5-7-8 construction; perhaps someone will point it out to me and I’ll add it back into this note.”
Well I’ve worked it out for myself.
Review: Trigonometry says that if we know sides a, b, and c and we want to know the angle B opposite side b, we can start with the law of cosines:
b^2 = a^2 + c^2 – 2*a*c * cos B
In the case where angle B is 60 degrees, cosine(60) is exactly 1/2, so this formula becomes:
b^2 = a^2 + c^2 – a*c
Now we can see why (a,b,c) = (3,7,8) forms a 60 degree angle opposite the 7 side:
7^2 = 3^2 + 8^2 – (3*8)
49 = 9 + 64 – 24
The numbers work; (3,7,8) is consistent with that equation and therefore the angle opposite the b side is a 60 degree angle.
It follows from this that if (a, b, c) is a solution then (c – a, b, c) is also a solution. To see this, work the substitution of c-a for a through the equation.
Thus once we know (3,7,8) has the 60 degree angle in it we could find (x, 7, 8) where:
x = c – a
x = 8 – 3
which gets us the (5, 7, 8) solution.
Some other integer solution pairs to this are:
- (5, 19, 21) and (16, 19, 21)
- (7, 13, 15) and (8, 13, 15)
- (11, 31, 35) and (24, 31, 35)
That last pair is huge but might be useful if divided in two using half-unit construction.
I built a nice looking 5,7,8 connection inside a 9-9-9 equilataral, all built using system bricks just as a demo.
I had to use “pylons” (visible in the bottom view) to make this work because, as usually happens with the system brick triangle constructions, the adjacent studs will interfere unless you build up using a pylon/riser construction.
Other Fractional Construction Techniques
Lego CAD tools that make use of the LDraw system define brick geometry in units called LDU (an “LDraw Unit” I suppose). A standard system brick is 24 LDU high and the stud-to-stud (or hole-to-hole in technic) spacing is 20LDU. So far we have been talking about constructions using an integer number of holes (20LDU) or half-holes (10LDU).
Because of the 24:20 (6:5) ratio of stud-spacing vs brick height, turning bricks on their side gives offsets other than just integers and halves. Bricks on their sides is called SNOT: “Studs Not On Top” http://www.brickwiki.info/wiki/SNOT_techniques
This allows resolution down to 4LDU for offsets, corresponding to 0.2 of the stud-to-stud distance. By using system bricks (with technic holes as required) we open up many more angle possibilities, including some new Pythagorean triples such as:
(6.6, 11.2, 13)
Of course, that triple was already “available” as an integer construction, with the original Pythagorean triple being an enormous (33,56,65). It could be reduced to (16.5, 28, 32.5) using half-unit construction, but the SNOT techniques allow us to realize that particular construction this way:
It may be easier to see this in an LDraw:
I left the pins out of this rendering just for clarity so you could see all the holes.
Look at the vertical element in more detail:
This shows the general technique for achieving fractional distance at 0.2 resolutions. Every “system” brick laid on its side adds 24 LDU (or 1.2 “holes”) to the distance. As with the technic beams you have to be mindful of the +1 factor because what matters is the center-to-center distance. So it takes a stack of four system bricks on their side to add a total of three times the height of a single system brick. The net of this construction is that the vertical segment is 6.6 units in length. The math for the other part of the construction works similarly.
Recap: Complete Pythagorean Triangle List
Taking all these techniques together, here is the complete list of Pythagorean Triangles you can make with the maximum dimension being less than 25 (there are more above that point), and the angle you get for the hypotenuse. The table is sorted by angle A, the smaller angle of the two non-right angles (the other angle is always 90 – A).
Table without using any SNOT techniques:
4.5 20 20.5 12.680 7 24 25 16.260 6 17.5 18.5 18.925 5 12 13 22.620 8 15 17 28.072 3 4 5 36.870 10 10.5 14.5 43.603
Table including SNOT triangles:
2.6 16.8 17 8.797 4.5 20 20.5 12.680 3.2 12.6 13 14.250 7 24 25 16.260 ... SNOT version: 2.8, 9.6, 10 6 17.5 18.5 18.925 8.8 23.4 25 20.610 5 12 13 22.620 7.2 15.4 17 25.058 8 15 17 28.072 6.6 11.2 13 30.510 3 4 5 36.870 10 10.5 14.5 43.603
Of course you can scale entries in the table. So, for example. you could make the SNOT version of (7,24,25) at (5.6, 19.2, 20) though at that point you probably might as well make it using the integer (7,24,25) construction.