Fold a butterfly down the middle and the two halves kiss. Spin a star and it clicks back into place. That hidden tidiness has a name — symmetry — and you can feel it with your own hands.
Start hereImagine you turn your back, a friend folds, flips, or spins a shape — and when you look again, you can't tell anything happened. The shape landed perfectly back on itself. That is symmetry.
There are two everyday kinds, and this whole page is about feeling both. Line symmetry is the folding kind: one half is the mirror of the other. Rotational symmetry is the spinning kind: turn the shape part of the way around and it looks exactly as it did. Many things you see every day — leaves, faces, snowflakes, logos, the very letters you're reading — are built out of these two moves.
Here's the simplest test in all of geometry. Take a shape, imagine a straight crease through it, and fold along that crease. If the two halves land exactly on top of each other — no bits poking out, no gaps — then that crease is a line of symmetry, and the shape has line symmetry (you'll also hear it called mirror symmetry or reflective symmetry, because one half is the mirror image of the other).
The crease itself is the mirror. Everything on the left is reflected to the right, point for point, the same distance from the line. That last bit is the secret rule of reflection: pick any spot on the shape, measure straight across to the mirror line, and you'll find its twin exactly the same distance on the other side, sitting directly opposite. The mirror line is the perfect halfway point between every pair of matching spots. A butterfly is the classic example: its wings are mirror images, the left wingtip and the right wingtip are equally far from the body, and the fold runs straight down the middle.
Why does this matter? Because it gives you a test you can actually run, even in your head. To check whether a line is a line of symmetry, you don't need scissors — you just ask, "If I flipped the shape over this line, would it land on exactly the same outline?" If yes, it's a real line of symmetry. If even one corner pokes out past the edge, it isn't. Real mathematicians, when they want to be precise, drop the word "fold" and say reflective symmetry instead, but it's the same friendly idea.
The dashed line is the fold. Tap the button and the left half (darker) and the right half (lighter) light up so you can see they're the same shape, flipped.
Pick a shape. In Mirror lines mode, drag across it to lay a line through its middle — the Lab checks whether the two halves really match. In Rotation mode, turn the shape and watch for the moments it lands back on itself.
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Equilateral triangle
Notice the rule the Lab keeps enforcing: every line of symmetry runs through the centre of the shape. You can't fold a shape onto itself along a line that misses its middle, because the two halves would have different sizes. That's why, no matter which way you drag, the Lab snaps your candidate line through the centre dot before it judges it. A square has four such lines; a rectangle that isn't square has only two; a plain slanted parallelogram has none at all — yet, as you'll feel in Rotation mode, the parallelogram still has a trick up its sleeve.
Take your time with both modes. In Mirror lines, try to guess how many lines a shape has before you find them — then press "Show all lines" to check yourself. In Rotation, drag the slider slowly and watch the little badge in the corner: it lights up "looks identical" at every angle where the spun shape lands back on its faded starting position. Counting those flashes is exactly how you find the order, which is what the next section is all about.
A shape can have more than one line of symmetry — or none. The number is a kind of fingerprint. Tap each shape below to reveal every line of symmetry it hides, plus its order of rotational symmetry (we'll get to that word in a moment).
Tap a shape to fold out its mirror lines.
A few results worth carrying with you. A square has 4 lines of symmetry: two through the middle of opposite sides, two along the diagonals. A rectangle that isn't a square has only 2 — the diagonals look tempting, but fold along one and the corners don't meet, because a rectangle's diagonals aren't mirror lines, they're just lines. The real surprise is the circle: any straight line through its centre is a line of symmetry, so a circle has infinitely many. There's no first one and no last one — you can keep finding more forever, which is exactly what "infinite" means here.
There's a lovely pattern underneath all this. The more "regular" a shape is — meaning all its sides are equal and all its angles are equal — the more lines of symmetry it has. A regular triangle has 3, a square has 4, a regular pentagon has 5, a regular hexagon has 6. In general, a regular shape with n equal sides has exactly n lines of symmetry, and a circle is what you get when n grows so large the sides melt into a smooth curve — so it makes sense that it ends up with infinitely many. Irregular shapes, where the sides and angles are all different, usually have none.
Spin a square a quarter-turn and it looks exactly the same. That's rotational symmetry: a shape has it if you can turn it less than a full circle and it lands back on itself.
The natural question is "how many times?" Spin a square slowly all the way around and it looks identical at a quarter turn, a half turn, three-quarters, and a full turn — four matching positions in one full spin. We say the square has rotational symmetry of order 4. The order of rotational symmetry is simply the count of positions where the shape looks unchanged during one complete turn.
Try it for yourself: jump back up to the Lab, switch to Rotation mode, and drag the turn slider — or hit "Spin a full turn." A triangle clicks back three times (order 3), a regular pentagon five times (order 5), a heart only once, at the very end (order 1). Order 1 means no real rotational symmetry — every shape returns to itself after a full turn, so order 1 is just the "boring" baseline that doesn't count as special.
