Parallelogram

A paral­lel­o­gram is a quadri­lat­eral whose oppo­site sides are pair­wise parallel. One of the few school defi­n­i­tions that we remember all our lives, and, as it turns out, for a good reasons — many objects around us work habit­u­ally due to the paral­lel­o­gram prop­er­ties.

The equality of the paral­lel­o­gram's oppo­site sides is already a conse­quence of the given defi­n­i­tion, i.e. the paral­lelism of the sides. And even if you build a hinged paral­lel­o­gram, for ex. using a child's building set, the sides will always remain parallel when the paral­lel­o­gram is skewed. By the way, this simple model demon­strates that, unlike a rectangle, the paral­lel­o­gram area is defined not by the sides' lengths alone: when skewed, sides' lengths don't change, but the area does.

The desk lamp has a solid base and lamp­shade connected by one, but more often by two paral­lel­o­grams. This design allows you to change the lamp posi­tion without changing it's tilt angle rela­tive to the table. The same behavior has, for example, panto­graph stands for studio micro­phones.

The hinged paral­lel­o­gram allows you to change the posi­tion of the lamp while main­taining the tilt angle: the side of the paral­lel­o­gram to which the lamp­shade is connected always remains parallel to the oppo­site side that does not change its direc­tion. But if the stand is made as a single paral­lel­o­gram, the move­ment of the lamp is prede­ter­mined — it may only move in a circle. A double (hinged) paral­lel­o­gram — two paral­lel­o­grams with a common side, or connected with a piece that doesn't change its geom­etry — allows you to main­tain the lamp tilt angle and still move it to an arbi­trary point in a fairly large area of space.

In 1669 at the Paris Academy of Sciences, Gilles Roberval demon­strated his balance whose read­ings remain inde­pen­dent of the item's posi­tion on its plates. By the way, it is the same Roberval who calcu­lated the area under the arch of the cycloid, bringing that area to the area under the sine wave.

And the core of the Roberval balance is the hinged paral­lel­o­gram yet again! Its sides, on which the plates are mounted, always remain vertical. Exactly due to the usage of the paral­lel­o­gram, the balance's read­ings are inde­pen­dent of where on the plates the items are placed. Roberval himself, being both a math­e­mati­cian and a mechanic, discov­ered this exper­i­men­tally, and the strict kine­mat­ical proof did not appear until the begin­ning of the 19th century. Balance, as an impor­tant tool in the life of society, has been improved numerous times, and there are now numerous designs of different complexity. But most of them, in one way or another, to this day still use the prop­er­ties of the paral­lel­o­gram.

From the begin­ning of the 20th century until the advent of computers, the primary drawing tool for engi­neers all over the world was the drafting tables. This device consists of a drawing board and a drafting machine.

And the core of the drafting machine is (yet again) the double paral­lel­o­gram. The double paral­lel­o­gram, rigidly attached by one side to the drawing board and fitted with a pair of scales mounted to form a right angle on an artic­u­lated protractor head, allows you to draw straight lines at any angle. In partic­ular, to draw parallel lines... of which the paral­lel­o­gram itself is composed.