Pythagorean Lego
The Pythagorean theorem says that with a right-angled triangle, the length of the diagonal, squared, is equal to the sum of the squares of the lengths of the other two sides.
This relationship allows certain solutions that are all perfect whole numbers, for instance, a triangle with lengths 3, 4, and 5 units works (32 + 42 = 52, 9+16=25).
The series of whole-number solutions are called the Pythagorean triples.
The sequence of triples starts, (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53) ...
A consequence of this is that there is a special sequence of Lego brick lengths that can fit onto a Lego grid at an angle, with the studs lining up perfectly.
3, 4, 5
In the case of the 3,4,5 triangle, a beam with five spaces between its end-studs – a six-stud beam - can sit on a pair of single-stud pillars and clip to a baseboard at an angle.
A six-by six square also works, as do multiples with further studs another five units apart ... so we can build a Lego model of a modern stilted concrete office building, measuring six by eleven, or an eleven by eleven building (or eleven by sixteen, etc.) and it will line up perfectly with a baseboard if we get the angle right
5, 12, 13
The next size up that can be laid at an angle is a beam with thirteen spaces between its end-studs – a fourteen-stud beam.