There's a neat shortcut hiding here. If a shape has rotational symmetry of order n, then the smallest turn that lands it back on itself is 360° ÷ n. A square (order 4) clicks back every 360 ÷ 4 = 90°. An equilateral triangle (order 3) needs 360 ÷ 3 = 120° each time. A snowflake (order 6) only needs 60°. So "order" and "smallest turn" are two ways of saying the same thing: a big order means lots of matches and tiny turns; a small order means few matches and big turns.
And here's the friendly surprise for regular shapes: their number of lines of symmetry and their order of rotational symmetry are the same number. A regular hexagon has 6 lines and order 6. A square has 4 and 4. That tidy match-up is special to regular shapes, though — as you're about to see, most shapes don't play so fair.
This is the part that trips people up, so it's worth slowing down. Line symmetry and rotational symmetry are separate properties. A shape can have lots of one and none of the other.
The parallelogram is the headline act. Hold a tilted parallelogram in your mind and try to fold it: along the long way, the slanted ends stick out; along the short way, the corners miss. No fold works, so no line symmetry. But pin it at its centre and rotate a half-turn — top-left swaps with bottom-right, top-right with bottom-left — and it lands exactly on itself. That's why the Lab reports the parallelogram as "0 lines, order 2." Having one kind of symmetry tells you nothing about whether the shape has the other.
It also works the other way around. An isosceles triangle (two equal sides) and a love-heart each have exactly one line of symmetry but no rotational symmetry at all — spin them and they only ever come home after a complete 360° turn. So when someone hands you a shape, get into the habit of asking two separate questions, not one: "Can I fold it?" and "Can I spin it?" The answers are independent, and a shape's full symmetry profile is really both numbers together — how many mirror lines, and what order of rotation.
The dark squares on the left are the given half. Tap squares on the right to draw the mirror image across the dashed line. When every right-hand square matches its reflection, the picture is symmetric.
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This is reflection made physical. A square at 2 across on the left must be matched by a square the same distance across on the right — the mirror line sits exactly halfway between them. Get every pair right and you've built a shape with a vertical line of symmetry, one tap at a time.
In plain block capitals, letters carry symmetry too. Some have a vertical mirror line (fold left-to-right), some a horizontal one (fold top-to-bottom), and some have rotational symmetry — turn them a half-turn and they read the same. Tap a letter to see which kinds it has.
Tap any letter above to reveal its symmetries.
Did you spot the quiet stars? H, I, O and X have it all — a vertical mirror, a horizontal mirror, and a half-turn that leaves them unchanged. S, N and Z are the sneaky ones: no mirror lines at all, yet spin them 180° and they snap right back. That's the same "one without the other" idea you met with the parallelogram, hiding in your keyboard. (It also explains a fun trick: words like SWIMS and NOON survive being rotated a half-turn.)
It feels like it should be true. Symmetry is so common that it's tempting to assume every shape can be folded in half somehow. It can't.
"Every shape has at least one line of symmetry — and a parallelogram, since it looks so balanced, surely has one."
Plenty of shapes have no line of symmetry: a scuffed scribble, a scalene triangle (all sides different), the letter F, and — the famous one — a slanted parallelogram. A parallelogram looks tidy because it has rotational symmetry (order 2), but its slant means no fold ever lines the halves up. Balanced-looking is not the same as fold-in-half. Always run the actual fold test in your head, edge against edge, before deciding.
Symmetry is a thing you check, not a thing you assume. When in doubt: try the fold, then try the spin.
Nature. Snowflakes grow with six-fold rotational symmetry because of how water molecules lock together as they freeze — every snowflake is a tiny order-6 sculpture. Flowers, starfish, beehive cells and the bodies of most animals (including you) are roughly mirror-symmetric, a layout called bilateral symmetry. Symmetry often signals health and good growth, which is part of why we find symmetric faces pleasing — though real faces are only nearly symmetric, never perfectly so.
Design and logos. Designers lean on symmetry because our brains read it as balanced, stable and trustworthy. A huge share of famous logos are built on a vertical mirror line or a rotational pattern — think of the three-pointed star inside a circle, or pinwheel-style marks that spin onto themselves. Symmetry makes a mark feel solid and easy to remember.
Building and making. Architects use symmetry to make buildings feel grand and steady; engineers use it so that forces balance and a structure doesn't twist. Tiles, wallpaper, quilts and Islamic geometric art repeat a single symmetric unit over and over to cover a whole surface with no gaps and no overlaps — that repeating-pattern trick is called a tessellation, and it leans on both kinds of symmetry at once. You did a tiny version of it yourself in the Mirror Painter, building a whole picture out of one mirrored half.
Science, all the way down. Symmetry isn't just a decorating choice; it's one of the deepest ideas in physics and chemistry. Crystals like salt and quartz grow into symmetric shapes because their atoms stack in repeating, symmetric arrangements. The shapes of molecules, the patterns of the elements, even the laws of nature themselves are studied through their symmetries. So the little game you just played — fold it, spin it, see if it lands on itself — turns out to be a doorway into how scientists describe the whole universe. Once you've trained your eye in the Lab above, you'll catch it everywhere you look.
If a crease folds the shape exactly onto itself, that crease is a line of symmetry. Count them.
If a part-turn lands the shape on itself, it has rotational symmetry. The order is how many times that happens in one full turn.
A shape can have one without the other. Check both — never assume.