Fun with Symmetry

1A fold is a symmetry test you can run

Fold along a candidate line. The two halves must land on each other. That landing is the test.

A tidy look without a fold is not the test.

Figure. A fold is a test, not a guess: the two halves must land on the same drawing. These two matching A panels share a dashed fold. If the halves disagree, that line is not a line of symmetry.

How it works

1. Pick a fold lineA candidate.
2. FoldHalves must meet.
3. Pass or failLanding, or a miss.

2Rangoli, mask, and building can pass or fail that fold

Each drawing gets its own fold-test. A rangoli that passes a vertical fold may fail a horizontal one.

A pretty mask is not a pass until the fold lands.

Figure. Rangoli, a mask, and a building facade can each pass or fail the same fold test. A matching pair of windows passes. A single window on one side only fails, even if the wall looks tidy.

How it works

1. Name the objectThis rangoli.
2. Run one foldOne line.
3. Record pass or fail for that lineDo not reuse another object’s result.

3A turn can be a symmetry test too

A half-turn around a centre can land on the same drawing. That is a turn-test, not a fold-test.

Do not call a fold-pass a turn-pass without turning.

Figure. A half-turn is a second test. This plus-sign lands on itself after a 180 degree turn about its centre. A drawing that only looks even, but does not land after the turn, fails.

How it works

1. Pick a centreA point.
2. Turn a half-turn (or the asked turn)Trace or think.
3. Ask if it lands on itselfYes: turn-symmetry.

4Almost even fails the fold

A bead that misses its partner by a little still fails. Almost is a miss.

Do not “give it” because it looks nearly even.

Figure. Almost even is still a fail. The two eyes sit at different heights, so the dashed fold does not land on the same drawing. Tidy is not the test; landing is.

How it works

1. FoldSee the miss.
2. If a part sticks outFail.
3. Name it not symmetric about that lineExact landing.

5A line of symmetry is a candidate until it lands

Drawing a dashed line does not make it a line of symmetry. The fold (or reflection) must land.

A candidate is a try, not a title.

Figure. A line of symmetry is only a candidate until the fold lands. The dashed median of this isosceles triangle does land: the two legs are equal, so that line passes. A line drawn just because it looks central still has to sit the test.

How it works

1. Draw the candidateA dashed line.
2. Run the testFold or match halves.
3. Title it only if it landsThen it is a line of symmetry.

6Copy one half to test the other

Trace one half, flip across the line, and see if it covers the other half. That copy is the test.

Copying and then doodling extra beads breaks the test.

How it works

1. Trace one halfHonest copy.
2. Flip across the candidate lineThe flip is the test.
3. Demand a coverIf it misses, fail.

7Two lines of symmetry are two tests, not one pretty picture

A square can pass a vertical fold and a horizontal fold. Those are two tests, two lines.

One pass does not gift you a second line.

Figure. Two lines of symmetry are two tests, not one pretty cross. A rectangle must pass the upright fold and the sideways fold separately. Passing one does not grant the other.

How it works

1. Run fold 1Pass or fail.
2. Run fold 2A new candidate.
3. Count only the lines that landedTwo tests.

8A bead string that does not match after a flip is not symmetric

Flip the string across the claimed middle. If a colour sequence does not match, it is not symmetric about that middle.

A lovely colour is not a match.

Figure. A bead string is symmetric only if the flip matches the original. Red-blue-green-yellow flipped is yellow-green-blue-red, which is a different string, so the flip fails.

How it works

1. Write the sequenceR-B-R.
2. Flip across the middleDoes it match?
3. If a colour is wrongNot symmetric about that middle.