2-D Rubik’s Cube
Imagine Green as the top, White as front, Blue as bottom, Yellow as back, Red as left, and Orange as right.
Image 1: rotation of top plane
Image 2: rotation of center plane (parallel to top)
Image 3: rotation of bottom plane
Image 4: rotation of center plane (parallel to sides)
Image 5: rotation of center plane (parallel to front)
Image 6: rotation of front plane
Image 7: rotation of left and right side planes [equivalent to rotation of center plane (parallel to sides)]
Three Flavors of Paradox
For my money, a paradox is a statement or a situation that doesn’t work out logically. It yields some sort of a contradiction, like concluding some fact must be both true and false.
Late philosopher Willard Van Orman Quine identified three types of paradoxes:
- Veridical - An acurate demonstration of something true that seems like it should be false.
- Falsidical - Accurate seeming, but flawed reasoning produces a result that is clearly false.
- Antimony - Correct reasoning produces a self-contradictory result.
I’d claim only the third type, Antimony, is a real paradox as I’ve learned the word. The others are cool party tricks and brain teasers, but real paradoxes have Philosophical implications.
- The Monty Hall Problem - This is a veridical paradox, meaning the result is true even though it violates intuition. In sketch: Monty Hall presents you with three doors, and behind one is a prize you might win. You select a door. Before opening it, Monty opens one of the non-chosen doors, revealing it as a no prize door, then he asks “Do you want to change your guess?”. The violation of intuition is that changing your guess at this point actually doubles your chances of finding the prize.
- Zeno’s Paradox - This is a falsidical paradox, so there’s a subtle flaw in the reasoning, so that the conclusion that seems false really is false. In sketch: Achilles is racing the Tortoise who gets a giant headstart, say 100m. Achilles sets off and reaches 100m, but by then the tortoise has run another 50m. Achilles makes up the 50m, but finds the Tortoise has run another 25m. It seems that no matter how many times Achilles catches up, he will never pass the Tortoise, even though he is running faster. The error in reasoning is to assume the infinite number of catch-ups cannot be made in finite time. In fact, they can.
- The Liar Paradox - An Antimony, and honest to goodness paradox. This is simply the statement “This sentence is a lie”. Assuming this is true yields a contradiction, as does assuming it is false.
I feel that only the third type are more than curiosities, but there are some other types of statements that I think could be part of a classification system along with paradoxes.
Some sentences are not paradoxical, but do make use of paradox and self-reference to guarantee their truth or falsehood. For example:
This sentence is not provable.
If one assumes this sentence is false, then a contradiction results, however there is no contradiction if we assume the sentence true. Therefore it is true (and for those who don’t know, the idea of this sentence is what’s behind nothing less that Godel’s Incompleteness Theorem.)
Programming has ruined my life ._.
Java is not coffee
See sharp is not a compliment
Python is not a snake
and loops are not for shoes
A stream does not have water
A library does not have books
Ruby doesn’t shine
Scripting is for the rooks
I do exactly what I’m told, nothing bad I guess
I do exactly what I’m told, that also means nothing less
Programming has ruined my life
I’ve slowly noticed that
Typing all the afternoon
I’m also getting fat ._.
The smooth motion of rotating circles can be used to build up any repeating curve even one as angular as a digital square wave. Each circle spins at a multiple of a fundamental frequency, and a method called Fourier analysis shows how to pick the radiuses of the circles to make the picture work. Decomposing signals like this lies at the heart of a lot of signal processing. [more] [code]
(!) Snake AI?
dude I saw this earlier today! My first thought was that someone just made an animation though, haha, so I guess I’m a bit skeptical. I know the game has been used for AI competitions…
Three Kinds of Metaphysical Nothing
The implementation of this text post has been left as an exercise for the reader.