Class 6 Mathematics Chapter 9: Symmetry

A fold that keeps its promise

Ananya folds her drawing of a kite straight down the middle. The left half closes onto the right, and not one corner pokes out. She opens the paper, folds it corner to corner instead, and this time the halves miss badly. One fold worked and one did not. Which line she chose is the whole question of this topic.

What symmetry means

Look at a rangoli, a moth, or a hibiscus flower. Something in each of them comes back again, and it comes back in a fixed pattern. That returning is what we mean by symmetry. A cloud has nothing that returns, so we call it asymmetrical. Symmetry is not a word for pretty. It is a word for a part that repeats.

The fold test

A line of symmetry is a fold line. Close the paper along it. If one half hides the other completely, the line has earned its name. If an edge or a corner pokes out anywhere, the line has failed. Nothing else settles the matter, so do not trust your eyes on their own.

Mirror halves

The two pieces either side of such a line are called mirror halves. Stand a small mirror upright along the line, look into it, and the reflection completes the figure exactly. That is a quick check when the paper is too stiff to fold, and it explains why the idea has a second name.

Some figures fold no way at all

Draw a triangle whose three sides are all different lengths. Try folding it any way you please. No fold ever brings the two parts together. Having no line of symmetry is perfectly ordinary. It is not a fault in your drawing, and plenty of everyday shapes are like this.

One line, and no more

An isosceles triangle has two sides of equal length. Fold it along the line running from the tip to the middle of the base, and it closes neatly. Every other fold fails. The capital letter A behaves in the same way, and so does a plain arrowhead.

A square folds four ways

Take a square of paper. Fold it in half from top to bottom, then open it out. Fold it in half from side to side, then open it out. Now fold along one diagonal, open it, and fold along the other. Four creases, four lines of symmetry. Hunt as long as you like; there is no fifth.

The rectangle surprise

A rectangle that is not a square folds only twice, through the middles of opposite sides. Its diagonal looks promising on the page. Fold along it and two corners hang over the edge, because a long side cannot land on a short one. A diagonal is a fold line for the square alone.

Counting is not guessing

Do not give a figure the number of lines it seems to deserve. An equilateral triangle has three. A regular hexagon has six. A rectangle has two. Every one of those counts comes from folding, or from picturing the fold with real care, and never from how balanced the figure looks.

Reflection says the same thing

Label the corners of a square A, B, C and D, going clockwise from the top left. Fold along the upright line through the middle. A lands where B was and B lands where A was. D and C trade places in the same way. The line has sent every point across to a partner.

A second name for one idea

Because of that trading of places, a figure with one or more fold lines is also said to have reflection symmetry. The two names describe one property. Your teacher may use either, and neither is more correct than the other.

Curved edges count too

Fold lines are not only for figures with straight sides. A plain heart shape has one. An oval has two, one along its length and one across it. A circle has more than anyone could list, because every diameter you care to draw folds it into halves.

Where to look at home

A window grill, a kolam at the doorway, the wings of a moth, a steel plate, the letter H on a signboard. Pick one today and settle it with a fold or a mirror. Then write down how many fold lines you found, and how you tested each one.

Scenario:

A class folds paper shapes to find lines of symmetry. A square gives four folds that match. Rohit tries a rectangle and claims four as well. Their teacher asks him to fold it along a diagonal. The halves do not match; one corner overhangs. The rectangle has only two lines of symmetry. Rohit sees why he expected four. He had assumed a shape with four equal angles behaves like a square, and the equal sides are what the diagonal fold needs. The class lists the folds for both shapes side by side and marks where the rectangle loses two.

Analysis:

Rohit generalised from the square. The diagonal fold needs equal sides, which a rectangle does not have.

Activity:

Fold paper cut-outs of a square, a rectangle and a circle. Record how many folds match for each.

The Taj Mahal was planned about a central fold line, which is why its two halves answer each other so closely in a photograph taken from the front.

