ADVANCED TENSOR ENGINE — PhD EDITION
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ADVANCED TENSOR ENGINE — PhD EDITION
Wickerson Studios · 2026 · Powered by Claude AI
www.wickersonstudios.com
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"Every field equation, every curvature tensor,
every quantum state — rendered as geometry."
For PhD students in Advanced Mathematics,
Theoretical Physics, and Computational Science.
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WHAT IS THIS?
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Advanced Tensor Engine is a Grasshopper C# Script component
for Rhino 7/8 that renders the mathematical foundations of
general relativity, quantum mechanics, tensor decompositions,
and advanced machine learning as interactive 3D geometry
directly in the Rhino viewport.
32 modes. 2,368 lines. 133KB. Zero external dependencies.
This is not an introduction to tensors. It assumes
familiarity with linear algebra and differential geometry.
For the introductory version (scalars → vectors → matrices
→ ML shapes), see WickersonStudios_TensorVisualiser.cs.
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WHAT'S INCLUDED
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WickersonStudios_AdvancedTensorEngine.cs
The Grasshopper C# Script component.
Drop into a C# Script node. Wire 14 inputs.
No NuGet packages. No DLLs. No installation.
WickersonStudios_AdvancedTensorEngine_Guide.html
Standalone interactive web companion.
32 animated canvas visualisations.
Live PyTorch / NumPy code for every mode.
Works fully offline after initial font load.
WickersonStudios_AdvancedTensorEngine_README.txt
This file.
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THE 32 MODES
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── GROUP A: TENSOR ALGEBRA (Modes 0–9) ─────────────────
Mode 0 — Einstein Summation Convention
─────────────────────────────────────────────────────────
Cᵢⱼ = Aᵢₖ Bₖⱼ (k is the dummy index — summed, vanishes)
Visualises the anatomy of tensor index notation:
free indices (label result axes), dummy indices (summed
over, disappear), upper contravariant and lower covariant.
Animated highlight sweeps through the contraction showing
which row of A and column of B combine for each output.
Recommended: Rows 4 Cols 4 AnimSpeed 0.025
Mode 1 — Covariant vs Contravariant Components
─────────────────────────────────────────────────────────
vᵢ = gᵢⱼ vʲ (metric lowers the index)
Shows the same geometric vector expressed with upper
(contravariant) and lower (covariant) indices. The metric
tensor gᵢⱼ converts between them. Param selects the metric:
0 = Euclidean δᵢⱼ 3 = Minkowski (−,+,+,+)
1 = Stretched coords 4 = 2-sphere S²
2 = Sheared coords 5 = Schwarzschild (BH)
Recommended: Rows 4 Param 0→3 AnimSpeed 0.025
Mode 2 — Metric Tensor gᵢⱼ
─────────────────────────────────────────────────────────
ds² = gᵢⱼ dxⁱ dxʲ
Renders the metric tensor as a matrix heatmap alongside
its inverse gⁱʲ and eigenvalue spectrum. The eigenvalue
signature directly reveals the geometry:
(+,+,+,+) Riemannian — positive definite
(−,+,+,+) Lorentzian — special and general relativity
Param cycles through six canonical metrics. Animated:
the dynamic metric (Param 4) updates each frame.
Recommended: Rows 4 Param 0–5 ShowEigen TRUE
Mode 3 — Tensor Contraction & the Trace
─────────────────────────────────────────────────────────
Aⁱᵢ = Tr(A) (full contraction → scalar)
Demonstrates how contraction reduces tensor rank by 2
(one upper + one lower index). Cycles through three cases:
full trace (all elements → scalar), partial contraction
(collapse one axis → vector), and Kronecker delta
contraction. Diagonal cells are highlighted to show which
elements contribute to the trace.
Recommended: Rows 5 Cols 5 AnimSpeed 0.02
Mode 4 — Symmetrisation & Antisymmetrisation
─────────────────────────────────────────────────────────
Tᵢⱼ = T₍ᵢⱼ₎ + T[ᵢⱼ] (unique decomposition)
Displays all three tensors simultaneously:
T (original — left panel)
T₍ᵢⱼ₎ (symmetric part — middle panel)
T[ᵢⱼ] (antisymmetric part — right panel)
The antisymmetric panel's diagonal is always zero, shown
with a pulsing highlight. For N=4: symmetric has 10
independent components, antisymmetric has 6.
