 IDEA: To present the number from 0 to 11 that gives the hour,
it suffices to specify its remainders after division by 3 and by 4.
To present the number from 0 to 59 that gives the minute (or the second),
it suffices to specify its remainders after division by 3, by 4 and by 5.
This is possible because of the Chinese Remainder Theorem.
The mathematics behind the clock is explained in some introductory slides.

VISUALIZATION: The vertices of a regular ngon describe the remainder after division by n.
The top vertex corresponds to the zero remainder, the next remainder comes clockwise.
The remainders for the hour are in the inner part of the display,
those for the minute (or the second) on the outer part of the display.
The dial shows the time of your electronic device,
so the timezone and the daylight saving time are not an issue.

DIGITAL: Check out the the digital Chinese Remainder Clock (and a variant of it) !
Paper & Slides
Summary of the paper.
We present an analog clock with five hands that illustrates the Chinese remainder theorem
and that can be understood also by nonmathematicians.
Moreover, we interpret the Chinese remainder theorem in terms of rotations
and prove it without equations.
Slides
Free App
Free app for Android (joint work with
John Perry).
A simple app to understand the Chinese Remainder Clock.
 There is an information sheet.
The manual version allows you to increase/decrease a given time or set a new time.
The settings include: showing or not the conventional time; displaying or not the seconds; 12h or 24h version.
 You can change the design (Archy/Bubbly/Linus/Ringy/Shady/Vertie) and the colors.

Available on Google Play.
Source files on Github.
Oncoming updates will have new features (stay tuned...). Your feedback is welcome!
Further material
 The exhibit. A Chinese Remainder Clock (by Markus J. Mühlbauer).
Based on an Arduino and with LEDs.
 The Sage activity. A Sage activity based on the Chinese Remainder Clock, which is part of the book Peering into Advanced Mathematics through
Sagecolored Glasses (by J. Harris, K. Kohl and J. Perry), freely available for download here.
 The animations. The JavaScript files: analog/digital/digital variant.
With a text on how to insert the animation in a webpage.
Many thanks to Ted Ridgway (American River College) for suggesting the variant of the digital design.