Private obsessions

I haven't posted much lately. The reason why is that my recent obsessions haven't been fit for public posting. Some ideas are private. I'll try to explain why.

I have a very active inner process. At any given time, a large amount of my cognitive resources are devoted to figuring out what I'm feeling and what that feeling means. The subjective experience of what goes on inside is difficult to describe. I'm aware of internal impressions and artifacts which I label "energy", "chakras", "meridians", "acu-points" and other names which come from new-age and eastern mysticism.

However, I do not talk about these things. For one, I don't adequately have the words to describe them. All of these words that I apply are in some sense bullshit. Chakras are supposed to be "whirling vortices of energy", Kundalini is supposed to be "latent energy which sits at the bottom of the spine", and Meridians are supposed to be "channels through which energy flows". However, these things absolutely do not correspond to physical structures in the body, and no imaging techniques can capture them. Even when I use words like "meridian" with myself, I put them in scare quotes.

For two, due to the subjective nature of the experiences I have, they are uninteresting at best to other people, and at the worst they are crack-pottery. I've heard other people ramble on about how their energy feels and what color their heart chakra is and it disgusts me. Even if another person had similar somatic experiences as I have, I feel like the true domain of knowledge of the experience is personal and is not to be shared.

For three, I cannot give any evidence of my experience. I cannot show another person what it is like to be me, and thus even if I could describe what I feel, it would be meaningless. Of course, it might have meaning if they felt the same things I did, but in that case there would be no need to speak about it since the same process would exist within each of us.

So, how much can I share about the things I've been obsessing over lately? Even if I wanted to, the answer would be "not much". They are all internal and they are all meaningless without adequate words and access to the first-hand experiences. I wish there were some way to share these experiences with others; I feel that I might be truly understood that way. Maybe someday technology will exist to allow such a link. But for now, like all mortal beings, I remain stranded on the island that is my body.

This is how you know it's an obsession

You know it's an obsession when you're unable to take time away from it to write about your obsessions in your blog which is solely designed to write about your obsessions. Thus was my last, which is thankfully no more. It's a stupid iPhone game called Galaxy On Fire 2. In the last couple of weeks I've spent 61 hours playing it. I know this because the stupid game keeps track. Anyway, I'm done.

My new thing is learning Spanish. I recommend getting the BrainScape app for learning anything. It's a flashcard app with a small twist, but the system is remarkably better than normal flashcards. When you answer a question, you rate your knowledge of the answer subjectively from 1 to 5. The lower your rate it, the shorter the time period before it comes up again. It's super effective.

Update on the xwords - my first puzzle will be published in the NYT on Friday 3/23. There may be a shortage of papers in San Francisco that day because I will be buying all of them.

Trainyard is so good

I close my eyes and see trains.

iPhone puzzle games

There are a bunch of great puzzle games on the iPhone. However, none compare to Trainyard. This is by far the best puzzle game I've played since SpaceChem.

I think I have a problem

Whitney Houston died, and my first thought was this:



On the plus side, I think I'm reaching the Bruce Lee stage of crossword construction. On the minus side, I'm losing my humanity.

Errata

How is it that I can't play NAZI in Words With Friends? Damned socialists.

Also, hummus is excellent on fish tacos.

Crossword for class

I wrote a crossword puzzle for my class. Long story. Here's the result. Enjoy!

Here's a printable file

Here's one you can solve on a computer

If you want to solve it on a computer, you can download the free software here.

Crossword puzzle that was published today

I guess it's ok for me to put this on the web now. Have a look!

Puzzle

Big week

So far this week, I have done the following:

• Cooked a prime rib
• Deep cleaned the bathroom
• Prepared and gave a 20 minute presentation in class
• Written three crossword puzzles
• DJed a 5-hour set at a party

Things I have not done this week:

• Get enough sleep
• Go to the gym
• Beat Erica at Words with Friends

DJ Playlist

The party last week was really fun, and the new music was a hit. I made a Spotify playlist! Enjoy.

