This is the mind voyage I traveled while thinking about clocks. 

First thoughts turned to “clockwise” and “counter-clockwise” rotation.  These are terms we use to denote direction of rotation around a circle: clockwise is the direction of hour markings on a standard clock face; counting down around the clock face in the reverse order: counter-clockwise.

Physics fact:  Thanks to the Coriolis Force, hurricanes rotate in a counter-clockwise direction north of the equator; you can see that rotational direction on TV weather channels.  Hurricanes’ counterparts south of the Equator are called cyclones or typhoons, and they rotate in a clockwise direction.  I was surprised to learn that there is narrow band about the Equator where such storms do not form.

Clockwise and counter-clockwise directions are the same everywhere, a rare example of a world-wide convention.  Not so with driving.  In the United States vehicles are driven on the right-hand side of the road, but in England the opposite is true.  Vehicles in England have the steering wheel on what we call the passenger side, so gear shifts are operated with the left hand.  Cars sometimes hit tourists crossing streets, because Americans are taught to look left for oncoming vehicles, while Brits are taught to look to the right.  Shouting “watch where you are going” at such tourists is not helpful.

Here’s a question for traffic engineers in the two countries:  to minimize vehicle collisions, should you direct traffic in a clockwise or counter-clockwise direction around traffic circles in the US, and around “roundabouts” in England?  Assume that exit and entrance ramps are “one-way” in both countries, and that such ramps connect to the “slow” lanes for both highways (expressways) and traffic circles (roundabouts). 

The ordering of hour markers around a clock face suggests to me that the original designers must have been right-handed.  The natural ordering for Southpaws would be reversed.  By the way, you can find wristwatches like this on the Internet as gag gifts for Lefties.  Left-handed faces reverse the conventional directions for clockwise and counter-clockwise.

That reversal made me think of mirrors.  We have all stood in front of a mirror holding up a right hand, and looked at our image holding up what appears to be a left hand.  Of course, if you happen to be one of about 1700 people with heterochromia, different colored eyes, you would observe another difference.

This raises the question of what mirror images of clock faces look like.  Hold up a standard clock face to a mirror, and see if the reflection of the hour and minute hands point to the same time on a “left-handed” clock.  Does mirror reflection preserve the angle between hour and minute hands?  Do the hour numbers look the same?

Angles seem to be preserved through left-right reversal, but only the numbers 1, 8, and 11 look exactly the same on left-handed clocks.  What properties do those symbols possess that 2, 3, 4, 5, 6, 7, 9, 10 and 12 do not?  Answer:  Vertical Symmetry and front-back symmetry.

Take a piece of translucent paper, and write the numbers 1, 2 and 3 down the center in ink.  Be sure to use just a vertical line for the number one. Hold up the paper to a mirror.  1 looks like a 1 in the mirror image, as does the back of the 1 when viewed through the back of the paper.  Not so with 2 and 3; mirror images look like those numbers when viewed from the back of the paper.  Mirror reflection preserves up and down, but reverses left/right and back/front.

Perhaps you have a clock face like the ones I remember from school, where Roman numerals were used to denote the twelve hours.  Those clock faces used only three symbols, I, V and X, and the Roman system for writing numbers.  A Roman numeral one placed after the symbols V and X means “add one to five and to ten, respectively; a Roman number I before these symbols means to subtract one. 

These are the Roman numerals for the numbers 1-12:

I; II; III; IV; V; VI; VII; VIII; IX; X; XI; XII. 

Of course, that numbering system could have used IIX for 8 instead of VIII, which saves one scratch mark.  Five Roman numerals look the same under mirror reflection, because they possess Vertical and Front-Back Symmetry:  I; II; III; V; and X. Note that IV and VI are each other’s mirror reflection.

Here’s a crazy question:  can you think of a way to write the numbers 1 through 12 with Roman Numerals I, V and X so that they have Vertical and Front-Back Symmetry? I have come up with just one answer, which requires repositioning the units that are added or subtracted.  Place units above to indicate addition and below to indicate subtraction, rather than on the right or left sides. 

If you make a clock face with these hour indicators, the mirror reflection should be a left-handed clock with identical hour indicators.  I haven’t done that experiment, so if I am wrong please let me know in the discussion box.

My final thoughts about Vertical Symmetry took me back to the days when TV was middle-aged and Star Trek was young.  One voyage of the Starship Enterprise was titled, “May This Be Your Final Battleground”.  In that episode two, two-toned aliens are engaged in unending combat.  They appear identical, except that one is black on the right side and white on the left side, while his antagonist is oppositely colored.  That distinction is lost on the Enterprise crew, because it seems irrelevant to anything that matters.

Of course, the most famous mental journey involving clocks is Albert Einstein’s Special Theory of Relativity.  Believe it or not, you can understand the basics using only simple Algebra.  Maybe there should be a lesson plan for that topic in Middle School.

So, this is the takeaway from our fun session:  daydreaming frees your mind to wander through the realm of ideas, even crazy ones.  That is a metaverse you can access without external devices.  Beam me up, Scotty, it’s time for Blog #4, “Strategizing with Daisy Petals and Childhood Chants”.