Observing Similarity in Change

The Digital Image Problem

Imagine you have a picture of a tiger.

• Image A (Original): Width 60 mm, Height 40 mm.
• Image C (Resized): Width 30 mm, Height 20 mm.

Why does Image C look like a smaller version of A, while other resizing attempts look distorted?

The Mathematical Reason

In Image C, both the width and the height were reduced by the same factor.

• Width: $60 \rightarrow 30$ (Halved or multiplied by $\frac{1}{2}$)
• Height: $40 \rightarrow 20$ (Halved or multiplied by $\frac{1}{2}$)

Why Subtraction Doesn’t Work

Consider Image B:

• Width: $60 - 20 = 40$ mm
• Height: $40 - 20 = 20$ mm

Even though we subtracted the same amount (20 mm) from both sides, the image looks distorted.

• Height factor: $20/40 = 0.5$
• Width factor: $40/60 \approx 0.67$

Since the factors ($0.5$ and $0.67$) are different, the proportions are lost.

Proportional Change

We say changes are proportional when quantities are multiplied or divided by the same constant. This preserves the “shape” of the object.