Awesome DJ music

These DJ hands will be spinning

DJing a party tonight! I found a bunch of cool music I like. I've learned that I'm a fan of a genre of music called "Complextro". Whatever.

Tomahawk - BT and Adam K

http://www.youtube.com/watch?v=8M_XETcoHE4&ob=av2e

Acid Wolfpack - Coyote Kisses

http://www.youtube.com/watch?v=SyORw8bee4o

Icarus - Madeon

http://www.youtube.com/watch?v=50KFz_wzZeQ

Crossword syndicate is a go

I have a puzzle running in the Harvard Crimson next week. The theme is "bodily fluids".

That's a joke.

Possible crossword syndication

I was recently contacted to be a part of a crossword making group that will be published in college newspapers. The guy could be genuine or full of it. I'm keeping my fingers crossed that he's not trying to get my bank account or something like that. If he's for real and you're in college, you might get to see more of my work.

The Hunger Games, again

Finished the third book. All I can say is that my early enthusiasm for book one turned to dread by the end of the third.

I'll try not to give too much away here, but there may be inadvertent spoilers.

The first book was tightly written with a believable, self-consistent plot device that drove the action forward. There was a logic to it and a sense of justice. It felt like the good could overcome the bad even in the most dire of circumstances.

By the end of the third book, that moral compass had been flipped on its head. The bad overcame the good just when you thought everything would be ok. So many likable characters ended up dead in such quick succession that I started questioning if there would be anyone around to finish the book.

To me, the third book had the same emotional landscape as a movie called "House of 1000 Corpses", directed by Rob Zombie. It's a torture porn movie (like "Saw" or "Hostel"). In the first half, you get to know a chummy group of 4 college age friends on a road trip. In the second half, you get to watch them be tortured to death.

The Hunger Games

I loved it. It sucked me in and was totally worth the entire day I spent reading it.

Structured Procrastination

Shout out to Teresa, who sent me this link: http://www.structuredprocrastination.com/

It's an amazing idea - we never get the thing done that's highest on our list of priorities. However, if we can structure our procrastinating to get done other things that are important, we will be endlessly productive.

I find that if I have a todo list, I'm never getting done the most important thing on the list. However, as long as I stick to items on the list and I'm getting one of them done, I'm being truly productive.

Words with Friends

This game has taken up about an hour of my day every day since I've got it. It's wonderful. I don't know how I never got in to scrabble when I was younger, but I regret it now since I'm getting beat left and right.

One thing I've noticed is that my opponents' personalities come through very clearly in this game. One is more flashy, one is more timid, one is more careful, and the words and strategy that follow are in tune with these characteristics.

Maybe there's room for a new type of personality test based on competitive/artistic strategy games.

Crosswords

I think I'm going to get back on the wagon with crosswords. My target is one per month. I have some friends who are pretty much stuck at home with a newborn. These two are probably the most pun-intensive people I know. They had a bunch of fun cluing up one that I gave them a few weeks ago, so I believe it will be a productive relationship.

Fructose

New obsession: avoiding fructose in all its forms. I watched a video last night about how the body processes fructose (only in the liver) and what gets created when it gets processed (really bad stuff), and I'm now scared out of my mind.

I'm planning on avoiding anything with fructose in it, including High Fructose Corn Syrup, table sugar, molasses, maple syrup, and honey.

Here's the video: http://www.youtube.com/watch?v=dBnniua6-oM

It's by Robert Lustig, an endocrinologist and pediatrician, given at UCSF in 2009. It's long. To sum up his conclusions:

Fructose metabolism leads to hypertension, diabetes, heart disease, obesity, and gout.

Fructose makes your body think it's starving, and so people will continue eating for longer if they are on a high fructose diet.

Glucose not so much.

I take that back

I rescind my statement that the iPhone is bad for me. It is probably the best piece of technology that I've purchased. I've had it for a week now and already I'm at the "how the hell did I live before this?" stage. Siri is a complete help in organizing my life. I'm more productive, more well organized, and I sleep better (since I set a reminder to tell me to go to bed at a decent time). No more missed meals due to forgetfulness. No more forgetting to take pills. I know the technology to make this happen has been there. But now that I can just say something into my phone and have it pop up on a to-do list, there's no barrier to organization.

