Archive for May, 2010

G4G9, Day 5: Balancing Laptops, Mobius Music, and Egg Cartons

This is the 6th and final post in a series of posts about Gathering For Gardner: 1 2 3 4 5

I started the last day of Gathering For Gardner 9 by waking up late.

As a result, I completely missed the first talk and entered the conference room in the middle of the second talk, by Karl Schaffer, about “Dancing Tessellations”. It consisted mainly of a few videos in which a normal dance would be reflected along certain axes so that it effectively makes a video tessellation. The next one was a short talk on extending the Side-Angle-Side (SAS) similarity theorem to three-dimensional shapes using the least possible number of measurements. One of the especially interesting talks in the first session was by Linda Zayas-Palmer, on why the infinite number, 0.999999… , is actually not just equivalent to, but greater than, 1. The talk directly after that was about some of Salvador Dali’s puzzles in some of his paintings, in which you have to find a certain phrase. Turns out his puzzles are rather easy.  My favorite talk in the session, though, was by Burkard Polster on how to balance your laptop on a bedside table such that it occupies the minimum amount of space on the table and doesn’t fall off. It starts out by a simple dissection of a square by Martin Gardner, and then exteds it to show how to balance the laptop just right so that it’ll balance, thus leaving room for puzzles or whatever you would put on a bedside table.

During the break, Mom and I went up to the exhibit room, where the exhibitors were packing up their exhibits. Most of the exhibits were already gone, but a few were there that weren’t the first time. For one of these, involving curious polyhedral sculptures made with egg cartons, Mom actually got to talk with Jeannie Mosely, the person who made them. In fact, she actually got a model of the octahedron which had fairly novel methods of holding together as an example to make more.

After the break, the talks continued with a talk on 3-dimensional packing puzzles made out of various spheres glued together to make polyomino-like shapes. Due to the spherical nature of the pieces, some of the puzzles need the polyspheres to “snap” together, something which doesn’t normally happen. A rather good talk in this session was by Ed Pegg, about the Wolfram Demonstrations Project, and some of the best demonstrations that had been posted, such as a program to find the smallest box that squares of size 1/n can fit in:

Or even how to solve the Orchard-Planting Problem, involving finding a certain number of lines that pass through a certain number of points, given the required points:

The talk after Ed Pegg’s talk was one of the most anticipated talks of the entire conference: Finding a single shape that covers the entire plane aperiodically, which was an unsolved problem. Joshua Socolar managed to find a single hexagonal tile (with matching rules, though) which creates a Sierpinski Gasket-like shape when made. He also made a 3-dimensional tile which acts the same.

The 2-d tile

Afterwards, Vi Hart did a great talk on making music with music boxes with the music scores in Mobius strips, or the music boxes arranged in such a way so that the music played by one music box is played by another music box seconds later.

After Vi Hart’s talk the lunch break started, but instead of going to lunch  I went immediately up to the Gift Exchange area, where I would wait in line to get a bag full of exchange gifts from nearly everybody, repeated for everyone, who would get one also.

Apparently everybody else had the same idea of waiting in line early. I ended up behind Bram Cohen, inventor of BitTorrent as well as a whole bunch of super-duper-hard twisty puzzles which have some rather ingenious ideas behind them. He happened to have a few non-twisty, but still hard, puzzles while waiting in line, and I managed to almost solve one of them (the Cast Rattle), which involved getting the right pieces in the right place (That isn’t a spoiler, is it?). The other one, Cast Marble, I couldn’t get immediately, but I might have been getting close. Soon, I got to the front of the line to get my bag of exchange gifts and got the huge bag (I could barely carry it) as well as a few other miscellaneous items from people at a few nearby tables, such as a perplexing wooden object which looked like a gear, or a few pictures from Caspar Schwabe (I presume) of large inflatable solids, including the huge 59th stellation of the icosahedron seen on day 3.

I brought the huge bag of gifts up to the hotel room to open, and even though the bag’s heaviness was an indicator of the number of things inside, it still felt like Christmas when I opened it. There were mathematical dice with the sides only using the number nine and a set of plastic rings with interchangeable art based on the Traveling Salesman problem; there was a CD containing rather high-quality pictures as well as a digital copy of the exchange book from G4G8; A set of pieces for an unsolved puzzle and a key to open doors with using a hammer; A book about formulae that changed the world and a second copy of  A New Kind of Science(Ah, so that’s where the heaviness came from);Even a mysterious back-scratcher and tapper. These are just a few of the things in the bag, and to list them all out would certainly lengthen this blog post by quite a lot.

