Norway’s Spiral in the Sky

Early yesterday morning, citizens in the town of Tromsø, Norway awoke to an amazing sight – a giant glowing spiral, taking up a huge portion of the night sky.

norway-spiral

Bulava

Thousands of people reported seeing it, and amateur pictures and video of the event quickly spread across the internet. What could this be? A prank using some powerful projector? Some military experiment? An intergalactic portal?

One plausible explanation was that it was a rocket, damaged during or early after launch. But where did the rocket come from? One report indicated that it was a RSM-56 Bulava submarine launched ballistic missle, which is currently undergoing testing and development. The notoriously unpredictable missle design experienced an issue when the third stage fired, and it began to spray fuel and spiral out of control.

So it this reasonable? Well, a video has appeared on YouTube with a particle simulation of the fuel dispersal due to a spinning third stage – judge for yourself!

Personally I think the explanation of dual jets off a spinning rocket stage fits the facts and is the simplest explanation. Additionally, a NAVTEX rocket launch warning was issued for an area in the White Sea.

ZCZC FA79
031230 UTC DEC 09
COASTAL WARNING ARKHANGELSK 94
SOUTHERN PART WHITE SEA
1.ROCKET LAUNCHING 2300 07 DEC TO 0600 08 DEC
09 DC 0200 TO 0900 10 DEC 0100 TO 0900
NAVIGATION PROHIBITED IN AREA
65-12.6N 036-37.0E 65-37.2N 036-26.0E
66-12.3N 037-19.0E 66-04.0N 037-47.0E
66-03.0N 038-38.0E 66-06.5N 038-55.0E
65-11.0N 037-28.0E 65-12.1N 036-49.5E
THEN COASTAL LINE 65-12.2N 036-47.6E
2. CANCEL THIS MESSAGE 101000 DEC=
NNNN

Tromsø is marked by a green house and the White Sea is marked as the blue anchor on the following map.

This location for the submarine makes sense for a northbound launch (launching it south into continental Europe would be a bit politically insensitive), and would correlate with a breakup later in flight visible from northern Norway. If, however, this is a prelude to the gates of hell opening up, feel free at that time to email me and gloat.

The Quaternion Bulb

My three dimensional unfolding of the quaternion Julia sets finally finished rendering. There are a fair bit of compression artifacts in the embedded version, click on the Vimeo button on the bottom right side of the video to watch it in full quality HD.

Since each quaternion can be described using four numbers, I unfolded these four dimensional quaternion Julia sets into three dimensional space, and animated the final coefficient.

xyzft

But once I did that I noticed some radial symmetry along the y-z plane – it looks like something that’s been made on a lathe. This means that we can “index” all these shapes in a more sensible manner by collapsing things along this axis of symmetry. While previously we could index all of our shapes with four coefficients a, b, c, and d.

abcd

We can now index them with four coefficients a, r, theta, and d after this transformation. But there’s a nice side effect now that our coordinate system reflects our symmetry – if we vary theta, the appearance of the Julia set doesn’t change, the object just appears to rotate about the a axis.

ard

So really we can index all possible shapes using only three coefficients – a, r, and d. This is awesome – it means we can use this symmetry to collapse a dimension and completely illustrate a discrete approximation of this four dimensional set in three dimensional space. The following images (click for 1080p full resolution images) illustrate the full set of these possible shapes – a is the horizontal axis, r is the vertical axis, and values iterate by 0.25. The grey sphere in the first image is the origin, and the images start at a d value of 0 and iterate upward by 0.25. We find that there exists additional symmetry with our d parameter – namely that d = -d, so we only need to illustrate the absolute value to see all shapes.

d = 0.00
juliacube-0.00

d = 0.25
juliacube-0.25

d = 0.50
juliacube-0.50

d = 0.75
juliacube-0.75

d = 1.00
juliacube-1.00

When d = 1.25 there are only a few bits of unconnected dust loops visible. This analysis only covers a single “slice” – namely the plane normal to (0,0,0,1). I’d be very interested to see if there are any other symmetries…

A Quaternion Fractal Chorus

Treating my last attempt at rendering quaternion Julia sets as a study, I wanted to move on to alternate methods of visualising the deep structure of these four dimensional objects. There’s a lot of complexity there which results in some compression artifacts – watch it in HD to get the full effect.