Stupid iPhone

Causing me to lose sleep. Too many apps.

By the way, Bjork's Biophilia app is awesome. So is Words With Friends. I'm ZebrafishHatchery.

Correction

I've been using my vocabulary wrong. Apparently, those "automata" that I've been creating have actually been Turing machines.

Another day, another automaton

Created another automaton to model an internal property of numbers when they go through the collatz function. There's a clear pattern that emerges. For every (3x + 1)/2 step, a 4 is appended to the end. Lemme explain.

The automata looks at the value k in base 3, where k = either x/2 or (3x + 1)/2, depending if x is even or odd, respectively. However, I insist that there are only even numbers in the base 3 representation of k, so every time there's a 1 or a 3, I subtract 1 from that spot and add 3 on the next trit down (trit = 3-valued bit). This continues until all values are even or a 3 squirts out the end. If the 3 squirts out (this is a technical term) then it means that x/2 isn't an integer, and therefore the next number in the sequence will be 3x + 1. In base 3, this is the same as adding a 4 to the end of the sequence, which is even, so we can proceed.

Now, every time that there is a 3x + 1 step, the 4 gets appended. However, the cool part is that any 4 that existed previous will stay there. So, a sequence which alternates multiplication and division steps might generate this sequence for k values:

2404

4244

20444

24444

44444

After this, there follow several division steps.

It seems that if a number gets to a sequence of 4s, then it will fall precipitously for the next several steps. There seem to be opposing forces at work here. If the number is getting bigger (by alternating multiplication and division steps) then this sequence of 4s builds up on the end. However, the 4s act like sandbags in a hot air balloon - the more there are, the more likely it will crash back to the ground.

Maybe this thing is provable yet.

Intuition

This is related to the Collatz conjecture.

I can't help but feel that these numbers are building something. The way that they move seems to suggest a deeper structure building up underneath them which allows them to fall back to 1. If I could only see that structure. I've tried visualizing these values in so many ways but I haven't yet found their true form, the one that causes the pattern to fall out like an elementary truth. Maybe it doesn't exist, maybe it's just noise. But the problem seems too simply stated for that.

Cellular Automata

Nerding out with a new Cellular Automata that I wrote to model the Collatz sequence by putting everything into the form 2k or 2k + 1 where k is written in base 3. There are tantalizing patterns, but nothing clear enough to hang a proof on.

New iPhone!

It's beautiful. I'd take a picture of it, but unfortunately the only thing I can take a picture with is the iPhone. Just spent two or three hours downloading apps. I missed dinner. This definitely counts as a new obsession.

Building on the base 3 findings

I started poking around with the maximum odd values in the sequence of numbers. An extremely robust pattern occurs. It turns out that the maximum odd values are always of the form 4*(sum of powers of 3) + 1.

For instance, the number 27 makes its way through the number 3077 before falling back to 1. 3077 = 4(3^0 + 3^1 + 3^2 + 3^3 + 3^4 + 3^6) + 1.

219 goes through 557 before falling.

557 = 4(3^0 + 3^1 + 3^4 + 3^4) + 1.

79 goes through 269 and then plummets.

269 = 4(3^0 + 3^1 + 3^3 + 3^3 + 3^4) + 1.

I must say, I'm excited to see this. If this is a canonical form which can work for both the multiplication and division steps and stay reasonable, then this might lead to something great.

Strange result

Just playing around with values. I started seeing how large the set was of numbers which peak underneath a certain value under the Collatz map. For instance, the number 27 eventually goes to 1, but it goes through the number 4616 to get there. There are 1171 numbers below 4616 with such a property, which is quite surprising, as 4616 seems to be a rather dull number. However when you look at it base 3, a surprising pattern jumps out. Other numbers with large sets below which pass through them seem to have a similar pattern.

