Mirror and Water Image

A word is a row of letters, so a vertical mirror does two jobs on it, and you must do both.

Job 1: reverse the order. The letter nearest the mirror stays nearest. If the mirror is on the right, the last letter of the word becomes the first letter of the image. FISH becomes H, S, I, F.

Job 2: reverse each letter. Every letter is turned left to right, like a stamp. Letters with a vertical line of symmetry (A H I M O T U V W X Y) look the same; the rest come out visibly reversed.

[[fig:0]]

In the picture, step 1 only changes the order and step 2 turns each letter. The last line is the mirror image of FISH.

Look at the second picture: TAXI is made only of symmetric letters, and each letter looks normal in the image, yet the word still reads backwards, as IXAT. So symmetric letters are never a reason to skip the order job.

[[fig:1]]

The same works for digits. 0 and 8 look the same. The digits 2, 3, 4, 5, 6, 7 and 9 come out reversed, and 6 and 9 are not swapped by a mirror: a reversed 6 is still a reversed 6.

Mirror image of a word: last letter first, and every letter turned left to right.

Sub-case 1: which letters and words do not change?

Some questions have no picture. They ask how many letters of a word look the same in a mirror, or which word looks exactly like its mirror image. Learn the mirror list as a pattern: A, HI, M, O, TUVWXY (11 letters).

[[fig:2]]

In the picture a letter is green if it is in the list and red if it is not. ALPHABET has four green letters by position. Count by position unless the question says "different letters".

A word looks exactly the same in a mirror only when both conditions hold: every letter is in the list, and the word reads the same backwards. WOW and OTTO pass. HOT has the right letters, but its image reads TOH, so it fails.

Sub-case 2: mirror writing and the rear-view mirror.

The word AMBULANCE is printed reversed on the front of an ambulance. The driver ahead sees the ambulance in his rear-view mirror, and the mirror reverses the writing once. Since it was printed reversed, the mirror reverses it a second time, and he reads it correctly.

[[fig:3]]

One reflection reverses the text. Two reflections give the correct reading.

To print a word so that it reads correctly in a mirror, print its mirror image: last letter first, each letter turned left to right. A mirror never turns the text upside down.

A vertical mirror swaps only left and right. The height of every part stays the same. So take any part of the figure, keep it at the same height, and carry it to the other side of the mirror line at the same distance.

Two things happen to every part.

1. It changes side. A part on the right of the figure appears on the left, and the top-left corner appears at the top-right corner.
2. It is reversed. A shape that is not symmetric comes out turned around. A right-angled triangle with the right angle at the bottom-right gives an image with the right angle at the bottom-left. An arrow that points right in the figure points left in the image. Symmetric shapes (circle, square, isosceles triangle) look the same; only their position changes.

[[fig:0]]

The orange dotted lines show the distance relation: a point and its image are at the same height and at equal distances from the mirror line.

The most common wrong option moves every part to the other side but does not flip it. At first glance it looks like a mirror image, but it is only a slide.

[[fig:1]]

In the picture, the middle triangle is the mirror image and the right one is only slid across. Check the right angle: it must change sides along with the triangle.

Left and right swap, up and down stay, and every lopsided shape must come out reversed.

Sub-case: parts inside a frame. Many questions put a symmetric frame (a circle, square or triangle) round a few small things: an arrow, a dot, a triangle, a letter. The frame does not change, and the small things decide the answer.

For each small thing, ask two questions, in this order.

1. Where does it go? Opposite side, same height for a vertical mirror; opposite height, same side for a flat mirror.
2. Does it turn? An arrow reverses its direction (a right-pointing arrow becomes left-pointing in a vertical mirror; an up arrow becomes a down arrow in water). A letter or a right-angled triangle is reversed. A dot, a circle or a square only moves.

[[fig:2]]

In the picture, the orange square and the arrow both change sides. The arrow also turns, so it now points left.

There is one more safeguard. The number of parts never changes in a reflection. An option with an extra part, or with one missing, can be struck out without any further checking.

A horizontal mirror, the water surface, swaps only top and bottom. Left and right stay where they are. Everything in the first type works again, but the roles of the two directions are exchanged.

Take any part of the figure, keep it at the same left-right position, and bring it to the other side of the mirror line at the same distance. A triangle that points up now points down. A part at the top corner goes to the bottom corner. An arrow that points up points down in the image.

[[fig:0]]

Think of the mirror as a lake under a hill. The hill stands on the shore, and its reflection hangs below the shoreline. The top of the hill is the part farthest from the water, so in the image it is the part farthest below the line.

[[fig:1]]

The orange lines in the second picture show the nearest-stays-nearest pattern. The small red square, close to the mirror, is also the closest part in the image. The blue dot, far from the mirror, is also the farthest part in the image.

A water image is not the same as a 180° rotation. Rotation also swaps left and right, so a part on the left would jump to the right. In a water image it never does.

Water image: top and bottom swap, left and right stay. A part on the left stays on the left.

Sub-case: parts inside a frame. The same two questions work for a flat mirror. Where does each small part go (the opposite height, on the same side)? And does it turn?

[[fig:2]]

The second picture shows a flat mirror. The triangle at the top-left goes to the bottom-left, with its right angle now at the bottom, and the arrow at the lower right goes to the upper right and points down.

