The fixed point of the cow within the cow… and time travel.

Consider a matryoska, russian dolls that you open up and have another doll that you open up, etc etc.

A matryoshka is a doll with a matryoshka inside.

Now, imagine this could go on forever, an infinite recursion of a doll within a doll within a doll…

This couldn’t happen in real life, after all, there will be a point that the doll will be made of so few atoms that it wouldn’t look like a doll anymore. In fact, this is sort of the idea that motivated Democritus, the greek philosopher from around 400BC, to propose the concept of atoms, atoms providing an end to the philosophical difficulties of making things smaller and smaller. But, time has passed, Democritus died, Newton was born, solved the mathematical difficulties of these kind of arguments, and then died too, and luckly now we feel empowered and gutsy enough to think about a doll within a doll within a doll within a doll… even if in real life this wouldn’t be possible.

In your minds eye imagine that point that is common to the center of all dolls, yes, all infinite number of them.  This point is called the fixed point.

A simpler, more visual example might be of help. Look at the label of a chocolate box, where a nun carries a tray with a chocolate box whose label has a nun that carries a tray…

A nun with a funky hat.

All those chocolate boxes within chocolate boxes have one point in common. If you where going to imagine the point where the next chocolate box will come from, that would be the common point, the fixed point.

The fixed point is a property of maps, a mathematical transformation, that converts something into a something within itself. Maps are very general kinds of transformations.

Mathematically, an object in the original “Something” (the domain) could be called $$X$$. The transformation $$M$$ would take $$X$$ to $$Y$$, which is also inside the “Something”. These are all related by: $$X rightarrow Y=Mleft(Xright)$$.

Any map that takes you inside of yourself has a fixed point.

The oval on the left gets mapped into another oval in the right, that is smaller and within the space occupied by the original oval. The red arrows indicate how a point at the top (bottom) of the oval gets mapped to another oval.

A fixed point would be a point $$F$$ such that $$F=Mleft(Fright)$$, that is, after the mapping nothing happens to it. In the diagram this is illustrated by the green arrow.

If you where going to map using $$M$$ the small oval on the right, $$Y$$, to another oval, $$Z$$, just like $$Y$$ was inside $$X$$, the oval $$Z$$ would be inside $$Y$$. Repeating this until you get sick of this (or reach infinity) would show that the fixed point $$F$$ is common to every single one of those ovals. There is a mathematical theorem that says that if a map maps something into itself, it will always have at least one fixed point.

Let me say that again to summarize: Maps have at least one fixed point.

One of my favorite applications of maps and fixed points is time travel. Imagine a person, let’s call him McFly. McFly’s person, life, memories, history, everything that he is and has been, will be labeled by $$X$$. McFly is friends with a mad scientist who gives him a time machine $$M$$.

Your friendly neighborhood mad scientist next to your average time traveler.

With it, McFly can travel back in time, and change his own history, becoming a new self $$Y=Mleft(Xright)$$. He can for example go back in time, deposit a penny in a bank account, wait for many years collecting interests, take the money out in the present and become rich. The new McFly, $$Y$$, is now rich. McFly could also go back in time, prevent his parents from ever meeting, and he never been born. The new McFly $$Y$$ doesn’t even exists.

These paradoxes are what make imagining time travel so much fun. The problem is that it also makes it seem implausible, violations of cause and effect being most unphysical. Except for one kind of solution called closed-timelike curves.

Closed-timelike curves would be like traveling back in time but not affecting the future at all, therefore, not violating causality. Closed-timelike curves is time travel that is consistent with itself. For example, a closed-timelike curve would be that McFly travels back in time to deposit the money in the bank whose collected interests will allow the mad scientist to build the time travel machine that will allow McFly to travel back in time to deposit the money… etc etc.

A closed-timelike curve is a fixed point $$F$$ in time travel $$M$$, such that $$F=Mleft( F right)$$.

A closed-timelike curve would not be one when McFly changes history in such a way he is unable to travel back in time to change history. For example, if McFly lost the penny he woudn’t be able to deposit it in the bank and thus the mad scientist won’t have any money to build the time machine creating a paradox.

The theorem that there must be a fixed point in any map implies that there will also be closed-timelike curves. Time travel in one of those wouldn’t violate causality, but might not be to useful either.

I leave you with a picture of a MinusTwoFish-approved cheese brand.

The cow within the cow within the cow within the cow…

Also, a very cool video of the effect of self maps and fixed points. Click it!
Flash Video of Droste Effect

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“A witty saying proves nothing.” –Voltaire