Who latched the door?

Here comes a little criminal sleuthing piece for you to think about during Christmas. Ten years ago something gruesome happend in Boulder, Colorado. Six year old JonBenet Ramsey was found dead on Christmas the 26/12-96 in the basement of her home. Please read the full background of the story here as I will only tell you about a small piece of the puzzle which I think has been overlooked through the years. The whole case is very strange, invloving things like the strangest ransom note ever and a very strange crimescene.

JonBenet was found in a room in the basement called the wine cellar. It was not used by the family for storing wine but more a kind of a storage room. This room was windowless and dark and the only entrance to the room was a wooden door. The interesting thing about this door was that it had a small wooden latch fitted at the top of the door. As the door was opening outwards the latch had no real use other than to keep the children out of the room when it was used for storing Christmas presents. Now, when the first oficer arrived at the house the winecellar door was latched. This led him not to search the room as he was securing the house for a possible intruder still lurking in the house. The latch told him that no intruder was in that room as it was latched from the outside. Later in the day the father did a search of his own and found the door still latched. He open it in found JonBenet on the floor. Assuming that it was the killer who placed JonBenet in the wine cellar he/she must have latched the door behind him/her. Think about it, who would be in the state of mind to latch this door?

An intruder would not only have to know about the latch but would also have to consciously latch it. As the latch didn't have any real use why would he. During later interviews the father expresses his astonishment that he found the door latched when he was looking for JonBenet. He even states that he was so surprised that the door knob slid out of his hand when he pulled on it the first time, then realizing that the door was latched. The mother said when interviewed that she sometimes latched the door. She did it during Christmas when she used the room to hide Christmas presents from the children and didn't want them to enter the room.

You can see the latch on the top left of the door in the picture.

The Baldwin Effect

Often in nature there seem to be a connection between learning and evolution. Some traits that are inherited today seem to first have been passed from parent to child by learning. First it was thought that genes actually could change as the result of learning, this is called Lamarckism, today we know this is not the case. The genes are the same through the life of an organism.

So how does it happen that learnt traits can enter the genome. It is really not that strange. If a behaviour starts to spread among a group of individuals and parents teach this behaviour to their children those children more adept at that behaviour will have a slight advantage. This is called the Baldwin effect.

This is interesting because when designing an evolutionary algorithm this should be accounted for. Introducing learning and teaching could have beneficial effects on the fitness landscape. The fitness of an organism should not be calculated on the genes but on the phenotype. Allowing the phenotype to change, by learning or by physically adapting, during it's life allows the genes to express their potential. The genes can be more general in their expression, you don't need a gene that says 'this creature climb trees' but instead say 'this creature is agile', 'this creature is strong' and 'this creature has claws'. The tree climbing part is part of the behaviour that is passed by learning and the physical traits increases the ability to fulfill that behaviour.

Introducing a phenotype gives other benefits. You want the genes to behave well during crossover but at the same time you want a fitness to be calculated from the genes in an easy way. Crossover works best when genes that cooperate is close in the genome. They want to get passed as a bundle. So allowing genes to be able to move within the genome, called transposition and occurs in real cells during recombination, allows genes that cooperate to move together and so increasing the chance that they will get passed together. And when you allow genes to move around it is often beneficial to introduce the concept of a phenotype that is easier to use for matching the organisms against each other.

Stirling meets Wankel

I have already talked about the Wankel engine. The beauty of one rotor spinning in the same direction all the time. The Wankel engine normally works by combustion though. This makes a quite inefficient use of fuel as the fuel doesn't burn optimally during combustion. The Wankel construction further worsens this by the shape of the combustion chamber, making Wankel engines consume even more fuel than their crank engine counterparts.

But there exist a different type of engine that makes much better use of the fuel. I am thinking about the Stirling engine or air engine as it was called by Stirling himself when he invented it 1816. This engine works by converting a temperature difference into rotational enegy. The basic principle is that a gas that is heated expands and a gas that is compressed increases in temperature. One part of the engine is hot and the other cold. Gas is expanded in the hot part and decompressed in the cold part. This construction usually make Stirling engines quite large as they need big cooling radiators.

The Wankel engine's drawback regarding fuel consumption and the bulkiness of the Stirling can maybe be overcome by combining them. We may not see Stirling engines in cars as the sole provider of energy but it could be used when acceleration is not needed. During constant speed on the highway for example. Stirlings have been used to great success in submarines where their quiet running and less need of oxygen is great benefits.

Some work has been done to fuse these two designs together. The QuasiTurbine and the Trochilic engine are good examples how the future may look like. I think there is much more to do in this field.

Wriggling on a hook

A worm of unit length is caught on a hook. He can only move in the plane i.e not in three dimensions. He can not exert any momentum against the hook which means that his center of gravity always will be on the vertical line from the hanging point to the ground. Suppose now he wriggles around as much as he can.

What is the area covered by the worm during his wriggling?

A value for the area and a short description of how you found it will suffice as an answer. Solve this puzzle and you will enter the "Fields of Gold" hall of fame together with Pooh (who solved the first cornfield puzzle and the fuse burning puzzle) and future master puzzle solvers.

Male and Female Issues

When two singles decides to move in together in a new apartment they usually have to decide who's stuff goes were. Who's TV are we going to use, who's dinner table and so on. Some of the stuff got to go to the landfill. Two hard heads may never be able to decide, they just love their stuff too much.

Funny thing is that solving that problem may just have been the cause of it! Bear with me, that sentence will clear up in a moment.

The first cells with cytoplasmic genes(genes not in the nucleus), had to solve exactly this problem,who's mitochondria are we going to use when we merge. Both cells are very eager to share and recombine their nucleus but what to do with the stuff that comes along.

The solution to this problem may very well be the introduction of gender. The female brings the cytoplasmic genes and the male has to ditch his. This is very plausible because there is a huge cost for having two or more genders. Having several genders means that you have to find a suitable partner when you want to have kids. Just mating with anyone would sure be easier. From an evolutionary perspective gender makes it harder to pass on the genes. There must be a huge benefit, and that may very well be the ability to have symbionts inside the cell.

Free the Red Queen

When you design an evolutionary algorithm(EA) you are approaching evolution in a different way than how evolution works in nature. Nature doesn't use evolution to solve a problem, evolution is an inherent property of life.

