Mixed Many Mathematical Dimensions

The ten CGB Mathematical Depositions, mixed. In each entry two or more formulas share one space: every formula keeps its own axes, and one red point carries all of them at once. The formulas and the live numbers are the same as on the Mathematical Animations page. Each formula on its own is on Mathematical Dimension (three axes) and Many Mathematical Dimensions (more than three). Free to view and open to everyone.

We rely on the matter to make the relation

Every entry on the Mathematical Animations page has two sides: a fact about numbers and a fact about the world, joined by a piece of matter. Here the formulas are set next to each other. Where two of them share a symbol, the entry says so and names what joins them. Where they share nothing but the page, the entry says that too: the numbers give the shape, and nothing in them says they must meet.

“These animations are free to view. Viewing them does not give you the underlying inventions or mathematics.”

Each plot turns slowly on its own. Drag it to turn it yourself. The button stops and starts it.

Harmonic Decay × Photon Chromosome EncodingLight

Harmonic Decay

Σ 1/n² = π²/6 ↔ ∫ e^−λtcos(ωt) dt ∝ λ/(λ²+ω²)

Photon Chromosome Encoding

C = ⌊log²(Δλ/δλ)⌋ + ⌊log²(2π/δθ)⌋ + ⌊log²(I_max/I_min)⌋ bits/photon

These are the two entries the page tags Light. One is a swing that fades in time; the other is a photon counted out in bits. Nothing in the numbers says they must meet. Here each keeps its own axes, and one point carries both at once.

Entropic Bridge × Zero-Point Fabrication × Quantum Counting ParadoxAbnormal

Entropic Bridge

S = −k_B Σ p_i ln p_i ≥ ∮ dQ/T ⇔ d²F/dT² = −C_v/T

Zero-Point Fabrication

E_zp = ½ℏω ⇒ Σ ½ℏω_k → ∞ ⇒ ζ(−1) = −1/12 ⇒ E = −ℏcπ²/720d³

Quantum Counting Paradox

2ⁿ states ≠ 2ⁿ computations ⇒ P = sin²(πk/4√N) after k = ⌊(π/4)√N⌋

These are the three entries the page tags Abnormal: disorder that only climbs so far, an endless sum that leaves a small real force, and a count of answers that cannot all be read. Each keeps its own axes, and one point carries all three at once.

Voxel Resonance × Golden Spiral Convergence × Dimensional FoldUnique

Voxel Resonance

Ψ(v) = ∏_k [α_k sin(2πf_k/N_k) + β_k e^−γ_kd²] · det(M)

Golden Spiral Convergence

lim F(n+1)/F(n) = φ = (1+√5)/2 ⇒ r_opt = φ⁻² ≈ 0.382

Dimensional Fold

V_n = π^n/2/Γ(n/2+1) · rⁿ ⇒ lim V_n = 0 ⇒ n* = 5 maximises V_n

These are the three entries the page tags Unique: a voxel that rings, a ratio that settles, and a volume that peaks and fades. Each keeps its own axes, and one point carries all three at once.

Recursive Growth Bound × Thermal Noise FloorCommons

Recursive Growth Bound

T(n) = a·T(n/b) + f(n) ⇒ N_gates(d)

Thermal Noise Floor

E_bit ≥ k_BT ln2 ≈ 2.85×10⁻²¹ J @ 300K ⇒ FLOPS_max = P/(k_BT ln2)

These are the two entries the page tags Commons: how fast a design can grow, and how little energy a bit can cost. Each keeps its own axes, and one point carries both at once.

Entropic Bridge × Thermal Noise FloorAbnormalCommons

Entropic Bridge

S = −k_B Σ p_i ln p_i ≥ ∮ dQ/T ⇔ d²F/dT² = −C_v/T

Thermal Noise Floor

E_bit ≥ k_BT ln2 ≈ 2.85×10⁻²¹ J @ 300K ⇒ FLOPS_max = P/(k_BT ln2)

Both formulas carry the same two symbols, k_B and ln 2. A two-state system at its most uncertain has entropy ln 2, in units of k_B. Erasing that one bit at temperature T costs at least k_B·T·ln 2. The peak of the first formula is the unit of the second, and the thing that joins them is whatever holds the bit.

