The power tower puzzle | Ep. 8 Lockdown live math

Q: Why does tetration require a specific evaluation order, and how is that order formalized?

Exponentiation is not associative, so $2^{2^2}$ differs from $(2^2)^2$. Tetration resolves this by evaluating “from the top down.” A recursive definition makes the order unambiguous: start with $a_1=2$ (or $a_0=1$ depending on convention), then set $a_n=B^{a_{n-1}}$. Unfolding the recursion produces a power tower of height $n$ with $B$ repeated $n$ times.

Q: What does it mean for the infinite power tower to converge, and what equation captures that?

Convergence means the iterates $x_{n+1}=B^{x_n}$ approach a finite limit $x$. Taking the limit on both sides gives the fixed-point condition $B^x=x$. For example, with $B=1.1$, the iteration rises and then stabilizes at a value $x\approx 1.111782\ldots$ that satisfies $1.1^x=x$.

Q: What does the cobweb diagram reveal that intersections alone do not?

The cobweb diagram iterates between $y=B^x$ and $y=x$: starting from an initial $x_0$, move vertically to the curve to get $x_{n+1}=B^{x_n}$, then horizontally back to $y=x$. Intersections indicate fixed points, but stability depends on the slope at the intersection. If the slope magnitude is less than 1, iterates contract toward the fixed point; if greater than 1, they move away and the sequence diverges.

Q: How is the critical base for convergence derived, and what is its exact value?

At the convergence boundary, the curve $y=B^x$ is tangent to $y=x$. Tangency means both $B^x=x$ and the derivative condition $\frac{d}{dx}B^x = 1$ at the intersection (since $y=x$ has slope 1). Using $\frac{d}{dx}B^x = B^x\ln(B)$ and substituting $B^x=x$ leads to $\ln(B)=1/x$. Solving yields $x=e$ and $B=e^{1/e}\approx 1.4447$.

Q: Why can’t the tower converge to any value between $e$ and infinity once $B$ exceeds the threshold?

Because the stability boundary happens at tangency: once $B$ passes $e^{1/e}$, the fixed point that would attract the iteration disappears as a stable attractor. The cobweb picture becomes “kissing then separating,” so the iterates no longer settle at any larger finite fixed point; they instead escape to infinity.

TL;DR

Tetration is repeated exponentiation evaluated from the top down, best defined via a recursion $a_{n} = B^{a_{n - 1}}$ .

Briefing Cornell Notes

Briefing

A single “power tower” question—how far repeated exponentiation goes before it either settles or explodes—turns into a full lesson on tetration, fixed points, and the sharp boundary between convergence and divergence. The central finding is that for towers of the form $B^{B^{B^{o b r e ak \dots}}}$ (height $n$ ), growth is not monotone in the way intuition suggests: once the base $B$ passes a critical threshold, the iteration stops converging and instead blows up to infinity. For real bases, that upper threshold is exactly $B = e^{1/ e}$ , about 1.4447—right around where the graphs of $y = B^{x}$ and $y = x$ become tangent.

The episode begins by defining tetration as “repeated exponentiation from the top down,” avoiding ambiguity caused by exponentiation’s non-associativity. Using a recursive sequence $a_{n} = 2^{a_{n - 1}}$ with $a_{1} = 2$ , the tower $2^{2^{2^{2^{o b r e ak \dots}}}}$ grows extremely fast: height 4 gives 65,536, and height 5 already reaches a number with roughly 19,000 digits. Height 6 is so large that storing it physically would be impossible—an intentionally dramatic way to motivate why mathematicians care about the *behavior* of the iteration rather than the raw size of its outputs.

To probe the “switch,” the lesson compares two growth regimes. Repeated multiplication by 1.1 (ordinary exponential growth) quickly reaches moderate sizes. But repeated exponentiation with base 1.1 behaves differently: the sequence rises at first and then locks into a fixed value $x$ satisfying $1. 1^{x} = x$ . That fixed-point equation becomes the key lens for everything that follows.

The next step is to find where convergence fails. For $B = 2$ , a tempting self-similarity argument suggests the infinite tower might converge to 4, but numerical iteration and a deeper graphical analysis show the tower actually approaches 2. The discrepancy comes from assuming convergence before verifying it. The cobweb diagram method—iterating $x \mapsto B^{x}$ while bouncing between $y = B^{x}$ and $y = x$ —reveals which fixed points are stable. Intersections alone are not enough: stability requires the intersection to have slope magnitude less than 1.

Using the tangency condition (the curve $y = B^{x}$ tangent to $y = x$ ), the critical base is derived by solving the system $B^{x} = x$ and $\frac{d}{d x} B^{x} = x$ at the tangency point. The result is $x = e$ and $B = e^{1/ e}$ . Past this value, the iteration “kisses and then separates,” never settling to a finite number—so the tower cannot converge to any value between $e$ and infinity. The episode closes by extending the idea to complex bases, where convergence/divergence patterns form fractal-like regions reminiscent of the Mandelbrot set, and by noting how even seemingly simple questions like whether $π^{π^{π^{π}}}$ is an integer remain unresolved.