There is a four dimensional Julia set for every four dimensional quaternion. We can label each quaternion using four numbers.

quaternion

I decided to “unfold” the first two values of the quaternion onto a plane and animate the last two values. The camera is centered at (0,0) and Julia sets are placed at intervals of 0.1 off to infinity for both axes.

grid

You can start to see the larger structure present more clearly. Perhaps a three dimensional unfolding next?

Save Face With Facebank

facebank-package

Japan’s consumer goods lived up to their surreal billing during a recent stay at Narita Airport in Japan on my way to IAC 2009. A collaboration between Japanese designer Takada and Bandai’s Banpresto division, the Facebank combines limited practicality with pure awesomeness.

The eyes are optical sensors which start the chewing motion as soon as your hand goes near (and as soon as you turn out the lights in the room as I also found out). Standing about 10cm tall, it takes 4 AAA batteries and retails for about ¥2000.

Quaternion Julia Fractals

What exactly is a quaternion Julia set? Well, it’s beautiful.

These shapes are animated projections of three dimensional slices of four dimensional objects known as quaternion Julia sets. The definition of a Julia set can get a bit complicated, but it can be thought of as an object that carves up four-dimensional space into two categories – belonging to the set, and not belonging to the set. How exactly the shape is carved depends on some very deep mathematics.

Now the big question – how do we look at a four dimensional object if we’re just mere three dimensional humans? Well, first let’s try to describe how we can look at a three dimensional object using only two dimensions.

When I think of two dimensions, I think of a flat sheet like a piece of cardboard. How could we use this flat sheet, or a lot of flat sheets, to make up a three dimensional object? Well, if we were very clever like Yuk King Tan, we could cut a huge number of cardboard sheets carefully and stack them up on top of each other. From far away it would look like a three dimensional object.

Tan-03

But if we look closely.

Tan-06

Very closely.

Tan-07

We can see that this is made up entirely of two dimensional objects cut into specific shapes, each shape cut perfectly to reflect the three dimensional object at a certain height. This is just like how an MRI machine takes “slices” of a three dimensional object (a human!) as it slowly moves upwards. The image below shows the 2D slices of the 3D skull starting just below the eyes.

mrislices

If we could only see two dimensions, we could flip through each one of these images in turn to get an idea of just what a three dimensional brain looks like. This is what doctors do – all of our current display technology, fancy HDTVs included, currently only display two dimensions. So they take many two dimensional slices and then compare and visualize them in relation to each other, in order to get some idea of what our three dimensional body is actually like.

So we can do the same thing with these four dimensional Julia sets. We can take many three dimensional slices, animate them, and then compare and relate these slices to each other in order to create some idea in our brain of just what this four dimensional structure is.

I See Sierpinski Shapes by the Sea Shore

I recently saw a very interesting photo of a sea shell on Flickr.

sierpinski_seashell

The patterns on the shell appear to be very similar to that of a mathematical structure called a Sierpinski triangle – and this is no coincidence.

ca_rule

A snail’s shell can grow only by adding on new material in a thin layer on the lip the shell. The pigmentation cells lie in a narrow band on this lip, and decide whether to switch on or off depending on the pigmentation of the area immediately around it. In short, the pigmentation patterns can be modelled as elementary cellular automata very accurately.

Several elementary cellular automata rule sets produce similar structures to that seen on the shell. Combine these basic rules with a little bit of noise due to nature, and you get these beautiful pattens with a bare minimum of computational effort.

The snail that grew the shell above is from the family Conidae. Other species have slightly different rules for pigmentation, but all produce their patterns by a method that can be modelled as cellular automata.

conidae_2

Color and Reality

When I was a kid, I used to wonder if everyone saw the world in the same way. We can all look at the same grass, but maybe the color I called green showed up in my brain as the color my friend called blue. Maybe all of our colors were shifted around to the point where all the colors were accounted for, but how we perceived them was shuffled up. I thought it would be remarkably exciting, and hoped that I could see the world through someone else’s brain to see if, in fact, this was true.

meadows

My eight year old self would be bitterly disappointed technology today has not progressed far enough to make that wish a reality. At the time, we had to settle the debate by another manner – asking an adult, a source of concrete and immutable knowledge. The answer I was given was that everyone sees the same colors of course (although why this was so obvious was never really clear) and if they didn’t it wouldn’t matter much since we couldn’t tell. Color was “real” – bits of light had a color (later I found out we could call it the wavelength of a photon), it hit our eyes, and our brains converted it to a beautiful image.

