Which letter looks the same in a mirror — A, or F?
Slide, flip, and turn are the three motions that preserve shape and size — two figures that overlap under them are congruent.
Experiment
Hands-on experiment
🔮 Predict first — of the letters A and F, which looks the same in a mirror?
🧩 Move the L-block
Use the three buttons and sort what changes from what doesn't.
original
rotation 0°
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Stamping seals, repeating patterns, cutting folded paper — each act raised the same question: how far can a figure move and still count as 'the same figure'?
The answer is three motions: sliding (translation), flipping (reflection), turning (rotation) — all preserving shape and size perfectly. Figures that overlap under them are congruent.
The moving eye then turns on the figure itself: shapes that overlap themselves (by folding — line symmetry; by turning — point symmetry) are the symmetric forms nature and design love.
Insight
Insights from the video
“Congruence is another name for 'movable onto each other'.”
Instead of measuring every side and angle, define sameness as 'overlaps after sliding, flipping, turning' — defining equality through motion is the root of modern geometry.
“A flip borrows an extra dimension.”
Flipping a plane figure requires lifting it into 3D. Left and right gloves, mirrored letters — flipped twins can never meet within their own plane.
Misconception
Common misconceptions
A flip can be reproduced by turning cleverly enough.
Not within the plane. A flipped L never overlaps the original however you rotate it — as a left glove never becomes a right glove by turning. Flipping requires 'lifting the paper': a fundamentally different motion.
Symmetry is one thing — matching left and right.
Folding onto itself (line symmetry) and turning 180° onto itself (point symmetry) differ. N fails the fold but survives the turn (point only); A is the reverse (line only); H does both.
Formula
Writing it as math
What the block experiment confirmed, in mathematical language.
The three motions
Translation, reflection, rotation — all three preserve lengths and angles (shape and size unchanged).
Congruence
Two figures brought to perfect overlap by combining the motions — all corresponding sides and angles equal.
Line vs point symmetry
Two ways of overlapping yourself — A has line symmetry, N point symmetry, H both.
In Real Life
Where you meet it in real life
Tiles and wallpaper
Every repeating pattern is 'one base piece + slides, flips, turns'. It's proven there are exactly 17 possible wallpaper patterns.
Nature's symmetries
Butterfly wings (line), starfish (rotational), snowflakes (6-fold) — the forms of life and crystals are classified in symmetry's language.
Car and aircraft design
Bodies are designed with left-right mirror symmetry, halving the work — draw half and reflection makes the rest. CAD's mirror tool IS the flip.
Game sprites
A character facing left isn't redrawn — the right-facing image is flipped. The classic memory-saving trick is a reflection.
Try Yourself
Test yourself
Q1Of H, A, N, F — which have line symmetry?Show answer ▾
H and A (vertical axis). H also folds on a horizontal axis (two mirror lines) and survives a 180° turn — point symmetric too. N is point-only; F is neither.
Q2How many 90° clockwise turns return a figure to the start?Show answer ▾
Four (90°×4 = 360°). A flip returns in two — the same structure as (−1)×(−1)=+1 from the negatives lab.
Q3In how many ways does an equilateral triangle overlap itself?Show answer ▾
Six — three rotations (0°, 120°, 240°) and flips over three mirror lines. Counting a shape's self-overlaps leads to university group theory.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
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Connection
Concepts connect
Previous concept
Polygons
Only after knowing sides and angles does 'unchanged under motion' become surprising.
← Polygons labLeads to next
Perimeter & Area
Motions that keep shape are mastered — now the two rulers for a shape's size: perimeter and area.
Go to the Perimeter & Area lab →Related
Labs worth exploring together
Related lab
The Coordinate Plane
Write motion in numbers — (x,y)→(x+a,y+b) — and movement becomes computation.
Go to the The Coordinate Plane lab →Related lab