I promised I would go back with a ruler and measure the distances between tick marks on the MC north-northeast number line, so I did. At the floor $3$ landing, we find the white-space between adjacent tick marks to be about $22$ cm. At the floor $1.5$ landing, that space is at least a few centimetres past the end of my $30$ cm ruler. That’s a difference of over $10$ cm! The painters really did an excellent job of making the transitions seamless.

Those (SandwichExpert) in disbelief that a number line could be irregularly spaced might find consolation in a few alternate explanations:

• All this stair-climbing has made me so physically fit that I accidentally made some of the measurements while walking at relativistic speed.
• MC is non-Euclidean.
• Black holes.
• Witches.

So far, we’ve covered the constants in the intervals $[0,1)$ and $[1,2)$; this issue we’ll cover $[2,3)$. It’s highly recommended that you actually go to the stairwell with this article in hand, for a fully-immersive tour.

Floor $2$

Puns aside, there are more constants to look at…

Floor $2.5$

You reach the top of the next flight of stairs (eleven steps) before you can read the next constant. Are there really no interesting numbers near $2$? (Quickly scouring Wikipedia, I couldn’t find any seriously notable constants around $2$.) Exercise: invent a constant worthy enough to fill the void near $2$.

You recall that the last constant we covered (in the previous issue) was around $1.6$ (the golden ratio), so this gap has lasted nearly two flights of stairs. Let’s see what constant broke the silence…

Golden Angle

$2.3999632297\dots$

(For more digits, see OEIS A131988.) A constant without a symbol? Let’s give it a symbol — how about $\theta_G$? It’s an angle, after all. But what is $\theta_G$ the angle of? Let’s do a thought experiment to find out.

Suppose you’re a flower, and your main goal is to look pretty. Evolution has told you that the prettiest flowers appear to have their petals evenly spaced. Unfortunately, you can only grow one petal at a time. Once it’s grown, you can’t move it. You also don’t know how many petals you will grow in your lifetime, so you better place them wisely. Even worse, plants don’t have much free will. The only thing you can choose is the angle $\theta$ between the last petal you grew, and the next.

So what should $\theta$ be? To ensure that your petals appear to be evenly spaced, you want to avoid having two petals that are close to one another. How might such a situation arise? Consider the $n$^th petal and the $(n + a)$^th petal, where $a > 0$. If they appear to be close to one another, then the angle between them, $a \theta$, is approximately a multiple of $2 \pi$: the number of radians in a full rotation. Then, for some integer $b$, $a \theta \approx 2 b \pi$. If $b = 0$, then $\theta$ must be very small. Thus, we want to avoid small values of $\theta$ (but intuitively, you probably already knew that). Otherwise, when $b > 0$, we can rearrange:

$$\tfrac{2 \pi}{\theta} \approx \tfrac{a}{b}$$

In English, this means that we’ll find two petals close to one another if $\flac{2 \pi}{\theta}$, the number of times $\theta$ fits into a full 360-degree rotation, is well-approximated by a ratio of two integers ($a$ and $b$). Aha! If we make $\frac{2 \pi}{\theta}$ a number with bad rational approximations, we can avoid that. Let’s make it the famous irrational number with the worst rational approximations. We’ve seen that number before in this stairwell: it’s the golden ratio $\phi$.

$$\begin{aligned} \tfrac{2 \pi}{\theta} &= \phi \\ \theta &= \tfrac{2 \pi}{\phi} \end{aligned}$$

However, if you evaluate $\frac{2 \pi}{\phi}$, you’ll actually get a number around $3.88$. To get the golden angle you must convert this to standard form: $3.88... - 2 \pi = -2.39...$. Since direction doesn’t matter here, that’s equivalent to $+2.39...$.

To recap, the golden angle $\theta_G$ is explicitly tied to the golden ratio $\phi$ by the relation $\theta_G = 2 \pi - \frac{2 \pi}{\phi}$. It’s the optimal angle (in radians) between consecutive items arranged in a circle, so that you minimize overlap.

Exercise: as much as we like our radians, most of us are better at visualizing angles in degrees. What’s $\theta_G$ to the nearest degree?

$K$

Khinchin’s constant
$2.685452001\dots$

(For more digits, see OEIS A002210.) Aleksandr Khinchin was a 20^th century Soviet mathematician who proved something remarkable about continued fractions. We’ve seen continued fractions before in this series — the golden ratio has the infinite continued fraction:

$$\phi = 1 + \frac{1}{1 + \tfrac{1}{1 + \ddots}}$$

In general, simple continued fractions can be finite or infinite, and are of the form

$$a_0 + \frac{1}{a_1 + \tfrac{1}{\ddots + \tfrac{1}{a_n}}}$$

where $a_i$ is positive for all $i > 0$. The finite continued fractions represent rational numbers, and the infinite continued fractions represent irrational numbers. Altogether, all real numbers can be expressed (almost uniquely) by a simple continued fraction.