Physical examples: gᵢⱼ and Rᵢⱼ are symmetric.
Fᵢⱼ (EM) and Lᵢⱼ are antisymmetric.
Recommended: Rows 4 Cols 4
Mode 5 — Levi-Civita Tensor εᵢⱼₖ
─────────────────────────────────────────────────────────
(u×v)ⁱ = εⁱʲₖ uⱼ vₖ det(M) = εᵢⱼₖ M¹ᵢ M²ⱼ M³ₖ
Renders all three i-slices of the 3D Levi-Civita tensor
simultaneously. Cells are colour-coded by value:
Tall (cyan) = +1 even permutation
Tall (rose) = −1 odd permutation
Wire outline = 0 repeated index
Animated highlight cycles through all 6 non-zero
permutations. The contraction identity
εᵢⱼₖ εⁱₗₘ = δⱼₗδₖₘ − δⱼₘδₖₗ is printed in Report.
Mode 6 — Kronecker Delta δⁱⱼ
─────────────────────────────────────────────────────────
δⁱⱼ vʲ = vⁱ (identity action) δⁱᵢ = n (trace = dim)
Draws the identity matrix as a tensor, annotated with
the index-replacement property and QM completeness
relation Σᵢ |i⟩⟨i| = 1̂. An input vector and the result
of δⁱⱼ acting on it are shown side by side — they are
always identical (demonstrating the identity property).
Recommended: Rows 5 Cols 5
Mode 7 — Exterior Algebra & Wedge Product ∧
─────────────────────────────────────────────────────────
(u ∧ v)ᵢⱼ = uᵢvⱼ − uⱼvᵢ |u∧v| = area of parallelogram
Displays the antisymmetric matrix u∧v with a geometric
parallelogram showing the area interpretation. The zero
diagonal (u∧u = 0) is highlighted. Maxwell's equations
in differential form notation dF=0, d⋆F=J are derived
in the Report output. The connection to the cross product
in 3D is shown explicitly.
Mode 8 — Hodge Star Operator ⋆
─────────────────────────────────────────────────────────
⋆Fᵤᵥ = ½ εᵤᵥρσ Fρσ (maps 2-form → 2-form in 4D)
Renders the full EM field tensor Fᵤᵥ and its Hodge dual
⋆Fᵤᵥ side by side. The dual reveals electromagnetic
duality: applying ⋆ swaps E↔B. An animated row sweep
shows which components of F contribute to each row of ⋆F.
Both Lorentz invariants E²−B² and E·B are computed and
displayed in the Report.
EnableDual TRUE recommended for full effect.
Mode 9 — Lie Bracket [X,Y]
─────────────────────────────────────────────────────────
[X,Y] = XY − YX [∇ᵤ,∇ᵥ]ψ = Rᵤᵥψ (curvature!)
Displays three matrices — A, B, and [A,B] — derived from
skew-symmetric generators. The zero diagonal of the
commutator is pulse-highlighted. The critical connection
to the Riemann curvature tensor is printed in Report:
curvature IS the Lie bracket of covariant derivatives.
su(2), su(3), and the canonical commutation relation
[x̂,p̂]=iℏ are discussed.
── GROUP B: TENSOR DECOMPOSITIONS (Modes 10–14) ────────
Mode 10 — Singular Value Decomposition A = UΣVᵀ
─────────────────────────────────────────────────────────
σ₁ ≥ σ₂ ≥ … ≥ 0 Rank-k approx: Aₖ = Σᵢ₌₁ᵏ σᵢ uᵢvᵢᵀ
Renders all three matrices simultaneously: U (left
singular vectors, colour coded by active column), Σ
(diagonal bars proportional to σᵢ), and Vᵀ (right
singular vectors). An animated highlight sweeps through
singular vectors showing which components contribute.
LoRA (Low-Rank Adaptation) is explicitly connected:
ΔW = BA is precisely a rank-r SVD approximation.
Recommended: Rows 5 Cols 4 ShowEigen TRUE
Mode 11 — CP / PARAFAC Decomposition
─────────────────────────────────────────────────────────
𝒯 = Σᵣ λᵣ aᵣ ⊗ bᵣ ⊗ cᵣ (sum of rank-1 outer products)
Shows D slices of the original 3-tensor (fading in depth)
alongside the rank-1 component terms that sum to
reconstruct it. Fitting via Alternating Least Squares
is described. The NP-hardness of computing tensor rank
(unlike matrix rank) is highlighted.
