I’m a huge fan of HBO’s series The Wire. Today, while I was watching the 4th episode of season 4, I saw a curious word on the strap hanging around Sgt. Landsman’s neck: HOWICIDE.
At first I thought this was a mistake or an easter egg. Why would they misspell homicide? But then I quickly realized that the letters H, O, I, C, D, and E all have the same symmetry—they are unchanged if they are reflected about a horizontal axis. M, on the other hand, becomes W in that circumstance. The yellow strap was actually the back side; it says HOMICIDE on the other.
So what we have are two mirror images:
Of course there are other letters that have this same symmetry (looking at capital letters only): B, C, D, E, H, I K, O, X.
We could use these to get all sorts of words that remain unchanged when reflected in a mirror in this way, for example:
I couldn’t think of any good words that contained an M which when you reflect it it became a new word with a W. This was the best I could do.
There are other symmetries among the letters too. A, H, I, M, O, T, U, V, W, X, and Y are all symmetric about the vertical axis. But if we wanted to create words from these that are mirror images of themselves we’d need to find palindromes, such as:
Or we could have words that were mirror images of each other like:
YAM | MAY
Finally, some letters have a rotational symmetry (rotate 180 degrees about its center): I, N, O, S, X, Z. Again, to find words that are the same when rotated, we must find palindromes using these letters, such as:
Just like our first example, M and W are paired—if we rotate one we get the other. Thus we can have other words that remain the same after rotation (I was especially proud of thinking of this one):
There are other possible symmetries (think O and X), but I think this exhausts the interesting possibilities (without including lower case letters).