“Almost all” comes with infinitely many exceptions. For starters, we’re pretty sure that no rational number has this property. (The irrationals far outnumber the rationals, so this is okay.) More interesting exceptions include the golden ratio, where $a_i = 1$ for all $i$. In that case, the geometric mean is obviously $1$.

Khinchin’s constant itself also has a continued fraction expansion. The coefficients go $2, 1, 2, 5, 1, 1, 2, \dots$ (for more terms, see OEIS A002211). However, we don’t actually know if their geometric mean is $K$. In fact, we don’t even know if $K$ is rational.

Exercise: here’s a throwback to the first stairway constant. What is the continued fraction for Liouville’s constant? Does the geometric mean of its coefficients work out to $K$?

$e$

Napier’s constant (Euler’s number)
$2.71828182845\dots$

(For more digits, see OEIS A001113.) Chances are that you recognized this number by its symbol, rather than its name. Maybe you were even waiting to see it on this number line. That’s how you know this is some constant. We don’t use $e$ for anything else, because that would be an insult to the one true $e$.

Why $e$? Euler was the first to use that symbol for this number, but his choice of $e$ was probably not in honour of himself. When you’re churning out papers like Leonhard Euler, you’ll gladly take the first letter you haven’t yet used for anything else. Before Euler, the number was also known as $b$. Nowadays, in honour of Euler, we use his notation. The fact that the notation happens to be the first letter of “Euler” and “exponential” is probably just a happy coincidence.

How’d he come up with that? Short answer: Maclaurin series. The Maclaurin series is also how Euler computed $e$ to 18 decimal places by hand! You can try it yourself:

$$e^x = \sum_{i=0}^{\infty} \tfrac{x^i}{i!}$$

A complete list of the properties of $e$ could probably fill a book thicker than this stairwell is tall. You yourself are familiar with many of them. Here’s a relatively accessible one from statistics you may or may not know: derangements. If you randomly shuffle a deck of $n$ cards, the probability that no card ends up in its original position approaches $\flac{1}{e}$ as $n$ increases to infinity. Of course, the proof of this result involves the Maclaurin series above. Exercise: fill in the details of the proof.

Floor $3$

Hiding above the door to the 3^rd floor is this article’s final constant.

$F$

Fransén-Robinson constant
$2.8077702420\dots$

(For more digits, see OEIS A058655.) This constant is relatively new. It seems to have emerged in Arne Fransén’s 1979 paper Accurate determination of the inverse gamma integral. Wikipedia mentions a Herman P. Robinson, which is the name of a late OEIS contributor and co-author of the report Mathematical constants. It’s likely, but I’m not sure he is the Robinson after whom $F$ is named.

As the title of Arne Fransén’s paper suggests, $F$ is the value of an integral of the inverse gamma function:

$$F = \int_{0}^{\infty} \tfrac{1}{\Gammap{x}} dx$$

In English, it’s the area bounded above by the graph of the reciprocal of the gamma function and bounded below by the $x$-axis. If you’re unfamiliar with the gamma function $\Gamma$, it is the continuous function where $\Gammap{n} = (n - 1)!$ for all positive integers $n$. Essentially, it’s what you get when you draw a curve of best fit through the points $(n, (n - 1)!)$. $\Gammap{n}$ grows really fast — it’s beyond exponential. (Next time someone uses the word “exponential” incorrectly, you can show them the gamma function and say that exponential growth is for babies.)

As a consequence of that growth, $\frac{1}{\Gammap{x}}$ approaches zero really fast. This is how $K$ can be finite; it’s the area between the $x$-axis and the graph of a function that pretty much kisses it.

Exercise: use lower Riemann sums to prove that $F \geq e$.

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Halfway through the term, we have reached just short of halfway up the MC north-northeast stairwell. However, we have covered more than half of the stairway constants. Why is that?

We’re not the first ones to notice. Simon Newcomb and Frank Albert Benford Jr. both observed this trend more than 80 years ago: numbers, in practice, tend to have small leading digits. This is known as the Newcomb-Benford law. It’s not a theorem, but rather a highly-applicable pattern for all sorts of real-world (“realistically-distributed”) data. In fact, the Newcomb-Benford law holds so reliably that it can and has been used to detect financial fraud!

Out of the numbers between $1$ and $7$, we’ve covered the leading digits $1$ and $2$ so far. According to the Newcomb-Benford law, almost $48\%$ of all “realistically-distributed” numbers should begin with $1$ or $2$ (on average, of course). The rest of the constants we have yet to cover in this stairway begin with $3$, $4$, $5$, or $6$; those digits account for less than $37\%$ of all “realistically-distributed” numbers. If you believe that stairway constants have a “realistic” distribution (whatever that means), then it should be no surprise that there are more of them on the lower floors.

Of course, there is also a bias towards smaller numbers in mathematics. We care much more about extremes, and many of the stairway constants are significant in the first place, because they are the smallest or the simplest number to satisfy some condition.

The only way to truly know why the stairway constants are distributed the way they are, is to ask the people who created this number line in the first place. Stay tuned for more developments.

Exercise: don’t take the elevator.

Footnotes