Foils

[sdi]

violetcabra mentioned hexafoils lately, which are a neat little geometric construction with evident spiritual properties:

I was curious what other kinds of "foils" there were. Turns out there's only two: a "tetrafoil" with four petals, and a "pentafoil" with five:

 

Let's consider why. First, a "foil" obviously connects the vertices of a regular polygon with circular arcs through the center of its circumscribed circle. Thus there no such thing as a "monofoil" or "difoil" since you can't have a polygon with fewer than three vertices. The "trifoil" is simply a degenerate hexafoil with half of its petals removed, and so there seems no reason to count it. (This isn't just pique, either: to construct a trifoil with a compass, one has to construct the full hexfoil first!) The "tetrafoil", "pentafoil", and hexafoil are shown above. The "heptafoil" cannot be constructed with a compass and straightedge, but even if it could, it's petals would overlap each other: this is because they are both wider and closer together than the hexafoil's. This is true for any number of petals greater than six, in fact, which means that the hexafoil is as high as we can go.

The "tetrafoil" is unusual compared to the other two in that arcs connect its adjacent vertices, rather than every other vertex. (If we connected every other vertex, the arcs would be of infinite radius—that is, they'd be straight lines—and we'd have a sun cross.) In this respect, the "tetrafoil" is more like a degenerate "octafoil," where we removed the extra petals that overlap.

The hexafoil is trivial to construct with a compass alone, but the others are quite complicated: the "tetrafoil" took me fourteen circles to make, and the "pentafoil," fifteen. (They're both pretty easy if you allow a straightedge, though.) I think this is reflected in their elegance: the hexafoil strikes me as by far the most orderly and beautiful of these, though the "pentafoil" reminds me of a sand-dollar, which has its charms...

Comments

[boccaderlupo]

Wonderful! Something going on here, spiritually speaking, with these numbers, I suspect...