Recommended: Rows 4 Cols 4 Depth 3
Mode 12 — Tucker Decomposition 𝒯 = 𝒢 ×₁A ×₂B ×₃C
─────────────────────────────────────────────────────────
𝒯ᵢⱼₖ = Σₚₒᵣ 𝒢ₚₒᵣ Aᵢₚ Bⱼₒ Cₖᵣ
Renders the core tensor 𝒢 and the three factor matrices
A, B, C in separate output channels. The compression
ratio is computed and printed. HOSVD (Higher-Order SVD)
and HOOI (iterative optimum) algorithms are described.
The containment hierarchy Tucker ⊃ CP ⊃ SVD is noted.
Recommended: Rows 4 Cols 4 Depth 3
Mode 13 — Tensor Train / Matrix Product State (MPS)
─────────────────────────────────────────────────────────
𝒯ᵢ₁…ᵢₙ = G⁽¹⁾ᵢ₁ G⁽²⁾ᵢ₂ … G⁽ᴺ⁾ᵢₙ
Renders L rank-3 cores chained horizontally with bond
dimension lines connecting them. The physical indices
drop downward from each core. Exponential compression
is computed explicitly. The quantum physics connection
(DMRG, area law of entanglement entropy) is explained:
MPS works for 1D gapped quantum systems precisely because
entanglement entropy scales as area, not volume.
Recommended: Depth 5 (chain length) AnimSpeed 0.02
Mode 14 — Tensor Network Graphical Notation
─────────────────────────────────────────────────────────
Nodes = tensors Edges = contractions Legs = free
Renders standard Penrose notation elements (scalar, vector,
matrix, 3-tensor) and a MERA-like network. The connection
between Transformers (as tensor networks) and the
holographic principle (AdS/CFT via MERA) is discussed.
The contraction ordering problem (NP-hard) is noted.
── GROUP C: GENERAL RELATIVITY (Modes 15–21) ───────────
Mode 15 — Minkowski Metric ηᵤᵥ (Special Relativity)
─────────────────────────────────────────────────────────
ds² = ηᵤᵥ dxᵤ dxᵥ = −c²dt² + dx² + dy² + dz²
Renders η = diag(−1,+1,+1,+1) as a heatmap with the
4-velocity uᵤ and its lowered form ηᵤᵥuᵥ shown as bar
charts. Param controls the boost velocity β (0→5 maps
to β ≈ 0 → 0.9), and γ is computed and displayed.
The norm uᵤuᵤ = −c² is verified in Report.
Recommended: Param 2 (β=0.3, moderate boost)
Mode 16 — Lorentz Transformation Tensor Λᵤᵥ
─────────────────────────────────────────────────────────
x̄ᵤ = Λᵤᵥ xᵥ det(Λ) = 1 Λᵀη Λ = η
Renders the 4×4 boost matrix with γ and γβ entries as
a dynamic heatmap (Param controls β). A rest-frame
4-vector (1,0,0,0) is shown being transformed to the
boosted frame. The SO(1,3) group structure and its Lie
algebra so(1,3) ≅ sl(2,ℂ) are discussed in Report.
Recommended: Param 3 (β=0.6, highly relativistic)
Mode 17 — Electromagnetic Field Tensor Fᵤᵥ
─────────────────────────────────────────────────────────
∂ᵥ Fᵤᵥ = μ₀Jᵤ ∂[ᵤFᵥρ] = 0
Renders Fᵤᵥ and ⋆Fᵤᵥ (its Hodge dual) simultaneously
with an animated row sweep. Param scales the field
strength. The two Lorentz invariants E²−B² and E·B
are computed numerically from the current field
configuration and displayed. Maxwell's four equations
as two compact tensor equations are given.
Recommended: ShowDual TRUE
Mode 18 — Riemann Curvature Tensor Rᵤᵥρσ
─────────────────────────────────────────────────────────
Rᵤᵥρσ = ∂ρΓᵤᵥσ − ∂σΓᵤᵥρ + ΓΓ − ΓΓ
Renders four (μ,ν) slices of the rank-4 Riemann tensor
as separate heatmap panels, with contraction to the Ricci
tensor shown as a bar chart. The five symmetries (20
independent components in 4D) are listed. The Bianchi
identities and the geodesic deviation equation are given.
