Sunset at the Sea of Primes
The pictures show a colorful visualization of a complex function, whose nulls are located at all primes on the real axis. The formula is shown below.
Links to related videos:
http://youtu.be/pkJ0GoFsBqs
http://youtu.be/trOkyzhb2RY
http://youtu.be/MDx5W4MaIcI
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/01_prim-25-25.png?itok=GqZCtUQJ)
Formel
- f(z) = \left| 2 - \sum\limits^\infty_{k=1} \frac{1}{k} \frac{e^{2 \pi i z}-1}{ e^{2 \pi i z / k}-1} \right|
Sunset at the Sea of Primes
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/02_prim40_3.png?itok=BbQs-izw)
Sunset at the Sea of Primes (detail)
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/03_one-sided40_3.png?itok=lxSCYIsu)
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/04_one-sided80_3.png?itok=Kru-YvwZ)
Ray of Primes 2
Primes at the right hand side can be identified by a blue “flame” overtopping the other integers.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/05_spiral-3-3.png?itok=0LYcNUEM)
Prime Spiral 1
Inspired by Ulam’s prime spiral: the ray of primes warped to an Archimedes spiral.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/06_spiral-10-10.png?itok=Go-GlYws)
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/07_spiral-20-20.png?itok=KlO_Sz_x)
Prime Spiral 3
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/08_spiral-40-40.png?itok=U3igNfz_)
Prime Spiral 4
Here, the same distance between numbers and between spiral arms was chosen. This leads to a chaotic distribution of primes.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/09_tspiral40_pi.png?itok=oTipJbMC)
Prime Spiral 5
Here, the distance between numbers are pi times of the distance between spiral arms. Now primes are forming interesting patterns.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/10_tspiral20_pi.png?itok=U6g33MG7)
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/11_tspiral20_pi3.png?itok=Ceb5ljQN)
Prime Spiral 7
Here, the distance between numbers are pi/3 times of the distance between spiral arms. Again, primes are forming interesting patterns.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/12_tspiral20_pi8.png?itok=pr45FDQ1)
Prime Spiral 8
Here, the distance between numbers are pi/8 times of the distance between spiral arms. Again, primes are forming interesting patterns.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/13_tspiral10_pi8.png?itok=MCBf5NBR)
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/14_tspiral050_0.3132.png?itok=hmYBEWoa)
Prime Spiral 10
Here, the distance between numbers are 3.19 times of the distance between spiral arms.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/15_tspiral025_0.3132.png?itok=2Zb7eCfy)