By the time I had gotten to the bottom of the bag, the third session was about to start so I had to rush down to see the talk by Gordon Hamilton (of the Magical Mathematics Museum) about having problems in K-12 classrooms involving unsolved problems such as the circle-packing problem (In how small a square can you pack n circles?), the 3n+1 conjecture (Do all Collatz sequences end in 4,2,1?), and others. Also in the same session was Solomon Golomb’s talk about the Pentomino Game on nxn boards. The “Pentomino Game”, is a rather interesting two-player game in which players alternate turns placing pentominoes onto an 8×8 board. The first player who can not place another piece loses. One of the most interesting talks of the session was “Fun with Egg Cartons” by Jeannie Mosely. In the talk,  she described how she made most of the Platonic Solids – out of egg cartons! The process of making these is pretty easy- just interleave strips of egg cartons at the vertices to make the edges- but the results are still interesting.

Immediately following was a talk not relating to mathematics at all (but still cool), about restoring various ancient text adventures. The talk was by Adam Atkinson, and it was about cross-compiling old text adventures (running on mainframes) so that you could play them on newer computers. Many of them are stored at ifarchive.org (the Archive for Interactive Fiction), including Acheton, probably the third text adventure ever made. Just don’t get eaten by a grue.

After the last short break, the last session of G4G9 began. Kate Jones started of with a philosophical talk involving pentomino puzzles,followed by Bill Mullins, who talked about Martin Gardner’s search for the person who wrote “The Expert at the Card Table”, probably the most important book ever on sleight of hand with cards, who wrote his name as “S.W.Erdnase”. It can be reversed to make E.S.Andrews , but from there it’s much harder. So far, they’ve found 5 suspects as to who S.W.Erdnase might be, 2 by Gardner and 3 by others, but the writer remains hidden. Hirokazu Iwasawa (also known as Iwahiro) then followed that with a talk about the subclass of “Hat-Team Puzzles” , how to solve them, as well as other variations on the problem. A bit later, Colin Wright did a double talk about “How far is the Moon?” and about notations for juggling. The former was left out (the pdf is available here) , but the talk about juggling was amazing. Not only did he juggle normally with up to 5 balls, but he also showed how to use a notation for juggling to make up your own tricks, some insanely complex and others trivial. Following that, George Hart talked about his new sculpture, “Comet!” which involves multiple smaller versions of the main model (a puzzle-like polyhedral-ish form) with different colors. It’s so big that it has to be hung on the ceiling of an atrium. After that, Mike Keith did the second-to-last talk about his book, Not A Wake, in which every word of the text- including the subtitle and the title- has the same number of  letters as the nth digit of pi does. The book goes on for 10,000 digits, with 10 stories followed by the digits of pi in that story, each story being a different style than the others.

After the last talk and closing notes, G4G9 was over. However, the fun(at least for me) continued. I was invited to dinner (along with Bill Gosper, Mom, Julian and Corey Hunts) by Dick Esterle to the Varsity Jr., an old-style fast food diner that had been operating for 45 years. Bill and Julian declined, but the rest of us went.

The diner had good food (especially the burgers) , and talking with Esterle was quite interesting. I brought a box puzzle (the same from the prelude) , and he managed to solve it in record time just by shaking it hard. He also, using the materials that were available, gave me 2 versions of the same puzzle. First, arrange 3 cups in an equilateral triangle such that a knife can reach from any cup to any other cup. Then, use the knives to balance a salt shaker in the middle of the triangle above the table. (This can be done using 2 knives) Then, set the cups so that they are just a bit too far for the knives to reach, and once again balance the salt shaker using 3 knives. Corey and I eventually solved it and put a few straw decorations on (from my solution to the problem using straws to extend the knives and only using 2 knives). To prevent spoilers, it’s at http://daftmusings.stattenfield.org/wp-content/uploads/2010/04/Neil-and-Corey-at-the-Varsity-copy.jpg .

After dinner, Dick Esterle drove us back to the Ritz-Carlton, where we went back up to our hotel rooms and played with puzzles until bedtime.

The day after that, I was waken up very early to get packed up for the airplane trip back to San Jose. We met Bill Gosper in the hotel lobby, and took a cab to the airport, at which point we waited until dawn-ish. On the airplane trip back (with an exchange in Chicago), I looked at all of the exchange gifts in the bag and Bill Gosper programmed on his laptop. Eventually, after having 2 breakfasts due to time zones, we landed in San Jose, drove over to my house, whereupon Bill drove back to his house. And because of time zones, I still had the rest of the day to play with puzzles.

Thus the epic of Gathering for Gardner 9 ended.

It was an absolutely great experience from before it even started to it’s end, and I met a lot of new people and saw a lot of puzzles and magic tricks and optical illusions. I would certainly go the next time it happens, and the time after that. Certainly, it was one of the best events that I attended ever, if not the best.

As an afterword, on May 22, Martin Gardner died. Hearing the news of this was incredibly saddening to me, as well as sudden. He was one of the most important people that ever lived, for mathematics and as well as for many other subjects. He helped popularize M.C. Escher and Godel,Escher,Bach , and introduced mathematics in a fun way to at least everybody at any of the G4Gs. He was truly amazing.