Recommended: Rows 4 (for 4D GR)
Mode 19 — Ricci Tensor Rᵤᵥ & Ricci Scalar R
─────────────────────────────────────────────────────────
Rᵤᵥ = Rρᵤρᵥ R = gᵤᵥ Rᵤᵥ
Displays the Ricci tensor for a 2-sphere S² with the
scalar curvature R animated as Param changes the radius.
The Weyl tensor (Riemann − Ricci, the traceless part)
is discussed as the carrier of gravitational wave content.
Vacuum Rᵤᵥ = 0 is contrasted with Rᵤᵥρσ ≠ 0 (spacetime
is still curved in vacuum — the key GR subtlety).
Recommended: Param 0–5 to see curvature vs radius
Mode 20 — Einstein Tensor Gᵤᵥ & Field Equations
─────────────────────────────────────────────────────────
Gᵤᵥ = Rᵤᵥ − ½gᵤᵥR = (8πG/c⁴) Tᵤᵥ
Renders Gᵤᵥ for a flat FRW cosmological model. Param
controls the scale factor a(t), showing how the Einstein
tensor changes as the universe expands. The contracted
Bianchi identity ∇ᵤGᵤᵥ = 0 → ∇ᵤTᵤᵥ = 0 (energy-momentum
conservation emerges from geometry) is highlighted. Known
exact solutions (Schwarzschild, Kerr, FRW, de Sitter)
are listed.
Recommended: Param 0–4 to animate FRW expansion
Mode 21 — Stress-Energy Tensor Tᵤᵥ
─────────────────────────────────────────────────────────
Tᵤᵥ = (ρ+p) uᵤuᵥ + pgᵤᵥ (perfect fluid)
Displays the perfect fluid stress-energy tensor alongside
one with shear (off-diagonal Param component). Param
controls the equation of state w = p/ρ cycling through
dust (w=0), radiation (w=1/3), and vacuum (w=−1).
The physical interpretation of each component (T₀₀ =
energy density, Tᵢⱼ = stress/pressure) is annotated.
── GROUP D: QUANTUM MECHANICS (Modes 22–26) ────────────
Mode 22 — State Vector |ψ⟩
─────────────────────────────────────────────────────────
|ψ⟩ = Σᵢ cᵢ |i⟩ P(i) = |cᵢ|² ⟨ψ|ψ⟩ = 1
Renders three simultaneous bar charts per basis state:
Row 1: Re(cᵢ) — animated with e^{iφt} phase rotation
Row 2: Im(cᵢ) — imaginary part (90° out of phase)
Row 3: |cᵢ|² — measurement probability (static)
The animated phase rotation shows quantum superposition
without any probability change — demonstrating that
global phase is unobservable (gauge freedom).
Recommended: Rows 6 AnimSpeed 0.03
Mode 23 — Density Matrix ρ̂
─────────────────────────────────────────────────────────
ρ = Σₙ pₙ |ψₙ⟩⟨ψₙ| Tr(ρ) = 1 ρ† = ρ ρ ≥ 0
The density matrix generalises state vectors to mixed
states. Param controls the mixing parameter:
0 = pure state (Tr(ρ²) = 1)
5 = maximally mixed (Tr(ρ²) = 1/N)
The purity Tr(ρ²) is computed and displayed. Von Neumann
entropy S = −Tr(ρ log ρ) is given in Report. ShowEigen
displays the eigenvalue spectrum of ρ.
Recommended: Rows 4 Param 0→5 ShowEigen TRUE
Mode 24 — Tensor Product of Hilbert Spaces H₁⊗H₂
─────────────────────────────────────────────────────────
cᵢⱼ = aᵢ · bⱼ (product state — Schmidt rank 1)
Displays a two-qudit product state as a coefficient
matrix cᵢⱼ with the individual subsystem vectors aᵢ and
bⱼ shown alongside. Schmidt rank = 1 (the matrix is rank
1, equals aᵢbⱼ). An entangled Bell state comparison is
given in Report (Schmidt rank 2, cannot be factored).
Exponential Hilbert space growth dim(H^⊗N) = 2ᴺ is noted.
