I’d call 120v single phase, not half phase.
Standard North American residential service is properly called “240v split-phase.” Your transformer has a primary winding attached to one phase of a 3-phase distribution network (the other two phases may go to the other side of your street, or the next street over, as mentioned upthread.)
The secondary winding is configured for 240 volts and has three taps: one on each end and one in the center. The center tap is grounded and forms the neutral, providing 240v from end-to-end, 120v from either end to neutral.
That would be an unusual panel. Any modern panel I have seen has phases on alternate lugs down both sides; a 120V breaker would attach to only one lug (neutral on that circuit would go the the neutral bar unless it’s a GFCI/AFCI breaker) while a 240V breaker is double the width and spans both lugs so it contacts the full 240V.
A three-phase panel has all three phases on alternating (in groups of three instead of two) lugs. A three-phase breaker in this case is triple-wide so it contacts all three phases.
This works because of the trigonometric identity (remember those?) that says that the sum of two sine waves of the same frequency but with arbitrary phases is also a sine wave of that same frequency.
My (work) panel has 3 rows: Red, White, Blue. Melbourne Australia 
Indeed. Here’s how I like to visualize split-phase and three-phase systems:
You’re both right. A typical split-phase panel has one leg down each side, with interleaving “fingers” that the breakers plug into, like this:
A 120v breaker plugs into one finger, a 240v breaker is twice as tall, and plugs into one finger from each side.
Yeah, I messed up a bit on my description. My point was that you can’t tell which phase a particular 120V breaker is on simply by looking at which side of the panel it is on.
BTW: Are you an electrician or an electrical engineer?
I’m familiar with the cost of multi-zoned HVAC, I’m in the planning stage of a remodel of a garden apartment that is supplied heat, air conditioning hot water and cooking gas by my unit on the first floor
At the same time I am also looking into the viability of grid-tied alternative energy production and backup power as my area in notorious power outages.
I like that, you use the memory circles (or whatever they are actually called) they take the edge of algebraic equations.
Actually you can, residential panels are (at least in the U.S) manufactured with the buss alternating left to right, vertically it would be every other breaker is on the same phase, left to rite every breaker is on the opposite phase of the breaker next to it.
The post I was replying to originally said all the breakers on the left are on one phase and all the breakers on the right are on the other. This is not correct. Left/right alone tells you nothing about phase, vertical position does.
Also, look at the photo friedo posted. Left to right the breakers in the same row are on the same phase.
I stand corrected, all modern residential US circuit panels that I have seen/encountered are arranged as I described and as was pictured.
I have never encountered a circuit breaker that bridges left to right across the buss bars, while it is possible, I have never encountered it.
I think you’re misunderstanding that picture. Each bus bar finger connects to two breakers, arranged horizontally. The red rectangles represent two breakers connected to the left bus bar, and the blue rectangles are connected to the right bus bar. If the panel is full, you can’t actually see the bus bars, but you’ll know that two breakers in the same row are connected to the same leg, and that adjacent breakers in the same column are connected to opposite legs. That’s why 240v breakers are twice as high - they connect to both bus bars.
Well, no it’s a single inline buss that does not allow for left to right side by side breaker placement so vertically every other slot is on the same phase/half phase.
Superficially in a closed panel you couldn’t tell what breaker is on what phase if the main is mounted separately or if it is a sub panel in which case you would remove the face plate and in the situation that all slots are filled you would cut main power and remove the bottom most 2 to 4 breakers to verify the bus layout;
Well actually cut main power and then remove the face plate, etc.
Oops, yup, you’re right, it does allow for side by side instillation.
The root element is still true in that you need to determine the topology of the bus layout but what can be determined with the face plate removed.
Yup, my bad I only glanced at the photo.