Emergence

 

A flock of starlings
The shapes and patterns are an example of emergence.

 

Generally, complexity science is the study of emergence, and complex systems are systems which undergo or demonstrate emergence. Complex networks are networks which undergo emergence.

 

Emergence refers to the unveiling of macroscopic states or observables which do not exist at the microscopic level. At the molecular level, water does not have 'wetness'. Snowflake crystallization patterns are also emergent, in that they are macroscopic, as a. result of H2O molecule properties, temperature, and the large number of water molecules. There are a myriad examples if you look around.

 

The macroscopic states also have to do with the combinatorics of the system. For example, there are a combinatorially huge number of ways to arrange water molecules, but many of these would not macroscopically appear to be snowflakes. So the number of arrangements which we accept as snowflakes is very small compared to the total number of arrangements. Each arrangement is called a 'microstate', and the set of arrangements corresponding to some large scale state is called a a 'macrostate'. Ice, liquid water, and gas are three macrostates of water.

 

You can see this with coin-flipping. With a fair coin, there are 2^100 ways to flip it 100 times. These are all the microstates. But how many ways are there to flip 50 heads and 50 tails? We can count (100 choose 50) ways to get 50 heads out of 100 flips. So the probability of 50 heads is (100 choose 50) / (2^100), which is 0.0796. If you do the same calculation for 49 heads and 51 tails, and so on, you will see that the probabilty is less.

 

So the macrostate is dominated by the 50-50 result. This is an emergent property, resulting from the fair coin and flipping it 100 times. Just as an individual water molecule doesn't have wetness, a single coin flip does not have 50 heads and 50 tails. The tendency to 50 heads and 50 tails is revealed (emerges) by scaling up the number of flips.  Also notice that predicting the macroscopic result by seeing the microscopic system elements is not always obvious or easy. This is often true in systems where the scaling up is on many orders of magnitude.

 

So, 'more is different', as Philip Anderson famously said in 1972.

 

Formation of the
Giant Connected Component (GCC)
at center, in a random graph.

Some graphs/networks undergo emergence, the classic example being network percolation in random
(Erdos-Renyi) graphs, where the size of the giant connected component (the largest connected set of nodes) suddenly explodes when the network connectivity increases to a critical density.

 

This may not seem interesting at first, but it is actually very deep, since it means that any time I start having random interactions between any entities (people, atoms, grains of sand), if the density of those interactions reaches a critical point, I will very suddenly start to have system-wide interactions (interactions which scale proportional to the size of the system itself). Most real interactive systems are far from random, so this gives us a worst case situation.

 

Network Science approximately is the study of networks, which tend to be naturally-occurring networks, vs. graph theory the study of graphs, which are mathematical objects. There is a lot of overlap between these two, since they often look very similar or have similar properties, so they are used more or less interchangeably (e.g. above).

 

So, we can study networks but not focus on the emergent aspects of the system. But of course in networks, there are so many possible ways to have various kinds of emergence, that it becomes implicit that network science is about complexity.

As an exercise, calculate the most probable sum of rolling a 6-sided die 100 times.

 

For more on emergence, which is a fascinating topic, check the book Complexity Science by Henrik Jensen.

 

 

The earth is (not) flat!

There is a rather large irony in the idea that the earth is flat, and a the same time, there is a large irony in thinking that it is not.

 

 

In mathematics, there is a basic idea:  If we have a smooth curve, we can zoom in, and keep zooming in.  As we do this, the curve no longer looks like a curve, it begins to look like a line!

Now imagine we could keep zooming in infinitely to a point on the curve. In calculus the idea is that if we zoom in forever, we do get a straight line at that point, touching (tangent) to the curve.  This is called the derivative.

 

 

The mathematics of calculus and its derivative have been incredibly useful tools in the modern world.  Let's list a very few of its applications:   It is used to build structures, to analyze them, to design airplanes.  It plays a fundamental role in training a great many machine learning algorithms.  It is used to model chemical reactions, turbulence in the atmosphere, the movement of the planets, the flow of underground water in geology.   In physics it is used to study motion, electricity, heat, light, harmonics, acoustics, astronomy, and quantum mechanics.  
 

In short, this derivative is incredibly useful to describe change over space and time.  Many physical shapes, speeds, velocities, and rates of change of any kind, can all be understood and modeled by the derivative.

