The Square Root of Three

Whole years have passed without my thinking about the square root of three. The square root of two has occurred to me more, perhaps because of the ancient scandal of its irrationality. I have not felt the absence of these thoughts of the square root of three as a loss. For what it's worth, the number is around 1.73.

Then recently a household member noticed in a design magazine the picture of a handsome round table. The top was supported on a structure that had rods rising up from an equilateral triangle, the latter supported by feet at the angles. The table was out of our price range; the question was whether we could get something like it built. Perhaps so, but it might be useful to reckon the specifications.

The first question was the size of the equilateral triangle at the top. That size depends on the size of the circle around it. It turns out that the length of such a side is equal to the square root of three times the radius of the circle. So for a circle with diameter 20", radius 10", one can fit in it an equilateral triangle with sides 17.33"

The next question was the size of the triangle at the bottom. An equilateral triangle with sides of length n will have its angles resting on the midpoints of the sides of an equilateral triangle with sides of length 2n. However, the picture shows the rods leaning out: the triangle twice the size of the top triangle is therefore larger than the actual supporting triangle. Well and good: how do we calculate the difference in length of sides, supposing that the sides of the larger triangle are at distance m from the sides of the smaller triangle?

I arrived at the answer by drawing a couple of right triangles with a side of length m opposite a sixty degree angle. The calculations necessarily involved the sine of sixty degrees, which is the square root of three, divided by two. The number I came up with was 6m divided by the square root of three, or roughly 3.46m. I will leave the derivation as an exercise for the reader, unless somebody asks me to show my work.

I found myself thinking of a paragraph from Kipling's story "The Impressionists":

“There's great virtue in that 'we,'” said little Hartopp. “You know I take them for trig. McTurk may have some conception of the meaning of it; but Beetle is as the brutes that perish about sines and cosines. He copies serenely from Stalky, who positively rejoices in mathematics.”

I don't know that I ever positively rejoiced in mathematics, but I can say that I wasn't as the brute beasts that perish even about trigonometric functions. I can also say that nothing I have read since my last math class, about 45 years ago,  has made me think as much about sines and cosines as the picture in a design magazine has.