This difference is very apparent when you consider parasites and co-evolution in general. In nature parasites are something bad that organisms need to be able to counter. Matt Ridley devised the Red Queen hypothesis with regard to this. The Red Queen concept is taken from Alice in Wonderland in which Alice meets the Red Queen. The Red Queen is running as fast as she can all the time. The unfortunate Queen is under the assumption that she must run as fast as she can in order to stand still. This is how evolution works in nature. There are several levels of evolution. You have to be able to survive and counter the static effects of nature. Secondly you have to be able to cope with the changing effects, you have to be able to adapt. Thirdly it is beneficial to be able to adapt to adapting counterparts like parasites. You see the point, this reasoning never stops and we are facing the same problem as the Red Queen. Evolution guides us to be able to better take advantage of and counter, evolution.

So designing an EA you have to design an artificial environment. The easiest way is to have a static environment. The function measuring the fitness is set beforehand and is equal from generation to the next. The next stage is to introduce competition among individuals in the population. This is good if you don't have a good understanding about what the goal actually is, but you know that there must be a best way to do it, like playing a game of chess. Here many designers of EA meet a tough challenge. They see that their population converge too fast. After a while all the individuals look the same and only mutations can make progress to better fitness. This is very suboptimal. So how can you keep you artificial population diverse?

Enter the Red Queen. You introduce parasites. Here parasites are good, they force the population to be diverse. The parasites quickly adapt to the host population and prey on the majority. If one parasite evolves that is able to prey on a major sub-population his children will have much easier to find suitable hosts and so they have a clear benefit.

Introducing parasites makes the fitness landscape change. Parasites are able to make valleys less deep and mountains less high in an ever changing way. They can never really turn the landscape inside out but they can make it much easier for sub-populations to escape a local minima and they keep the whole population from settling in one valley.

So, should you introduce the parasites in the same way as you design the fitness landscape? Ideally no. In nature the parasites appear just by themselves because the Red Queen is loose. Co-evolution sometimes give rise to symbiosis like in the case of the mitochondria and sometime give rise to parasites. Ideally you should set the Red Queen free.

Burn the fuse

I have gotten a hint that my puzzles have been to analytical so far. So by the following puzzle I hope I can engage the more natural numbered readers.

You have two separate cords of dynamite fuse. Both cords will take one hour each to burn but they don't burn at even rate i.e. half a cord won't necessarily take half an hour to burn.

How should you burn the fuses to measure 45min?

Solution: Found by Pooh. See comments.

Is the coin about to flip?

Today I became aware of that my blog was linked to by someone I didn't know. I got curious of course and looked it up. The site was about flippism, that didn't make me any wiser. But looking deeper I found the key and I felt really good about it.

Looking around a bit more I found out that flippism seems populated by Icelanders. Now I felt even better being an avid reader of the Icelandic sagas. When I was younger I spent one month driving around Iceland looking up placed from the sagas. I met a professor in archeology there and tried to get him to talk about the sagas and the remains, unfortunately he was not into flippism and didn't even acknowledge that the location of Bergthorsknoll was known, the earth moves he said. I found it on the map though and I was not disappointed by the view there, just beautiful.

Driving around Iceland was adventurous. I was driving a Ford Escort which had never seen anything but paved roads. As I had set my mind on going to Snaefellsjökull I had to take the car out on the gravel. It didn't take long before the gas tank had sustained to much punishment. Gasoline started to leak out and I had to rush back to civilization consuming 1 gallon a mile.

During my high school years earning the epithet of 'flippig' was good. 'Flippig' is a Swedish word which I don't have a good translation for but it was assigned to the somewhat uncontrollable and creative. Seems like there is a connection there. So far I'm only on the watching list but who knows, maybe I will finally reach the realm of flippism.

Spinning triangle

Most engines today use the piston and crankshaft system. Explosions in the cylinders make the crankshaft turn. Is it possible to construct an engine without a crankshaft? There is one very neat design, the Wankel engine. It doesn't have pistons but instead just one rotor formed like a Reuleaux triangle. This shape is by the way the solution to the second cornfield puzzle. It is a curve of constant width. You can construct such curves from any odd edged equal sided polygon, Britain had some 20p and 50p coins that had these shapes. They had the neat property that slot machines could recognize the width easily while the coins contained less metal than a circular coin of the same width.

The neat thing about the Wankel engine is that intake, expansion,compression and exhaust happens in different parts of the housing allowing the rotor to turn continuously. A gear in the middle of the rotor turn the drive shaft. Mazda seems determined to make it practical and their Renesis engine looks very promising. I hope we will see more development of this beautiful engine in the future.

Our common friend

Animals, plants and fungi have a common friend. Inside our cells is the remnant of another creature. This little friend provides us with power, converts the food we eat to ATP which is a molecule the cell can use when energy is needed. This friend contains it's own DNA and has it's own protein plant.

Sometime in the distant past two different cells bonded. One cell absorbed the other by endocytosis and allowed it to live inside it. The symbiosis was very successful and gave rise to most of the advanced life forms we see today.

I'm talking about the mitochondrion. It divides by itself outside of the normal cell cycle, it basically lives it's own life inside the cell. This means that the mitochondrial DNA is not inherited in the same way as the nucleus DNA. We humans get our mitochondria from our mothers. Their genes mutate and evolve and allows us to study our maternal heritage. This shows us that all living humans descend on the female side from a woman living at about 150000 years ago, sometimes called mitochondrial Eve.

The interesting thing is that the genes of the mitochondria compete in the same way as the nucleus genes. What happens when genes compete? The genes are selfish as Richard Dawkins puts it. All genes try to be the one that gets passed on to the next generation. Some do it the honest way by being good and doing something useful and some try to fool the system. An interesting thing about the mitochondrial genes is that they do not compete on the same terms as the nucleus genes as they only gets passed on the female side. Being in a male is not useful for a mitochondrial gene, it can never be passed on from a male.

So here is a radical idea. In a very selfish sense it would make sense for such a gene to cause male homosexuality. If present in a mother the gene would be passed along to all grandchildren. As we humans inherit wealth this would increase the chances for the grandchildren as they don't have to share the wealth with offspring from the male siblings of their mothers.

Searching a tree

What games are interesting to us? They should not be too easy and not too hard. By that I mean, if it's too easy one player would always win, if it is too hard nobody would have any clue were to move. Examples of too easy games are abundant as many games for children are in this category. Too hard games are not around as nobody finds them interesting.

A grown up playing Nim with a child finds it quite boring as he knows what to do to win while the child still finds it interesting. We say that the game of Nim is solved. For the general version of Nim with several piles there exists a nice theory for how to play it perfectly. Nim is solved in the strong sense which mean that we know how to play the game perfectly from every possible position.

Some games are weakly solved. Weakly solved means that a player can win the game if the game is played from it's starting position. Played in this way many positions never arise and can still be unknown.

