Consider the family of curves

|x|ᵖ + |y|ᵖ = 1

The value of p controls the shape of the curve.

When p = 2, the equation becomes

x² + y² = 1

which is a circle.

As p increases beyond 2, the curve becomes increasingly square-like. The reason can be understood by examining what happens to numbers between 0 and 1 when they are raised to higher powers.

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1. Start with the Circle: p = 2

When p = 2:

x² + y² = 1

Consider a point where:

y = 0.5

Substituting this value gives:

x² + 0.5² = 1

Since:

0.5² = 0.25

we get:

x² = 0.75

Therefore:

x ≈ 0.866

So the circle passes through approximately:

(0.866, 0.5)

The curve is therefore noticeably rounded between the top and right sides.

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2. Increase the Power to 4

Now consider:

x⁴ + y⁴ = 1

Again take:

y = 0.5

Then:

x⁴ + 0.5⁴ = 1

Since:

0.5⁴ = 0.0625

we get:

x⁴ = 0.9375

Therefore:

x ≈ 0.984

The corresponding point is approximately:

(0.984, 0.5)

Compare the two results:

p = 2 → x ≈ 0.866

p = 4 → x ≈ 0.984

The point has moved much closer to (1, 0.5).

Consequently, the curve becomes flatter.

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3. Increase the Power to 10

Now consider:

x¹⁰ + y¹⁰ = 1

Again:

y = 0.5

Since:

0.5¹⁰ = 0.0009765625

we get:

x¹⁰ = 0.9990234375

Taking the tenth root gives:

x ≈ 0.9999

So the corresponding point is approximately:

(0.9999, 0.5)

It is now extremely close to:

(1, 0.5)

The curve is therefore becoming almost vertical at this location.

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4. Comparing the Three Curves

At the same height, y = 0.5:

Power	x
2	0.866
4	0.984
10	0.9999
Very large	Almost 1

The point therefore moves:

(0.866, 0.5)
→ (0.984, 0.5)
→ (0.9999, 0.5)
→ (1, 0.5)

This movement toward x = 1 is what produces a flatter side.

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5. Why Does This Happen?

The key is the behavior of numbers between 0 and 1.

Take the number 0.5.

As the power increases:

0.5² = 0.25

0.5⁴ = 0.0625

0.5¹⁰ ≈ 0.001

The value rapidly becomes smaller.

Now return to:

|x|ᵖ + |y|ᵖ = 1

Suppose:

y = 0.5

For p = 2:

y² = 0.25

So y makes a substantial contribution to the total of 1.

But for p = 10:

y¹⁰ ≈ 0.001

Now y contributes almost nothing.

Therefore, x¹⁰ must provide almost the entire 1:

x¹⁰ ≈ 0.999

which forces:

x ≈ 1

This is the fundamental reason the curve becomes square-like.

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6. What Happens Inside the Square?

Consider the region:

−1 ≤ x ≤ 1

−1 ≤ y ≤ 1

Every coordinate inside this region has an absolute value less than or equal to 1.

For values strictly between 0 and 1, higher powers make them rapidly smaller.

Number	Power 2	Power 4	Power 10
0.2	0.04	0.0016	0.0000001024
0.5	0.25	0.0625	0.0009766
0.8	0.64	0.4096	0.1074
0.9	0.81	0.6561	0.3487
1.0	1	1	1

There is a crucial difference between numbers below 1 and the number 1.

As the power becomes large:

numbers less than 1 → approach 0

while:

1 → remains 1

Therefore, in

|x|ᵖ + |y|ᵖ = 1

the smaller coordinate increasingly contributes very little.

The larger coordinate has to do most of the work.

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7. How Does This Produce the Four Sides of a Square?

Consider a point such as:

(0.7, 0.4)

Both coordinates are less than 1.

For a very large p:

0.7ᵖ → 0

and:

0.4ᵖ → 0

Therefore:

0.7ᵖ + 0.4ᵖ → 0

rather than 1.

Such a point cannot remain on the boundary as p becomes very large.

Now consider:

(1, 0.4)

Here:

1ᵖ = 1

regardless of how large p becomes.

Meanwhile:

0.4ᵖ → 0

Therefore:

1ᵖ + 0.4ᵖ → 1

So points along the line

x = 1

remain on the limiting boundary.

The same reasoning applies to:

x = −1

y = 1

y = −1

These four lines form a square.

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8. Why Does the Smaller Coordinate Become Less Important?

The equation

|x|ᵖ + |y|ᵖ = 1

requires the two powered coordinates to add up to exactly 1.

When p = 2, both coordinates can contribute significantly.

For example:

0.866² + 0.5² = 1

Both coordinates matter.

But for a large power:

0.9999¹⁰ + 0.5¹⁰ ≈ 1

The contribution from 0.5 is almost negligible.

The coordinate close to 1 provides almost the entire total.

Therefore, the boundary gets pushed toward:

x = 1

or:

y = 1

depending on which coordinate is larger.

This creates the flat sides characteristic of a square.

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9. The Limiting Formula

As p approaches infinity, the equation

|x|ᵖ + |y|ᵖ = 1

approaches:

max(|x|, |y|) = 1

This means:

The larger of |x| and |y| must equal 1.

Therefore either:

|x| = 1

or:

|y| = 1

These are precisely the four sides of the square.

The progression is therefore:

p = 2 → circle

p = 4 → rounded square

p = 10 → very square-like

p → ∞ → exact square

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Conclusion

The transformation from a circle to a square can be explained by one simple mathematical fact:

For numbers between 0 and 1, raising them to increasingly high powers makes them rapidly approach zero.

In the equation

|x|ᵖ + |y|ᵖ = 1

this means that, as p increases, the smaller coordinate contributes less and less.

The larger coordinate must therefore become increasingly close to 1.

As a result, the curve is pushed toward the four lines:

x = 1, x = −1, y = 1, y = −1

which form a square.

Thus:

Circle → rounded square → nearly square → exact square in the limit

The remarkable result comes from nothing more complicated than the behavior of powers between 0 and 1.