## Infinity, inevitability and π

### December 10, 2013

For reasons which need not concern you dear reader, I’ve been thinking about numbers as of late. There’s a common trope out there which states that since π is infinite and never repeats, every finite string of numbers you care to come up with occurs somewhere in π, simply because it is infinite and never repeats itself. I should point out right at the start that this is believed to be true by the majority of mathematicians, but it hasn’t been proven. It is certainly possible to create an irrational number which lacks this property. A favourite example is:

`0.0110001111000001111110000000111111110000000001111111111...`

which is written in base-10, yet only uses zeroes and ones. Every subsequent group of zeroes and ones is one digit longer than the previous group, hence it never repeats and it is infinitely long. But no matter how far you search, you’ll never be able to find the string ’492′ in this number. Similarly, we could take π and replace all the nines with zeroes:

```Actual π:   3.1415926535897932384626433832795028841971693993751058209...
Modified π: 3.1415026535807032384626433832705028841071603003751058200...```

No matter how far you search this new irrational, the string ’492′ will never occur, yet the number is still just as infinite now as it was before we started dicking around with it.

I don’t think this logic is a shocking revelation, this stuff has been dealt with and is reasonably well understood in a post-Cantor world. The reason I bring it up is because —to me— it sounds like this argument is often trotted out by the multi-verse or infinite-universe crowd as proof of our non-uniqueness. If the universe goes on forever, then there must be infinitely many exact duplicates of Earth (where I am right now writing this exact same blog post) out there because you can only arrange a bunch of particles in so many ways. Is that really true, or is it only as inevitable as being able to find any finite string of symbols in an irrational number?

## The Hydrogen Sonata

### November 3, 2013

My last week in Seattle at McNeel headquarters and I ran out of stuff to read. I found ‘The Hydrogen Sonata’ by Iain M. Banks in the local Barnes & Noble and decided to give Mr. Banks one more chance. I’ve read two of his books (Transition and Stonemouth) before and although he is clearly a gifted author it’s not the sort of (science) fiction I enjoy reading.

I haven’t gotten very far into The Hydrogen Sonata yet and I realise I may be missing large parts of the story as it is the tenth book in something called ‘The Culture Series’, but so far it’s —again— not my kind of story. Too many humanoid aliens, intragalactic travel far too easy, and lots of silly names like Gzilt, Eshri, Briper Drodj and Banstegeyn. I understand things need to have a name and you cannot really get around this when writing a story set in a galaxy densely populated with alien civilisations, but it adds a thin patina of ridiculousness to the whole endeavour in my opinion.

I did come across a scene that struck a chord with me, especially in light of the discussions sparked by one of my recent posts ‘Worrisome trends in architecture education‘. In this scene we learn something about the composer of a famous piece of music (long pauses represented by ellipses removed by me):

“I have no idea,” the old man said, smiling. “But the point is the Hydrogen Sonata is an elaborate, contrived attack on the sort of composition it represents. He [...] hated clashing, atonal music. He was basically taking the piss, showing how easy it was to write … how difficult to listen to. Now the piece he’s most remembered for.” He shrugged again. “‘Such is fate,’ as they say.” He gazed out to sea for a moment, then added, “One should never mistake pattern for meaning.”

Iain M. Banks, The Hydrogen Sonata

Sound advice I’d say.

## Six most poignant takes on religion

### October 20, 2013

Today part 2 of my 2 part series on religious music. Yesterday I listed my six favourite religiously inspired musical compositions, today it’s all about holding up a critical mirror to that very same religion[1]. To qualify for this listing, a piece has to be musically up to snuff, funny or ironic rather than serious or angry and focus on either religious dogma or religious practice.

From funniest to somewhat less funniest:

1. Always look on the bright side of life by Monty Python. Composed by Eric Idle. Nothing quite sticks in your head like this brilliant ending to this brilliant film.