Cellular Automata

Cellular automata are simulations on a linear, square, or cubic grid on which each cell can be in a single state, often just ON and OFF, and where each cell operates on its own, taking the states of its neighbors as input and showing a state as output.

One of the simplest examples of these would be a 1-dimensional cellular automaton in which each cell has two states, ON and OFF, which are represented by black and white, and where each cell turns on if at least one of its neighbors are in the ON state. When started from 1 cell, this simply creates a widening black line. When the layers are shown all at once, though, you can see that it makes a pyramidal shape.

All layers at once

For example, in the figure above, the second line is generated from doing the rule for all cells in the first line, the third line from the second line, and so on. More complicated figures can be generated from different rules, such as a cellular automaton in which a cell changes to ON if either the cell to it’s top left or top-right is ON, but not if both are on. This creates a Sierpinski Triangle when starting from a single cell:

Stephen Wolfram developed a numbering system for all cellular automata which base only on themselves, their left-hand neighbor, and their right-hand neighbor, often called the elementary cellular automata, which looks something like this for the Sierpinski Triangle automata (Rule 18):

This code has all possible ON and OFF states for three cells on the top, and the effect that it creates on the cell below them on the bottom. Using this system, we can find that there are 256 different elementary cellular automata. We can also easily create a number for each automaton by simply converting the ON and OFF states at the bottom to 1s and 0s, and then combining them to make a binary number (00010010 in the Sierpinski Triangle example). Then, we convert the binary to decimal and so get the rule number. (128*0+64*0+32*0+16*1+8*0+4*0+2*1+1*0= 18 for the example).  We can also do the reverse to get a cellular automata from a number. Using this method, we can create pictures of all 255 elementary cellular automata:

Some of these are rather interesting, such as Rule 30 and Rule 110:

Rule 30

Rule 110

Whilst some are rather boring, such as Rule 0, which is just white, or Rule 14, which is a single diagonal line.

There are many variations on this basic cellular automata type, such as an extension of the code where next-nearest neighbors are also included. This results in 4294967296 different cellular automata, a few of which appear to create almost 3-dimensional patterns such as the 3D Tetrahedrons cellular automata (rule 3283936144 ) which appears to show certain tetrahedral-ish shapes popping out of a plane.

There are also totalistic cellular automata, which are created by basing the next cell somehow on the average of the top-left, center, and top-right cells above it. These can have more than two states, and sometimes produce very strange-looking patterns, such as Rule 1599, a 3-state cellular automata:

As well as all these, there are continuous-valued cellular automata, which, instead of having cells that can only be in certain states, have the cells have real-number values. Then, at every step a function is applied to the cell that is to be changed as well as it’s neighbors. A good example of this is the Heat cellular automaton, in which the function is ((left_neighbor+old_cell+right_neighbor)/3+ a number between 0 and 1) mod 1). It produces a “boiling” effect, in which it resembles a pot of water slowly boiling on an oven.

There are tons more 1-dimensional cellular automata; Stephen Wolfram filled most of an entire (1200 page) book with these. However, there are essentially only 4 classes of cellular automata. The first type is the most boring; it is where the cellular automata evolves into a single, uniform state. An example of this would be the Rule 254 elementary cellular automata (the first example), which eventually evolves into all black. The second type, repetition, is a little more interesting, as it does not evolve into a single state but is instead repetitive. This can include a single line, simple oscillation, or fractal-like behavior, an example of which would be Rule 18. The third type is simply completely chaotic behavior- not very interesting, but definitely more than the previous two- such as in Rule 30.   The last type, type 4, is where there are many individual structures that interact, sometimes passing right through, other times blowing up. An example of this would be Rule 110. This type is probably the most interesting to watch, as the eventual outcome is unknown.

These 4 types cover nearly any cellular automata, except for the ones which appear to be midway in between the types.

We can easily go past 1-dimension and study two-dimensional cellular automata. Probably the most famous of these is Conway’s Game of Life, invented by John Conway in 1970. In it, clusters of cells appear to grow, and then collapse as “gliders” move across the screen. It only uses 4 rules, and easily falls into the category of Class 4 cellular automata.

The rules are:

1. Any live cell with less than 2 neighbors dies. (starvation)

2. Any live cell with more than 3 neighbors dies. (overcrowding)

3. Any live cell with 2 or 3 neighbors stays alive.

4. Any dead cell with three live neighbors becomes alive (birth)

Here, the neighborhood of a cell is defined as the 8 cells that surround it.