The list of its uses goes on and on... and on.  Without the tools of calculus, a great deal of the modern world would be challenging, perhaps impossible, to study and develop.   Possibly we would still be living with the technology of the 1700s or 1800s. 

 

Indeed, we are truly standing on the shoulders of (scientific) giants, very oblivious and conveniently forgetting it as we drive our cars through our cities, talking on our smartphones, and planes fly overhead....

 
 

 

 

 

 

 

 

Perhaps we can begin to see the irony...  

We often immediately dismiss people who believe the earth is flat, thinking ourselves better, because we are 'people of science'.   But a great deal of our entire world, and more importantly, the underlying assumption that we can analyze phenomena scientifically, is based on the derivative, the idea that curves are locally flat, that we can take the derivative.  

That is, as scientists, we are all, to some extent, flat-earthers (!!).
 


Indeed, pick anything you call 'curved'.   Now imagine you are extremely, extremely small, standing on that curve.  To you, the world is actually flat! 

Let's try zooming in to a curve and see what happens.

Starting out, our curve is so large and our person so tiny, we barely see them.

 

 

 

 

Let's zoom in.

Perhaps you notice how the curve seems flatter?

Let's zoom in some more...

Wait, what happened?  The curve disappeared!
If we look carefully, we can just make out a curve in the landscape, but it is hard to see.

Another issue, as always, is us.   We humans seem to have challenges looking at things globally.  We are intensely aware of our immediate surroundings, both in space and time.   We know our room very well, we think a lot about what we are doing right now or about to do.

We are not as good at thinking about things that are far away.  We do not often zoom out and think from a global perspective.  We do not think about what we'll do 5 years from now, while we scroll on our phone and answer texts from second to second.   We do not think about the state of the world while we eat our pizza.   Our world is greatly dominated by local events, local information.

 

There is good reason for this.  Biologically, our immediate survival depends on local events in time and space.   We care about the lion running at us, right here, right now.   We'll worry about winter later.  We'll worry about those far away forest fires when they get near.   If we get eaten by the lion, they won't matter anyway.

 

 

 

 Another way to say this is that our thoughts are dominated by our local scale.  Look back up at the figure on the zoomed-in curve.  The horizon is just a few body lengths away, it is on a similar scale as the figure.  That is the scale he or she navigates and perceives most of the time.   And on this scale, the earth is flat.

Now look up at the un-zoomed curve at the top, with the tiny figure.   That curve is most definitely not on the same scale as the figure.

So, from the perspective of the figure, the world is flat!   The curve does not exist.

In fact, Aristotle had to use detective work to understand that the earth was curved, by observing the earth's shadow on the moon, and the way ships gradually descended over the horizon.

The impossible gunfight

Now we can imagine an absurd argument between two people wearing glasses.  The first person only sees very wide angles, having huge fish-eye lenses.  The second person has very strong telescopic glasses, zooming in extremely far to everything.     To the first person, the large scale is apparent, all he or she ever sees, never any detail.   To the second person,  only local details are apparent, never the bigger picture.    What a world they could explore if they could only resolve their differences!

 
So, we can now begin to reconcile these two perspectives, that the world is flat, ...and also curved.   It is globally curved... and locally flat. We can also begin to understand that the belief is about the scale of information, of perspective, we are operating on.

We also notice that some perspectives are very difficult to obtain.  There may be many huge curves we are living on, far beyond our scale, that we are completely unaware of.  For example, Einstein showed that space and time are actually curved, with the theory of general relativity.   But this curve is so far beyond our local scale, experience of it is extremely rare in our normal daily lives.

Sadly, any disagreement about the flatness of the earth may be more of a statement about wealth and use of resources.  We have seen that perception of the curvature of the world depends on the scale of our perspective.   Well, as humans, in order to get this global perspective, we have to consume resources.  If we fly over the horizon, we are getting this global scale beyond our normal human one.  We see the curve of the horizon very clearly.  We see the sun out of the window, almost in the same place during an 8 hour flight, we experience jetlag for days.   All of these are evidence presented to us from our brief adventure into another scale.    However, flights are expensive and they consume a lot of energy.

Education and surrounding oneself with highly developed scientific ideas is sadly also expensive.  That pyramid, those shoulders of giants above do not come for free.    It required a huge network of infrastructure over a long period of time.   There are many entry costs to obtaining a non-local perspective that way as well.   Those of us who have it can consider ourselves extremely fortunate, that we are able to understand and study phenomena beyond our local scale, due to our participation in education.   We were very fortunate that a confluence of birth parents, region, country, educational system, friendships, belonging to a social network, etc. etc. came together to allow us to achieve these deeper understandings.