The only problem is that this is wrong.

Color as Wavelength

Well, alright. Before you get upset, it isn’t completely wrong. We were all taught about Sir Isaac Newton who discovered that a glass prism can split white light apart into its constituent colors.

pink-floyd-dark-side-of-the-moon-crop

While we consider this rather trivial today, at the time you’d be laughed out of the room if you suggested this somehow illustrated a fundamental property of light and color. The popular theory of the day was that color was a mixture of light and dark, and that prisms simply colored light. Color went from bright red (white light with the smallest amount of “dark” added) to dark blue (white light with the most amount of “dark” added before it turned black).

Newton showed this to be incorrect. We now know that light is made up of tiny particles called photons, and these photons have something called “wavelength” that seems to correspond to color. Visible light is made up of a spectrum, a huge number of photons each with a different wavelength our eyes can see. When combined, we see it as white light.

visible_light_spectrum

So this appears to resolve my childhood debate. Light of a single wavelength (like that produced by a laser) corresponds to a single “real” color. The brain just translates wavelengths into colors somehow, and that is that. There’s just one problem.

We’re missing a color!

Color as Experience

To find out just what we’re missing, we have to consider how we can combine colors. For instance, you learned some basic color mixing rules as a kid. In this case, let’s use additive color mixing since we’re mixing light.

Additive_color_mixing

Let’s find two colors on the spectrum line, and then we can estimate the final color they’ll produce when you mix them by finding the midpoint.

Red and green make yellow.

red-green-yellow

Green and blue make turquoise.

blue-green-turquoise

Red and blue make…

red-blue-green

Green? What? That doesn’t seem to make any sense! Red and violet make pink! But where is pink in our spectrum? It’s not violet, it’s not red – it seems like it should be simultaneously above and below our spectrum. But it’s not on the spectrum at all!

So we’re forced to realize a very interesting conclusion. The wavelength of a photon certainly reflects a color – but we cannot produce every color the human eye sees by a single photon of a specific wavelength. There is no such thing as a pink laser – two lasers must be mixed to produce that color. There are “real” colors (we call them pure spectral or monochromatic colors) and “unreal” colors that only exist in the brain.

A Color Map

So what are the rules for creating these “unreal” colors from the very real photons that hit your eye? Well, in the 1920s W. David Wright and John Guild both conducted experiments designed to map how the brain mixed monochomatic light into the millions of colors we experience everyday. They set up a split screen – on one side, they projected a “test” color. On the other side, the subject could mix together three primary colors produced by projectors to match the test color. After a lot of test subjects and a lot of test colors, eventually the CIE 1931 color space was produced.

CIE-1931

I consider this to be a map of the abstractions of the human brain. On the curved border we can see numbers, which correspond to the wavelengths in the spectrum we saw earlier. We can imagine the spectrum bent around the outside of this map – representing “real” colors. The inside represents all the colors our brain produces by mixing – the “unreal” colors.

So let’s try this again – with a map of the brain instead of a map of photon wavelengths. Red and green make yellow.

cie-red-green-yellow

Green and blue make turquoise.

cie-blue-green-turq

Blue and red make…

cie-blue-red-magenta

Pink! Finally! Note that pink is not on the curved line representing monochromatic colors. It is purely a construction of your brain – not reflective of the wavelength of any one photon.

Is Color Real?

So is color real? Well, photons with specific wavelengths seem to correspond to specific colors. But the interior of the CIE 1931 color space is a representation of the a most ridiculously abstract concept, labels that aren’t even labels, something our brain experiences and calculates from averaged photon wavelengths. It is an example of what philosophers call qualia – a subjective quality of consciousness.

I later learned that my childhood argument was a version of the inverted spectrum argument first proposed by John Locke, and that the “adult” perspective of everyone seeing the same colors (and it not really mattering if they didn’t) was argued by the philosopher Daniel Dennett.

