Understanding Lego Geometry

Essentially this chapter gets at the bare bones, or bricks in this case, behind Lego construction. In the world of Legos, measurements and sizes of bricks are noted by the number of studs on a Lego brick, also known as Lego units. When stating the sizes of Lego bricks, it is stated in dimensions in the order of, width, length and finally height. Width is determined as being the shorter of the two horizontal sides of a Lego brick when it is laying 'studs up.' For example, the smallest Lego brick has dimensions of 1 x 1 x 1 Lego units, however this does not mean that it is a cubular shaped brick. In the world of Legos, one 'stud' has a ratio of 6:5 when comparing height to width in milimeters, thus explaining why the one stud Lego 'cube' isn't really a cube.

When comparing Lego bricks to Lego plates, Lego plates are approximately 1/3 the height of one stud.

However, bricks and plates are for those of you that like to kick it Oldskool, as the NXT Mindstorms sets use the newer Lego Technic pieces. The Technic pieces are essentially studless versions of the Oldskool blocks that are more weight efficient and less cumbersome when it comes to precise and lightweight building, that retain the same strength.
With the studless beams and liftarms in the Technic set, capabilities such as diagonal crossbracing become a possibility. When diagonally crossbracing with studless beams, one can use Pythagorean's Theorem to calculate if a certain beam can be used to crossbrace a structure.
Pythagorean Theorem essentially is a formula that relates the measurements of the hypotenuse of a right angle to being equal to the squared values of the other arms...
In other words...a^2 + b^2 = c^2, where c is the hypotenuse.
For example a structure with a base of 15 units, and a height of 8 units can be crossbraced because:
15^2 + 8^2 = 17^2, c in this case equals to a number whose square is a whole number, meaning a studless beam could be used to crossbrace.

I apologize as this is a rather long-winded post...however to conclude my summary of the chapter, one can also use liftarms to brace Lego structures. The key difference here is that liftarms arent straight beams, meaning that one can use them regardless of the Pythagorean Theorem to brace a structure from impacts in totally unique ways.