Which images look similar and which ones look different?
Answer
Images A, C and D look similar. Images B and E look different — even though every one of the five shows the same tiger.
A → 60 mm × 40 mm → 3 : 2 C → 30 mm × 20 mm → 3 : 2 D → 90 mm × 60 mm → 3 : 2 B → 40 mm × 20 mm → 2 : 1 E → 60 mm × 60 mm → 1 : 1
Widths and heights in mm, drawn to scale. The blue frames all reduce to 3 : 2; the red ones do not.
Why it happens: Looking similar has nothing to do with being the same size. A, C and D are of three different sizes — C is half of A in each direction, D is one and a half times A — yet all three share one number, the width-to-height ratio 3 : 2. That single number is what decides the shape of the frame, and therefore whether the tiger inside is drawn faithfully or squeezed.
Q2.
Do images B and E look like the other three images?
Answer
No. They are distorted. In B the tiger appears stretched and elongated; in E it appears compressed and fatter.
B → 40 : 20 = 2 : 1 (wider than 3 : 2 for its height) E → 60 : 60 = 1 : 1 (not wide enough for its height)
A frame that is 2 : 1 is relatively wider than a 3 : 2 frame, so the same tiger has to be pulled sideways to fill it. A 1 : 1 frame is relatively taller, so the tiger has to be squashed sideways — it looks short and fat.
Q3.
Why?
Answer
Because in B and E the width and the height were not changed by the same factor.
A → B : width 60 → 40, factor 40⁄60 = 2⁄3 ; height 40 → 20, factor 20⁄40 = 1⁄2 A → E : width 60 → 60, factor 1 ; height 40 → 60, factor 60⁄40 = 3⁄2
In each case the two factors are different, so one direction was stretched more than the other.
Why the obvious explanation is not enough: It is tempting to say “E looks odd because it is a square, and the others are rectangles.” But B is a rectangle too, and it still looks wrong. So being rectangular is not the test. The test is whether the two factors of change agree.
Q4.
What makes images A, C, and D appear similar, and B and E different?
Answer
A, C and D were obtained from one another by multiplying both the width and the height by the same factor. B and E were not.
From A to
Width factor
Height factor
Same?
C (30 × 20)
30⁄60 = 1⁄2
20⁄40 = 1⁄2
Yes — similar
D (90 × 60)
90⁄60 = 3⁄2
60⁄40 = 3⁄2
Yes — similar
B (40 × 20)
40⁄60 = 2⁄3
20⁄40 = 1⁄2
No — distorted
E (60 × 60)
60⁄60 = 1
60⁄40 = 3⁄2
No — distorted
Why it happens: Look at how B was actually made: 20 mm was taken off the width and 20 mm off the height. The difference is the same, yet the picture distorts. The reason is that 20 mm is one-third of a 60 mm width but half of a 40 mm height — the same subtraction eats a different share of each side. Shape is about shares, so it is preserved by multiplying, and destroyed by adding or subtracting. Changes to width and height that use the same factor are called proportional.
Q5.
Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?
Answer
Both factors are 3⁄2, so yes, they are the same.
Width: 60 × f = 90 ⇒ f = 90⁄60 = 3⁄2 Height: 40 × f = 60 ⇒ f = 60⁄40 = 3⁄2
D is A enlarged one and a half times in every direction. That is why D, though the biggest of the five, still looks exactly like A.
Check it yourself: Go the other way — from D to A the factor is 2⁄3 in both directions (90 × 2⁄3 = 60 and 60 × 2⁄3 = 40). Reversing a proportional change gives another proportional change.