Combining three images taken about 30 seconds apart as the moon moves produces a slight but noticeable camera artefact on the right side of the Moon. Because the moon has moved in relation to the Earth between the time the first (red) and last (green) exposures were made, a thin green offset appears on the right side of the moon when the three exposures are combined. This natural lunar movement also produces a slight red and blue offset on the left side of the moon in these unaltered images.
It is known that this doubling can be done with 5 pieces, but not with less than 5 pieces. The pieces cannot be cut with a knife – it involves assembling sets of points for each “piece” using an uncountable number of operations to choose the points. Thus this theorem depends on accepting the axiom of choice. Interestingly, this doubling can be accomplished in 3 or higher dimensional spaces, but not in 1 or 2 dimensions. Essentially the group that describes rotations and translations in 3 or higher dimensions is complex enough to allow this doubling, but in 1 or 2 dimensions the groups are too simple to support doubling a line segment (for 1 dimension) or a disk (for 2 dimensions).
According to Wikipedia :
The Banach–Tarski paradox is atheorem in set-theoretic geometrywhich states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e., subsets), which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not “solids” in the usual sense, but infinite scatterings of points. A stronger form of the theorem implies that given any two “reasonable” solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated colloquially as “a pea can be chopped up and reassembled into the Sun.”
The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. “Doubling the ball” by dividing it into parts and moving them around byrotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations preserve thevolume, but the volume is doubled in the end.
Unlike most theorems in geometry, this result depends in a critical way on the choice of axioms for set theory. It is usually proven using the axiom of choice, which allows for the construction of nonmeasurable sets, collections of points that do not have a volume in the ordinary sense and for their construction would require performing an uncountably infinite number of choices.
It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.”
One of the big fantasies people have about wealth and money is that it all exists somewhere, piles of gold and cash stashed away in vaults.
That’s not true. A lot of wealth is ownership. Stocks are ownership. Whether a stock is worth 10$ or 100$ is due to a lot of factors weighed and considered by “the market” . The market is people interested in buying and selling.
As an analogy, let us consider a peaceful tropical island in which there is no money, but there are coconuts and seashells. We can eat coconuts, so we tend to value everything in how many coconuts we are willing to trade for them. You have 10 very pretty seashells, and seashells like that are usually worth about 5 coconuts. We tally up your wealth, including counting your pretty seashells, and we determine you have 100 coconuts of wealth- half of it in sea shells.
Then, something changes. For some reason, people just don’t want sea shells as much as they used to. Maybe they found something better. Maybe times are harder, and seashells are pretty, but you cannot eat them, so people think they are less important. For whatever reason, the seashells that traded yesterday at 5 coconuts each are now bringing in only 3 coconuts in the island market.
Again, we tally up your wealth, including the 10 pretty seashells you own. This time, you have 80 coconuts worth of wealth.
You still have the same number of seashells. You still have your hut, your fishing pole, and some coconuts. Nothing has changed. But 20% of your wealth has vanished!
Ah, but then you say “But that was just people’s subjective evaluation. All a matter of opinion! I haven’t actually lost anything.”
That’s all the market is. A consensus of opinion. But it remains the very best way of determining the relative value of pretty sea shells and coconuts. “
It may be easier to describe chaos theory than to define in a manner that is clear.
Take the image below. There is an equation that will define this shape, ie, if you plot the graph of the equation, this is what you get:
What is interesting is that the line continues to swoop back on itself. In one instance it is in the lower left corner, and the very next, it is in the upper right corner, than lower right, then upper left, and finally back to lower left, and the cycle repeats. This is the butterfly of the so called “butterfly effect”.
Now for a thought experiment, imagine that the above image is a weather prediction model. Down is cold, Up is Hot, Left is Dry and Right is Wet. What makes this system chaotic is that if you plug in your variables (maybe pressure, wind speed, time of day, etc etc) into the equation, and you spit out a point in the lower left, so Cold and Dry is coming….. But wait… your variables were off by .0000001, hmm better plug them back in just to be safe. But low, now you’ve got a point in the upper right, so Hot and Wet….
So which is it? cold and dry, or hot and wet?
That is chaos. Tiny, almost imperceptible changes to the input not only change the answer, but they change it categorically. It doesn’t just go from cold to being slightly colder… it completely flips cold to hot, dry to wet, or both!
So what is the result of all this? to get reliable weather prediction we not only need a really accurate model/equation, we need to know our input variables to an incredible level of precision. This is why we can’t accurately predict weather out past a fairly short amount of time and why we still get it wrong sometimes when predicting things only mere moments away.”