Kolam and rangoli artists work outwards from a grid of dots, so the fold lines of the finished pattern are decided before a single curve is drawn.

Testing a line of symmetry — Fold it, do not eye it. — Eyes are generous and paper is honest. If the halves really match, the crease will show it in a second.

Fold lines of a square — Two through the sides, two through the corners. — That is the whole set of four for a square, and the corner pair is exactly what a longer rectangle loses.

A blot that came out tidy

Ishaan tips a drop of blue ink onto one half of a folded sheet, presses the halves together and opens it again. Out comes a creature with two matching wings that he never planned. He did not draw the second wing. The crease did the work for him, and it did the work exactly.

Why the crease does the drawing

Whatever sits on one side of a fold is pressed onto the other side. Every mark lands at the same distance from the crease, on the opposite side. That is the reflection you met earlier, carried out by the paper itself. So the crease becomes a fold line of whatever you have made.

Ink blots at home

Fold a sheet in half and open it flat. Drop a little ink or thick paint on one half only. Press the halves together, rub gently, and open the sheet. The pattern you get is symmetrical about the crease, whatever shape the drop happened to take.

Where the fold line hides

Given a finished blot, you can find the crease without being told. Look for the line that the pattern turns about. Every feature on the left must have a twin on the right at the same distance. Where that works for the whole picture, you have found the fold.

Folding and cutting

Fold a sheet in half, cut a shape out of the folded edge, and open the sheet. The hole you get is symmetrical about the crease, and so is the sheet around it. A cut made anywhere near the crease is really two cuts, one on each layer.

Predicting before you cut

The useful habit is to predict first. Picture the cut, picture its reflection across the crease, and draw what you expect. Then cut and open the sheet. Predicting turns a craft activity into a piece of mathematics, and mistakes become interesting rather than annoying.

The punching game

Fold a square sheet in half and punch one hole through both layers. Open it and you have two holes, placed like mirror twins about the crease. Punch a second hole in the folded sheet and you get four in all, still in matching pairs.

Two folds, four holes

Fold the sheet in half and in half again, so there are four layers, then punch once. Opening the sheet gives four holes. Three folds give eight layers and eight holes. The count doubles with every fold, since 2 x 2 x 2 = 8 after three of them.

Reading the holes backwards

You can also work the other way. Given a sheet with holes already punched, ask where a fold would place each hole on top of another. The line that does this for every hole is the line the paper was folded along, and there may be more than one.

A single straight cut

Some surprising shapes come from one straight cut on a cleverly folded sheet. A square hole in the middle of a sheet needs two folds and one cut across the corner. Try it with rough paper first, since the folding is the difficult part and the cut is easy.

Paper lace for a festival

Fold a strip of thin coloured paper several times, snip small pieces from the folded edges, and open it out. You get a repeating border with several fold lines. This is how paper decorations for a festival are made, and a few folds give a long pattern quickly.

Completing half a figure

Squared paper lets you finish a half drawing so that a given line becomes a fold line. Count the squares from the line to each marked point, then count the same number on the other side and mark the twin. Join your new points in the same order.

Two given lines at once

If two fold lines are given, do the same job twice. Reflect the drawing in the first line, then reflect everything you now have in the second line. Check the finished figure by folding it in your mind along each line in turn.

What all of this shares

Ink, scissors, a punch and squared paper look like four different activities. They are one idea in four costumes. A fold copies whatever is on one side onto the other, and a copy at equal distance is exactly what a line of symmetry means.

Scenario:

A class makes symmetric shapes by folding paper and cutting. Nita cuts a shape on the fold and gets a butterfly. Imran cuts his away from the fold and gets two separate pieces. Their teacher asks the class what the fold was doing. It was the line of symmetry, so whatever stays joined across it becomes the mirror pair. Imran cut it away, so nothing joined the halves. He tries again keeping part of the fold uncut and gets one shape with two matching sides. The class writes the rule down: keep some of the fold, or you have made two shapes rather than one symmetric one.

