Step 1: The Square and the Inscribed Circle
First, imagine a square with side length
2
�
2r, and inside it, you inscribe a circle. The circle touches the midpoints of the square’s sides. Now, this is the largest possible circle that fits perfectly inside the square.
Step 2: Cutting the Corners (Infinite Cuts)
If you start cutting the corners of the square to form a polygon (octagon, decagon, etc.), you’ll get closer and closer to a circle. But no matter how many cuts you make, the shape is still fundamentally bound by the square. So, the perimeter remains tied to the square's boundary.
Step 3: Perimeter to Radius Ratio
As you keep cutting, you might get closer to the circle's smoothness, but the perimeter of the shape you’re approximating will always be a function of the square. For any polygon, the ratio of the perimeter to the radius will approach 4. This relationship doesn’t change with more cuts.
Step 4: Digital Circle
Now, let’s take this idea to your digital screen. The pixels on your screen form a grid, and when you try to draw a circle, it’s just an approximation made of discrete pixel boundaries. The more pixels, the finer the approximation — but it’s still discrete.
This is where it gets crazy: No matter how high the resolution, the ratio of the perimeter of the pixelated circle to its radius (divided by two) will always stabilize at 4. This is because the fundamental shape remains tied to the grid, and it cannot achieve the smoothness of a true circle.
Step 1: The Square and the Inscribed Circle First, imagine a square with side length 2 � 2r, and inside it, you inscribe a circle. The circle touches the midpoints of the square’s sides. Now, this is the largest possible circle that fits perfectly inside the square.
Step 2: Cutting the Corners (Infinite Cuts) If you start cutting the corners of the square to form a polygon (octagon, decagon, etc.), you’ll get closer and closer to a circle. But no matter how many cuts you make, the shape is still fundamentally bound by the square. So, the perimeter remains tied to the square's boundary.
Step 3: Perimeter to Radius Ratio As you keep cutting, you might get closer to the circle's smoothness, but the perimeter of the shape you’re approximating will always be a function of the square. For any polygon, the ratio of the perimeter to the radius will approach 4. This relationship doesn’t change with more cuts.
Step 4: Digital Circle Now, let’s take this idea to your digital screen. The pixels on your screen form a grid, and when you try to draw a circle, it’s just an approximation made of discrete pixel boundaries. The more pixels, the finer the approximation — but it’s still discrete.
This is where it gets crazy: No matter how high the resolution, the ratio of the perimeter of the pixelated circle to its radius (divided by two) will always stabilize at 4. This is because the fundamental shape remains tied to the grid, and it cannot achieve the smoothness of a true circle.