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.)

In the 1970’s, a student of Maoist inclination asked [Columbia philosopher Sidney Morgenbesser] if he disagreed with Chairman Mao’s saying that a proposition can be true or false at the same time. Dr. Morgenbesser replied, ‘I do and I don’t.’

From Morgenbesser’s 2004 New York Times obituary.

Drill a hole straight through the center of a sphere, leaving a band in the shape of a napkin ring. Suppose the height of this band is h. Now take a sphere of a different size and drill a hole through that, contriving the hole’s width so that its depth is again h. Remarkably, the two napkin rings will have the same volume.

As the sphere expands, the band must grow both wider and thinner, and it turns out that these two effects exactly cancel one another. It’s called the napkin ring problem.

Just a striking image: German astronomer Johann Friedrich Julius Schmidt prepared a plaster moon for the Field Columbian Museum in 1898. From the Field Museum Library.

Students beginning with the compass learn to draw this rosette, sometimes called the Flower of Life. If the arcs and the circle have the same radius, a, then the area of one petal is

and the unshaded area of the circle is

Remarkably, though the area we sought is bounded entirely by the arcs of circles, the final expression is independent of π.

Related: When two cylinders of radius r meet at right angles, the volume of their intersection is 16r³/3 — again, no sign of π.

(J.V. Narlikar, “A Pi-Less Area,” Mathematical Gazette 65:431 [March 1981], 32-33.)

10/01/2024 UPDATE: Some deft rearranging shows that the unshaded area of the circle is just the area of an inscribed regular hexagon:

So the absence of π isn’t that surprising. Thanks to readers Catalin Voinescu (who sent this diagram) and Gareth McCaughan for pointing this out.

MIT computer scientist Gerald Sussman offered this example to show the importance of sophisticated planning algorithms in artificial intelligence. Suppose an agent is told to stack these three blocks into a tower, with A at the top and C at the bottom, moving one block at a time:

It might proceed by separating the goal into two subgoals:

1. Get A onto B.
2. Get B onto C.

But this leads immediately to trouble. If the agent starts with subgoal 1, it will move C off of A and then put A onto B:

But that’s a dead end. Because it can move only one block at a time, the agent can’t now undertake subgoal 2 without first undoing subgoal 1.

If the agent starts with subgoal 2, it will move B onto C, which is another dead end:

Now we have a tower, but the blocks are in the wrong order. Again, we’ll have to undo one subgoal before we can undertake the other.

Modern algorithms can handle this challenge, but still it illustrates why planning is not a trivial undertaking. Sussman discussed it as part of his 1973 doctoral dissertation, A Computational Model of Skill Acquisition.