I have come no closer to resolving my question from long ago of “individual spectrums” – but for the future, I vow to pay more attention to the idle questions of children.

The Mystics and Realists of Quantum Physics

It is said that when the 20th century is long gone, it will be remembered for two revolutionary theories – those of relativity and quantum physics. While both have led to a deeper understanding of our world, quantum physics stands alone in its profound weirdness – the ability to accurately predict totally counter-intuitive aspects of the physical world. From the simple indisputable oddity of the double slit experiment to the philosophical implications of Schrodinger’s cat, it becomes clear that we still understand very little of the actual mechanics of our world.

When explanations are lacking, the mystical is often brought up to fill the void. This has degenerated today into complete pop-psychology crap such as The Secret or What the Bleep Do We Know, but the role that human consciousness plays as an “observer”, if any, was considered very early by the founders of these theories. These arguments brought forth by some of the finest thinkers of our time echo to this day.

Niels Bohr

Niels_Bohr

Winner of the Nobel Prize in Physics in 1922, employed by the Manhattan Project, and father of the Bohr model familiar to every high school student, Niels Bohr was first accused by Einstein of introducing “mystic” elements in his explanation of quantum physics – mystic elements which in Einstein’s view had no place in science.

This was part of the famous Bohr-Einstein debates, and was perhaps not a fair criticism. Bohr appeared to not worry excessively about the “reality” underpinning the equations of quantum theory, and was simply more concerned about the equations of quantum theory rather than their implications. He rejected the hypothesis that the wave function collapse requires a conscious observer, insisting that “It still makes no difference whether the observer is a man, an animal, or a piece of apparatus”.

His view is perhaps best summarized in the following quote recalled by Heisenberg:

This argument looks highly convincing at first sight. We can admittedly find nothing in physics or chemistry that has even a remote bearing on consciousness. Yet all of us know that there is such a thing as consciousness, simply because we have it ourselves. Hence consciousness must be part of nature, or, more generally, of reality, which means that, quite apart from the laws of physics and chemistry, as laid down in quantum theory, we must also consider laws of quite a different kind. But even here I do not really know whether we need greater freedom than we already enjoy thanks to the concept of complementarity.

In short, if the numbers work out, don’t worry too much.

Wolfgang Pauli

Wolfgang_Pauli

But some did worry. Pauli was a skeptic’s skeptic – a man so dedicated to rationality it led him down a strange path. In 1927 the Solvay Conference was busy reaching consensus that Bohr’s approach was the best way to regard quantum physics (the Cophenhagen Interpretation), but Pauli was equally confident in a different interpretation. He tried to trace out just what part of consciousness it is that seems to prevent an in-depth, rational understanding. Deeply influenced by Schopenhauer’s The World as Will and Representation, Pauli appropriated his concept of a will “which breaks through space and time”.

He viewed that the acquisition of knowledge in mathematics or quantum physics “gives rise, however, to a situation transcending natural science” that can even acquire a “religious function” in human experience. This is not a belief in the religions of old, but as Pauli states “I do not believe in the possible future of mysticism in the old form. However, I do believe that the natural sciences will out of themselves bring forth a counter pole in their adherents, which connects to the old mystic elements.”

Perhaps the most interesting viewpoint on Pauli was that of Heisenberg, who viewed Pauli’s paradigm as even more rational than Bohr’s equation-focused approach because only he acknowledged that the scientific evidence pointed to something pre-rational or ‘mystical’. Pauli claimed that consciousness was presented philosophically by mystics and studied scientifically by psychologists. With the advent of quantum mechanics, physicists should then be able to unify both approaches. Unfortunately, we continue to wait.

Albert Einstein

Einstein 1921

Einstein was a scientific superstar, with fame not equalled to this day. One day, a quote was making the round in British newspapers that Einstein subscribed to the theory that “the outer world is a derivative of consciousness”. His response was swift and critical.

No physicist believes that. Otherwise he wouldn’t be a physicist. Neither do [Eddington and Jeans]. . . . These men are genuine scientists and their literary formulations must not be taken as expressive of their scientific convictions. Why should anybody go to the trouble of gazing at the stars if he did not believe that the stars were really there?