Harmonic Decay × Zero-Point FabricationLightAbnormal

Harmonic Decay

Σ 1/n² = π²/6 ↔ ∫ e^−λtcos(ωt) dt ∝ λ/(λ²+ω²)

Zero-Point Fabrication

E_zp = ½ℏω ⇒ Σ ½ℏω_k → ∞ ⇒ ζ(−1) = −1/12 ⇒ E = −ℏcπ²/720d³

Both formulas hang on a sum over the whole numbers. Harmonic Decay adds 1/n² and the total settles at π²/6. Zero-Point Fabrication adds the whole numbers themselves, a total that grows without limit, and the page prints the value −1/12 that stands in for it. One sum closes; the other has to be tamed by two metal plates.

Golden Spiral Convergence × Recursive Growth BoundUniqueCommons

Golden Spiral Convergence

lim F(n+1)/F(n) = φ = (1+√5)/2 ⇒ r_opt = φ⁻² ≈ 0.382

Recursive Growth Bound

T(n) = a·T(n/b) + f(n) ⇒ N_gates(d)

Both formulas describe growth by a rule that feeds on its own earlier values. Each Fibonacci number is the sum of the two before it, and the step-to-step ratio settles on φ ≈ 1.618. The chip tree on the page doubles at every level, a ratio of exactly 2. Side by side they are two growth rates, one found by nature and one chosen by a designer.

Quantum Counting Paradox × Dimensional FoldAbnormalUnique

Quantum Counting Paradox

2ⁿ states ≠ 2ⁿ computations ⇒ P = sin²(πk/4√N) after k = ⌊(π/4)√N⌋

Dimensional Fold

V_n = π^n/2/Γ(n/2+1) · rⁿ ⇒ lim V_n = 0 ⇒ n* = 5 maximises V_n

Both formulas are about spaces with many directions. A register of n qubits has 2ⁿ states, a number that doubles with every qubit. A ball in n dimensions has volume V_n, a number that peaks at five dimensions and then fades toward zero. As n climbs, one count explodes while the other empties out.

Voxel Resonance × Harmonic DecayUniqueLight

Voxel Resonance

Ψ(v) = ∏_k [α_k sin(2πf_k/N_k) + β_k e^−γ_kd²] · det(M)

Harmonic Decay

Σ 1/n² = π²/6 ↔ ∫ e^−λtcos(ωt) dt ∝ λ/(λ²+ω²)

Both formulas multiply a swing by a fade. In Harmonic Decay the fade runs along time, e^−λt. In Voxel Resonance it runs along distance, e^−γd². Put on the same axes they are one picture: a ringing that dies away the longer you wait and the farther you stand.

All ten at onceLightAbnormalUniqueCommons

Harmonic Decay

Σ 1/n² = π²/6 ↔ ∫ e^−λtcos(ωt) dt ∝ λ/(λ²+ω²)

Entropic Bridge

S = −k_B Σ p_i ln p_i ≥ ∮ dQ/T ⇔ d²F/dT² = −C_v/T

Voxel Resonance

Ψ(v) = ∏_k [α_k sin(2πf_k/N_k) + β_k e^−γ_kd²] · det(M)

Photon Chromosome Encoding

C = ⌊log²(Δλ/δλ)⌋ + ⌊log²(2π/δθ)⌋ + ⌊log²(I_max/I_min)⌋ bits/photon

Recursive Growth Bound

T(n) = a·T(n/b) + f(n) ⇒ N_gates(d)

Zero-Point Fabrication

E_zp = ½ℏω ⇒ Σ ½ℏω_k → ∞ ⇒ ζ(−1) = −1/12 ⇒ E = −ℏcπ²/720d³

Golden Spiral Convergence

lim F(n+1)/F(n) = φ = (1+√5)/2 ⇒ r_opt = φ⁻² ≈ 0.382

Quantum Counting Paradox

2ⁿ states ≠ 2ⁿ computations ⇒ P = sin²(πk/4√N) after k = ⌊(π/4)√N⌋

Thermal Noise Floor

E_bit ≥ k_BT ln2 ≈ 2.85×10⁻²¹ J @ 300K ⇒ FLOPS_max = P/(k_BT ln2)

Dimensional Fold

V_n = π^n/2/Γ(n/2+1) · rⁿ ⇒ lim V_n = 0 ⇒ n* = 5 maximises V_n

One axis for each of the ten formulas. Each axis carries the live number that formula shows on the page, and one point carries all ten at once. The numbers give the shape. Nothing in them says they must meet; on this page they are simply put side by side.

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