Sunday, June 16, 2013

Tilting the LEGO World: Diagonal Bracing

Hello, LEGO Mindstorms NXT hobbyist! Who said that the LEGO beams must connect at a right angle to each other? The very nature of LEGO is to produce squared things, but with the advent of studless parts, diagonal connections are mainstream now, making our world a bit more varied and interesting, and giving us another tool for problem solving. 

Lego Mindstorms NXT hobbyist, this posting relate past posting about squaring lego world vertical bracing. You now know that you can cross-connect a stack of plates and beams with another beam. And you know how it works in numerical terms. So how would you brace a stack of beams with a diagonal beam? You must look at that diagonal beam as though it were the hypotenuse of a right-angled triangle. Continuing from the previous sample, Figure 1.8 adds a cross-brace to support the structure and provides a sample for this next bit. Now proceed to measure its sides, remembering not to count the first holes, because we measure lengths in terms of distances from them. The base of the triangle is eight holes. Its height is six holes: Remember that in a standardized grid, every horizontal beam is at a distance of two holes from those immediately below and above it. In regard to the hypotenuse, it counts 10 holes in length.

For those of lego mindstorms NXT hobbyist who have never been introduced to Pythagoras, the ancient Greek philosopher and mathematician, the time has come to meet him. In what is probably the most famous theorem of all time, Pythagoras demonstrated that there's a mathematical relationship between the length of the sides of right-angled triangles. The sides composing the right angle are the catheti~let's call them A and B. The diagonal is the hypotenuse~let's call that C. The relationship is:
Now we can test it with our numbers:
This expands to:
(8 x 8) + (6 x 6) = (10 x 10)
64 + 36 = 100
100 = 100

Yes! This is exactly why the example works so well. It's not by chance; it's the good old phythagorean theorem. Reversing the concept, you might calculate whether any arbitrary pair of base and height values brings you to a working diagonal. This is true only when the sum of the two lengths, each squared, gives a number that's the perfect square of a whole number. At this point, you're probably wondering whether you have to keep your pocket calculator on your desk when playing with LEGO, and maybe dig up your old high school math textbook to reread. Don't worry; you won't need either, for many reasons:
  • You won't need to use diagonal beams very often.
  • Most of the useful combinations derive from the basic triad 3-4-5. If you multiply each side of the triangle by a whole number, you still get a valid triad~by 2: 6-8-10, by 3: 9-12-15, and so on.These are by far the most useful combinations, and they are very easy to remember.
  • We provide a table in Appendix B with many valid side lengths, including some that are not perfect but so close to the right number that they will work very well without causing any damage to your bricks.
We suggest Lego Mindstorms NXT hobbyist take some time to play with triangles, experimenting with connections using various angles and evaluating their rigidity. This knowledge will prove precious when you start building complex structures.

OK... next, we discuss about TECHNIC Liftarms: Angles Built In

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