Recommended: Rows 4 Cols 4
Mode 25 — Pauli Matrices σₓ σᵧ σᵤ (su(2) algebra)
─────────────────────────────────────────────────────────
[σᵢ, σⱼ] = 2iεᵢⱼₖ σₖ σᵢσⱼ = δᵢⱼI + iεᵢⱼₖσₖ
Renders all three Pauli matrices (real and imaginary parts
separated) plus an animated Bloch sphere showing a qubit
state precessing. The Clifford algebra relation connects
Pauli matrices to both the angular momentum algebra and
the rotation group SU(2). The rotation operator
R(θ,n̂) = exp(−iθ n̂·σ⃗/2) is derived in Report.
Mode 26 — Entanglement & Schmidt Decomposition
─────────────────────────────────────────────────────────
|ψ⟩ = Σₖ λₖ |uₖ⟩⊗|vₖ⟩ S = −Σₖ λₖ² log₂(λₖ²)
Param controls the entanglement angle α from 0 (product
state, S=0) through π/4 (Bell state, S=1 ebit maximum).
Schmidt coefficients λ₁, λ₂ are shown as bar charts with
the entanglement entropy S computed in real time. The area
law of entanglement entropy (S ∝ perimeter, not volume)
for 1D gapped Hamiltonians is discussed — this is why
Matrix Product States / Tensor Trains work.
Recommended: Param 0→5 AnimSpeed 0.03
── GROUP E: ADVANCED MACHINE LEARNING (Modes 27–31) ────
Mode 27 — Jacobian Tensor ∂yᵢ/∂xⱼ
─────────────────────────────────────────────────────────
VJP: (vᵀJ)ⱼ = Σᵢ vᵢ ∂yᵢ/∂xⱼ (backpropagation)
Renders the full Jacobian matrix J ∈ ℝ^(m×n) with an
animated column sweep showing the active partial
derivatives. The vector-Jacobian product (VJP = reverse
mode = backprop) and the Jacobian-vector product (JVP =
forward mode) are contrasted. The key result — reverse
mode costs O(1 × forward pass) regardless of n — is
derived explicitly.
Recommended: Rows 5 Cols 6 AnimSpeed 0.03
Mode 28 — Hessian Matrix ∂²L/∂θᵢ∂θⱼ
─────────────────────────────────────────────────────────
Newton step: Δθ = −H⁻¹∇L κ = λ_max/λ_min
Renders the symmetric Hessian matrix with its eigenvalue
spectrum displayed as bars. The condition number κ is
computed. High κ means SGD converges slowly (oscillates
in ill-conditioned directions). Second-order methods
(Newton, L-BFGS, Gauss-Newton), the Fisher information
connection, and K-FAC (Kronecker-Factored Approximate
Curvature) are discussed. Param scales the curvature.
Recommended: Rows 5 Param 0–5 ShowEigen TRUE
Mode 29 — Multi-Head Attention
─────────────────────────────────────────────────────────
A = softmax(QKᵀ/√dₖ) V [B, H, T, T] tensor
Renders H parallel attention weight matrices (T×T each)
as heatmaps, with an animated query-row sweep showing
which token is attending to all others. The full tensor
contraction form is given. Flash Attention (reorders
computation for cache efficiency) and the O(T²) vs O(T)
complexity trade-off are explained.
NOTE: For a natural attention matrix set Rows = Cols
(square: sequence length × sequence length).
Recommended: Rows 6 Cols 6 Depth 4 (heads) AnimSpeed 0.03
Mode 30 — Convolutional Layer as Tensor Contraction
─────────────────────────────────────────────────────────
y[b,c',i,j] = Σ_{c,k,l} x[b,c,i+k,j+l] · W[c',c,k,l]
Renders the input feature map with the sliding kernel
position animated (moving highlight shows the receptive
field), the 3×3 kernel, and the output map. The im2col
algorithm — which converts convolution into a single GEMM
(General Matrix Multiply) call — is explained. This is
why convolutions run on exactly the same hardware path
as linear layers.
Recommended: Rows 6 Cols 6 AnimSpeed 0.04
Mode 31 — Fisher Information & Natural Gradient
─────────────────────────────────────────────────────────
Fᵢⱼ = E[∂ᵢ log p · ∂ⱼ log p] θ ← θ − η F⁻¹ ∇L
Renders the Fisher information matrix F alongside its
inverse F⁻¹ and eigenvalue spectrum. Natural gradient
descent uses F⁻¹∇L (invariant to reparametrisation).
The Cramér-Rao bound Var[θ̂] ≥ F⁻¹ (MLE achieves this)
is stated. K-FAC approximation F ≈ A⊗G and information
geometry (statistical manifold as a Riemannian manifold)
are discussed.
