Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock."

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There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational.

Anyway, this started me wondering about how to improve the design of this clock from a mathematical standpoint. Here's my formulation of the problem:

Find 12 numbers that have been proved to be irrational to place around the outside of a clock.
Each of eleven of the numbers must approximate as closely as possible one of the integers from 1 through 11. The 12th can either be just smaller than 12 or just larger than 0.
The numbers must have expressions that are as simple as possible (in the spirit of - or even more simple than - those in the clock given in the picture here). Thus, for example, no infinite sums, no infinite products, and no continued fractions. Famous constants and transcendental functions evaluated at small integers encouraged.
Expressions should be as varied as possible. Better answers would include at least one use of trig functions, logarithms, roots, and famous constants.

Obviously, goals 2, 3, and 4 act against each other. And, as Jonas Meyer points out, "as closely as possible" and "as simple as possible" are not well-defined. That is intentional. I am afraid that if I tried to define those precisely I would preclude some answers that I might otherwise consider good. Thus, in addition to the mathematics, there's a sizable artistic component that goes into what would be considered a good answer. Hence the "soft-question" tag. I'm really curious as to what the math.SE community comes up with and then what it judges (via upvoting) to be the best answers, subject to these not-entirely-well-defined constraints.

Note that the designer of the clock given here was not trying to approximate the integers on a clock as closely as possible.

Finally, it's currently New Year's Day in my time zone. Perhaps a time-related question is appropriate. :)

Note: There is now a community wiki answer that people can edit if they just want to add a few suggestions rather than all twelve.

Solution 1:

Here's a list of irrational numbers which almost fulfill your criteria.

Each were chosen to be accurate to within ±0.1, so that no two of them implicitly express the same mathematical approximation, and so that none of them "cheats" in order to fudge an exact result involving integers to obtain a slightly inexact, irrational result. Only the last number fails to meet your criteria, as it is slightly larger than 12.

$\ln(3)$
$7\pi/11$
$\sqrt 2 + \frac\pi2$
$7/{\sqrt[3]5}$
$\mathrm e^\phi$
$\sec^2(20)$
$4\sqrt3$
$5\phi$
$2\pi+\mathrm e$
$\sinh(3)$
$\pi^3-20$
$\csc^2(16)$

I would prefer not to use cosecant, integers greater than 12, or more than two additions/subtractions — it is too easy to get results if you rely on these — but I think I've spent enough time on this diversion for now. :-)

[EDIT: revised the formula for 4 two times now: this first to change the formula for 4 from √3 + √5 — which is too close to √4 + √4 — and the second time to correct the formula as I somehow copied a result which was not approximately 4.]

Added: Since it seems that I can't sleep tonight, here is a list of approximate values:

$\begin{align*} \ln(3) & \approx 1.0986 \\ 7\pi/11 & \approx 1.9992 \\ \sqrt 2 + \tfrac\pi2 & \approx 2.9850 \\ 7 / \sqrt[3]5 & \approx 4.0936 \\ \mathrm e^\phi & \approx 5.0432 \\ \sec^2(20) & \approx 6.0049 \\ 4\sqrt3 & \approx 6.9282 \\ 5\phi & \approx 8.0902 \\ 2\pi+\mathrm e & \approx 9.0015 \\ \sinh(3) & \approx 10.018 \\ \pi^3-20 & \approx 11.006 \\ \csc^2(16) & \approx 12.064 \end{align*}$

Solution 2:

Here are my meager suggestions:

For a new "zero", the symobl $\epsilon$, which although not representing a specific number, is nearly universally used to represent a very small positive quantity.
Alternatively, for zero one could use $\frac{1}{\omega}$, which is one of the most canonical infinitesimals in the surreal numbers.
For 2, one could use $|1+i|^2$. (I realize this is exact and hence rational, but the expression involves non-rational numbers.)

As a mathematician who is also an antique clock collector, I love this question, and I would be interested if after some good answers are submitted, they are collected and made into a suitably attractive pdf image that could be printed and actually used for a clock face. My suggestion would be to adopt an old-style clock typography, if this is possible, since it might also help the clock face look more like a clock face.

Solution 3:

This is intended to be a community wiki answer that people can edit if they just want to give a few numbers rather than all twelve. I'll start with suggestions currently in the comments. Feel free to add more!

For 1: $ -\sin 11$

For 3: $e + \sqrt{\frac{5}{63}}$

For 7: $22/\pi$

For 12: $\log_{1922}(1782^{12}+1841^{12})$ (Approximately 11.99999999996, featured on The Simpsons.)

Solution 4:

Suggestions

11: $e^{\pi i}$

4: ${\sqrt 5}^{\sqrt 3}$ (aprox. 4.03019240+ )

While, it's no match for the previous entry, I can't resist giving one more for

11: $\frac{\sqrt[4]{104 + e^{\pi \sqrt{58}}}}{6^2}$

Solution 5:

Despite the large constant, I like tanh(2011) for 1. It is very close and shows the year.