Sacred Geometry – 1

Flow Lights are a 3D Sine Wave 

As you view the Flow Machine, you are seeing a sine curve passing in front of you.  The sine curve and sine function are fundamental geometric and trigonometric entities. 

Figure A1. Intersection of a right circular cylinder with an oblique plane. The resulting curve is an ellipse. When the cylindrical surface is unwrapped into the plane, this ellipse maps to a sinusoidal curve.

Sinusoidal Curves as Unwrapped Elliptical Sections of a Cylinder

DISCAIMER: AI came up with this proof as I’m not a mathematician.  If you are, and you see an error, let me know! 

Let C⊂R3C \subset \mathbb{R}^3C⊂R3 be a right circular cylinder of radius R>0R > 0R>0 whose axis is aligned with the yyy-axis, defined by

C={(x,y,z)∈R3:x2+z2=R2}.C = \{(x,y,z) \in \mathbb{R}^3 : x^2 + z^2 = R^2\}.C={(x,y,z)∈R3:x2+z2=R2}.

Consider the plane P⊂R3P \subset \mathbb{R}^3P⊂R3 given by

P={(x,y,z):z=my},P = \{(x,y,z) : z = m y\},P={(x,y,z):z=my},

where m≠0m \neq 0m=0 determines the inclination of the plane relative to the cylinder axis. The intersection curve E=C∩PE = C \cap PE=C∩P satisfies

x2+(my)2=R2,x^2 + (m y)^2 = R^2,x2+(my)2=R2,

which may be written equivalently as

x2R2+y2(R/m)2=1.\frac{x^2}{R^2} + \frac{y^2}{(R/m)^2} = 1.R2x2​+(R/m)2y2​=1.

Thus, EEE is an ellipse with semi-major axis a=Ra = Ra=R and semi-minor axis b=R/mb = R/mb=R/m.

Introduce cylindrical coordinates on CCC via

x=Rcos⁡θ,z=Rsin⁡θ.x = R \cos \theta, \quad z = R \sin \theta.x=Rcosθ,z=Rsinθ.

Restricting to the plane PPP yields

Rsin⁡θ=my,so thaty=Rmsin⁡θ.R \sin \theta = m y, \quad \text{so that} \quad y = \frac{R}{m} \sin \theta.Rsinθ=my,so thaty=mR​sinθ.

If the cylindrical surface is unwrapped into the plane by identifying the angular coordinate θ\thetaθ with a linear coordinate uuu, the elliptical intersection curve EEE is mapped to the planar curve

y(u)=Rmsin⁡u,y(u) = \frac{R}{m} \sin u,y(u)=mR​sinu,

which is a sinusoid. Consequently, a sine wave may be understood as the unwrapped representation of an elliptical cross-section of a cylinder. Sinusoidal motion and elliptical geometry are therefore equivalent descriptions of the same underlying structure, related by a change of coordinates.