Triple-mirror magic

Most people realise that a simple mirror is not so simple at all. A famous conundrum is why a mirror swaps left and right sides but not up and down. We have already given our take on that puzzle (which differs from many explanations). So, if we all agree that a mirror is strange, it will be no surprise that three mirrors combined is much stranger.

We make a triple-mirror by joining three mirrors to give the corner of a cube, with the reflective sides on the inside. As does a normal mirror, a triple-mirror shows your reflection. However, as the picture above illustrates, the mirror image will be standing on its head! That’s pretty weird, and it gets weirder.

Close one eye and look into the triple-mirror. No matter the direction from which you look, the reflection of your open eye will be exactly at the corner, where the three mirrors meet. For the same reason, a photo of the triple-mirror will show the camera at exactly the corner point; if you use the camera’s flash, it will flash directly back at you from the corner.

This may all seem new and surprising, but the reflection properties of triple-mirrors make them extremely useful. The reflectors on bikes and cars and roadways are simply arrays of tiny triple-mirrors. They are designed this way so that a car light shining on the reflector will shine right back at the driver: it is exactly why reflectors are so easily seen.

Similarly, triple-mirrors make excellent reflectors for radar. Intersecting three metal plates at right angles creates a configuration of eight triple-mirrors, which will always reflect a radar signal back to its source. This is ideal for little boats that want to be sure they’re noticed by big oil tankers. Triple-mirrors were also planted on the surface of the Moon by the astronauts; lasers from Earth were then beamed at these mirrors, permitting an extremely precise measurement of the Moon’s orbit.

This is fascinating and very useful, but why does a triple-mirror work in this way? To understand it, first make a double-mirror: stand two mirrors vertically on a table and join them at right angles: viewed from above, the double-mirror makes a right-angle on the table, as in the following diagram.

Now shoot a light ray parallel to the table. The key point is that the incoming and outgoing rays will be parallel. To see that is true, we just have to realise that the two green angles are equal, the two orange angles are equal, and that the green and orange angles add to 90 degrees. We’ll leave you to chase all the angles around the path.

So, any horizontal light ray beamed into a double-mirror will return the way it came, with the emerging ray slightly offset. The closer the ray shines into the corner, the smaller the offset.  

A triple-mirror works exactly the same way: we just have to think of a triple-mirror as three double-mirrors and then combine the effects. It’s a little awkward to detail it, but at its heart it is no more than the double-mirror diagram above.

Our goal has been to understand triple-mirrors, but double-mirrors are also a lot of fun. Below is a picture of a vertical double-mirror, which shows you exactly as other people see you, with left and right sides not swapped: lift your right hand and your mirror image also lifts his right hand. Very weird things also happen when you start rotating this mirror: make one and try it out!

Puzzles to Ponder: 

 1) We’ve actually cheated: the double-mirror and triple-mirror photos of our junior maths master were manipulated in Photoshop. Why is it impossible to capture these images with a camera?

 2) Can you chase the angles through the double-ray diagram, to show that the incoming and outgoing rays really are parallel? (If you need some assistance, you can check out a more helpful diagram here.)

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

www.qedcat.com

Copyright 2004-∞ All rights reserved.