8.4 How To Make a Numerical Quilt

— Fourier and Ulam —

The Fourier way

More than a century before Ulam published his spiral there was another math-magician named Joseph Fourier, and he discovered that all seemingly chaotic vibrations could be split up into an infinite number of very regular sines and cosines. And all sines and cosines can be represented by a circular motion.

The field of mechanical engineering is a grateful customer of Fourier since certain violent sines can be isolated and terminated by introducing a sine to the system that is equal to the one that needs to be annulled, except that it is the opposite of it. This way the violent sine and the counter-sine cancel out and the system is safe.

And that is why large engines and machines have big idle chunks of steel hanging from the main shaft. They're not idle; they're creating a counter sine and prevent the machine from shaking apart.

Primes on the cob

Imagine every prime number in the number sequence to be a circular disk with a diameter that corresponds with that prime number (3 is a disk with a diameter of 3; 17 has a diameter of 17, etc.). And all those disks are being driven by a conveyer belt that in turn is driven by a disk with diameter 1. This way every disk turns slower than the 1-sized disk.

Here's the trick: Each disk is equipped with a cam that opens a switch when the disk comes in the position in which it was first activated. The 1-disk has a little tire and with it we roll the entire contraption along diverging circles; each circle is 8 positions larger than the prior one (because the Ulam Spiral is not a spiral but closed circles that get larger like peels of an onion).

Whenever all switches are closed two things happen:

1. A paintbrush is activated that paints the spot we're on.
2. An automatic disk-making device constructs a disk with a diameter equal to the location we're on, and adds that, complete with switch, to our Prime-O-Graph.

Then we let it loose. A simple device, especially when the disks are not actually there but somewhere on the hard disk of a computer. It becomes easy to predict the next prime, since this is where no disk is in the position of birth, and all disks turn with a rotation speed that is related to the 1-disk. Not having to roll along the entire number sequence but simply jumping to the next place to paint is convenient especially when numbers get large and the density of primes becomes low.

So, where's the randomness? Every disk rotates with absolute precision. Even at numbers near googolplex our little 3-disk still rotates the way it always did. Same with the 5-disk. Same with the 2^1398269-1 disk. They all rotate with absolute precision. And absolute precision doesn't exist in nature...

The number sequence depends on fixed values and rigid rules but the universe depends on freedom and sovereignty. Much like it would take infinite power to become almighty, or infinite knowledge to become omniscient, the number sequence tries to reach the freedom on which the universe is based by being infinitely large. The universe maintains freedom by not predicting what comes next. The number sequence tries to imitate that by summing up the infinite number of outcomes. Creation allows finite variety and achieves freedom; the number sequence sums up infinite variety but instead of achieving freedom, it achieves infinity. Or rather, does not achieve freedom.

The number sequence needs perpetual maintenance, perpetual additions in the form of prime numbers to maintain the continuity it depends on. The universe works in quite the opposite fashion. Everything that has ever existed, or will ever exist, was already there in that awesome event some call The Beginning and others The Big Bang. And any system that requires a) infinity and b) constant input of new pulses cannot represent the universe. And if you think that description fits solely the number sequence you are in for a rather grim surprise.