Art core problem solutions

Solution to Problem 1 – Resonant color mixing#

We are given

$$C = X(f_1 + f_2 + f_3), \quad X = F_3 \cdot T_f.$$

Saturation is proportional to $$C$$. To double saturation, we need to double $$C$$. Assuming the effective composite is being modulated through resonant-time $$τ_r$$ (via how $$X$$ is realized in time), we require

$$C' = 2C.$$

Because $$C$$ scales linearly with the effective factor controlled by $$τ_r$$, doubling $$C$$ corresponds to scaling $$τ_r$$ by a factor of 2.

Answer: $$τ_r$$ must be doubled.

Solution to Problem 2 – Sculpture stress harmonics#

The sculpture’s stress pattern is

$$S = D_6(τ_r) \cdot ΛΘ.$$

If $$Θ$$ is increased by 20%, the new value is

$$Θ' = 1.2Θ.$$

The new stress is

$$S' = D_6(τ_r) \cdot ΛΘ' = D_6(τ_r) \cdot Λ \cdot 1.2Θ = 1.2S.$$

Answer: The harmonic stress $$S$$ increases by 20%.

Solution to Problem 3 – Light installation timing#

Brightness is

$$B(t) = F_3 \sin(T_f t),$$

but the artist controls the effective elevated frequency via resonant-time:

$$T_f' = \frac{T_f}{τ_r}.$$

For a sinusoid, the period is

$$T_{\text{period}} = \frac{2\pi}{T_f'}.$$

The artist wants $$T_{\text{period}} = 4$$. Thus,

$$4 = \frac{2\pi}{T_f'} = \frac{2\pi}{T_f / τ_r} = \frac{2\pi τ_r}{T_f}.$$

Solving for $$τ_r$$:

$$τ_r = \frac{4T_f}{2\pi} = \frac{2T_f}{\pi}.$$

Answer: $$τ_r = \dfrac{2T_f}{\pi}$$.