ternary-coded decimal numeral

Question

Is it possible to create a ternary clock, similar to this one, using LEDs to express ternary values?

Answer 1

There are two major considerations you need to take into account.

The first is whether you want to convert the time into ternary digit-by-digit (i.e. each of the six digits) or number-by-number (i.e. the three numbers: hour, minutes, seconds). In the diagram above, it does the binary conversation digit by digit. In ternary, a digit by digit conversion would require you to have $1,3,9$ as your powers of three (since each digit in a decimal base is less than $10$). In a number-by-number conversation, you would need $1,3,9,27$ since the numbers representing the hours, minutes, seconds go up to $59$.

Secondly, in binary, there are only two "symbols" used to represent the number. Usually we write those symbols as $0$ and $1$; in this picture however, the two symbols are represented by the two different states of the circle (empty or filled). In ternary, there are three symbols (usually written $0,1,2$) and thus each circle would need to have three states. There are many ways to do this such as having three different colors or having a half-filled circle as an additional state. The different ways of visually representing the three symbols however is more of a artistic design decision and not a mathematical one, so the possibilities are endless.

Answer 2

You need three colors to represent $0,1,2$. An LED that can be off/red/green can do that. Then instead of the $1,2,4,8$ stack you have $1,3,9$. No problem.