The Lost Clock Emporium: details
Here’s a few details about the weird clocks in the previous post The Lost Clock Emporium.
The clocks are all regular analogue clocks (rotation aside), with the exception of Hades’ Timepiece, which has numbers and hands going counter-clockwise (so a mirror image of a normal clock).
The entire clocks are rotating clockwise at these rates (per day):
| Clock | Clock rotations PD |
|---|---|
| Minute P.O.V. | -24 |
| Hour P.O.V. | -2 |
| Devil’s Clock | -13 |
| Hades’ Timepiece | 26 |
About Hades
Here’s a side-by-side view of what’s going on with Hades Timepiece (click ‘Start’):
Comparing Hades to the normal clock on the left:
- the hands are effectively swapped around in Hades’ clock (wrt a regular clock showing the same time); but the mirrored and rotating numerals mean the Hades is showing the correct time in its own frame of reference (i.e. entire clock rotating and with backwards numbers)
- an imaginary hand half-way between the hour and minute hand (on both clocks, and in both directions) points at the place the numerals are mirrored around on the other clock
- the clocks look identical at 12:00 am/pm
If you run Hades at real-time speed and look closely, you can see the clock numerals rotating a little with every tick, which I quite like.
Hades clock as final boss: the Fuster Cluck Clock
We can merge a normal clock and Hades clock into one by using just two (altered) hands if we allow two numeral dials.
The two hands straddle the centre of the clock, and each points at both an hour and a minute:
To read this clock, just use the green hands against the green numbers, or red hands against the red numbers.
The general idea
Analogue clocks with two hands are bi-valid1 under a number dial mapping that allows rotation and mirroring.2
That holds for any pair of hands you choose – e.g. seconds and hours – or for arbitrary hand pairs that go at any ratio you like.
This also applies to a clock showing any number of hours on its face – 24, 10, 2, even 13.
Challenge / interview question from hell
Does the Hades idea apply to clocks with 3 or more dials, i.e. are they bi-valid? Under what conditions?
Spoiler – click to expand
I believe a clock with any numbers of hands is bi-valid if the hands all move with a palindromic hand rate delta sequence.
A clock with two hands has only one number in the hand rate delta sequence, which is always a valid palindrome.
Practically speaking, for three handed clock with hand rates \(r_0, r_1, r_2\), it’s bi-valid if \(r_1 - r_0 = r_2 - r_1\). So a normal analogue clock with three hands (hour, min, second) isn’t bi-valid because it has rates \(1 : 12 : 720)\). Shame! But that’s down to the usefulness of practical clocks, where we want divisions into smaller parts with something more like an exponential/geometric development, rather than linear.
A bi-valid clock made with 3 hands just wouldn’t be very useful for practical purposes – the hands are too similar in relationship.
When hands cross
While we’re here: the interview question which I think started this train: how often do the hour and minute hands cross on a clock face?
The hour and minute hands on an analogue clock with \(n\) hours on its face cross every \(n/(n-1)\) hours. Their crossing points mark the hours of a clock with \(n-1\) hours on its face (with the maximum hour where the 12 usually is), i.e. a regular \(n-1\) polygon.
Example: regular clock with \(n = 12\). Its hands cross every \(12/11\) hours (approx 65 min 27 secs).
I made up bi-valid for want for a better word. Let me know if you have something more apt ↩︎
we can think of the non-obvious version of the clock with the possessed moving numerals as the ghost clock ↩︎
for clocks with an unusal numbers of hours on them (not 12), you have to decide what your ‘hours’ represent: actual hours, or is it a division of a half-day, or a day? ↩︎