Einstein’s opposition to Bohr’s view or more “mystical” approaches is often cast as the great divide between the philisophies of idealism and those philosophies based on realism. Pauli often referred to Einstein’s “philosophical prejudice” assuming that reality is independent of any mind. In fact, his approach and objections were far more subtle. His major concern was the incredibly subjective aspect of consciousness introducing an unmeasurable “geist”, and this clash with the precise and well defined philosophical principles of physics such as locality or determinism.

This led to Einstein’s famous attempt at “breaking” quantum physics, the EPR paradox. At first a thought experiment which appeared to demonstrate quantum physics violating the seemingly well established principle of locality, later experiments showed that quantum physics instead proved locality false.

Violations of locality and determinism seemed to bother Einstein greatly, and this can be seen in his famous quote objecting to the randomness involved in wave function collapse under Bohr’s interpretation, that “God does not play dice”.

Bohr, summing up the debate perfectly, replied “Einstein, stop telling God what to do with his dice.”

John von Neumann

JohnvonNeumann-LosAlamos

The “last of the great mathematicians”, von Neumann solved one of the great problems of quantum theory. While the theory itself was established and experimentally verified, it lacked a “deep” mathematical understanding based on an axiomatic approach. He treated the world as a Hilbert space, an infinite dimensional structure.

While classical mechanics approached the world as a collection of points with six different characteristics (position and momentum along the x, y, and z axis), von Neumann’s approach considers a quantum system as a point in infinite dimensional space, corresponding to the infinite amount of possible outcomes. This led to very interesting implications in terms of “measurement”. While the “measurement” of a classical system simply involved finding one or more of those six values, the “measurement” of a quantum system involved mathematical operators acting on an infinite amount of values to produce a finite measurement.

The interesting conclusion arises when we consider the “real” interpretation of these mathematical operators. While we may say that an scientific instrument has caused wave function collapse, we run into the problem that no physical system (and a scientific instrument is a physical system completely described by quantum mechanics) can cause wave function collapse. We can describe the entire ensemble perfectly as a Hilbert space. But we do not experience this Hilbert space – we measure and experience only finite values.

The conclusion von Neumann reached is that consciousness, whatever it is, appears to be the only thing in physics that can ultimately cause this collapse or observation. This does not mean that consciousness is “required” for the universe to work, but that wave function collapse appears to be caused by consciousness and we observe only a tiny slice. We are therefore an “abstract ego” acting as a measurement device on the infinite values of true reality.

Today

Today, the argument has largely died down, a combination of practicality and lack of any suitably shocking experimental results. The majority of physicists today take the approach of “it works”, namely that quantum physics produces accurate predictions of the real world and that the mathematical formalism is just that – a mathematical formalism that produces accurate results.

It may not reflect the true reality of the world (whatever it is), but it is suitably accurate to any level of precision that we are physically able to obtain. One may stay awake at night wondering “why”, but one will not get much work done with this approach. Perhaps more clarity lies in the future, but in the meantime – we will have to tolerate crap that tells us we can “will” our way to riches with quantum mechanics (and coincidentally make the authors rather rich, will indeed) instead of a rational approach dedicated to the pursuit of truth.

Human Evolution and Frameshift Mutations

How did humans evolve from early primates? How did “human like” traits such as a smaller jaw relative to apes and hairlessness pop up when they don’t appear in the wild in any real frequency? The typical explanation for why humans have smaller jaws than early primates is that our diets changed and our brains got bigger, pressures that caused a smaller jaw. But there’s another way to look at this – what if our diets changed and our brains got bigger due to proto-human society dealing and adapting to an increasingly frequent and nearly catastrophic mutation of the jaw?

Myosin Heavy Chain 16

The human and chimpanzee genomes have both been mapped, so we are able to make comparisons between them. This is extremely useful, as chimpanzees and humans shared a common ancestor, but genetic lines split apart approximately 7 million years ago. So examining the differences may tell us something about how humans evolved.

myosin

There is a protein called myosin heavy chain 16 (aka MYH16) which in chimpanzees and other non-human primates is expressed almost exclusively in their powerful jaw muscles. These strong jaws are an adult trait – a logically complex one that would be more sensitive to random mutations.

