ambidextrous tetrahedrons

Congruence versus Chirality

As math teachers, we tend to spend a lot of time with congruence. Two triangles are congruent if and only if they may be superimposed on one another. Part for part, angle for angle they match up.

Rather than translate the triangles (i.e. moving them), we have those theorems all trig students learn: SAS, SSS, ASA, AAS and HL. If two sides and an included angle are the same, the two triangles are congruent. Ditto if two angles and an included, or non-included, side are the same. And so on.

That’s all well and good until we get to the issue of chirality, or handedness. Is my left hand congruent with my right hand?

Typically, a planar figure is said to be congruent with another planar figure if you can superimpose one on the other even if you need to flip one of them over to do so.

Consider the game Tetris. We all know, if we’ve played that game, that a right facing L shape does not rotate into a left facing L shape. In the planar world, we have two L shapes.

However, if we allow flipping, which Tetris does not, the Left L and Right L may be superimposed, and are thereby shown to be congruent.

Likewise in space, one has no way to turn a left hand into a right hand. Through inside-outing, a left-handed glove may become a right-handed glove, but isn’t inside-outing a fairly radical transformation?

Appeals to a higher dimensional space need to be made explicit, for congruence to work as a concept. A and B are congruent with the proviso that we allow “flipping”.

Where congruence is concerned, we always do, whatever “flipping” means.

However, as math teachers, we also need to remind students that “handedness matters”.

For example in chemistry, the “same” molecule may, often does, have very different properties if it’s “flipped” from left to right, or right to left.

Superimposing left-handed and right-handed molecules is not possible in the space of chemical operations, short of breaking them down into elements and reassembling them.

If the textbook you’re using has insufficient caveats, when it comes to the congruence concept, consider mining Youtube for some appropriate reminders.

Metachirality? by Vihart

Lots online.

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