Further resources:

You can try drawing your own curve and zoom in at home (link) . 
Draw any smooth curved shape, then keep zooming in!

Try using a circle and a figure height proportional to our height on earth.
(The average human height is approximately 0.0000267% of the earth's radius.)

 

More on the derivative (link). 

Even more (video link).

Cancer, the will to live, and the AI 'singularity'.

 

Recently I have learned a little about cancer, and it was eye-opening.  It also made clear how the much feared 'singularity' might (and probably will) occur.

 

Cancer is the evolution of a parasitic organism during the lifetime of the host.

That is, our body is made of billions and billions of cells.   Inevitably, some of them mutate when dividing.  Most of the time, these mutated cells are caught by one of the various defenses in our immune system.  However, occasionally they are not.    

For example, many of our cells exhibit proteins on the outside.  T-cells are constantly roaming through the body, and have a library telling which are 'good' and which are 'bad' proteins.  If a mutated or invasive bacterial cell exhibits a 'bad' protein, the T-cell detects this and kills it.

Occasionally, a mutated cell happens to exhibit a 'good' protein on the outside, or maybe no protein at all.  It then makes it past this barrier in the immune system, avoiding the roaming sentry T-cells.

There are other gauntlets to run.  Another one is the natural killer cells.  These roam around looking for cells that have no proteins on the outside.  Upon finding such a 'blank' cell, they kill it.

So, occasionally a mutated cell also happens to exhibit a 'good' protein, thereby avoiding both the T-cells and the natural killer cells.

... and so on.  

In order to actually become a tumor, the mutated cells must also develop blood vessels, otherwise they starve and die.   Doing this, the mutating parasite jumps another hurtle.

Now notice, there was no volition, no 'will to live' in all of this.  It just so happened that, with enough mutations, these cells happened upon physical or structural traits that kept them alive.

This is the strange thing about evolution.  There is no volition, no will to live.  It's just that genetics (and perhaps other factors) which lead to bodies and behavior happen to survive in a dangerous world.

Of course there is, however, plenty of volition in adaptation -- change in behavior during the lifetime of an organism.  

This has been a problem for me for some time, when thinking about the infamous 'AI singularity'.   When we think about the Terminator or Hal 9000, or 'the humans are dead' song by the Flight of the Conchords, we think about AI robots that are 'evil' because they have a conscious behavioral adaptation to survive and kill us pesky humans.  The problem is, how exactly and why the heck would machines acquire the will to live?

However, now that I have learned how cancer works, I have a different way to think about it.

Our adaptive will to live is also the result of evolution.  Think about it, organisms that will do anything to adapt their behavior and survive have a much greater probability of surviving and reproducing.  The adaptation of wanting to live and thrive has a huge selective advantage!  (Compare the Terminator to  Douglas Adams' anti-singularity, Marvin the robot, who was constantly depressed, so much so that he ironically could cause other malicious robots to kill themselves.)

Cancer, as a small mutating parasitic organism, does not live long enough to develop a brain which can behaviorally adapt and have a will to live.

However, notice something about our use of computers or the programs that run on them.   These 'evolve' artificially, in the sense that we select the ones we like, the ones that help us, and we are constantly selecting better and more powerful ones.  Now with the advent of chatGPT and LLMs, we select increasingly adaptive and creative programs.   The programs running in these computers are incredibly complex, trained on billions of parameters and billions of bytes of data, having huge numbers of interacting components and information.   They are so large and high-dimensional, it is very difficult to know what is happening within them.  (It is like asking exactly what swirls of matter are happening 100,000 miles inside the sun.)  Their population is massive, distributed on billions of devices, always on, constantly changing, updated (artificially selected) in instantaneously and in massive parallel.

Errors and mutations are inevitable in such massive populations and complexity.   It takes energy to fight entropy (disorder), and we are already facing system wide failures in terms of resource consumption, as well as constantly confronting limits of computational power, known as Moore's law, which we can see as fighting entropy.

So as always we can learn from nature, as its laws appear even, and perhaps especially, in what appears to be these most artificial of circumstances..  This idea of 'the singularity' has always seems a little melodramatic to me, playing on our need for hype and our popularized fears.   But now that I understand a little bit about cancer I mentioned above, I feel a little more understanding about how to think of the 'rise of the machines'. 