There exists some games were it's known who will win the game but no one knows how to play it that way, Hex is such a game. These are called ultra-weakly solved.

How do you go about solving a game? One way is to list all possible positions and just test them if they are winning, loosing or draws. This technique only works on very simple games as harder games contain just too many possible positions. Solving harder games you need to search the game tree.

So, how do you search a game tree effectively. You need to prune the tree as you go along. There is the common alpha-beta search method which has the advantage that it does not take much memory. It is also fairly good at pruning the tree. You should also take advantage of how the game is structured. In the game of Qubic for example there are lots of forcing sequences. For this type of games the Proof Number(PN) search method is very rewarding.

Since these search algorithms were derived computers have become faster and we have almost solved Checkers and Othello by using the same techniques. But what is needed to go further. Chess seems out of reach and Go is not even on the horizon. When Victor Allis solved Go-Muko and Connect 4 he incorporated a great amount of knowledge into the algorithm. For harder games this is too troublesome, there is just too much knowledge needed.

I think game solving could actually teach us about what we mean by knowledge. Somehow we must make the computer gather this knowledge as he runs through the game tree and we must provide algorithms where this is possible. Ideally the algorithm is independent of the problem. PN-Search is very close. It tries to prove nodes. At every moment it chooses the most proving node and tries to prove it. If that fails it reevaluates the tree and possibly chooses another node to prove.

Here I see a parallel to how mathematics works. You set up definitions and prove theorems. The theorems sometimes lead to dead ends, mathematics which never get used. Sometimes several theorems from different branches of mathematics converge and you can prove something new, the recent proof of Fermat's Last Theorem is a good example of this. To prove something it's not always best to go straight for he solution, actually taking side paths is sometimes beneficial. Who decides which proofs we should go after? Some conjectures are deemed more important than others, like the Riemann hypothesis. Why is this hypothesis important? Because even as it's not proven, mathematics has continued to develop assuming it is true. As time passes more and more conjectures are depending on it. Why have the Riemann hypothesis been given this treatment? Because it is very likely true, and it is powerful. Maybe this technique could be something to consider designing a new tree search algorithm?

A mixed hypothesis

Here follows a mixed hypothesis I have devised out of some previous posts, Running Man, Toba Catastrophe and Aquatic Mind.

Studying the conditions of the Japanese pearl divers gives good insight about how life could have been for an early human species living near the coast foraging from the sea. Interestingly the Japanese considers women better suited for diving as they can withstand the cold better with their extra layer of fat, also they were considered to be able to hold their breath longer.

Considering an early sea foraging hominid, maybe it was the women that were the primary sea foragers, diving for abalone and shellfish, and men kept more to the land?

When diving just adding a few feet to the maximum depth can make a huge difference regarding what can be foraged from the sea. This would probably cause their fur to gradually flatten and align itself nicely along the body to minimize drag and allow deeper dives.

Water drains heat to a much greater extent than air and keeping heat in is a major concern for marine mammals. Most marine mammals have kept their fur, and developed body fat. Same thing happened to the hominid group, they developed subcutaneous fat. To be able to dive and hold their breath for extended periods of time they needed to ditch panting as heat regulating mechanism. As they still spent great deal of time on land their eccrine glands developed into a water cooling mechanism.

We are now at ~70000 years ago. The supervulcano Toba erupts and throws many species into population bottlenecks. Life is really hard, especially at the top of the food chain. Many of the human species are decimated so bad that they eventually will go under, they were too dependant on finding prey inland. The semi-aquatic group were lucky though. Their extra income from the sea made them get through the most terrible years and turned them from being a very minor hominid group to the major group, about 2000 individuals.

Many years after the disaster the earth starts to recuperate. The suffocating clouds of ash and sulphur dioxide abides and the sun brings new life to the planet. Now many species find great opportunities arising. The semi-aquatic hominids find that hunting inland is easy. They find that they even can catch prey just by being stubborn and keep running. The prey gets overheated before they do because they have a more sophisticated cooling mechanism.

Spending less and less time in water they find that they don't need their fur anymore. Their fat keeps them warm at night and they can dissipate more heat without the fur when running. The little fur that is left shows the same streamlined pattern as when it was designed for gliding through water.

The gender differences regarding the sea foraging remains and is further accentuated by sexual selection.

Regarding the population bottleneck of the human species 70000 years ago we see that tracing the Y-Chromosome back in time we arrive at that Y-Chromosomal Adam lived between 60000 and 90000 years ago. This is also an indicator of a population bottleneck at this time. Fossil finds also seem to strengthen this hypothesis about a bottleneck.

Meet the Straw Man

Parallel with science there is bad science(BS). With BS I mean the collection of articles that for some reason tries to look like science but don't follow through. You can spot BS quite easily, these articles use 'Theorem' and 'Fallacy' instead of 'I am right' and 'You are wrong'. I'm not sure if they use it to deceive the real scientists or people who don't know science but recognizes the buzz words.

In mathematics you prove a theorem using definitions and axioms. If you disagree with the definitions or axioms then all is fine, if you agree the theorems follow as a consequence. You will spot BS theorems by the fact that the proof part is replaced by rambling and the definitions are nowhere to be seen.

Most sciences don't use theorems at all, mainly because nothing actually can be proven. In those sciences you often have an abundance of theories and ideas and the correctness is decided by majority of vote. The more vague the field the more theories. Physics for example don't state theorems, it's based on empirical knowledge. Physics state equations. A proper equation is generally a formula that states observed facts and is thus empirically true. And a formula is something that obeys the rules of a syntax. In this sense, E = mc^2 is a formula, but so is E = mc^3. Consequently physics has several theories explaining the same things, physicists are not bothered(too much...) about using several different theories in parallel. Sometimes light is treated as a wave and sometimes it is a particle depending on what you want to calculate.

An observation regarding this is that the more vague the field, the more problems the scientists seem to have with accepting several theories at once. I think this has to do with pragmatism. Physics is used in practice to get things done and at the same time provides answers to existential questions. When a field has no practical use only the existential aspect is left which leads to a mentality of belief. Regarding beliefs it is often hard for an outsider to understand what a specific disagreement is about. There is one God but your God is the wrong God because you call him A when he actually is called B. Saying there is no God at all is not as bad as giving him the wrong name, it is only when you enter the election and gather voters that you are a threat.