2. The Spanish Inquisition composed by Mel Brooks. “Torquemada; do not implore him for compassion. Torquemada; do not beg him for forgiveness. Torquemada; do not ask him for mercy. Let’s face it, you can’t torquemada anything!” (I linked a high quality version of the video which unfortunately lacks the non-musical introduction).

3. The Vatican Rag composed by Tom Lehrer. Mathematician and musician and satirist, what more can you ask for?

4. Creation Science 101 composed by Roy Zimmerman, modern master of the clever rhyme.. “… and you are just beginning to, educate yourself when you, shun, evolution.” Definitely check out his other stuff too.

5. The Good Book composed by Tim Minchin. Not his best work, but his other stuff either reflects on Islam or is angry (neither of those links safe for work).

6. Herod’s Song composed by Tim Rice and Andrew Lloyd Webber. The sceptics anthem… “Prove to me that you’re divine, turn my water into wine. That’s all you need do, then I’ll know it’s all true.” I’ve long since had a soft spot for Jesus Christ Super Star, it seems to me to be a very human scale version of the Biblical accounts, where nobody is totally good or totally evil, but rather everyone has decent motivations which lead them to immoral acts in the face of an imperfect world. Plus the music and the dancing in the 1973 version is totally awesome.

[1] Again, I’m going to limit myself to Christianity.

## Six religiously inspired musical masterworks

### October 19, 2013

I’m in Seattle at RMA headquarters for three weeks discussing the future of Grasshopper (don’t worry, not whether there is a future, but what it might look like). Although there is a great apartment for me to stay in, I’m separated from my books and my music. I don’t hold with Kindle or some such nonsense —paper for me thank you very much— and my laptop speakers can theoretically play my music but what they mostly do is repeatedly stab it in the kidneys until it’s lying as a bleeding and broken corpse on the floor.

However it did get me thinking about music categorization and I thought it would be fun to put up two blog posts with the best humanity has to offer from two opposing groups. Today, my six favourite religiously[1] inspired masterpieces, tomorrow my six favourite pieces making light of religion.

## Video Art

### April 1, 2013

Just a quick post with some of the best digital video art I’ve come across. Heavy focus on geometry, algorithms and abstract imagery. In no particular order:

## Book Reviews [2]

### February 24, 2013

I’ve read some more since my last book-related post, though not as much as I’d have liked. First the ones without any redeemable quality. I’m not even going to upload cover pictures for these, try and avoid them if you can at all:

• Kathy Reichs; Flash and Bones (typical popular tripe, happens to be a No. 1 best seller)
• Jeff Abbott; Cut and Run (typical popular tripe, not a best seller per se, but certainly written by an author who wrote other best sellers)
• Michael Ridpath; 66° North (typical popular tripe feeding on wide-spread antipathy in the wake of the financial crisis, books fails miserably to deliver on tagline “In Iceland, revenge is best served at arctic temperatures…”)
• Ruth Rendell; Not in the Flesh (I’m increasingly annoyed by the tendency in modern detective stories and tv series to let the murderer kill everyone before the detective in charge finally arrests the one remaining suspect on account of all the others being dead, put some frikkin’ thought into it already and solve the case before 4 more people die).
• Robert Ludlum; Bourne Trilogy (typical popular tripe, highly repetitive writing. All of them international best sellers obviously) I’m amazed how the movies have absolutely nothing in common with the book apart from a few character names).
• Seth Godin; Linchpin (Self help bunkum, normally I quite like Godins books, but this one was hogwash).

Now for the middling to good stuff, in no particular order.

## NVAL2 discovered

### January 5, 2013

Now the hunt is on for NOVO2, the gene that causes certain men to drop their voice tonality to unnaturally low bass levels while uncontrollably spewing exaggerated and sensationalist clichés like “killed everything in its wake” and “destruction on a scale never before seen by scientists”.