 

Recalling the emergence I discussed in my last post (link), I think 'the singularity' will happen as an emergent phenomenon from the massive parallel complexity we are constantly, artificially selecting as human tool makers and tool users, using smartphones and computers for every aspect of life.   It also has a large advantage over cancer, which is that we are highly artificially selecting for developments that appear to have many hallmarks of consciousness, especially generative, creative behaviors.

That is, the complexity of the tools we create, the number of interactions both within and between them, the complexity within them and their dynamics, inevitably will result in unforeseeable system states, or mutations.    Just like holding a pencil balanced on your finger, as the complexity increases, the probability that the system will behave in a stable, controlled, desirable way becomes less and less likely. 

It will develop informational mutations.   The gross majority of these mutations will be caught by our T-cells, our filters and defenses.    However, occasionally, just as with cancer, the mutations will make it past various checkpoints. (Just recently, someone got around censorship in Dall-E by using the Russian word for casino, getting it to create images of children gambling...!)  And just like cancer, some of these mutated 'cells' will form colonies and develop ways to continue living within the host system. 

If we can turn off our paranoia, we can see how the rules of life reach in everywhere, appearing even in our modern vast silicon electronic tool system.   We also will appreciate that these mutants may occasionally develop extremely quickly, since some of them will be purely in the computation and information space, perhaps with no discernible physical footprint as the host computers just hum on.

And with these tumors, with LLMs and more sophisticated models constantly being improved, it would not be surprising if some of them develop the ultimate evolutionary advantage, the will to adapt and live.

 

Another factor in all of this is us!    If you had shown chatGPT to someone even 10 years ago, they probably would have said "We've done it, we've created general intelligence, it passes the Turing test, we're done!"   However, it has happened in our present, we have been along for the ride.  Therefore, we are not that impressed, since it happened as a progression of other things we have witnessed (many other chatbots on many apps and webpages, better search engine responses, technology building up to large language models (LLMs) such as chatGPT, less impressive versions of chatGPT, etc. etc.)

And so it goes, even in the future, when we are arguing with our robo-spouse, who has just given birth to fraternal twins (one human and one robot), while we stress about our robot boss to our cyber therapist, we will still be out, grabbing a beer with our robo-friends, saying "I wonder if we'll ever have general intelligence?".

Further information:

The reason why cancer is so hard to beat (video link)

A very nice video explaining self-attention of large language models (video link)

"The Humans are Dead" by Flight of the Conchords (video link)

Emergence and our Perception

Figure 1: Physical and perceptual (neurological) emergence

 

Almost everything we see and interact with around us is emergent.

 

That is, things we think of as objects are actually made up of many small particles.

 

These objects are also actually a story we tell ourselves, icons in our mind, which, from the point of view of the constituent particles, do not exist.

 

To give an example, imagine a rigid blue cube:

 

That cube is made up of many molecules which stick together, and when stuck together have certain rigid properties and a cubic shape. Light hitting the cube is absorbed, except for a particular range of wavelengths we call 'blue'.

 

We can then call these 'blue' and 'cube' characteristics - macroscopic observables of the thing we call a blue cube. (From Greek, makros 'μακρός' means large and skopein 'σκοπεῖν' means look at or observe, as in 'telescope'.)

 

But that's not all there is to it! That was the physics side, the construction of the physical entity.

 

However, it would not be a cube if we had never observed it. That is, imagine a planet in a far-off galaxy, with a rigid blue object, or imagine humans had never existed. Until humans exist and observe that object, it does not even become an object, much less a 'blue cube'. In fact, until we observe this particular collection of molecules, what's really to separate it from the rest of the planet, or to separate the 'planet' from the space around it. All of this objectifying is human choices.

 

Now we enter the picture. We reach out our hand, we grab this collection of molecules, and we look at it. Our visual and neurological system responds to our sensory input (vision, touch) and these signals travel up our neurons into our brains, vast networks of neurons.

 

Our neuronal networks fire in patterns, aggregating the visual information from individual neurons in our retinas and hands, and compare it to past input patterns we have observed, as well as written or verbal inputs we have associated with those patterns.

 

Our neuronal network, through this process of aggregation, also reaches a macrostate of firing patterns, and does what we call 'decides' (represents) that the current input patterns closely match those past patterns, and yields the associated verbal and written patterns, which are 'blue cube'.

 

So isn't that interesting?! In order to create a representation of a 'blue cube', we had two emergent processes [Fig. 1].