It is in this circus of vote gathering that the practice of pointing out fallacies has become popular. The formal use of the term fallacy comes from the field of logic. For example stating 'if A then B' and then observing B, it is a logical fallacy to conclude that A must be true. If it rains the grass get wet, observing that the grass is wet and then concluding that it has rained is false. This also applies to set theory, when you say 'All cats have four legs', and then after observing a creature with four legs concluding that it must be a cat is a logical fallacy. Statements are either true or false in logic(although there are paradoxes as in all complete systems as shown by Gödel).

The language of logic appeal to everyone involved in arguments and wish they could throw the fallacy argument at their opponent in a discussion. Detecting a fallacy is not that easy though, normally it is not as straightforward as the cat example above. So what do you do? You expand the list of fallacies and make them more usable. This reminds me of how New Agers have taken over the word energy. When a word gets credibility just hijack it for your own purposes. Therefore the list of fallacies has increased beyond logic. Many scientific fields have gotten their own favorites. When used right those fallacies can be very powerful but unfortunately many fallacies are very vague in their description, the consequence when you are moving away from the formal language of logic.You can find many articles relying on those fallacies in trying to debunk other theories by using them, this is very easy if you get your mind to it and use them in a sloppy way.

Creationists(Believers of a creator) tries to debunk evolution by using the fallacy of "Begging the question" or "Petitio principii" which is it's fancy name. They argue that saying "survival of the fittest" is a clean cut example of the fallacy as fittest means he who actually survive. Very true, a clean cut example of circular reasoning. Is this bad news for evolution? Not really, an interesting thing about "Begging the question" is that this does not say anything about the validity of a statement, only that it is meaningless taken out of it's context. Using a fallacy may very well be a fallacy itself.

These type of fallacies must be used with reason because there is no formalism behind them. Used in a bad way it becomes deceiving rhetoric. Sometimes you see a long list of fallacies that are supposed to apply to arguments by proponents of a certain theory, no examples, just an appeal to bow before the power of logic. Using the language of logic and the language of mathematics in many cases is just a desperate effort to lend credibility to arguments the writer thinks needs some artificial leverage. Just putting Theorem in front of a statement does not make it a theorem. So, next time you encounter the Straw Man there is a good chance you are reading BS.

The Running Man

When time comes for the Olympics many scientific magazines likes to show how inferior humans are compared to other animals. High jumpers would not stand a chance against the Snow Leopard. Long jumpers would be an easy match for the kangaroo. Speed runners would be crushed by the cheetah and swimming fast certainly is not something humans do well compared to the Tuna. But there is one thing that humans do really well that would earn a medal at the animal Olympics.

Humans have great stamina and can outrun almost any animal over long distances. This is actually one of the benefits you get from being bipedal. Bipedalism is more energy conserving than walking on four legs. We have abundant sweat glands(eccrine) for cooling, those glands also have an endurance feature as they can keep on producing sweat without any recharging phase. This thermal eccrine system in humans is quite unique. Our Achilles tendons, our big knee joints and our muscular glutei maximi makes us very well designed for long distance running.

How would a man fair against a horse over a distance of say 22miles? Actually there is a yearly competition between man and horse in the Welsh village of Llanwrtyd Wells. In June 2004, for the first time ever, the human won.

Among the people who rejoiced at this outcome were University of Utah biologist Dennis Bramble and Harvard University paleoanthropologist Daniel Lieberman. Bramble had noticed that humans certainly are not adapted for speed, humans are comparably pitifully slow, a chimp can run at 60km/h while humans tops at 30km/h. So we certainly didn't go bipedal for speed. Also, bipedal speed runners all have tails. The tail is a major balancing organ for those animals.Bramble says.

In the whole history of vertebrates on Earth—the whole history—humans are the only striding biped that's a runner that's tailless.

So when did we become marathon runners? Most scientists agree that the chimp like Lucy, the 3.2-million-year-old hominid, couldn't have been a good endurance runner. Homo erectus(40.000-1.8million years ago) on the other hand had much longer legs, he had the disposition to run, but did he have the eccrine thermal regulation of modern humans?

Many animals use panting for heat regulation, this means increasing the breathing frequency to vent out heat. Experiments on chimpanzees show that they increase their breathing frequency when temperature increase, and sweat very little. This process has two major disadvantages. It needs muscular work, which itself increases heat, and it causes an excess loss of carbon dioxide from the lungs which in turn cause alkalosis. Heavily panting animals regularly become severely alkalotic. Cooling by sweating has the disadvantage that you loose precious water. This means that animals that sweat a lot need access to water to a greater extent than panting animals.

For long distance running the thermal regulation issue is very important, it is overheating that causes many animals to stop running. Suppose hominids started to travel long distances. Moving long distances gives you the advantage to adapt to changing environments i.e. move to areas that provide better opportunities for survival.But in this case, why give up panting? Wolves wander long distances and use panting for heat regulation which reduces their need for water.

Many animals can not control their breathing, studying panting animals you see that they follow a certain pattern, the respiratory system is elastic and has a natural frequency of oscillation. If it were not so the panting mechanism would generate more heat than it dissipates. Humans can control their breath, evolving from apes, this could mean that humans abandoned panting in favour of breath control if there were any benefits to gain. What benefits are there to be able to control your breath? It allows you to dive and it allows you to speak. Maybe it was the path through controlled breathing that led to a change in heat regulation in human evolution. This in turn led to humans, who already had the disposition to run, being able to exploit the niche of distance running and further develop in that direction.

The Toba Catastrophe

Around 70–75,000 years ago the Toba caldera in Sumatra Indonesia erupted. This was a supervulcano eruption, probably the largest eruption within the last two million years. The climatic effects of this event led to a decrease in average global temperatures by 3 to 3.5 degrees Celsius for several years. This massive environmental change made life very hard for many species and created population bottlenecks. This accelerated differentiation of the isolated human populations, eventually leading to the extinction of all the other human species except for the branch that became modern humans. Genetic evidence tells us that all today living humans descend from a population of about 1000-10000 individuals living at the time of the Toba event.

Population bottlenecks increase the rate of genetic drift. The Polar Bear for example could have evolved in this way. A Brown Bear group could have ended up on an arctic island at the end of the last Ice Age. Beneficial traits such as color, size, swimming ability, cold resistance, and aggressiveness could develop in just a few generations.

What kind of environment were these humans living in after the Toba disaster? Finding prey inland could be hard if many species were decimated. Could it be that it was this occasion that led humans to seek food predominately in coastal areas and those humans who were best suited for this survived? Maybe a strand of humans had already embarked on the aquatic path and went from being a very minor group to the one that became our ancestors.

The chords

When I chose the name for this site I thought "Fields of Gold" was quite nice as I indented to write about many topics which I find interesting. The song by Sting being one of my favourites also may have had something to do with it. Now that the site has gotten a few hits on google I see that all of them tried to find chords for the song. So I will not disappoint them. The chords below was set by a dear friend of mine.