Cornell Notes

Tetration is defined as repeated exponentiation evaluated from the top down, producing power towers like $B^{B^{B^{\dots}}}$ . For real bases $B$ , the infinite tower converges only when the iteration $x_{n + 1} = B^{x_{n}}$ approaches a stable fixed point $x$ satisfying $B^{x} = x$ . Graphical cobweb diagrams show that intersections of $y = B^{x}$ with $y = x$ are not sufficient; stability depends on the slope at the intersection having magnitude less than 1. The convergence-to-divergence boundary occurs when the curve is tangent to $y = x$ , yielding the exact critical base $B = e^{1/ e} \approx 1.4447$ . Above that threshold, the tower diverges to infinity rather than settling to some larger finite value.

Why does tetration require a specific evaluation order, and how is that order formalized?

Exponentiation is not associative, so

2^{2^{2}}

differs from

(2^{2})^{2}

. Tetration resolves this by evaluating “from the top down.” A recursive definition makes the order unambiguous: start with

a_{1} = 2

(or

a_{0} = 1

depending on convention), then set

a_{n} = B^{a_{n - 1}}

. Unfolding the recursion produces a power tower of height

n

with

B

repeated

n

times.

What does it mean for the infinite power tower to converge, and what equation captures that?

Convergence means the iterates

x_{n + 1} = B^{x_{n}}

approach a finite limit

x

. Taking the limit on both sides gives the fixed-point condition

B^{x} = x

. For example, with

B = 1.1

, the iteration rises and then stabilizes at a value

x \approx 1.111782 \dots

that satisfies

1. 1^{x} = x

How can a self-similarity trick fail when analyzing $2$ -based towers?

A common move assumes the infinite tower equals a target value and then uses self-similarity to solve an algebraic equation. For

B = 2

, that reasoning suggests a limit of 4 by setting the tower equal to 4 and deriving

x^{2} = 2

. But that assumption implicitly requires convergence to 4, which is false. The correct behavior is that the tower approaches 2, because the relevant fixed point is stable while the “would-be” limit is not. The lesson: verify convergence/stability before trusting the algebraic substitution.

What does the cobweb diagram reveal that intersections alone do not?

The cobweb diagram iterates between

y = B^{x}

and

y = x

: starting from an initial

x_{0}

, move vertically to the curve to get

x_{n + 1} = B^{x_{n}}

, then horizontally back to

y = x

. Intersections indicate fixed points, but stability depends on the slope at the intersection. If the slope magnitude is less than 1, iterates contract toward the fixed point; if greater than 1, they move away and the sequence diverges.

How is the critical base for convergence derived, and what is its exact value?

At the convergence boundary, the curve

y = B^{x}

is tangent to

y = x

. Tangency means both

B^{x} = x

and the derivative condition

\frac{d}{d x} B^{x} = 1

at the intersection (since

y = x

has slope 1). Using

\frac{d}{d x} B^{x} = B^{x} ln (B)

and substituting

B^{x} = x

leads to

ln (B) = 1/ x

. Solving yields

x = e

and

B = e^{1/ e} \approx 1.4447

Why can’t the tower converge to any value between $e$ and infinity once $B$ exceeds the threshold?

Because the stability boundary happens at tangency: once

B

passes

e^{1/ e}

, the fixed point that would attract the iteration disappears as a stable attractor. The cobweb picture becomes “kissing then separating,” so the iterates no longer settle at any larger finite fixed point; they instead escape to infinity.

Review Questions

For a given base $B$ , what fixed-point equation must the limit $x$ satisfy if the infinite power tower converges?
In cobweb diagrams, what slope condition determines whether an intersection is stable or unstable?
What exact value of $B$ marks the real-number threshold between convergence and divergence, and how does tangency to $y = x$ relate to that threshold?

Key Points

1
Tetration is repeated exponentiation evaluated from the top down, best defined via a recursion $a_{n} = B^{a_{n - 1}}$ .
2
The infinite power tower converges exactly when the iteration $x_{n + 1} = B^{x_{n}}$ approaches a stable fixed point $x$ satisfying $B^{x} = x$ .
3
Intersections of $y = B^{x}$ and $y = x$ are not enough; stability requires the slope magnitude at the intersection to be less than 1.
4
For real bases, the convergence-to-divergence boundary occurs when $y = B^{x}$ is tangent to $y = x$ , giving the critical base $B = e^{1/ e} \approx 1.4447$ .
5
Once $B$ exceeds $e^{1/ e}$ , the tower does not settle to any finite value larger than the attracting fixed point; it diverges to infinity.
6
A self-similarity algebra trick can produce the wrong target if it assumes convergence without checking stability.
7
Extending $B$ to complex numbers produces intricate convergence/divergence fractal patterns, echoing Mandelbrot-like behavior.

Highlights

Height-5 towers of 2 already produce numbers with about 19,000 digits, and height-6 is beyond practical storage—so the focus shifts from size to stability.

For B=1.1, the iteration converges to a fixed point x solving 1.1x=x, showing that repeated exponentiation can “lock in” instead of exploding.

The 2​ case illustrates a key warning: algebraic self-similarity can suggest a limit (4) that cannot actually be reached because the assumed convergence is unstable.

The exact real threshold for divergence is B=e1/e, derived from the tangency condition between y=Bx and y=x.

With complex bases, convergence regions form fractal-like structures, and even questions like whether ππππ is an integer remain open.

Topics

Tetration
Power Towers
Fixed Points
Convergence Threshold
Cobweb Diagrams