An interactive tool that applies a Discrete Fourier Transform (DFT) to a spectrum of personality orders indexed from −n to +n, and maps the result through the Hopf fibration η: S³ → S² for topological visualization.
Each personality dimension is assigned a Likert score on a five-point scale:
Scores 1–5 are mapped to personality orders k ∈ {−n, …, 0, …, +n}, forming a discrete signal x[k].
The Hopf S³ mode uses the following five schizoaffective symptom categories as named personality orders:
| Order k | Dimension | Color |
|---|---|---|
| −3 | Negative Symptoms | #ce93d8 |
| −2 | Hallucinations | #4dd0e1 |
| −1 | Delusions | #009688 |
| 0 | Disorganized Thinking | #ffb300 |
| +1 | Depressive Episodes | #90a4ae |
| +2 | Manic Episodes | #ab47bc |
| +3 | Cognitive Impairment | #4fc3f7 |
X[m] = Σ_{k=0}^{N-1} x[k] · e^{−i 2π mk / N}
Each coefficient X[m] captures the strength and phase of the m-th oscillation pattern across the personality spectrum.
Each personality order score x[k] maps to a unit complex number on S¹:
z_k = e^{i · 2π(x[k]−1)/4} ∈ S¹ ⊂ ℂ
Adjacent pairs (z_k, z_{k+1}) are normalized to a point on S³ ⊂ ℂ²:
q_k = (z_k, z_{k+1}) / ‖(z_k, z_{k+1})‖ ∈ S³
The Hopf map η projects each S³ point to S²:
η(q_k) = (2Re(z̄_k z_{k+1}), 2Im(z̄_k z_{k+1}), |z_k|² − |z_{k+1}|²) ∈ S²
Each personality order defines a circle S¹_k. The wedge sum ⋁_k S¹_k identifies all base points at the neutral score:
X ∨ Y = (X ⊔ Y) / ∼ (x₀ ∼ y₀)
H₁(⋁_k S¹_k) ≅ ℤᴺ
The S² base space is computed via the homotopy lifting property:
H(z_k, t) = e^{i·t·α_k} t ∈ [0, 1]
Path γ_k(t) = η(H(z_k, t), H(z_{k+1}, t)) on S²
At t=0 all base points are at the south pole. At t=1 they reach their final positions determined by the personality scores.
| Panel | What it shows |
|---|---|
| Hopf Fiber Torus | Villarceau circles on the torus — each fiber is a great circle on S³ stereographically projected to ℝ³. Drag to rotate. |
| S² Base Space | Homotopy lifting paths γ_k(t) on S². Use the t-scrubber to animate. |
| Wedge Sum Bouquet | Petal radius = raw x[k] score. Animated breathing pulse. |
| Symptom Radar | Raw x[k] scores (solid) vs. DFT magnitude |
The Hopf fiber torus canvas is modelled after this rainbow spiral torus, where each hue band corresponds to a personality dimension:
| Mode | Input type |
|---|---|
| Discrete | Integer Likert sliders (1–5) per order, with scale-type selector |
| Continuous | Drawable waveform canvas per order over 64 time samples with calendar date range |
| Survey | JSON questionnaire loader with preset OCEAN / Dark Triad / MBTI questionnaires |
| Real ℝ | Continuous real-valued slider per order (configurable range ±1 to ±100) |
| Hopf S³ | Schizoaffective symptom dimensions mapped to Hopf fibration geometry |
React 19 · TypeScript · Tailwind CSS 4 · Recharts · Framer Motion · Canvas 2D API