4616 = 2022222 in base 3

19682 = 222222222 base 3

47978 = 2102210222 base 3

Each of these values has hundreds of lower values which pass through it. There are other examples as well, and they all end in long sequences of 2s when looked at mod 3.

Thoughts from last night

I wanted to get these thoughts down quickly, as I tend to forget.

These relate to the Collatz conjecture.

1. There might be some surprising results when looking at large numbers of elements simultaneously, studying the graph the same way one might study a gas. There could be some analogue to pressure and temperature which stays constant as a set of elements evolves through time.

There might also be an analogue to entropy. If one could demonstrate that entropy must increase over time as a well-ordered set evolves (say for instance all integers from 1 to 2^n), then the existence of a loop might disturb this entropy.

2. There is an almost-but-not-quite predictive relationship between the binary expansion of a number and its parity sequence. By parity sequence, I mean the sequence of ups and downs that happen to an element x under the accelerated Collatz function. For instance, the number 3 has an up (to (3(3) + 1)/2 = 5) then another up (to (3(5) + 1)/2 = 8) then a down (to 8/2 = 4). If we label ups with 1s and downs with 0s, then the parity sequence of 3 starts with 110. 3s binary expansion is 011. In fact, for all numbers from 1 to 4, the binary representation is a mirror image of the first two terms of the parity sequence. However, as you start examining more terms, random errors start cropping up. For instance, the parity sequence of 5 is 100, but the binary expansion is 101. At any given step, only a fraction of elements exhibit a new parity error. For the life of me I can't find a pattern. If I could demonstrate that the sequence of elements that develop a new parity error is random (i.e. irreducibly complex), then I could demonstrate that a proof of the conjecture would be impossible.

There are two sequences that never develop errors. Anything with ...000000 as a binary expansion always has parity sequence 000000..., and anything with ...1111111 as a binary expansion always has parity sequence 1111111... It appears that parity errors happen more regularly on elements with binary expansions with roughly equal numbers of 0s and 1s.

Density

If you are unfamiliar with the Collatz conjecture, see here: http://en.wikipedia.org/wiki/Collatz_conjecture

In my previous post, I spoke of the density of numbers which pass through a given number under iteration of the Collatz function. I wanted to clarify what I meant by that.

Let C(x, n) be the set of all numbers y which are less than n for which repeated iteration of y under the Collatz function yields x. For instance, if x = 5, then C(x, 12) would include 10 (since f(10) = 5), and would also include 3 (since f(f(3)) = f(10) = 5). However, it would not include 20, since 20 > 12.

Let d(x) (for density) be defined as the limit of the size of C(x, n)/n as n goes to infinity.

I have a new conjecture, and I'm working on a proof for it.

Conjecture: if x is equal to 0, 2, 3, or 4 mod 6, then d(x) = 0.

if x is equal to 1 or 5 mod 6, then d(x) > 0.

If I can prove this, then it shows that if there exist counterexamples to the Collatz conjecture, then they have a non-zero density. This is because it is possible to show that any even number will iterate to an odd number, and any odd number will iterate to a number which is 1 mod 6 or 5 mod 6. If x is a counterexample, then it is possible to generate y which is also a counterexample such that y = 1 or 5 mod 6. If there exists such a y, then every element of C(y, n) is a counterexample, and so the density of counterexamples would be greater than 0.

However, there is some evidence that counterexamples must be non-dense. These statements together could form a contradiction, thus proving that the Collatz conjecture is true.

Evidence (probably not proof) that the Collatz conjecture is true

If you're not familiar with the Collatz conjecture, see here: http://en.wikipedia.org/wiki/Collatz_conjecture

The best evidence I can come up with that the conjecture is true has to do with the density of counterexamples in the integers.

Restricting the Collatz map to the odd integers produces an interesting result. Let G(x) be the set of odd numbers which pass to an odd number x without going through any other odd numbers. For instance, G(5) would include 3, since 3(3) + 1 = 10, and 10/2 = 5. G(5) would also include 13, since 3(13) + 1 = 40, 40/2 = 20, 20/2 = 10, and 10/2 = 5.