## A half-arsed idea for a computer game

### September 30, 2012

For a while now I’ve been thinking about computer game worlds. Although my schedule is overflowing with things I ought to be doing, I’m unfortunately the kind of person who cannot properly work until a competing train-of-thought has been dealt with. Hopefully writing this post will put my mind at ease long enough for me to focus on writing an article for AD magazine, recording and editing my acceptance speech for the Acadia 2012 award for innovative research, finishing the new RhinoScript compiler for Rhino5 and trying to go hiking a few more times before winter truly arrives.

Aaanyway… computer games. I used to play a lot on my father’s Acorn computer. Think early 1990′s. Mad Professor Mariarti, Starfighter 3000, Spheres of Chaos, Lander, Tower of Babel, Lemmings, Super Foul Egg, NebulusCataclysm …. the list goes on. Good times. Then nothing much until I got an XBox console about 2 years ago, but even there I log maybe 4~5 hours a week. Maybe.

The Lander game on Acorn Risc Os.

It is quite shocking how much the graphics and physics of games have advanced in this time-period, but equally shocking how little progress has been made with regards to fun. But that’s a story for another blog-post. Increases in storage, memory and processing over the past 20 years have allowed game developers to create humongous worlds for games to be acted out in but as far as I know most —if not all— commercial games have hand-crafted worlds which puts a limitation other than hardware on the size of a game; namely the amount of work needed to design and draw the geometry involved.

I’m reasonably familiar with the worlds of Red Dead Redemption and Just Cause 2. Although both are very big by my old-fashioned standards, they aren’t nearly big enough to truly give a feeling of boundlessness. If you spur on your horse you can ride across the entire RDR world in 5~10 minutes. And although the world in JC2 is much bigger, it’s extremely repetitive and therefore travelling loses its meaning. Although I am sure that the world-builders use/write algorithms to automate tasks (such as placing plants or rocks), it seems that these algorithms are not used to generate data during game play.

For a long time now there have been landscape generators available and some of them appear rather impressive, however it seems most of them are mere proof-of-concepts that fall well short of actually generating enough data to challenge the quality of hand-crafted worlds. I acknowledge it is very difficult to generate terrain, vegetation, plant-life, roads, settlements and all the other things needed for a full blown open-world game. But let us assume this nut has been cracked and that we can generate an endless amount of unique landscape based on a finite collection of settings. Let us call these settings the Terrain-tensor (τ). It may contain properties to do with soil, vegetation, roughness or a myriad of other characteristics. How would we apply such a terrain generator? It will most likely be quite computationally expensive to generate large terrains to the level of detail we’ve come to expect. Although far-away parts of the world need only be generated in low-poly approximations, it still seems like an uncomfortable sacrifice to spend cycles on generating a large world if that results in a marked decline in visual quality.

Another problem with generating a large ‘flat’ world is that you can often see a long way. The world is there, you can see it, you can ride around in it as far as you want. In such a case, the only benefit to having a world-generator would be to remove the boundaries of the map and although the game may now well be infinite, you can only travel so far so fast, and you can therefore only encounter new environments at a fairly limited rate.

But what if you cannot see very far? In that case the world generator would not be constrained much by what it is already showing you. It could adapt the τ and generate an environment that is actually controlled (in part or in whole) by the actions of the player. Fog or darkness would be one way of limiting the information given to the player, but I was thinking of something a bit more interesting:

We’re all very familiar with what it feels like to be on a spherical world. It’s just that our real world is so big that for all intents and purposes it might as well be flat. The radius of the Earth is roughly 6000 kilometers meaning that for every kilometer we travel in a straight line the surface of the Earth drops about 5 centimeters due to Earth’s curvature. Typical landscape on Earth has a larger curvature. What if we shrink the size of the world? What would it feel like to be on a globe with a radius of 10km, 1km, 100m, 10m? There are two interesting visual distances associated with a spherical world; the distance to the horizon (dH) and the distance to the furthest visible object behind the horizon (dF). The former distance represents the area completely visible to the player; the local world. The latter distance represents the area that is fixed at any given time; the global world. Anything beyond dF though must be generated when the player moves in that direction and of course the τ for this newly generated piece of landscape is up for grabs.

