====== Lines of Force ======

Faraday, Maxwell, Einstein

field, force, wave, particle

shape, pattern of force field

energy to matter, simple to complex

attraction, repulsion

Faraday coined the term //lines of force//. \\
Maxwell used the term //tubes of force// and compared electromagnetic force to fluid dynamics.

=== python, blender ===

blender force fields: force, charge, magnetic, harmonic, collision, gravity 

Particles, two types: hair and emission.

===== Torus =====

torus - a geometry surface generated by revolving a circle about an axis, 
creating a donut-shaped tube.

parts of a torus:
  * tube
  * central void
  * axis of revolution
  * circle, revolved around the axis, creating the tube

a torus is defined by two radii:
  * $R =$ the major radius, from the axis to the center of the circle
  * $r =$ the minor radius, the radius of the circle

types of torus, based on the aspect ratio between the two radii:
  * $R > r =>$ torus of revolution - normal donut shape, axis does not touch the circle
  * $R = r =>$ horn torus - axis is tangent to the circle
  * $R < r =>$ spindle torus - self-intersecting, axis passes twice through the circle

{{https://upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/220px-Toroidal_coord.png}}

  * poloidal direction, red arrow, toward the poles, ring around the surface
  * toroidal direction, blue arrow, parallel to lines of latitude

Note that:
  * $R+r =$ radius of the outer rim of the tube 
  * $R-r =$ radius the inner rim of the tube

==== Torus Spiral ====

There are two ways to draw a spiral on the surface of a torus.

  - poloidal winding, a cylindrical helix drawn around the tube
  - toroidal winding, a closed helical trajectory drawn perpendicular to the axis

=== poloidal winding ===

{{https://i.stack.imgur.com/yLq6t.jpg}}

where $n = $ number of windings

[[https://math.stackexchange.com/questions/324527/do-these-equations-create-a-helix-wrapped-into-a-torus
| StackExchange: Do these equations create a helix wrapped into a torus? 
(math for poloidal winding]]

=== toroidal winding ===

"closed helical trajectory drawn along the surface of a torus"

{{https://i.stack.imgur.com/aTgOW.gif}}

[[https://math.stackexchange.com/questions/921333/torus-helix-radius-change-equation |
Torus helix radius change equation (math for toroidal winding)]] \\
[[https://www.youtube.com/watch?v=eLUNm8KFC6Q
| YouTube: toroidal winding built in Grasshopper ]]

==== Applications ====
=== Chiral Swimmer ===

{{https://www.researchgate.net/profile/Benjamin-Friedrich-5/publication/26859348/figure/fig1/AS:601681855250464@1520463541897/A-A-helical-path-r-can-be-described-as-the-trajectory-of-a-point-on-the-circumference-of.png}}

[[https://www.researchgate.net/publication/26859348_Steering_Chiral_Swimmers_along_Noisy_Helical_Paths
| Steering Chiral Swimmers along Noisy Helical Paths]]


=== Electron Model ===

{{https://www.researchgate.net/profile/Richard-Gauthier-2/publication/228626249/figure/fig1/AS:301891599781898@1448987974663/The-closed-double-looped-helical-trajectory-of-the-electron-models-transluminal-energy.png}}

[[https://www.researchgate.net/publication/228626249_Transluminal_Energy_Quantum_TEQ_Model_of_the_Electron 
| Transluminal Energy Quantum (TEQ) Model of the Electron]]