Symmetrical Designs

1A letter cutout can use a fold as a line of symmetry

Fold a cutout letter. If the halves land, that fold is a line of symmetry for that letter.

A dashed line you drew is only a candidate until it lands.

Figure. A paper A folded on the dashed midline lands on itself. That fold is a line of symmetry for the cutout.

How it works

1. Pick a letter cutoutThis A, or this H.
2. Try a foldVertical or horizontal.
3. Title the line only if it landsThen it is a line of symmetry.

2Vertical and horizontal lines are two different tests

A vertical fold and a horizontal fold are two tests. One pass does not gift the other.

H may pass both; A may pass only vertical. Test each.

Figure. The vertical fold of A matches. The horizontal fold does not: the peak and the flat base are different tests, not one “plus-shaped” fold.

How it works

1. Run verticalPass or fail.
2. Run horizontalA new test.
3. Count only the lines that landedTwo tests.

3Some letters need a quarter fold

A letter might fail half-folds you tried and still have a different fold (or none). A “quarter fold” is another candidate — test it; do not assume.

Needing a new candidate is not a pass.

Figure. A plus (or H) matches only after a second fold at right angles to the first. That quarter fold is two tests, not one generous “looks even”.

How it works

1. If half-folds failTry another candidate if the task asks.
2. Fold that new lineLanding is still required.
3. If none land, the letter has no that-line symmetryHonest fail.

4A and H do not share the same folds

A typically passes a vertical fold and fails a horizontal one. H can pass both. Do not copy H’s folds onto A.

Each letter is its own test-sheet.

Figure. A has only the upright fold. H has the upright fold and the cross fold. Copying A’s fold list onto H (or the other way) fails one of the tests.

How it works

1. Test A on its ownWhich folds land?
2. Test H on its ownWhich folds land?
3. Refuse a shared cheat-sheetDifferent letters, different passes.

5A windmill firki is a turn test

A firki can land on itself after a quarter or half turn around its centre. That is turn-symmetry, not a fold unless you also fold.

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

Figure. A firki is a turn test, not a fold test. A quarter turn sends blade 1 to blade 2. If a blade is a different colour, the turn fails.

How it works

1. Pick the centreThe pin.
2. Turn a quarter or halfThe asked turn.
3. Ask if it lands on itselfYes: turn-symmetry.

6Almost even fails the fold

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

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

Figure. The two halves look near even, but the right cheek is shorter and the eye sits lower. The fold test fails. Almost is not a line of symmetry.

How it works

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

7Copy one half to test the other

Trace one half, flip across the line, demand a cover. Extra doodle beads break the test.

The copy-flip is the evidence.

How it works

1. Trace one halfHonest copy.
2. Flip across the candidateThe test.
3. Demand a coverIf it misses, fail.

8The birthday wall is the story; the fold is the maths

Letters on a wall are the picture. Fold and turn tests are the maths.

Do not invent a party-only symmetry.

Figure. The wall is a story. The dashed midline is the maths: after a fold, do the tiles land on tiles? Decoration does not grant a line of symmetry.

How it works

1. Keep the settingA wall of letters.
2. Run fold or turnThe test.
3. Use ordinary landingThe party is not a new rule.