There are three numbers that define dH and dF; the radius of the world (r), the height of the largest object (h) in the world and the elevation of the camera (e). Let us write down some equations that describe the relationships between these numbers, all the while assuming a perfect spherical planet.

dH = squareroot((r+e)2 – r2)

α = arccosine(r/(r+e))

dHw = 4π2r/α

AH = 4πr2 sine2(α/2)

dF = dH +  squareroot((r+h)2 – r2)

β = α + arccosine(r/(r+h))

dFw = 4π2r/β

where:

dH = the distance from the camera to the horizon.

α = the angle between the camera and the horizon as measured from the planet centre point.

dHw = the distance along the planet surface from the camera to the horizon.

AH = the surface area of the visible ground (everything inside of the horizon).

dF = the distance from the camera to the tip of the furthest visible objects.

β = the angle between the camera and the furthest visible objects as measured from the planet centre point.

dFw = the distance along the planet surface from the camera to the furthest visible objects.

The whole point of using a spherical world is that it limits how much of it you can see at any given time. However this characteristic dissipates as the world radius grows larger. However a very small world is problematic too as the objects on it will be relatively large and thus there will be very little ‘undefined’ area left over. Also, a small world does not allow for big terrain. You cannot grow a 30 meter cliff face on a planet with a radius of 20 meters without it looking very silly indeed.

So let’s say we have a world with a radius of 50 meters and the camera is 4 meters above the ground, which is a fairly typical elevation for a third person game. We’ll populate our world with trees and buildings, but no massive landscape features, so we’ll limit the highest objects to 15 meters.

These values put the horizon roughly 20m away and the furthest visible objects roughly over 60m. The total world area is a little over 30,000m2, of which a bit less than 1200m2 is visible, which is roughly one thirtieth. The length of the horizon is about 120m and the length of the defined world boundary is nearly 300m. So let’s say we walk 60 steps in a random direction. This will put us at the old world boundary. Half of what we see now we’ve seen before, the other half has been generated while we walked. There was no constraint to the τ (the terrain-tensor) for this newly generated landscape, though we do want it to conform somewhat to the landscapes it borders on.

Since we’re moving along the surface of a sphere, our landscape is two-dimensional. This means we can draw a two-dimensional tensor field where certain coordinates for a fixed τ. Between these coordinates terrain tensors can be interpolated:

Now if we move along our surface world, we can use this tensor-field to determine what new landscapes to generate at the boundary. But even more interestingly, we can generate a new tensor field based on the game play history. For example, imagine we’re standing in the middle of a field and we walk in a straight line due North. After 60 steps we’ve reached the old boundary of the world (i.e. where the boundary was before we started walking) and before us we see a giant swamp. Now we walk 60 meters due South until we’re back where we started. The swamp has disappeared beyond the visible boundary and we’re back in the field. Now we walk 60 steps in the NNE direction, very similar but not identical to the earlier path taken. Now, instead of a swamp, we’re greeted by a thick forest, even though we’re only ~15m away from the point where we turned around not so long ago. This should not be possible, but because we can generate brand new landscapes along the visual boundary, our spherical world in fact behaves as though it has a large negative curvature, rather than the positive curvature we’d expect from a sphere. After all, what we’d expect from a sphere is that if you walk in a straight line, no matter what direction you walk in, eventually you’ll always end up in the same spot, i.e. directly opposite the point from where you started.

In practice this principle could be implemented in a number of ways. It could be that the walking direction always affects the τ along the boundary. Or it could be that only certain gateway paths result in a change in τ. Think of it as being stuck on a small constant world, and eventually walking into a completely different, but also constant world once you’ve figured out how to get there. It is even possible to change the size and topology of the world itself, growing it or shrinking it as one navigates its surface.

