#093 – The Padovan Count
What am I supposed to do, the day I hit Forty-Nine?
In past and modern wisdoms, there’s no fucking account
Of how it should carry and flow, this life, which is mine!
By large I should know now, the horses and women I can mount.
Any time left at all, to run fearlessly to the horizon line?
A few more deserts to cross, will I find any fount?
Shall this wary aching body keep enjoying the ruby red wine?
And will my worn eyes still tell the harlot from the bount?
Only one assurance, frigid and sealed, that’s quite fine,
That no one else will elude this bloody Padovan count!
It’s a birthday today, my dear brother turns 49, just a few months after me.
One would argue that every day is a milestone in one’s life. And so is every year, a little more.
What’s the meaning of this? It’s up to us really … haven’t we been designed to give things a name, life a purpose, and the world a meaning?
Unless … there’s some deeper mystery conveyed by numbers around us, those tiny yet essentials bricks of the universe, and atoms of human conscience, that might well be driving our lives … ?
49 … 7×7 … a powerful number if any, some would say magical or mystical … not a prime, but a closure, of the magnificent 7 on itself.
49 … the first square whose digits are square themselves, 2×2 and 3×3
Further research reveal that 49 appears in the Padovan sequence, as the 16th iteration in the recurrence relation that says:
P(0) = P(1) = P(2) = 1 and then P(n)=P(n-2)+P(n-3)
The first few values of P(n) are: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, …
I thought I would play with this idea, and a visualisation of this recurrence in the form of a triangle spiral:
Interesting to note also that the Padovan sequence is related to the Plastic Number, a sort of cousin of the Golden Ratio, and the only solution to the equation x³ = x+1. It’s usage in L-systems and recurring geometry leads to beautiful spiral and tree-like shapes such as the one below:
Jumping into Processing, I code the algorithmic loop able to draw the spiral up to number 49, and I add some shaders to give is a friendly visual aspect:
The rest is really about fiddling with colours in FX Photo Studio Pro.
And adding a few variants of recursive spiral trees generated from the Padovan and Fibonacci sequences with various parameters.
I find this visualisation very powerful, as the synthetic visual representation of a man’s life: The year 49 marks the beginning of the period in orange, then followed by another one starting at the age of 65, and a final one starting at 86, to end somewhere before the next number, known as 114. The diagram tells in a powerful manner that, in spite of physical decay, human life is a continued growth journey, in which each period is only richer than the previous ones accumulated: P(n)=P(n-2)+P(n-3)
Wisdom of the day 🙂