02BEYOND THE FLAT PAGE

THE CUT
GOES DEEPER.

Every extra feature adds a new direction. The boundary grows with it—but its job never changes: separate one region from another.

w₁x₁ + w₂x₂ + ··· + wₙxₙ + b = 0
INTERACTIVE / DIMENSION MACHINE

One rule.
Any number of dimensions.

2DA LINE CUTS A PLANE
DECISION LINE
x₁ →x₂ ↑x₃ ↗
03 / SEEING THE INVISIBLE

The shape changes.
The operation does not.

2 FEATURES

Line

With height and weight, each person is a dot on a flat chart. One straight line becomes the classifier’s boundary.

3 FEATURES

Plane

Add age and every person needs three coordinates. The separating line expands into a flat sheet through the cloud.

4+ FEATURES

Hyperplane

Add income, location, history, or pixels. We cannot draw the space, but the same weighted equation still divides it in two.

Think of it as
a smart knife.

The data is the loaf.

Every feature is a new direction inside it.

The weights choose the angle.

Large weights make a feature influence the cut more.

The bias chooses the position.

It slides the cut without changing its orientation.

The sign chooses the class.

Positive score on one side, negative score on the other.

04 / WHERE ONE CUT FAILS

Some patterns need more than one cut.

A single perceptron can only make a straight boundary. If classes curve around each other—or alternate like XOR—no single line can separate them.

This is why neural networks stack perceptrons: each neuron makes a simple cut; later layers combine those cuts into corners, curves, islands, and intricate regions.
XOR / TWO CUTS CREATE THE NEEDED REGIONS