The \(x^x\) spindle

The image below shows the \(x^x\)-spindle, as described by Mark Meyerson.

import {make_spindle} from './components/make_spindle.js'

viewof n = Inputs.range([0, 15], {
  value: 15,
  step: 1,
  label: "Thread pairs"
})

spindle = {
  const plot = make_spindle();
  invalidation.then(() => plot.destroy());
  return plot;
}

Explanation

For real \(x\), \(f(x) = x^x\) can be viewed as a function \(f:\mathbb{R}\to\mathbb{C}\). In fact, there are infinitely many such functions given by the various branches. To see this, let’s write \[ x^x = e^{x\log(x)}. \] Now, the complex log has many branches defined by \[ \log(x) = \ln(|x|) + i(\text{Arg}(x) + 2k\pi). \] Each choice of \(k\) yields a new branch. Those separate branches are illustrated by the separate threads in the plot.

spindle.update(n)

Crossing the real plane

In a calculus class, it’s common to define \[x^{p/q} = (\sqrt[q]{x})^p.\] In this definition, we’re assuming that \(p\) is an integer and \(q\) is a positive, even integer. Thus, when \(x\) is negative, \(x^{p/q}\) is

• well defined and positive, when \(p\) is even,
• well defined and negative, when \(p\) is odd, and
• not defined otherwise.

We can plot this graph for our calculus students like so:

import {real_intersection_plot} from './components/real_intersections.js'
real_intersection_plot(15, {width: Math.min(width, 800)})

As is often the case, the more complete view of complex variables sheds like on the real viewpoint. Indeed, the dots occur where one of the threads puncture the \(x\)-\(\text{Re}(x^x)\) plane. For negative \(x\), in fact, branch \(k\) has imaginary part \[ |x|^x\sin\!\bigl(\pi x(2k+1)\bigr). \] Thus the crossings occur exactly when \(x(2k+1)\) is an integer.