Analysis:

Both children cut carefully and got different objects. The fold is the mirror line, so cutting all of it away removes the only thing joining the halves.

Activity:

Fold and cut three symmetric shapes. Mark the line of symmetry on each and say what would happen if you cut it away.

Title:

Predict the holes, then open the sheet

Safety:

Ask an adult to work the punch with you, and keep your fingers clear of its jaws.

What You Notice:

The count runs two, four and then eight, doubling at every fold rather than growing by one. The holes never land at random. They sit in matching pairs about each crease. The hole punched on the crease is the odd one out, because it opens as a single hole lying across the fold line. A point already on the line has nowhere else to go.

Paper cut-outs folded and snipped in this way decorate doorways at festivals across India, and the folds decide the pattern before a single snip is made.

A blot pressed between folded paper is used in some old picture games precisely because nobody can plan what the second half will look like.

Counting holes after folding — Each fold doubles the holes. — One fold turns one punch into two holes, a second fold turns two into four, and a third turns four into eight.

Completing a half drawing — Count across, mark the twin, then join. — Count the squares from the fold line, mark the matching point at the same distance, and only then draw the joining lines.

The windmill that will not fold

Devika pins a paper windmill to a stick and holds it still. She tries to fold it down the middle and the halves refuse to meet, because every blade leans the same way. Then she turns the whole windmill a quarter of the way round. It looks exactly as it did before. Folding failed and turning worked.

A second kind of symmetry

A figure has rotational symmetry when turning it about a fixed point leaves it looking unchanged. That fixed point has a name, the centre of rotation, and for the windmill it is the pin. Nothing is folded here and no mirror is used. The figure simply turns.

Angles of symmetry

A turn that leaves the figure looking the same is called an angle of symmetry. For the windmill, a quarter turn of 90 degrees works. So does a half turn of 180 degrees, and a three quarter turn of 270 degrees. A turn of 40 degrees does not work at all.

The full turn always works

Turn any figure right round through 360 degrees and it comes back to exactly where it began. Nothing has moved. So 360 degrees is an angle of symmetry for every figure ever drawn, tidy or untidy. That one fact does more work than it looks as though it should.

Order of rotational symmetry

The order of rotational symmetry is the number of angles of symmetry in one full turn, counting the full turn itself. The windmill has 90, 180, 270 and 360 degrees, so its order is 4. The order is simply how many positions the figure has that all look the same.

Order is never zero

Since the full turn always works, every figure has at least one angle of symmetry. So no figure can have order 0. A figure with nothing better than the full turn has order 1, and saying it has order 1 and saying it has no rotational symmetry mean exactly the same thing.

The angle from the order

The angles of symmetry share a full turn equally, so the smallest of them is 360 divided by the order. Order 4 gives 90 degrees, since 4 x 90 = 360. Order 2 gives 180 degrees, since 2 x 180 = 360. Order 1 gives 360 degrees itself, since 1 x 360 = 360.

Watching a corner travel

Label the corners of a square A, B, C and D. Turn the square 90 degrees about its centre. Corner A moves to the seat B was in, B moves to C, C moves to D and D moves to A. The outline has not changed at all, yet every corner has moved one place round.

A figure with order 1

Draw a long strip patterned in a way that leans, so that the left end does not match the right. Turn it 180 degrees and the pattern is upside down, not the same. Only the full turn restores it. Its order is 1, and it has no rotational symmetry.

The parallelogram

Now take a parallelogram that leans, with no right angle in it. Turn it 180 degrees about the point where its diagonals cross and it fits its own outline exactly. So its order is 2 and its smallest angle of symmetry is 180 degrees. Nothing about this depends on folding.

And yet it will not fold

Try every fold line you can think of on that same parallelogram. Upright, across, along either diagonal, slanted at any angle you like. Not one of them brings the halves together. The figure has rotational symmetry of order 2 and not a single line of symmetry.

