2

I've rediscovered the fun of geometry recently and found the beautiful and (to me at least) unexpected result that the two diagonal lengths of a regular unit heptagon are related by:

$$\frac1a + \frac1b = 1$$

Does this sort of relationship have a name? Eg I'd like to be able to say to a fellow maths geek that "the two diagonal lengths are ..." without finishing the sentence with algebra.

I was thinking something along the lines of "harmonic complements" but Google doesn't give any results for that.

IanF1
  • 1,499
  • For anyone who is interested and doesn't know, the heptagon result follows by applying Ptolemy's theorem of cyclical quadrilaterals to ABCE, where the heptagon is ABCDEFG. – IanF1 Nov 07 '14 at 06:36
  • Conjugate exponents. But the term is usually used in some other context (functional analysis). – Quang Hoang Nov 07 '14 at 06:44

2 Answers2

3

Positive real numbers $p$ and $q$ in this relationship with each other are called Hölder conjugates. This relationship is important in functional analysis, where it describes duality of $L^p$-spaces. I don't think I've heard anyone use the term outside of functional analysis.

Qiaochu Yuan
  • 419,620
2

Qiaochu has the name you are after. To think about the relationship geometrically, $\frac{1}{x}+\frac{1}{y}=1$ is equivalent to $xy-x-y=0$, so the shape of the curve is a certain conic section.

To find exactly which one, note that the equation is equivalent to $(x-1)(y-1)=1$,

so this conic section is the hyperbola $xy=1$ shifted one unit right and one unit up.

2'5 9'2
  • 54,717