## Interactive Visualizations

Explore how the ⵟ-product differs from traditional similarity measures

### Similarity Measure Comparison

Drag the anchor point (⭐) to see how each measure responds. Notice how the ⵟ-product creates a localized potential well.

Anchor W: (3.0, 4.0) ε: 1.0
Dot Product
w · x
Euclidean Distance²
‖w − x‖²
ⵟ-Product
(w·x)² / (‖w−x‖² + ε)
Cosine Similarity
w·x / (‖w‖‖x‖)

### Gradient Vector Fields

The gradient fields reveal optimization dynamics. The ⵟ-product creates a vortex-like attractor around the weight vector.

Dot Product ∇
Euclidean ∇
ⵟ-Product ∇
Cosine ∇

### XOR Problem: Single Neuron Solution

The classic XOR problem cannot be solved by a single linear neuron. The ⵟ-product's intrinsic non-linearity enables a single unit to solve it.

#### Linear Neuron

Cannot separate XOR

#### ⵟ-Product Neuron

Solves XOR with w = [1, -1]

Input x | XOR Output | w·x | ⵟ(w,x)
--- | --- | --- | ---
(0,0) | 0 | 0 | 0
(0,1) | 1 | -1 | >0
(1,0) | 1 | 1 | >0
(1,1) | 0 | 0 | 0

### Decision Boundary Comparison

Linear neurons create hyperplane boundaries. ⵟ-product neurons create vortex-like territorial fields around learned prototypes.

#### Linear Model Boundaries

Unbounded half-space partitions

#### ⵟ-Product Boundaries

Localized vortex territories

### Loss Landscape Comparison

The optimization landscape for the XOR problem. Dot product has a spurious minimum; ⵟ-product creates exploitable valleys.

#### Dot Product + Sigmoid

#### ⵟ-Product