In a water image the mirror lies flat, so the work on a word is different from the vertical mirror, and in one way simpler.

The order does not change. The first letter stays first and the last letter stays last. A flat mirror does not push letters from the right end to the left end.

Each letter is turned upside down. Letters with a horizontal line of symmetry (B C D E H I K O X) look exactly the same upside down. The rest come out visibly flipped.

[[fig:0]]

In TRAIN only I has a horizontal line of symmetry, so I looks normal and T, R, A and N are upside down.

A word made only from the water list does not change at all. DECK is such a word: D, E, C and K all have a horizontal line of symmetry, so its water image looks exactly like DECK.

[[fig:1]]

Digits follow the same idea. 0, 3 and 8 look the same in water. Most of the rest come out upside down (1 depends on the style of print).

Keep the two mirrors apart. With a vertical mirror the order reverses. With a horizontal mirror it never does. If an option shows the letters of the word in the reverse order, it is not a water image.

Water image of a word: same order, every letter turned upside down.

Sub-case: which words do not change in water?

Some questions have no picture. They ask which word, or how many words, look exactly the same in water. Learn the water list as a pattern: BCDE, HI, K, OX (9 letters). Water keeps the order, so a word is unchanged whenever every letter is in the list. No backwards-reading test is needed here. DECK and HIDE pass; TAXI fails, because T and A are not in the list.

[[fig:2]]

Some new-pattern questions draw the mirror at 45 degrees, along a diagonal. The idea is the same: each point goes to the other side of the mirror line, at the same distance, straight across (at right angles to the line). What changes is how the picture looks, because the mirror is tilted.

Think of a square box of side s (here 6 by 6) with the origin at its bottom-left corner, and a mirror along the diagonal from the bottom-left to the top-right (the line y = x). A point (x, y) goes to the point (y, x): the two coordinates swap. Then:

• A horizontal line becomes a vertical line, and a vertical line becomes horizontal.
• A point on the mirror line stays where it is.
• A point near the top-left corner goes to the bottom-right corner, and the reverse.

[[fig:0]]

In the picture the black parts are the original and the blue parts are the image. The blue dot is as far from the dashed line as the black dot, on the other side.

For the other diagonal, from the top-left to the bottom-right, the line is x + y = s and the formula is (x, y) goes to (s − y, s − x), that is (6 − y, 6 − x) here. These formulas are for a square only; for a non-square rectangle, reflect each point directly. The opposite corners swap: the bottom-left corner goes to the top-right and the reverse.

A diagonal mirror image is not a quarter turn. In the second picture, the middle figure is the diagonal mirror image and the right one is a 90° rotation. They are different pictures.

[[fig:1]]

Diagonal mirror: horizontal becomes vertical, and each part crosses the line to the opposite side. It is a flip, not a quarter turn.

Hold a clock in front of a vertical mirror. The mirror swaps left and right, so a hand that points to 3 now points to 9, and a hand that points to 2 now points to 10. In general a hand pointing to the mark x moves to the mark 12 − x. The numbers on the dial also look reversed, but we only read the hands.

[[fig:0]]

In the picture the left clock shows 8:20. The right clock is its mirror image: the minute hand has moved from the 4 to the 8, and the hour hand from a little past the 8 to well past the 3, close to the 4. Read in the ordinary way, the image shows 3:40.

There is a quick way to get there without drawing anything.

Mirror time = 11:60 − actual time.

Here 11:60 is just 12:00 written so that you can subtract the minutes. For 8:20: minutes 60 − 20 = 40; hours 11 − 8 = 3; the answer is 3:40.

This works because the minute hand turns to 60 − m, and the hour hand, which moves half a degree for every minute, turns to the matching hour position 11 − h, with 60 − m minutes.

If the clock shows 12 o'clock something, treat 12 as 0: the mirror of 12:30 is 11:60 − 0:30 = 11:30. And the mirror of 11:30 is 12:30. If the time is given in 24-hour form, convert to a 12-hour time first.

[[fig:1]]

The picture shows the special pair 12:30 and 11:30. Doing the mirror twice brings back the original time, so the method also works backwards: the real time is 11:60 minus the time seen in the mirror.

Water image of a clock. Lay the mirror flat below the clock. The horizontal line through 3 and 9 is now the fold line, and a hand at the mark x goes to the mark 6 − x: 12 goes to 6, 1 to 5, 2 to 4, while 3 and 9 stay.

[[fig:2]]

The flip of an exact time is not a true clock time, because the two hands no longer agree with each other. So the exam asks in one of two ways. If the options are dials, flip the hands and match the long hand first. If the options are times, a convention is used: water time = 6:30 − actual time (4:15 gives 2:15, and 8:20 gives 10:10). It matches the minute hand exactly and the hour hand only roughly, so never use it on a dial picture.

Dial with dots or without numbers. Here nothing is calculated; you only see what the mirror does to the two hands. The big dots are 12 at the top, 3 on the right, 6 at the bottom and 9 on the left. The long, thin hand is the minute hand and the short, thick hand is the hour hand.

[[fig:3]]

In a vertical mirror each hand moves to the other side of the 12-6 line, at the same height. In a flat mirror each hand moves to the other side of the 3-9 line and keeps its own side. A hand lying on the mirror line does not move. Vertical mirror: a hand changes side, not height. Flat mirror: a hand changes height, not side.