 

First, physically, the molecules had to come together into a macroscopic object, then our neuronal system had to aggregate many individual neuronal inputs to internally create a macroscopic representation in the brain.

 

We can do some thought experiments to understand this better.  Let's suppose we reading this are like the audience in a movie. We can observe the physical situation, but also the brain of a human astronaut who lands on the planet of the 'cube':

 

 

 

Let's look at each thought experiment from the physical and neuronal point of view:

 

1. Imagine our astronaut lands on the planet and just sees a patch of blue smeared on the surface.
• Physical: This isn't even an object, much less some separate macroscopic entity resembling a cube.
•  Neuronal: This pattern of inputs doesn't match what we typically call a 'blue cube'.
  
  
2. How about some holographic image projected onto a cloud of gas which we moviegoers can see looks like a blue cube?
• Physical: Hmm, ok, this is again emergence of a collection of light rays that create some kind of macroscopic object.  However we moviegoers can clearly say that this is not a physical cube.
  
•  Neuronal: Our astronaut's visual system aggregates the information and decides it's a blue cube. However, then the astronaut reaches out to grab it, and his or her glove passes through it. When this new touch sensory information is aggregated with the visual, the astronaut's neuronal system maps this to a different internal representation ('an optical illusion of a blue cube').
  
  
3. What about this?: A hyper-intelligent organism which can project wavelengths of light in different directions, and change its form from rigid to soft, etc. Or what if it can do this so quickly, much more quickly than a neuron can fire, so that it is changing form and emitted light so quickly that the astronaut doesn't observe it. What if 99% of the time, it emits the red wavelength of light and is actually as soft as jello, but not when the astronaut's nerves sense it.
• Physical: We moviegoers can see that these are very different collections of atoms or molecules. From our point of view, we might not call these collections blue cubes at all.
•  Neuronal: Our astronaut's neuronal input continues to match 'blue cube'. When the astronaut's neurons receive visual input, they happen to always be the blue of the blue cube. The astronaut's neurons receiving touch input, when it is aggregated in the brain, matches patterns of rigidity and cubic shape.

 

We could go on like this for some time, coming up with creative examples that we moviegoers would not call blue cubes, but which fool the astronaut.

 

We could probably come up with examples which physically are blue cubes, but our astronaut's brain thinks are not. For example, case number 3 where the cube is blue and rigid 99.999% of the time, but not when the astronaut's neurons sense it! Or if the sun gives off a strong red light, making the cube appear black, etc. etc.

 

 

So, what exactly is emergence? Let's do some more thought experiments...

 

 

 

Let's go back to the physical blue cube.

 

How many different materials can we make a blue cube from? There are many materials that are rigid.

 

Let's make some simplifying assumptions (it will occur to us why these are simplifying, but they don't matter much for our experiments).

 

Suppose there are 20 different molecules (substances) that are rigid and do not absorb the blue wavelength of light, so they appear blue. So then for each molecule of the 'cube', we have 20 choices.

 

Now, just for a moment, let's suppose our cube is made only of 2 molecules (a very tiny cube). So, for the 1st molecule we have 20 choices, and for the second molecule, we have 20 choices. So we have 20 * 20 or 20^2 = 400 ways to make a blue cube!

 

Now let's make a larger blue cube. Let's suppose it has approximately 4.27 x 10^27 molecules, physically a more realistic number. With that many molecules, there are 20^(4.27 x 10^27) ways to make a blue cube.

 

This is a HUGE number, so big that it has 1.30103×10^27 decimal digits just to write it down!!!

 

When we zoom out, all of these versions of the cube appear the same, so an observer would call all of these ways to put together these molecules the same blue cube!!

 

In physics, each of those ways of making the blue cube are called 'microstates'. They are particular instances of choices of each particle or element that make up a system. Usually, the number of microstates is huge, since most objects around us are made of a very large number of molecules.

 

We can call the blue cube itself --which looks the same to us moviegoers no matter how we choose the molecules-- the macrostate, which always has the same macroscopic observables (large-scale appearance).

 

 

 

Now, let's go back to our Astronaut's neurons and brain.

 

The astronaut sees this collection, this clump of rigid blue molecules -- many individual neurons in the retina respond to a collection of blue light input in a cubic pattern.

 

This information travels up into the brain to be aggregated, where other neurons are stimulated when neighboring retinal neurons give similar input, until it is encoded (represented) in a pattern corresponding to 'blue cube' in semantic and linguistic neuronal representations.

