I’ve written three posts in the past judging geometric shapes. For this post, I’m going back to that series, this time for irrational numbers. As per the conditions of the series, we are judging them as fine, lame, awesome, overrated (not as awesome as they seem), and underrated (more awesome than they seem).

Fine: π (pi) = 3.14159…

Pi is the ratio between a circle’s circumference and its diameter. It’s pretty famous as far as irrational numbers go, and it’s kind of an icon of mathematics. The main problem with pi is that the diameter of a circle isn’t a very natural way of representing the size of a circle mathematically – you really want its radius. So there’s a reasonable portion of the math community that has switched allegiance to τ (tau) ≈ 6.28318 instead, which is equal to 2π, and is the ratio of a circle’s circumference to its radius. I’m not taking sides on the conflict in this post, but the fact that the conflict even exists suggests to me that pi is just fine.

Lame: A (Mills’ Constant) = 1.30637788…

OK so imagine you’re William Harold Mills and it’s 1947 and you just proved that there’s a smallest constant $A$ such that $A^{3^n}$ rounded down is always prime for any whole number $n$. Sounds cool, right? Except that the only way to actually calculate it is to find the primes manually, and even then you need the Riemann Hypothesis to be true. It’s extremely disappointing! I guess it’s not trivial that there is such a constant and that there’s a smallest one out of the infinitude of irrational numbers, but it’s still lame.

Awesome: φ (Golden Ratio) = 1.61803…

I’ll admit the golden ratio is kind of overhyped, but it’s definitely not overrated. It’s awesome, but not for the reasons that most people think it’s awesome. People think it’s awesome because it’s the key to beauty and art and spirals or whatever. But the real reason it’s awesome because it’s mathematically elegant.

The point of the golden ratio is that φ-1 = 1/φ. If you turn that into a quadratic and solve it, you’ll get φ = (1+√5)/2. Technically there’s also a second solution at (1-√5)/2 but that other one is just -1/φ ≈ -0.61803 so it’s basically just φ again. The awesomest part of φ in my view is that it’s the most irrational number, in that it’s the hardest to approximate with a rational number that has a small denominator. For comparison, pi can be approximated well as 22/7 and Champernowne’s constant (below) as 10/81.

Overrated: $C_{10}$ (Champernowne’s Constant) = 0.123456789101112…

Champernowne’s Constant was constructed in 1933 by D.G. Champernowne. It’s just the digits of all whole numbers in base 10, in increasing order, stuck together after the decimal point. The reason it’s supposed to be interesting is that it’s “normal”, in that every subsequence of a given length is equally likely to occur as every other. It’s one of the few numbers that we know is normal for sure, but we expect that most irrational numbers are normal, hence the term. So Champernowne’s constant is kinda disappointing and overrated.

Underrated: δ Feigenbaum Constant = 4.6692016…

I am very slightly cheating here because we haven’t proven that the Feigenbaum constant is irrational, but c’mon, literally 100% of numbers are.¹ Among other things, it’s the limit of the ratio between the sizes of successive bulbs of the Mandelbrot set, but it also shows up in all kinds of chaotic maps that involve quadratics somehow.

Check out the “bifurcation diagram” below: it shows the values you approach when you repeatedly apply the function $rx(1-x)$ to its own output.  Depending on the value of r, you either:

• Approach a single value (like on the left where the graph is one line)
• Oscillate between a finite number of values (like in the middle where the graph splits)
• Just jump around chaotically (like on the right where the graph goes all over the place).

And the ratio between the successive splits on the diagram is somehow always the Feigenbaum constant; both for this function and a whole bunch of others.  It’s literally order found in chaos, and I think it’s completely underrated.

Bifurcation diagram: classic chaos with a hidden order.

Coming Soon: Going Back: In Time