The two kinds are independent

So the two symmetries travel apart as easily as together. The letter S turns but does not fold. The letter A folds but does not turn. The letter H does both and a plain scalene triangle does neither. Never settle one question by answering the other.

Finding the centre

The centre of rotation is not always the middle of the page. For a square or a parallelogram it is where the diagonals cross. For a windmill it is the pin. Put a pencil point there, turn the paper, and watch whether the outline stays put.

Testing with a tracing

Trace the figure onto a thin sheet, lay the tracing exactly over the original and press a pin through the centre. Turn the tracing slowly and count the positions where it covers the figure again before you reach a full turn. That count, together with the full turn, is the order.

Scenario:

A class turns a paper fan with three equal blades and counts how often it looks the same in one full turn. The answer is three. Meena then tries a square and gets four, and a rectangle and gets two. She says the number must be the number of sides. Their teacher hands her an equilateral triangle, which has three sides and matches three times, and then a circle. The circle matches at every angle. Meena's rule breaks on the circle. The class rewrites it. The order counts how many positions look identical, and that is not always the side count.

Analysis:

The side rule held for three shapes and failed on the circle. Order counts matching positions, which is a different thing from counting edges.

Activity:

Turn four objects a full circle and count how often each looks unchanged. Compare the counts with their number of sides.

A three bladed ceiling fan looks the same after a turn of 120 degrees, which is why a slow fan seems to hang still for a moment as it passes each blade.

The playing cards in a standard pack are printed with order 2 in mind, so a hand of cards reads the same whichever way round a player picks it up.

Order can never be zero — A full turn always fits, so one is the floor. — Every figure returns to itself after 360 degrees, so every figure has at least one angle of symmetry and at least order 1.

Finding the smallest angle — Share 360 among the order. — The angles of symmetry split a full turn evenly, so the smallest of them is 360 divided by the order of the figure.

A figure made of arms

Draw a dot, then draw four straight arms out of it like the spokes of a small wheel, each with a little flag at its end. Nothing here has a boundary to fold along, so folding is beside the point. Turn the drawing instead, and the question becomes how far you must turn it before the arms line up again.

Four arms, and the angles between them

If the four arms are spaced evenly, the angle between one arm and the next is 90 degrees. Turn the figure 90 degrees and every arm takes the place of the arm in front of it. The figure looks untouched, so 90 degrees is an angle of symmetry and the order is 4.

The spacing is what matters

The arms need not be any particular length for this to work. What matters is the angle between neighbouring arms. Make those angles unequal and the figure stops fitting itself at a quarter turn, even though it still has four arms sticking out of one point.

Making a figure of order 2

Take the four arm figure and push two of the arms closer together, keeping the pattern opposite each other. Now a quarter turn fails but a half turn still works. The order has dropped from 4 to 2, and its smallest angle of symmetry has risen from 90 degrees to 180 degrees.

Three arms that do not work

Draw three arms out of one point at angles you have not measured, say a wide gap, a narrow gap and a medium one. Turn the figure as much as you like. Only a complete turn brings it back, so its order is 1 and it has no rotational symmetry at all.

Why the three angles must be equal

Suppose a three armed figure is to fit itself after a turn. Call the three gaps between the arms A, B and C. The turn must carry A onto B, B onto C and C onto A. Three angles that cover each other must be equal, so A, B and C are all the same size.

Sharing the full turn

Those three equal gaps go right round the point, so together they make 360 degrees. Each gap is therefore 360 divided by 3, which is 120 degrees. Now the figure fits after a turn of 120 degrees, and 3 x 120 = 360 checks the sharing.

The same argument for any number

Nothing in that reasoning depended on the number three. Five evenly spaced arms give gaps of 72 degrees, and 5 x 72 = 360. Six arms give 60 degrees, and 6 x 60 = 360. Eight arms give 45 degrees, and 8 x 45 = 360. The order is the number of arms.