This set of numbers can be generated in this case by the formula G(5) = {(4^k * 2 * 5 - 1)/3 where k is a non-negative integer}. If we choose k = 0, we get 3. k = 1 produces 13, and k = 2 produces 53.

However, not every number has the same odd inverse generating function. For instance, if we look at the number 7, the numbers that map to it without passing through other odd integers include 9, 37 and 149. G(7) = {(4^k * 4 * 5 - 1)/3 where k is a non-negative integer}. k = 0 produces 9, k = 1 produces 37, and k = 2 produces 149.

The functions are similar, but not identical.

If we look at the number 3 in the same fashion, another interesting result occurs. There are no odd numbers which map to it.

It turns out that the general result for G(x) is determined by the value of x mod 6. If x = 1 mod 6, then G(x) = {(4^k * 4 * x - 1)/3}. If x = 3 mod 6, G(x) is empty. If x = 5 mod 6, then G(x) = {(4^k * 2 * x - 1)/3.

Another interesting result comes from looking at the mod 6 values of G(x). If G(x) is non-empty, then the values of G(x) cycle between 1, 3, and 5 mod 6.

Putting this all together, it implies that for all odd numbers which are not of the form 3 mod 6, the tree which is generated by looking at successive iterations of G has the same shape. What this means is that if we have a number x which is not in the set of values which converge to 1, then the tree generated by iterations of G on x is the same shape as the tree which converges to 1. Since anything that maps to x must also not map to 1, the density of numbers which don't map to 1 should be non-zero over the integers.

However, we can show that if we take a set of numbers A and apply the Collatz function to all of them, as long as there are roughly as many even and odd values, the average of f(A) will be about 3/4 the average of A. Similarly, after two iterations of f, the average of values will be about 9/16 what they were when they started. As long as we can assume that the results of the set after an iteration will be approximately equal numbers of even and odd (which has not been proven, but seems likely) we can show that the average of the set will decrease regularly unless the majority of the elements are less than 4. Thus, the set of numbers which don't converge to 1 must be small in the integers. Otherwise, it would follow that this average would stop decreasing well before all of the elements dropped to a sufficiently low level.

So, there's one argument that counterexamples, if they exist, must be dense. And another that says that if there are counterexamples, they must be non-dense. The only possibility then would be that either the arguments are flawed (totally possible, since they are not rigorous proofs) or there are no counterexamples.

Attempting to prove that the Collatz conjecture is unprovable

For those of you unfamiliar with the problem, see here: http://en.wikipedia.org/wiki/Collatz_conjecture

Latest tack on the problem: trying to prove it can't be proven true.

My thoughts are as such. What does it mean to prove a statement for all elements in a set? Let's say you've got a statement K that you want to prove for all elements in a set A. Now, let's say that you can pick any element x in that set and through some method establish that K is true about x. Does this mean you have a proof?

No, it does not. Proofs must be finitely written and must generate results in finite time. If A is an infinite set and in your "proof" every element of A requires at least one extra bit of information to be written, then you can't say it's a proof because it is not finite.

Similarly, if you can establish that K is true about every element in the infinite set A, but every element requires at least one extra operation to establish its truth, then you don't have proof.

So, I would establish these as necessary for the existence of proof of a statement K over all elements of a set A:

1. There exist sets A1, ... , An such that the union of these sets is A. There is a finite number of these sets, and they are all definable in finite space. It is possible to show that the union of these sets is A in finite time.

2. For each i from 1 to n, all elements of Ai have a property Li which implies K. Li can be checked for the entire set Ai in finite time.

The Collatz conjecture seems to fail at the intersection between steps 1 and 2. I've tried multiple ways of defining sets of integers that imply the conjecture, but the requirement of finitude in space and time prevents a "proof" from existing. If there is a finite number of sets which cover all integers, then each set takes an infinite amount of time to establish the conjecture on. Conversely, if I can find a set sharing an easily checked property which implies the conjecture, that set is non-dense in the integers, and so it would require an infinite number of such sets to cover the integers.

My current view of the conjecture is that it is true but unprovable in finite space and/or time.