Cornfield revisited

The pilot has found the best route out of the cornfield.

Arriving to his own farm he decides to fence his own cornfield. His cornfield is not rectangular but he decides to fence it in with a rectangle. While he is doing this he finds that independently of how he tries to build the fence using as little wire as possible he always ends up with a square with a side of 100m. He curses his bad luck as his cornfield is area wise the smallest possible cornfield that has this property.

What is the area his cornfield?

PS: The cornfield is compact, it does not contain any patches of grass or anything else.

Solution: See Spinning triangle.

Unlur, a gem among games

In 2002 there was held a competition for abstract game developers. The main objective was to develop a game of unequal forces which means that the objective for the players are different. In Chess for example the goal for both players is to mate the king and both players follows the same rules, this leads to one player always have the same advantage at the start of the game. To make a game interesting you don't want this advantage to be too big. In Go which is a game of points you can adjust this advantage by giving one of the players a point handicap. In a game like Hex which is severely biased to the starting players favour this bias is negated by the second player having the option to swap colors. This swapping rule forces the first player to make a move that is not too good and at the same time not too bad, this adjust things very good but still in this case the second player has a tiny advantage.

In the competition Jorge Gomez Arrausi entered with the game of Unlur. This game has many similarities with Hex. It is played on a hexagon-shaped tessellation of hexagons. The game is played by placing white and black stones inside the hexagons, there is no restriction where to place the stones as long as the hexagon is not previously occupied. The goal of the game is different for the two players. White has to connect two opposing sides of the tessellation and black has to connect 3 sides that are not adjacent. After the first black move the next player has the option to either play a white piece, pass or swap color i.e. take the black pieces. By passing the black player places another stone and the next player again has the choice to play,pass or swap. After the second player chooses not to pass anymore the colors are decided. This nice contract opening of the game makes it possible to make the game very even.

In my opinion this game has a lot of appealing qualities. The board itself is very nice, it has six symmetry lines and six rotation symmetries, this is more than both Chess and Go. The game can't end in a draw, by stopping one player from creating his objective you automatically fulfill your own objective. The rules are very simple but the game has great depth. It is easily scalable, just increase the size of the board and you have the same game, just more complex.

There is a Spanish site called Ludoteka which allows you to play Unlur for free, unfortunately it is hard to get a good game as few plays it, but have patience and you might get lucky. Try this game out, you won't be disappointed.

The mating game

Among males and females there is an asymmetry regarding the care for the child. For most animals it is the female that give birth and take the responsibility for the child. A major concern for her is to get the male to stay and help out raising a family. If we view this in a game theoretic way we say there is a cost for raising a child and a gain which is the child itself. Initially there does not seem to be a game going on, the female chooses a man and he helps out raising the child. The thing is that this setting is not stable in an evolutionary sense because soon there will appear males that just leaves after the child is born. In a game theoretic sense it makes sense for them not to pay the cost and at the same time gain the benefits. How should the females respond to this? They want to protect themselves from the philandering males by testing the presumptive male beforehand, they start insisting on a period of courtship before mating. This makes it harder for the philanderers, if they meet a coy female(females insisting on courtship) there will be no babies. Now we have four players in our mating game. The so called fast females who don't insist on courtship, the coy ones, the faithful men who court the females and finally the philandering males who leaves after the mating and don't spend time courting. If we study this in a game theoretic sense we assign some values to the gains and the cost involved.

Suppose the payoff to each parent of babies is +15, and the total cost of raising babies is −20. Suppose the cost of a long courtship is −3 to each player. What would this figures lead to? If a coy female meets a faithful male there would be an equal net gain of 15-3-(20/2) = 2 points for each player. If a coy female meets a philanderer nothing happens and both gets zero points. If a fast female meets a faithful man both get 15-(20/2)=5 points. If a fast female meets a philanderer the female will gain 15-20= -5 points while the philanderer gains 15 points.

Looking for equilibrium, what would the relation be between the number of faithful and philandering males? This fraction can be calculated from the fact that a mixed strategy would be optimal when the expected payoff from either coy fast females are equal. Suppose the males are faithful with probability x, we would then have a probability (1-x) for the philanderers. Setting up the expected gain for the females we would have

Coy: 2*x + 0*(1-x) = 2x
Fast: 5*x + (-5)*(1-x) = 10x-5

From this we get x = 5/8 and (1-x)=3/8. So for every 8 males there would be 5 faithful ones and 3 philanderers.
Doing the same calculation for the expected gain for the men with x being the probability to encounter a coy female and (1-x) the probability to encounter a fast one

Faithful: 2*x + 5*(1-x) = 5-3x
Philanderer: 0*x + 15*(1-x) = 15-15x

From this we get x = 5/6 and (1-x) = 1/6. So for every 6 females there would be 5 coy ones and 1 fast one.

The interesting thing with this result is that it is not Pareto-optimal with regard to males and females as a whole. Not Pareto-optimal means that both parties can change strategies and gain more. With the figures above the net gain for the females is 5/4 points and 5/2 points for the males. If all females were fast and all men were faithful the parties would have a net gain of 5 points. Both males and females would struggle to get out of this mess. It is tough to be a bird.

More in depth info about this can be found in 'The selfish gene' by Richard Dawkins.

Evolving a Nim player

In the post Most Human Human I suggested that you could program a computer randomly and actually make the computer do something you didn't explicitly program it to do. I propose here an example of how this could be done.

First I will introduce you to the game of Nim. Nim is an impartial two player game in which you alternate taking matches from one or several piles of matches. The goal of this version of the game is to be the player not taking the last match from the board. In this example I will use the simplest version where you only have one pile of 21 matches and you are forced to take one, two or three matches when it is you turn.

How should we design a scheme that a Nim player can follow which is appropriate for random programming? Every position in this version of Nim can be expressed by a number 1,2,3,...,21. Some of these positions will be loosing and some will be winning. So lets use a scheme where every number is assigned a 1 if it is loosing and 0 if it is a winning position for the player to move. A program could look like the sequence below where the value of a pile with 1 stick is represented by the number to the left, and the value of a pile with 21 sticks is the right most number.

010010010101001100100

A player could use this scheme to know which move to make in every position. The example above tells the player that the initial position of 21 sticks is a winning position as the last number in the sequence is a zero. How does he know how many sticks to take when it is his move. He looks at the scheme and sees that the three following positions are numbered 010. This means that if he takes two sticks he will give the opponent a position with value 1 which he would think was a loosing position. If he have multiple options he chooses randomly between those. Note that this representation is not always consistent, but let evolution take care of that.