Listing all the angles

A figure of order 3 does not stop at 120 degrees. Turn it 120 degrees again and it fits once more, at 240 degrees, and a third turn brings it to 360 degrees. So its angles of symmetry are 120, 240 and 360 degrees: the multiples of the smallest one.

The pattern in the lists

Order 2 gives 180 and 360 degrees. Order 3 gives 120, 240 and 360 degrees. Order 4 gives 90, 180, 270 and 360 degrees. Every list is the multiples of its smallest angle, and every list ends at the full turn. That is not a coincidence; it follows from the sharing.

The smallest angle divides 360

Since the full turn is always on the list, and the list is the multiples of the smallest angle, the smallest angle must go into 360 a whole number of times. So a smallest angle of 24 degrees is possible, because 15 x 24 = 360. A smallest angle of 25 degrees is not.

When the angle is not a whole number

Nothing forbids an order that leaves an awkward answer. Sixteen evenly spaced arms have a smallest angle of 22 and a half degrees, because 360 shared among 16 is 22.5. The rule has not failed here. The angle is simply not a whole number of degrees.

Checking a figure you have drawn

Trace your figure onto a thin sheet and cut it out. Hold the original still, pin the cut out at the centre and turn it slowly. Count every position where the cut out covers the drawing, up to and including the complete turn. That count is the order.

Reading the question the other way

You may be told the smallest angle and asked for the order. Divide 360 by the angle. A smallest angle of 40 degrees means order 9, since 9 x 40 = 360. You may also be given one angle of symmetry that is not the smallest, in which case look for factors.

Scenario:

A class designs a wheel pattern with arms spread evenly round a centre. Arjun draws five arms and is asked where each should sit. He guesses. Their teacher asks the class to work it out instead. A full turn is 360 degrees, and five equal arms means 360 divided by 5, which is 72 degrees apart. Arjun redraws with a protractor and the pattern now looks even. The class checks the order by turning the drawing, and it matches five times. Arjun sees that evenly spaced was a measurable instruction, not a matter of judgement, once the arms were counted.

Analysis:

Evenly spaced sounds like a judgement and is a division. Three hundred and sixty divided by the number of arms gives the angle exactly.

Activity:

Design a radial pattern with six arms. Work out the angle between them and draw it with a protractor.

The wheels on many bullock carts carry twelve spokes, which spaces them 30 degrees apart and makes the wheel fit itself twelve times in one revolution.

Snowflakes grow six arms because of the way water molecules join, which gives almost every snowflake a smallest angle of 60 degrees.

Order from the smallest angle — Order and angle multiply to 360. — If you know one of the two, divide 360 by it to find the other. Order 9 and 40 degrees belong together, since 9 x 40 = 360.

Testing a proposed smallest angle — It must go into 360 without a remainder. — A whole number smallest angle has to divide 360 exactly, which is why 24 degrees can be one and 25 degrees cannot.

The shape that will not be counted

Priya sets a paper disc on her desk, puts a pencil point at the centre and turns it a little. The disc looks exactly as it did. She turns it a little less, and it still looks the same. However small the turn she tries, the disc never gives her away. Every other figure in this chapter had a smallest turn. The circle has none.

Every angle works

Every turn there is fits a circle, whatever its size. One degree works, and so does half of one degree, and so does any fraction you care to name. That is why we cannot give a circle an order. An order counts positions, and here the positions never run out.

Every diameter folds it

Now take a point anywhere on the rim and join it to the centre. Carry the line on to the far rim and you have a diameter. Fold the disc along it and the two halves meet exactly. Every diameter is a line of symmetry, and there are endlessly many diameters.

Both kinds, without limit

So the circle is the one figure in this chapter with unlimited amounts of both kinds of symmetry. It has more lines of symmetry than can be counted and more angles of symmetry than can be counted. That is what makes a wheel roll so smoothly, whatever position it is caught in.