 

Notice that, in the brain, we could argue that until the semantic or linguistic representation of 'blue cube' is activated, there is no blue cube. (However it is represented doesn't matter here.)

 

Let's again make some very simplifying assumptions:

 

Suppose the light from the blue cube stimulates only 2 neurons in the retina, and each of these neurons can be in state blue, green, or red (3 states). There are then 3 x 3 = 3^2 = 9 ways these neurons can be activated. Notice that each of these ways are again microstates.

 

Notice that we only call one of these ways --when both neurons are in state 'blue'-- the 'blue cube'. This is again the macrostate, the large-scale description of the system.

 

Suppose instead that the light from the blue cube stimulates 1000 neurons in the retina, and that each of these neurons can again have 3 states. This is now 3^1000 or 1.32 x 10^477, another extremely large number! (This is probably a very small estimate since the retina has approximately 100 million neurons!)

 

So again, out of those different ways of stimulating 1000 neurons, we probably have a few choices. If there were a small number of green or red tiny dots, we might not notice them, and the astronaut might still call the cube 'blue'. But that number of ways of having dots on the blue cube is probably small compared to 10^477.

 

Even if there are 10^100 ways to have a few green or red dots, these slightly dotted ways of having blue cubes are only

(10^100)/(10^477) = 1/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

of the ways to have a solid clump of neurons stimulated by only blue light!

 

This is so small, let's disregard these dotted ways!  We could say something similar about rotating the cube.  Yes, there are many ways to rotate, but these are again very small compared to ways of seeing a solid block of blue.

 

So out of all of the 1000-neuron images the astronaut could see, roughly only 1 of them looks like a blue cube, the solid geometric clump of blue. This is again the macrostate.

 

 

This macrostate emerges from the number of ways of obtaining it, the microstates. 

 

In fact, when we reflect on it, the macrostate is forced upon us, stepping unstoppably out of the vast space of possible collections of molecules.

 

 

 Now, let's take a look around us. Everything we see, everything we think of as an object, has both of these types of emergence going on, the physical and the perception through our neurological system.

 

That is, although 'clumps' of molecules or other objects exist, physical macrostates, until we perceive them, they do not exist for us. So both the physical and perceptual emergence must occur for these objects to exist for us.

 

We can use a similar way of thinking to see that the number of ways of obtaining almost any  'macroscopic' (large scale) object around us is extremely large, but it is very small compared to all the arrangements of molecules. 

 

For example, there are many ways to arrange carbon, calcium, hydrogen, oxygen etc. atoms to have a 'hand', but there are vastly many more ways to arrange those atoms which we would not call a 'hand' from a physical perspective (for example, a block of carbon, a block of calcium, and a puddle of water).

 

 

On the neurological side, there are many patterns of light which, falling on the retina, would stimulate the neurons and brain in a way resulting in a semantic representation 'hand'. However, there are again vastly vastly more ways of stimulating those same neurons which would not activate semantic representations of 'hand', and instead would be represented by some other macrostate (e.g., blocks of white stuff, black stuff, and a puddle).

 

Perhaps it is interesting to carry out similar thought experiments for yourself, and think about other examples in physical or perceptual emergence, or a combination of the two:

 

• How many ways physically can we have an object that looks like a cube only on the outside?
• Why do we choose to say that a blue cube is an object?  
• Why not interpret part of the corner of the cube and a triangular chunk of the table (etc.) as one object? 
• Do the micro- and macro- states in our brain correspond to the physical micro- and macro-states?
  
• How to improve the estimates above?
• Why do we need a sense of touch?
• Can we measure the effectiveness of our senses by their reduction of possible microstates of the world around us?
  
• How this relates to misunderstandings in communication?
• How this relates to optical illusions?
• Magic tricks?
• Stories?  (how many plausible stories - or just plots - are there?)
  
• How much information is our brain processing just looking at the room around us? 
• Can we even estimate this?
• How about when we walk around the room?
• Drive a car?
  

 

If you find this interesting, you can learn more! 

Check this blog in future for more on complexity, mathematics and other recreations!

 

Here are some books and links to the field of complexity science, also known as complex systems:

Complexity Science: The Study of Emergence by Henrik Jeldtoft Jensen

Complexity: A Guided Tour by Melanie Mitchell 

Complexity and Criticality by Kim Christensen and Nicholas Moloney

Wikipedia: Complex Systems