41, 43 … wait, where is 42?


XLII: A superabundant journey (static)

XLII: A superabundant journey (panimating)

Milestone 41!

Milestone achieved: 41 artworks printed in A3 format and gathered in my first book. It’s been an exciting 9 months journey since June last year. An incredible mental discipline, an unexpected psychological journey, spent opening new doors and scouting new grounds I never dared to stride before.

IMG_0177  IMG_0178

My first book, including 40 A3 prints in 20 clear pockets, plus a cover one.

41 is a special number to me, fond that I am of mathematics in general, and of prime numbers in particular. It is the first prime number in the famous series of primes discovered by Leonhard Euler, and the last of  “Euler’s 6 lucky numbers“: 2, 3, 11, 17 and 41. Considering this equation:

x² + x + 41 = N

For X an integer from 1 to 49, this equation produces 40 prime numbers N:

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, … and so on, up to 1601 (Full list here: A005846)

What’s fun is that you just have to add +2 to the increment every time:

  • 41+2 = 43
  • 43+4 = 47
  • 47+6 = 53
  • 53+8 = 61
  • … and so on, at least until X = 40, since then we meet the 41*41 singularity.
  • Note that the list continues beyond X=41 though.


So where to after opus #41, which was my encounter with the Ghost of Blaise? … Shall I tackle 43 then? But what about 42? That is certainly not a number I would dare to skip over mindlessly.


Let’s talk about 42 …

I can already those who know their classic english letters screaming it:

42? Come on … This is the Answer to the Ultimate Question of Life, the Universe, and Everything!!!

That’s at least what we can draw fromThe Hitchhiker’s Guide to the Galaxy, by Douglas Adams.  The story says that it took the enormous supercomputer Deep Though a jaws-dropping period of 7.5 million years to compute it. Only problem was that after all this time, the people who built the computer kind of forgot what exactly was the original question. To ascertain the matter, they decided to build in turn another device to compute the Ultimate Question. To do so, a special computer the size of a small planet was built from organic components and named “Earth”.

What if … it were true? 42 is definitely an “odd” number despite it’s clearly even nature. Beyond the unbridled imagination of Douglas Adams, let’s just examine a few mathematical facts first:

  • 42 is the 2nd sphenic number, after 30: In number theory, a sphenic numberis a positive integer that is the product of three distinct prime numbers. The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. 42 comes next by being  2 x 3 x 7
  • 42 is the 3rd primary pseudo-perfect number, after 2 and 6.
  • 42 is the 4th Catalan number, after 2, 5 and 14. Consequently, 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered binary trees with six leaves, the number of ways in which five pairs of nested parentheses can be arranged, etc.
  • 42 is the 6th pronic number: A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n(n + 1).  It comes after  2, 6, 12, 20, 30.
  • 42 is the only known value that is the number of sets of four distinct positive integers a, b, c, d, each less than the value itself, such that ab − cd, ac − bd, and ad − bc are each multiples of the value. Whether there are other values remains an open question.
  • 42 is the resulting number of the original Smith number (4937775 = 3 × 5 × 5 × 65837): Both the sum of its digits (4 + 9 + 3 + 7 + 7 + 7 + 5) and the sum of the digits in its prime factorization (3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)) result in 42. Albert Wilansky of Lehigh University named this number  after he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith. Mind blowing.
  • 42 is a repdigit in base 4: 222(4). In base 2, it is 101010.
  • In geometry:

Gyroelongated pentagonal bicupola

So far so good? Not convinced yet that 42 is nor only remarkable, but a very special number? Let’s continue looking in other fields:

  • In sciences, 42 degrees is the critical angle from which a rainbow appears.
  • Theoretically, it would take 42 minutes in free fall to go across the Earth through a tunnel joining the antipodes.  This was theorised and demonstrated in 1966 by mathematician Paul Cooper. Remarkably enough, the same idea and value was proposed, without calculation by Lewis Carroll in 1893, in his book Sylvie and Bruno concluded!
  • Talking about Lewis Carroll, the original edition of Alice in Wonderland contains exactly 42 illustrations by John Tenniel.
  • 42 is the number with which God creates the Universe in Kabbalistic tradition.
  • In the Book of the Dead of ancient Egypt, there are 42 questions asked of persons making their journey through Death.
  • The Guttenberg bible, known as the first book ever mass-printed by Man, is also known as the 42-line bible, or B42 for short. All pages contain exactly 42 lines.

So what do we do with all that? Sitting between Alice, Arthur Dent, Euler, and a pile of ancient books?

Seriously, that’s not a number one can ignore, and I had to make something special about it.

Hence this new panimating, and a superabundant one! The initial storyboard was pretty loose on paper:

XLII - Storyboard
XLII – Storyboard

This animated version will take another few hours to render in Cinema4D and Final Cut, but at least I can already release a still, with a few elements of explanation:

XLII explained

Techniques: Shadertoy, Particle Designer, Processing, GlyphsCinema4D, Final Cut Pro, Pixelmator

Making of:

Gyroelongated pentagonal bicupola ship
Gyroelongated pentagonal bicupola ship in Cinema4D


Some good readings
Some good readings in Cinema4D


A random cockpit from the 3D warehouse
A random cockpit from the 3D warehouse


The 42 non crossing partitions of a pentagon
The 42 non crossing partitions of a pentagon (+10 crossing ones)


Making an alien font
Making an alien font in Glyphs


Animated texture in Processing
Animated texture in Processing


A plasma jet in Particle Designer
A plasma jet in Particle Designer


Alice's leaflets falling in Particle Designer
Alice’s leaflets falling in Particle Designer
A pseudo tunnel effect in Shadertoy
A pseudo tunnel effect in Shadertoy

Tweaking in Cinema4D
Tweaking in Cinema4D
Compositing in Final Cut
Compositing in Final Cut

And that’s about it … just a few more modelling, texturing lighting and tweaking in Cinema4D, Final Cut and Pixelmator.

A great journey full of learnings and insights!

XLII explained