And that’s exactly what seems to have happened. Non-human primates have DNA that codes for the complete MYH16 protein. The corresponding part of human DNA is missing a random chunk – which causes a frameshift mutation.

Frameshift Mutations

What is a frameshift mutation? Well, first let’s find out how we build proteins. We have a strand of messenger RNA (imagine a long tape with letters on it) which a ribosome (hell, imagine a tiny elf) uses to produce proteins. The critical thing to consider is that a ribosome builds a protein by reading three nucleotides at a time, and these three nucleotides code for a certain amino acid. These amino acids are chained together to produce proteins. Some combinations of three nucleotides can also act as “punctuation marks”.

"Wait, did you say there's three million more pages after this?"

So our wee elf looks closely at the long tape of letters, and starts off with the first three. His “frame”, the little chunk he works on, is three letters long. This frame is an instruction to build a certain amino acid, which he makes. He then goes along the tape, three letters at a time, making an amino acid each time that he sticks onto the last. This will eventually create a long chain of amino acids that we call a protein. But each frame doesn’t need to code for just an amino acid – it can also code for other instructions (those “punctuation marks”) starting or stopping this chaining process.

Now you may have guessed what a frameshift mutation is by now – it’s where a single letter in our tape disappears, or a new random one gets thrown in, causing our frame to get shifted slightly. This means that the resulting triplets after this error will be horribly wrong. It’s like the difference between

HEY MAN HOW ARE YOU BRO and
HEY MAN HWA REY OUB RO_ or HEY MAN HOQ WAR EYO UBR O__

if one were to speak in sentences containing only three letter words. The first sentence makes sense if we parse three letters at a time. The two others have a random letter removed, and a random letter added in. If we parse them three letters at a time, the sentence turns into garbage halfway through! The resulting nonsense (or malformed protein) is a result of a random insertion or deletion of information (nucleotides) and our “frame”, the manner in which we interpret it.

Consequences

So a frameshift mutation occured in early humans that affected the production of the protein MYH16. This protein is involved in the strong powerful jaws that primates have, but not humans. We often think of mutations as a simple little “blip” in the genetic code, but the way our bodies parse this code can cause cascading effects. Instead of MYH16 having a slightly different amino acid in a random spot from a random mutation, the specified amino acids after the mutation will change completely!

So you might think that we’ll have some odd protein that’s mostly normal, and the parts after the mutation affected by the frameshift will be wonky. But – and this is an important but – the triplets code for “punctuation marks” too, remember? In this MYH16 mutation, it turns out that this frameshift caused a punctuation mark (aka a stop codon) to just pop up – so the protein is cut off far sooner than it should be! Not too good for any traits relying on that protein.

Look at the differences between these gorilla and human skulls below. The large bony ridges on the gorilla skull on the left are where the larger jaw muscles attach – otherwise they would literally tear off of the skull. You can also see how the gorilla skull seems “empty” on the sides – that’s because it is filled with large jaw muscles, reducing space available for the brain. The red tinted parts are where the jaw muscles attach – you can see how much more “anchoring” a gorilla’s jaw muscle requires.

human_gorilla_skulls

And this is where it gets interesting. This mutation in our human ancestors happened approximately 2.4 million years ago. Right before our ancestors stopped looking like primates and started looking like us. If you lacked the protein that operated a powerful jaw muscle, you could not carry a large jawbone around and use it effectively. If you can’t carry a large jawbone around, there is strong selection pressure for those with smaller jaws to survive. If your jaw gets smaller, then the loading of the jaw on the skull decreases – bony ridges disappear, and the skull can get larger and lighter since it doesn’t need to be as strong. A larger and lighter skull can accommodate a bigger brain.

It appears that a random mutation, flipping a single bit of genetic information, has beautifully complex cascading results. Viewing the world as a hostile agent of noise and fury, winding down to an eventual death by entropy is wrong. You can fold a piece of paper, give it to a child, and have them cut crude holes in it with cheap scissors – and when you unfold it, the snowflake is beautiful.

So too can randomness be folded and twisted by logical structures in biology and physics – and the result is our amazing world.