Recommended: Rows 5 ShowEigen TRUE ShowDual TRUE
------------------------------------------------------------
INPUTS (14)
------------------------------------------------------------
Mode int 0–31 Which concept to visualise.
Use the 5 group tabs in the
companion HTML guide to browse.
Param int 0–5 Sub-parameter. Meaning changes
per mode:
· GR modes: physical quantity
(velocity β, radius, scale
factor, field strength)
· QM Mode 23: mixing 0=pure→5=mixed
· QM Mode 26: entanglement angle
· Most Algebra modes: unused
Rows int 1–8 Size of dimension 0. Ignored
by all 4×4 spacetime modes (15–21)
which are always fixed 4D.
Cols int 1–8 Size of dimension 1.
For attention (Mode 29): set
Rows = Cols for square [T×T].
Depth int 1–5 Size of dimension 2.
· CP/Tucker/TT: number of slices
or chain length
· Attention (Mode 29): num heads H
· Ignored by most other modes
CellSize float 0.2–4 Size of each 3D box in Rhino units.
0.8–1.0 for everyday work.
2.0–3.0 for projection / teaching.
Gap float 0–1.5 Spacing between cells.
0.10–0.15 recommended.
Increase if cells feel crowded.
Animate bool toggle Enables 80ms animation loop.
Requires Timer component (see
Animation Setup below).
AnimSpeed float 0.005–0.3 Animation speed per tick.
0.020–0.030 recommended.
Slower = easier to follow.
Faster = better overview of pattern.
Reset bool button Clears static state (_af, tensor
data). Rewinds all animations.
ShowIndices bool toggle Draws Einstein index notation
lines and summation path arrows.
Most useful for modes 0, 3, 5.
ShowEigen bool toggle Adds eigenvalue bar charts.
Most useful for modes 2, 10,
19, 23, 28, 31.
ShowDual bool toggle Renders dual tensor geometry
(contravariant, Hodge dual,
inverse metric).
Most useful for modes 1, 8, 17.
Precision int 1–4 Decimal places in Report text.
3 recommended for most modes.
4 for GR (metric components).
------------------------------------------------------------
OUTPUTS (7)
------------------------------------------------------------
TensorGeo Main 3D box geometry (height = magnitude).
Connect to Preview. Assign light grey or
white material.
DualGeo Dual / contravariant / Hodge dual geometry.
Connect to separate Preview. Assign
contrasting colour (amber, gold).
EigenGeo Eigenvectors, singular vectors, Schmidt
coefficients as bar geometry.
Connect to separate Preview. Assign
accent colour (cyan, green).
IndexGeo Summation path lines, brace annotations,
and Einstein index notation geometry.
Connect to separate Preview. Assign muted
grey at 50% opacity.
OperationGeo Arrows, connecting lines, operator symbols.
Connect to separate Preview. Assign bright
accent colour (amber, gold).
Report Full PhD-level educational text for the
current mode. Connect to a Grasshopper
Panel. Enable "Wrap Text" and use a
monospaced font for best readability.
Formula Clean mathematical formula for the current
mode (LaTeX-compatible notation). Connect
to a small Grasshopper Panel.