Now lets create an initial random population of creatures who's genome are the same as the scheme above. Cross breeding two creatures would be done by choosing a random position in the genome and swap the genes just like nature does it. Below '-' marks the crossoverpoint.

P1: 10011001 - 1010100101001
P2: 01001001 - 0101001100100

gives two possible child creatures.

C1: 10011001 - 0101001100100
C2: 01001001 - 1010100101001

Sometimes we also will change a bit randomly, this is called a mutation and allows new traits to emerge in a stagnant population. When the initial population is in place we can start matching up creatures against each other playing Nim for their life, losers will be replaced by offspring generated from parents chosen randomly from the population. After a few generations we will see that almost all creatures will have the same gene in the left most position. The total gene pool can be described like this

1XXXXXXXXXXXXXXXXXXXX

where the 1 means that every creature now know that when there is only one stick left you have a loosing position, you are forced to take it and therefore loose. The creatures that had a failing gene here would succumb fast to the more sophisticated ones. After running the program for a few more generations you will see that consensus have emerged with regard to a few more genes, like this

1000XXXXXXXXXXXXXXXXX

Now evolution has arrived at the notion that when there are 2,3 or 4 sticks left these positions must be winning as a player faced with these values always have the possibility to leave the next player with the loosing position of one stick left. After running some more generations all the creatures will look the same(except for a few random mutated ones that always will be a minority). The final dominant genome will look like this.

100010001000100010001

So, the computer have randomly, guided by evolution, arrived at a strategy to play a perfect game of Nim. The interesting thing here is that the programmer of the genetic algorithm may very well be unaware of this optimal strategy beforehand. This is different than the algorithms used in chess programs which are designed to search deep, evaluating the board from preset patterns designed by humans. If on the other hand the evaluation function was designed by an evolving population of competing chess programs we maybe could arrive at a chess program who has better understanding of chess than any human.

Declaration of Digital Rights

One application of Digital Rights Management(DRM) is to supply digital content such as text, music, video or executable, and preventing that this content is used in an inappropriate way. Copying and sharing could be prohibited and even how and when you have access to the data. The basic ingredient here is to encrypt the media using a bulk cipher. A bulk cipher is an algorithm that is fast and appropriate for encrypting large amount of data. Often you also want the decryption to be able to start at an arbitrary position in a file, then you will have to use a so called block cipher. Today the most common such algorithm is called AES which is a symmetric key(encryption key is the same as the decryption key) cipher available in CBC(Cyclic Block Cipher) mode and stream cipher mode. No asymmetric fast algorithms are known, RSA, the most common asymmetric cipher, is about 1000times slower than AES in most implementations. If an asymmetric cipher is invented that is fast enough and available for block and stream ciphering this could make many cryptographic protocols much easier. The advantage of the asymmetric algorithms is that the key transportation issue becomes easier, you can share the public part of the key. Therefore RSA and other asymmetric algorithms such as El Gamal and Diffie-Hellman are used to transport symmetric keys to be used for the actual bulk data decryption, this is done in a similar way as the one described in the sharing a secret post.

So, assuming a symmetric encryption, you have an encrypted media file, and you somehow must get hold of the key to decrypt the media. Passing the key can be done in three ways. Either you send the key along with the content and hope that the device that receives the content won't allow anyone to read the file. It has a great flaw in that if someone catches the file before it reaches a safe environment the key can be read by anyone. Another way is to have a method for generating the key based on a password. When you purchase the file a password is supplied and you will receive a specially designed content file just for you. A problem with this scenario is that the security here is based on obscurity, which means that once the key generation method is discovered the whole system breaks. This is basically what has happened to MS-DRM and forces MS to supply patch after patch of new key generation methods. Some solutions allow the user to choose a password himself. The flaw with this is that a malicious user can distribute his password and then everyone can use the file. The eReader DRM tries to circumvent this by using the credit card number of the user as input for the key generation method. This is quite smart as most users are not keen on giving their credit card numbers out to everyone. Still the flaw of security by obscurity remains. The third solution to the key transportation issue is to deliver the key separately in a license file. This allows the content provider much more flexibility and can be made very safe. This is used in OMA V2 DRM and to some extent in OMA V1, but in OMA V1 the licenses just contains the keys in plain text so it is not really safe. OMA V2 describes a much more sophisticated way of building and acquiring the license. This system requires a protocol called ROAP to be followed. This protocol is quite similar to SSL/TLS protocols used for safe transactions on the Internet, using digital certificates for authentication. These solutions relies heavily on asymmetric ciphers for key transportation.

The system with a license catches the essence of DRM. The license describes the rights granted by the issuer. This means that scenarios where the content itself plays a minor role appears. You can focus on the rights and use the license on its own to prove your rights. I think many speakers who have put forward the notion that DRM actually should be called 'Digital Restrictions Management' have totally missed this important fact about DRM.

Content providers using the simpler versions of DRM still have one more card up their sleeve. They can use digital watermarking. If a user cracks a certain file and this file is distributed unencrypted the content provider can track the file to a certain user. A digital watermark can be described as changing the original digital content in a way that does not distort the content. Take a picture for example. Every pixel in the picture is described by a number. Usually this number can range from 0 to 65535 or more. This number describes the color and intensity of the pixel. Two numbers that are close are very hard for the human eye to distinguish. This means that some of the pixels can be changed my a minute amount and no one can tell the difference. The content provider who has the original picture can compare the two digitally and see the difference, the hidden code will point to a specific user.

Eve the eavesdropper

Thin king and fat king solved their problem with the untrusted messenger by the ingenious trick of using two padlocks. They felt safe for a while but one day thin king heard that one of his gifts hadn't reached fat king, in fact fat king never even got the chest to apply his own padlock on. Thin king soon understood that his messenger once again had tricked him. He had sent his messenger away with the locked chest and soon the messenger had arrived back with another padlock on the chest. He removed his own and sent the messenger away again. In hindsight he remembered that he hadn't really gotten a good look at the second padlock, probably the messenger had created his own padlock and just applied that one instead of allowing fat king to apply his. From that moment on thin king and fat king had to find a way of being sure that the padlocks were indeed theirs and not the messengers. They found out that to do this they had to involve a third party which both of them trusted, this third party could apply non-forgable stamps on the padlocks and he promised never to apply the same stamp to the messengers padlocks.