Wheels, fans and flowers

Look for turning symmetry around you. A ceiling fan with three blades has order 3 and a smallest angle of 120 degrees. A cycle wheel with 36 equally spaced spokes has order 36 and a smallest angle of 10 degrees. A hibiscus with five petals has order 5 and a smallest angle of 72 degrees.

A circle that has been coloured

Colouring takes the symmetry away again. Divide a circle into 12 equal sectors, each measuring 30 degrees, since 12 x 30 = 360. Colour them in a repeating group of three shades. The pattern now returns every three sectors, which is 3 x 30 = 90 degrees, so the coloured disc has order 4.

Choosing the order you want

Change the size of the repeating group and you change the order. A repeating group of four shades returns every 4 x 30 = 120 degrees, so that disc has order 3. Only group sizes that divide 12 will work, because the pattern has to close up neatly as it comes right round.

Tiles on a floor

The same game can be played with square colour tiles. Lay sixteen of them in a four by four block, colour some of them and leave the rest plain. Where you put the colours decides the symmetry of the block, and you can plan it before you lay a single tile.

A design with exactly one fold line

To get exactly one line of symmetry, choose an upright line down the middle of the block and place every coloured tile with a partner across that line. Then break the other possible symmetries deliberately, for instance by colouring a tile near the top that has no partner below it.

A design with exactly two fold lines

For two fold lines, match your tiles across the upright line and across the line running from side to side. Every coloured tile then belongs to a group of four. Such a design usually turns out to have rotational symmetry of order 2 as well, which is worth checking.

Sketching to order

You can be asked for a figure with one kind of symmetry and not the other. For rotational symmetry without a fold line, a leaning parallelogram will do. For a fold line without rotational symmetry, a triangle with two equal sides will do. Keep one example of each in your head.

Regular figures are the tidy case

A regular polygon with n sides has n lines of symmetry and order n. A square has four of each. A regular hexagon has six of each. A circle behaves like the end of that list, with both counts beyond any number, which is exactly how it behaves when you turn it.

Designing a rangoli

Decide the order first, then draw. For order 4, draw a quarter of the pattern inside a right angle at the centre and copy it three times, turning 90 degrees each time. If you want fold lines too, make your quarter pattern symmetrical about its own middle line.

What to carry away from the chapter

Two separate questions, asked of every figure. Does it fold onto itself, and how many ways? Does it turn onto itself, and at how many positions? A figure may answer yes to both, to one, or to neither, and the circle answers yes to both without limit.

Scenario:

The pupils of a school in Madurai must lay a rangoli at the gate for the harvest festival. Selvi draws a chalk circle one metre across. She divides it into twelve equal parts with straight lines from the centre. Her plan is that the rangoli should look the same from four sides. A visitor walking round it would then see one design rather than four. Her friend Karthik wants a different colour of rice flour in each of the twelve parts. Selvi says that would spoil the plan. The art teacher asks them to settle it with a calculation. Nobody may open a packet of colour until they have.

Analysis:

The order the pupils want fixes how often the colour pattern may repeat, and that in turn fixes the size of the repeating group.

Activity:

Draw a circle, divide it into twelve equal parts and colour it with a repeating group of three shades. Then check the order by turning the paper.

A potter shapes a bowl on a turning wheel precisely because a circle looks the same at every angle, so the clay is pressed equally all the way round.

Rangoli patterns at festivals are often planned for order 4 or order 8, which is why an artist can draw one quarter and copy it round.

A manhole cover is round so that it can never fall through its own hole, whichever way it is turned.

The circle — Every turn, every diameter, no order. — Both kinds of symmetry are unlimited for a circle, which is exactly why no order can be written down for it.

Designing symmetry — Decide the order first, then draw one piece. — Draw a single piece of the pattern, then copy it round the centre by the smallest angle until the design closes up.