RECOMMENDED COLOUR ASSIGNMENT:
TensorGeo White / light grey
DualGeo Gold / amber
EigenGeo Cyan / turquoise
IndexGeo Muted grey, 50% opacity
OperationGeo Bright amber or orange
------------------------------------------------------------
RECOMMENDED INPUT VALUES
------------------------------------------------------------
GROUP A — TENSOR ALGEBRA START
Mode 0 Rows 4 Cols 4 AnimSpeed 0.025
ShowIndices TRUE
MINKOWSKI / LORENTZ BOOST
Mode 15 or 16 Param 2 AnimSpeed 0.015
ShowDual TRUE
EINSTEIN FIELD EQUATIONS
Mode 20 Param 2 AnimSpeed 0.015
Precision 4
RIEMANN CURVATURE
Mode 18 Rows 4 Cols 4 AnimSpeed 0.020
ShowEigen TRUE
EM FIELD TENSOR + HODGE DUAL
Mode 17 or 8 AnimSpeed 0.020
ShowDual TRUE
QUANTUM STATE / DENSITY MATRIX
Mode 23 Rows 4 Param 0→5 AnimSpeed 0.025
ShowEigen TRUE
BELL STATES / ENTANGLEMENT
Mode 26 Param 0→5 AnimSpeed 0.025
SVD / LOW-RANK APPROXIMATION
Mode 10 Rows 5 Cols 4 AnimSpeed 0.025
ShowEigen TRUE
MULTI-HEAD ATTENTION
Mode 29 Rows 6 Cols 6 Depth 4 AnimSpeed 0.03
HESSIAN / LOSS LANDSCAPE
Mode 28 Rows 5 Param 2 AnimSpeed 0.020
ShowEigen TRUE
NATURAL GRADIENT / FISHER
Mode 31 Rows 5 AnimSpeed 0.020
ShowEigen TRUE ShowDual TRUE
PRESENTATION (large cells, slow)
CellSize 2.5 Gap 0.35 AnimSpeed 0.010
------------------------------------------------------------
ANIMATION SETUP
------------------------------------------------------------
1. Params > Util > Timer → set interval to 80ms
2. Connect Timer output → C# Script component (triggers
Grasshopper to re-solve on each tick)
3. Boolean Toggle → Animate input (set TRUE to run)
4. Button → Reset input (click to restart)
5. Number Slider → AnimSpeed (start at 0.025)
6. Number Slider → Mode (0–31)
7. Number Slider → Param (0–5, mode-dependent)
IMPORTANT: Without a Timer, animation will not run even
if Animate = TRUE. The Timer is the clock source.
------------------------------------------------------------
TIPS FOR THE RHINO VIEWPORT
------------------------------------------------------------
· Use Arctic or Shaded display mode for clean rendering.
Arctic (white background) works especially well for
the mathematical geometry.
· Four-view layout: Top view shows matrix/tensor
structure most clearly (cells are flat). Perspective
view reveals the height dimension (magnitude encoding).
· For the spacetime modes (15–21), top view gives the
standard matrix representation. Perspective lets you
see relative magnitudes as varying heights.
· For Group D (Quantum), isometric perspective captures
both the matrix structure and the probability height.
· The Report output is most readable in a Grasshopper
Panel set to Wrap Text with JetBrains Mono or Consolas
at 10–11pt. Use a wide panel (400px+).
· Disable the Timer when not actively using the script.
The 80ms interval generates continuous Grasshopper
solves — fine during exploration, but unnecessary
when comparing static outputs.
· To compare two modes: duplicate the C# Script node,
set different Mode inputs, place outputs side by side
in Rhino. Wire separate colour previews for each.
· For publication screenshots: set AnimSpeed to 0.001,
advance frame by frame using manual slider nudges.
------------------------------------------------------------
LEARNING PATH SUGGESTIONS
------------------------------------------------------------
PURE MATHEMATICS PATH:
0 → 1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9 → 10 → 11
GENERAL RELATIVITY PATH:
0 → 2 → 3 → 15 → 16 → 17 → 18 → 19 → 20 → 21
QUANTUM MECHANICS PATH:
0 → 6 → 22 → 23 → 24 → 25 → 26 → 13
DEEP LEARNING PATH:
10 → 27 → 28 → 29 → 30 → 31
TENSOR DECOMPOSITIONS PATH:
10 → 11 → 12 → 13 → 14
FIELD THEORY PATH:
5 → 7 → 8 → 17 → 9 → 18 → 19 → 20
------------------------------------------------------------
WHAT YOU WILL UNDERSTAND AFTER USING THIS
------------------------------------------------------------
After working through all 32 modes you will have
developed geometric intuition for:
· Why repeated indices in Einstein notation mean
summation, and what free vs dummy indices represent
· Why the metric tensor is the central object of both
Riemannian geometry and general relativity — and how
it differs from the Kronecker delta in curved space
· Why the Levi-Civita tensor is the unique completely
antisymmetric object, and why it appears in determinants,
cross products, and Hodge duality simultaneously
· Why the Riemann curvature tensor is exactly the Lie
bracket of covariant derivatives — and what that means
for the equivalence of flat geometry and commuting
derivatives
· What the 20 independent components of the Riemann
tensor represent, and why the Ricci contraction
destroys information (Weyl tensor = what remains)
· Why the Einstein field equations have the form they do,
and why ∇ᵤTᵤᵥ = 0 (energy-momentum conservation)
is not imposed but emerges from Bianchi identities
· Why a quantum state is a tensor product of subsystem