In the digital world the same problem occurs. Here the solution is called digital certificates. These certificated are used to transport the public part of a public/private key pair. A certificate contains the public key, and some extra useful stuff like validity period, name of the issuer and name of the owner of the transported public key and finally a digital signature. The certificate is created by a trusted third party, a so called CA(Certificate Authority). The CA digitally signs the certificate with his private key and everyone can verify it using the CA's public part, which everyone is supposed to know and trust.

Natural optimization

Often we try to find the best solution to a problem. It can be, lowest price for a specific item, the loan with lowest interest or the stock with the highest expectations. Sometimes we even bring out our calculator and try to use maths to get to the answer, for example, how should I construct a can that can hold a certain volume using the least amount of material. Optimization like this can be thought of like finding the highest hill in a vast landscape. If you are lucky there is only one hill and you will get to the top just by walking upwards. If you are unlucky there are many hills and mountains. The problem turns into finding the highest peak in the Alps just by walking about. Just going upwards won't help you as you will just end up on the top of the hill closest to where you started.

How does nature do it? Think of Newton and his apple. The apple will fall to the ground because of gravity. This phenomenon can be thought of as an optimization problem. Nature wants to find the lowest potential energy. In this case nature doesn't do any better than hill climbing, it just finds the closest valley.

But sometimes nature can do better. Think of a cup of water you have put in the fridge. It will freeze. When liquids freeze the molecules line up very neatly in a configuration holding the least energy. How does nature find this configuration? It is done in a process called annealing.Using the mountain range analogy this process can be imagined by replacing Newtons apple with an extremely restless child. It will skip across the mountains randomly and slowly loose energy. The more tired the child becomes less likely it is to go uphill. Finally the child will be totally out of energy and tumble down into the closest valley. This process often comes close to a minimal energy but often the final crystal will have some flaws indicating that a local minimum has been reached.

And then we have the process of evolution. In the mountain range analogy we will have millions of creatures scattered all over the mountain range. The creatures on the hilltops will be called inferior and the creatures in the valleys are superior(or vice versa). Evolutionary selection will remove the creatures on the hilltops to a greater extent and the creatures in the valleys will mix, share traits and create a new creature that will get a position(genes determine position in the landscape) somewhere between the two parents. This process will lead to communities being formed in some of the best valleys, breeding between those will sometimes create a new creature located perhaps in an even better valley and a new community will be formed there. Finally the majority of the creatures will live in the same, possibly deepest valley.

Throw me!

We have all seen the amazing javelin throwers at the Olympics. They throw the javelin so far that they have to redesign it every other year to keep it inside the stadium. But what object has the record for being thrown the furthest by a human without any artificial help, is it the javelin?

It's a Frisbee. The record today is a quarter of a mile. The object was a specially designed Frisbee in the form of a ring, the Aerobie Pro ring. But how does a Frisbee work, what makes it fly?

A Frisbee is a cross between a wing and a gyroscope. Giving the Frisbee speed, it cuts through the air like a wing, and giving it spin it gets stability. Without spin the disc would turn over and fall to the ground. How much the disc spins and the particular design can give the disc very different flying characteristics termed 'understable' and 'overstable'. Understable discs need a lot of spin while overstable needs less.

I heartly recommend the sport of discgolf where you can play with the physics of flying and get a moderate exercise at the same time.

Lost in a cornfield

A man was flying over his neighbours lands one dark night when the plane all of a sudden started to have problems. He pulled out his parachute and jumped. Soon he landed in his neighbours cornfield, the only cornfield in the neighbourhood. He knew one thing about this particular cornfield and that was the exact layout of the field. It formed a perfect rectangle, 100m wide and very long, several kilometers. Unfortunately he had lost all sense of direction gliding down the night sky.

As he stood there among the corn he thought to himself, how should I walk to be sure to get out of the cornfield as quick as possible, in a worst case scenario?

He is able to walk along any curve he chooses but he has no means of knowing his initial direction. Also, the cornfield is so thick that even if he stands on the very brink of the cornfield he is not aware of it, he must actually step out of it to notice he has succeeded.

Solved by anonymous: Optimal walk is ~227.8m.

The solution is a so called 'yurt' curve.


The optimal value can be found by minimizing


If you place one leg of the yurt in origo and the other leg in u you arrive at the formula above.

Aquatic mind

Most people agree that humans have evolved from apes. There are many differences though between apes and humans. Many of these differences is quite subtle and not everyone is aware of them and some are quite striking. At first glance we notice the nose is different. We have bigger lips and fattier tongues, which allows us to speak with extra help from the larynx, among other things. We are not as hairy and our hair is aligned nicely and it's not furry. We walk on two legs and we have subcutaneous fat just like the whales.

Have you heard about the aquatic ape hypothesis? It states that man during his evolution at some time spent a great deal of time close to, and partly in water. This hypothesis explain all the differences above and many others very nicely. Our nose is turned downward because this prevents water from entering the nose while wading and swimming. Big lips and fat tongue help while eating sea creatures such as mussels and oysters. We lost hair and developed body fat just like the whales. We have a quite strong diving response which can even rival sea creatures like sea lions. We have grown used to water cooling and therefore sweat when warm, our cousin apes don't sweat. Our bipedalism can be a consequence of initially being 'waders'.

I don't claim here that this theory must be correct, just that it's gives quite good answers. Which other theories give plausible answers to these traits?

Some evidence even suggests that humans gave birth in water during this period. Human newborns are coated with vernix, an oily substance produced by the skin. No other land mammal, including the apes, produces vernix-coated neonates, some sea mammals do. Also a baby which is born under water will swim to the surface on its own.

For me diving and free diving in particular gives a great sensation of being calm and free. Is this the aquatic ape inside me giving me the thumbs up?

Truly parallel

Today it has become common with multicore CPU's. Much discussion has erupted about the average users benefits from this and if it's just a hype. An obstacle in they way of making use of all the cores is that the majority of the programs today is not meant to be running on machines like this, they can't themselves make use of the extra power. This is a quest for the operating system which also have a hard time achieving this as the programs are written in languages which has not been designed for the issue. Sure, Java supports threads and Unix systems allow C programs to fork but it is up to the programmer of every program to design how this is going to be used. As these systems also should be able to run on platforms with only one CPU, it is thus hard for a programmer to make use of the power, the programmer does not know how many cores the system will run on, the operating system will decide how and when the code will run.

I would like to see a new programming paradigm. A paradigm more similar to how nature makes use of parallelism. Take a human cell for example. Every cell contains the exact same program(the DNA) but still the cells can form a complex being out of a single cell. Why does not the human cells act like bacteria that just multiply, making copies of itself.

