Ghost leg is a method of establishing random pairings between any two sets of equal size. For example, it might be used to assign chores randomly to a group of people. The names of the participants are listed across the top of the diagram and the chores across the bottom, and a vertical line is drawn connecting each name to the chore below it. Then the names are concealed and each participant adds a “leg” to the diagram. A leg is a segment that connects two adjacent vertical lines (it must not touch any other horizontal line).

When the legs have been drawn, the names are revealed and a path drawn from each name to the bottom of the diagram. Each path must follow each leg that it encounters, jumping to the adjacent vertical line and continuing downward. When it reaches a chore at the bottom, it establishes a link between a name and a chore.

The benefit of this method is that it will work for groups of any size, reliably establishing a 1:1 correspondence between their elements. And it will work no matter how many horizontal lines are added. In Japanese it’s known as amidakuji.

From Lee Sallows:

A feature common to many geomagic squares is that the set of shapes they employ reveal an atomic structure. That is, they are built up from repeated copies of a single unit shape. Examples of this are piece sets composed of polyominoes, the unit shape then being a (relatively small) square.

For the would-be geomagic square constructor, a key advantage of the atomic property is that the shapes concerned are each describable in terms of the positions of their constituent atoms. Or, to put it another way, they can be represented by a set of numbers. Hence, unlike non-atomic shapes, they are readily amenable to analysis and manipulation by computer.

Take, for example, an algorithm able to identify and list each of the different ways in which a given planar shape can be tiled by some specified set of smaller shapes. Such a program might be challenging to write, but provided the pieces concerned are composed of repeated units, implementation ought to be straightforward. But could the same be said in the case of non-atomic pieces? Without a set of numbers to describe piece shapes, how are they to be represented in a digital computer?

This is worth noting since, as inspection will show, the shapes employed in the square above are plainly non-atomic. In line with this I can confirm that the only computer program involved in deriving this solution was a vector graphics editor used to create the drawing seen above.

(Thanks, Lee.)

Peter Nicholson’s Carpenter’s New Guide of 1803 contains an interesting technique:

To find a right line equal to any given Arch of a Circle. Divide the chord ab into four equal parts, set one part bc on the arch from a to d, and draw dc which will be nearly equal to half the arch.

Apparently this was an item of carpentry lore in 1803. In the figure above, if arc ad = bc, then cd is approximately half of arc length ab.

Nicholson warns that this works best for relatively short arcs: “This method should not be used above a quarter of a circle, so that if you would find the circumference of a whole circle by this method, the fourth part must only be used, which will give one eighth part of the whole exceedingly near.”

But with that proviso it works pretty well — in 1981 University of Essex mathematician Ian Cook found that for arcs up to a quadrant of a circle, the results show a maximum percentage error of 0.6 percent, “which I suppose can be said to be ‘exceedingly near.'” He adds, “[I]t would be of interest to know who discovered this construction.”

(Ian Cook, “Geometry for a Carpenter in 1800,” Mathematical Gazette 65:433 [October 1981], 193-195.)