simulations can teach

Inspiration

This animation draws a curve

z(θ) = exp(iθ) + exp(i𝝅θ),

where θ ∈ ℝ, z ∈ ℂ.


1 Harmonics with an irrational ratio of frequencies:

exp(iθ) and exp(i𝝅θ) are periodic functions that draw cycles on the complex plane. But since the ratio of their frequencies 𝝅θ and θ is irrational 𝝅θ / θ = 𝝅, their phases relation never repeats. By the time the first harmonic exp(iθ) completes one full turn (θ = 2𝝅), the second harmonic exp(i𝝅θ) has been rotated by 2𝝅² radians.

This irrationality (transcendentality is not necessary actually) of frequencies' ratio (namely of 𝝅 in this example) explains why the curve z(θ) from the animation is quasiperiodic and never closes.


2 Self-intersecting projection of not-self-intersecting curve:

Observing that the curve z(θ) never closes, I thought that there might be some other curve mapped in z(θ) that lives in another space of higher dimension and never intersects itself.

(I remembered the Lorenz curve that is not-self-intersecting and thought that there might be some chaotic features found in a transcendentality of 𝝅. But there are not. Actually the behaviour observed in the animation is quasiperiodic and not chaotic.)

With the help of GPT I found out that the not-self-intersecting curve I was looking for is

θ↦(exp(iθ), exp(i𝝅θ)) ∈ ℂ²
and
z(θ) = exp(iθ) + exp(i𝝅θ) is its ℂ¹ projection.

In order to somehow visualise the curve (exp(iθ), exp(i𝝅θ)) I considered a torus that it wraps around. exp(iθ) and exp(i𝝅θ) trace out circles in two orthogonal complex planes. The first circle revolves with frequency θ, while the second - with 𝝅θ. These frequencies can then be used as angles for parameterization of torus in ℝ³ and it's visualisation.

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165 viewsMay 24 at 12:47

Inspiration

This animation draws a curve

z(θ) = exp(iθ) + exp(i𝝅θ),

where θ ∈ ℝ, z ∈ ℂ.


1 Harmonics with an irrational ratio of frequencies:

exp(iθ) and exp(i𝝅θ) are periodic functions that draw cycles on the complex plane. But since the ratio of their frequencies 𝝅θ and θ is irrational 𝝅θ / θ = 𝝅, their phases relation never repeats. By the time the first harmonic exp(iθ) completes one full turn (θ = 2𝝅), the second harmonic exp(i𝝅θ) has been rotated by 2𝝅² radians.

This irrationality (transcendentality is not necessary actually) of frequencies' ratio (namely of 𝝅 in this example) explains why the curve z(θ) from the animation is quasiperiodic and never closes.


2 Self-intersecting projection of not-self-intersecting curve:

Observing that the curve z(θ) never closes, I thought that there might be some other curve mapped in z(θ) that lives in another space of higher dimension and never intersects itself.

(I remembered the Lorenz curve that is not-self-intersecting and thought that there might be some chaotic features found in a transcendentality of 𝝅. But there are not. Actually the behaviour observed in the animation is quasiperiodic and not chaotic.)

With the help of GPT I found out that the not-self-intersecting curve I was looking for is

θ↦(exp(iθ), exp(i𝝅θ)) ∈ ℂ²
and
z(θ) = exp(iθ) + exp(i𝝅θ) is its ℂ¹ projection.

In order to somehow visualise the curve (exp(iθ), exp(i𝝅θ)) I considered a torus that it wraps around. exp(iθ) and exp(i𝝅θ) trace out circles in two orthogonal complex planes. The first circle revolves with frequency θ, while the second - with 𝝅θ. These frequencies can then be used as angles for parameterization of torus in ℝ³ and it's visualisation.

BY simulations can teach