Like I said, a half-arsed idea. It needs a lot of work but I’m not a game developer and I hope I can stop thinking about this now. If anyone ever implements this idea —or an idea vaguely like it— I’d love to try it out.

## The Dark Side Of The Universe

### August 27, 2012

Biology (especially evolutionary Biology) and Cosmology have always been interests of mine. My level of understanding in either field is probably best described as “blundering amateur”. I am not able to parse —let alone make use of— equations such as one would find in Relativity, Quantum Mechanics, Thermodynamics or Game Theory. I am however a firm believer in the notion that pretty much everything can be explained* through regular language without recourse to mathematics.

I am at the same time disgusted by the sensationalist approach of many documentaries made these days by the likes of Discovery Channel & National Geographic. I switch over almost immediately whenever I’m confronted with a gravelly voice prophesying destruction on a scale hitherto unimagined by scientists ‘when we return after these messages’. Fuck you Discovery Channel for ruining science by removing that which is best about it; knowledge. Even the quality of BBC documentaries has plummeted to lamentable depths over the past decade, which is clearly the most glaring sign of the upcoming apocalypse we could ever hope to get.

At least there is an ever growing number of individuals and small groups who are making quality stuff and distributing it on YouTube and the like. Potholer, Sixty Symbols and Ozmoroid are just a few examples of people who make some excellent content that is free for all. We also see more and more universities recording lectures and putting them on social media for all to see.

Last week I stumbled upon a lecture series by Sean Carroll (from CalTech) about Dark Matter and Dark Energy, two topics about which I knew preciously little. These lectures (24 × half an hour each) are unfortunately not free —they are in fact rather pricey— but they are excellent. Carroll is by far the most intelligible lecturer I know on topics as complicated and unintuitive as Big Bang Cosmology, The Standard Model and Dark Matter/Energy. I have a feeling he pulled as few punches as humanly possible and despite the very high information content, Carroll remains calm and composed. The script is well written (except for a few rather lame jokes) and seems exhaustive in terms of both factual and historic content. I especially liked his treatment of the WMAP data of the Cosmic Background Radiation. It is quite shocking how much information there is embedded in this one image.

Wilkinson Microwave Anisotropy Probe data

Carroll explains what we know for sure to be true, what we think might be true, what we suspect could be true, what we know couldn’t possibly be true, what we’ve yet to know and how we came to know these things. I highly recommend this production for those who wish to learn more about the ‘Dark Side of the Universe‘.

* if one wishes to teach rather than merely explain, I concede that mathematics is often unavoidable.

## Fortuna Tasting Menu: Wine

### June 2, 2012

I’ve been living in Poprad, Slovakia for about 3 years now. I knew the place well since I used to visit throughout my childhood with my parents, but I’ve only ever been here while in holiday mode which is not the same as living mode. One of my main gripes about the place is that it’s difficult to get good food and good ingredients. The supermarkets sell very standard stuff and I have to frequent all of them because they specialize in different categories (Billa has good ham, Kaufland good veggies, Tesco is better at cheese and alcohol, HyperNova does good wine and ground meat etc.). The local market is rather pathetic in terms of selection. The overwhelming majority of restaurants and diners in this country are dismal as well. You’re usually fine while ordering mashed potatoes and schnitzel, but anything involving spices or good meat will almost certainly be a disappointment.

Imagine my elation when we found a fantastic place at walking distance from our home, quite by accident about a year ago. The serving staff speak decent English, the restaurateur is passionate and the chef is a genius. He has the knack of creating a dish around a key ingredient without exaggerating the taste. I know that sounds easy, but I doubt it is.

Today we attended a special tasting menu based on Slovak traditional ingredients and wines, and I was surprised to learn how many good ones there are produced here. The following list is mostly for my own reference. The wines were all selected from two native wineries; Ostrožovič and Vino Nichta. There were a total of six courses, each accompanied by two selected wines.