Hilbert spaces, and why entanglement is precisely
the failure of that product to factorise (Schmidt
rank > 1)
· Why the density matrix is necessary for mixed states,
and why Tr(ρ²) < 1 captures genuine quantum uncertainty
rather than ignorance
· Why backpropagation is reverse-mode automatic
differentiation — a sequence of vector-Jacobian
products — and why it costs O(1 × forward pass)
· Why a convolutional layer is a tensor contraction,
and why im2col reduces it to the same GEMM hardware
path as a linear layer
· Why the Fisher information matrix is the Riemannian
metric on the statistical manifold, and why natural
gradient descent is coordinate-invariant while SGD
is not
· Why tensor networks like MPS/TT achieve exponential
compression for systems satisfying the area law of
entanglement entropy
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TECHNICAL NOTES
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· Rhino 7 or 8 required
· Grasshopper C# Script component (built-in, no install)
· No external NuGet packages or DLL dependencies
· All tensor data seeded deterministically (seed 42)
for reproducible values across sessions
· Static state (_af, _A, _B, _sv, _U, _Vt) persists
between Grasshopper solves during animation
· SVD is approximated via randomised power iteration
(sufficient for visualisation; not numerically exact)
· Eigenvalues are approximated via power iteration with
deflation (good for display; use scipy for computation)
· The Schwarzschild metric (Param 5, Mode 2) uses the
standard exterior Schwarzschild solution. No interior
continuation or Kruskal extension is implemented.
· Reset button clears all static state and reseeds
tensor data from the fixed seed
· Timer drives animation at ~12.5fps (80ms interval)
· CellSize and Gap are in Rhino document units;
adjust to match your working scale
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RELATION TO WICKERSON STUDIOS TENSOR VISUALISER
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WickersonStudios_TensorVisualiser.cs (Level 1, beginner)
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15 modes covering: scalars, vectors, matrices, rank-3
and rank-4 tensors. Elementwise operations, matrix
multiply, transpose, reshape, broadcasting, reduction,
dot product, softmax, outer product. Real ML shapes:
RGB images, image batches, text embeddings, attention
matrices, weight matrices.
Requires: basic linear algebra.
WickersonStudios_AdvancedTensorEngine.cs (Level 2, PhD)
─────────────────────────────────────────────────────────
32 modes covering: full index calculus, differential
geometry, general relativity, quantum information, tensor
decompositions, and advanced ML theory.
Requires: differential geometry, quantum mechanics,
or advanced ML research background.
Start with the Tensor Visualiser if you are new to
tensors as computational objects. Move to the Advanced
Tensor Engine once you understand shape, indexing,
and basic operations.
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COMPANION HTML GUIDE
------------------------------------------------------------
WickersonStudios_AdvancedTensorEngine_Guide.html
Covers all 32 modes with:
· Animated canvas visualisation for every mode
· PyTorch / NumPy code examples
· Key properties and physical interpretation
· Interactive mode browser with group tabs
· Input guide with mode-specific Param documentation
Open in any modern browser. Works fully offline
after Google Fonts load on first open.
------------------------------------------------------------
CREDITS
------------------------------------------------------------
Script Wickerson Studios · 2026
AI Engine Claude (Anthropic)
Website www.wickersonstudios.com
Mathematical sources and notation conventions:
· Misner, Thorne, Wheeler — Gravitation (1973)
· Penrose & Rindler — Spinors and Space-Time (1984)
· Nielsen & Chuang — Quantum Computation (2000)
· Orús — Practical Introduction to Tensor Networks (2014)
· Amari — Information Geometry (2016)
· Goodfellow, Bengio, Courville — Deep Learning (2016)
------------------------------------------------------------
QUOTES
------------------------------------------------------------
"The special theory of relativity owes its origin to
Maxwell's equations of the electromagnetic field."
— Albert Einstein
"God used beautiful mathematics in creating the world."
— Paul Dirac
"Shut up and calculate."
— N. David Mermin (attr.)
"The unreasonable effectiveness of mathematics in the
natural sciences is a wonderful gift which we neither
understand nor deserve."
— Eugene Wigner
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www.wickersonstudios.com
Wickerson Studios · 2026 · Powered by Claude AI
Every wall is a polynomial.
Every monster learns.
Every tensor has a story.
Enter at your own risk. ☠
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