Up By The Root

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The adventures of a cute little doodle!  by ELYXtroqman | Instagram

Here is “ELYX“, a cute little doodle wandering through Paris, France as well as various parts of the world through the sketchbook of YAK, its creator, who draws his mascot for more than 3 years! Some adorable, funny and creative adventures to follow on his Instagram account or on his website: ELYX.

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anyone who’s been following this blog knows that we’re highly susceptible to popular memes  

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Maybe I’m getting tired – I can’t think of anything but nights with you. I want them warm and silvery.
- Zelda Fitzgerald, “Letter to F. Scott Fitzgerald,” May 1919
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'Dad's on a hunting trip, and he hasn't been home in a few days'



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You know what feels really amazing? Mutual love and respect. Lol who would have thought


Type of Spirals: A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas:The radius r(t) and the angle t are proportional for the simplest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form:  x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation:  x²+y² = a²[arc tan (y/x)]².

You can make a  spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point. 
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn.
(3) A spiral as a curve comes, if you draw the point at every turn(Image).

Figure 1: (1) Archimedean spiral - (2) Equiangular Spiral (Logarithmic Spiral, Bernoulli’s Spiral).
Figure 2 : (1) Clothoide (Cornu Spiral) - (2) Golden spiral (Fibonacci number).

More Spirals: If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and  f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0.

Figure 4:  If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat’s spiral.
Figure 5: If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don’t reach it. There is a pole.  Spiral 2 is called the Lituus (crooked staff).

Figure 7: Spirals Made of Line Segments.

Source:  Spirals by Jürgen Köller.

See more on Wikipedia:  SpiralArchimedean spiralCornu spiralFermat’s spiralHyperbolic spiralLituus, Logarithmic spiral
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral
Hermann Heights Monument, Hermannsdenkmal.

Image: I shared at Spirals by Jürgen Köller - Ferns by Margaret Oomen & Ferns by Rocky.

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