The secret is that when a human cell divides the program counter, the DNA decoding device, is not placed at the beginning of the code. The child cells will run another part of the code even thought it contains the exact same code. Tom Ray made some initial promising studies on this in his Tierra project. Also he added another piece of the puzzle to his evolving code snippets. He designed a computer language that was very robust to random changes, something the CPU in your computer in front of you is not. Make a random change to a PC program and the program will break.

But isn't this exactly what a fork in a C program does, dividing the code in two and setting the program counter to a new place for the child? Yes, it is, the difference is how the processes communicate. In a PC one process writes to memory and another reads, this has to be controlled very strictly, every process knows what the other processes are supposed to do, and who will listen to a specific message. In a living being the child processes are on their own, no message(chemical) is meant for a specific cell. This approach adds hugely to the evolvability of a system. An interesting thing is that there is nothing in the computers design that keeps us from doing it in natures way, it is just very hard to write programs like these with the languages and operating systems we have today. Ask a Linux kernel developer what he thinks about inter process communication, I think he would welcome something new.

Most human human

You must work very hard to convince the judges that you're human, You shouldn't
have any trouble doing that - because you are human.

This is what Robert Epstein told a panel of five humans who's task was to compete against computers for being most human. The inspiration for this event dates back to the earliest days of computing. In 1950, pioneer Alan Turing proposed that if a computer could successfully impersonate a human being during a free-form exchange of text messages, then for all practical purposes, the computer should be considered intelligent. To really make the humans try hard there was a little side award for the most human human.

You can argue that this "Turing test" really is not the core of the issue about computers being able to think. Many argue that all computers has to be programmed, which is true, and that this implicates that computers never really can think, which is false. The interesting thing here is that a computer really does not need to be programmed by a human. A computer could do the programming. But does this really mean anything? Yes, because programming a computer to do the programming of the second one you can allow the first one to do his job randomly. But, putting a monkey at the keyboard hacking away randomly would certainly not result in a program exhibiting thinking qualities. Yes that is correct, but then you round up a thousand, a million or more of these little programming computers and make them compete in an evolutionary manner. I bet this article would not give me any award for being the most human human, but is is really written by me? Going down for maintenance.

The Oak Island Enigma

Have you ever heard of a real buried treasure? This is a story about a real treasure hunt that has not yet come to an end. It started one summer day in 1795 when Daniel McGinnis was wandering about Oak Island outside Nova Scotia in Canada. He came across a curious circular depression in the ground and leaning over the depression was a tree which had been cut in a way which looked like it had been used as a pulley. He went home but came back the next day with two friends and started digging. After 10 feet they ran into a layer of oak logs spanning the pit. The same thing awaited them at 20 and 30 feet. At this point they were not able to dig deeper but returned 8 years later. They continued digging to 90 feet finding oak logs every 10 feet. At this point the pit suddenly filled up with water. The gang had sprung a booby trap.

Later it was found that the pit was connected with sloping tunnels to a bay on the island and the whole bay had once been dammed. It seemed there was a vault down in the pit and the oak logs had been placed to take the load of the ceiling of the vault. If someone dug close to the vault the pit would fill with water. Up to this day the treasure in the vault has not been recovered even though much work has been put into it. What do you think lies buried here? Who put so much work into hiding something on this island?

Law abiding words

Does words follow any law? Yes they do, and it is called Zipf's law. This law states that if you count the number of occurrences of words in a text of a natural language these numbers must relate to each other in a certain way. If you order the words in a row sorting by the number of occurrences you will find that the frequency of any word is roughly inversely proportional to its rank in the table.

For the English language you will see that the word 'the' is most common and occurs about 7% in a given text. Second comes the word 'of'' with an occurrence of 3.5% and on third place comes 'and' with an occurrence of 2.3%. From this follows also that the major part of all the words occur only once in a given text.

What fun can we have with this knowledge? Actually this is a good tool in cryptography. If you are given a text that you don't recognize you can start counting the words. If you find that they obey Zipf's law the text is probably not encrypted and you just have to find somebody who understands it.

Now this does not always help you. The picture above is taken from a book called the Voynich manuscript. This documents has a very strange history and no one has so far been able to understand the script. Some have wondered if it is actually encrypted, but when you count the words you see that the text obeys Zipf's law. So is this written in a forgotten language seen nowhere else? Maybe you can solve the mystery...

The lucky dozen

Have you wondered why the are twelve notes in the scale? Why not thirteen? This is actually just a lucky mathematical coincidence. The individual frequency of the tones are really not important(in Mozart's days an A was not the same as today), what is important is the relative frequency of the notes. All frequencies does not sound good together, there need to be harmony or chords striving towards harmony. Harmony is achieved when the frequencies relate to each other in sequences of natural numbers like 3, 4, 5 or 4, 5, 6, 7. When the frequencies double up we think they sound so similar so we say they are the same tone just in another octave. When you try to fit these relations together within an octave you will arrive at 12(or 19,31,72 or more) different tones for the tempered scale. Twelve is normally chosen because it accomplishes what is needed and is the least amount that does it. The tempered 12 tone scale is a good compromise because 2^(4/12) is very close to 5/4, 2^(5/12) is very close to 4/3 and 2^(7/12) is very close to 3/2 for example. You can also chose to tone your instruments for the exact chords(Just Interval) but then some of the tones outside the chord or in another octave will sound strange. The interesting thing is that it is not really agreed upon which tuning is the best, there are several rivals 'equal tempered'(most common), 'Just interval', 'Well tempered', 'Mean-tone' or 'Pythagorean'.

This is something to tell those who claim to have the ability to recognize 'absolute pitch'. They simply have a very good memory for frequencies and there isn't really an absolute pitch to remember. Mozart who is said to have had the ability would not recognize the 'A' of today's tunings.

Broom with a wroom

The witches have upgraded their brooms to spaceships. Centuries ago people were abducted by trolls or the devil himself. In those days the woods were actually scary and you could get lost, strange sounds heard were attributed to fantasy creatures. Today we in the west really cant believe in those things anymore, but we still get abducted, just not by the fairies of the woods but by aliens. Space is black and vast and who listens to scientists when you can explain it so much more interesting. Actually scientist have a very plausible explanation why people experience these things, in Japan they call it 'kanashibari', in the west we call it sleep paralysis. Sleep paralysis is not well known in the west, people experiencing it associate it with an abduction like taken away forcefully on a broomstick or taken by aliens, this is what we have been tought. Recent studies in Canada, Japan, China and the United States have suggested that it may strike at least 40 percent or 50 percent of all people at least once. I am still waiting for my first, maybe I just have to upgrade my broom.