When the Game of Life was first shown, tons of people went crazy writing programs for simulating it  on computers, and supposedly thousands of hours of computer time were “wasted” simulating these patterns. One worker at a company even installed a “Boss” button for switching the display from Life to whatever he was supposed to be working on when his boss walked by!  Conway had offered a $50 dollar prize to whoever could find a pattern that expands infinitely. This could be a sort of glider gun, which shoots out gliders, a puffer, that leaves a trail of debris, or a spacefiller which expands out in all directions. The prize was claimed by Bill Gosper when he discovered the Gosper Glider Gun.

Since then, lots of new patterns have been discovered in the Game of Life, such as a puffer train, a hexadecimal counter, a fractal-generator, and even a “computer” which will do practically anything it is programmed to do.

Parts of the Life Computer

There are many other 2-dimensional cellular automata, which can be written in a certain notation which tells with which neighbor-numbers the dead cell turns alive, and for what neighbor-numbers the live cell stays alive. For example, Conway’s Game of Life could be written as B3/S23 . Many other cellular automata can be written using this notation. Some of the more interesting ones are Fredkin’s automaton (B1357/S02468) , which replicates any starting pattern. That’s all it does, no exceptions, so there’s no possibility of making anything like an adder in it.  Another interesting one is the “Maze” rule (B3/S12345) , which produces maze-like patterns. Changing the rule to B37/S12345 creates dots that move through the shape. One of the most interesting of these, though, is 2×2 Life (B36/S125) , a rule that is similar in character to Life but has much different patterns. Gliders are also a bit more rare, although there are a lot of interesting oscillators.  In rules like these, such as Day & Night (B3678/S3478) it makes almost no difference whether the colors are reversed. Day & Night also, at the end of patterns, has lots of oscillators.

Naturally, you can extend this form to allow multiple states. Brian’s Brain (/2/3) is an example of this, in which there are three states,  and in which gliders and glider guns are very much common. In fact, Still Lifes are almost nonexistent! The notation above means that a cell in state 1 (and only in state 1) stays alive if  it has (null) neighbors, that a dead cell becomes a state 1 cell if it has 2 neighbors, and that there are 3 states (0,1,2) .

A typical simulation

There are many modifications of this rule, one which causes scaffold-like structures to form, and even one which combines with Conway’s Game of Life!

You can easily make your own rules by simply choosing numbers to put in. Many of them appear to just be chaotic, but you can find rules which create rather interesting patterns. A good one is the Star Wars cellular automaton, 345/2/4 , which starts out like the Brian’s Brain rule but soon creates structures which shoot out gliders. A fun thing to do in this rule is to make “Train tracks” which let 1×3 rectangles move around them in both directions. Of course, you can also simulate all of the Life-ish rules by changing the number of states to be 2, so that there are only ON and OFF states.

As if all this weren’t enough, there’s even a generalization of the previous into arbitrarily many rules for arbitrarily many states, as a rule table. Basically, the rules are based on a large table that tells the cell in a certain state to change to a different (or the same) state if it has <this> many live neighbors. The different rules for each state makes it easy to get the cellular automaton to do exactly what you want it to do.  A good example of this type of rule is the Wireworld cellular automaton, invented by Brian Silverman, in which electrons travel down wires simulating the connections in a computer. It’s easy to make a 1-way gate, an AND gate, a clock, a NOT gate… and nearly everything you’d need to create a computer.  In fact, Mark Owen even made a wireworld computer that calculates and displays the prime numbers!

Amazing when actually run.

Rudy Rucker has also made a lot of Rule Table cellular automata, one of the most interesting being his Cars cellular automaton, which produces racing cars in several types, not usually something you’d expect to see from a cellular automaton.  The cars also crash into each other, and, in the process, make rather strange cars.

I have also made an interesting cellular automaton, which only uses 2 states, but still shows interesting behavior on wrapped grids, called SkyscraperMakers. In it, large structures are easily made, and there is a very simple puffer which requires only 6 cells. Signals also appear to transfer through the structures, but mostly just lower the towers.

There are also cellular automaton rules where only 1 cell is actually active at any one time. An example of this is the Langton’s Ant cellular automaton, in which the moving cell has two rules:

1. If you are on a white square, turn right, flip the color of the square from white to black, and move forward one square.

2. If you are on a black square, turn left, flip the color of the square from black to white, and move forward one square.

Although this seems very simple, when the cellular automaton runs on a blank grid the pattern produced is rather chaotic. In fact, you have to wait around 11,000 steps until the “ant” produces a “highway” in which the ant repeats the same pattern over and over.

The first 200 steps of Langton's Ant

Naturally, there’s a generalization to multiple states and different rules, in which you simply tell the ant what to do when it touches a certain state. It is usually expressed using a string of Rs and Ls to show what direction the ant takes when it touches a certain-colored cell. For example, the classic Langton’s Ant rule could be expressed as RL, meaning that it turns right when it touches a cell of state 0 (white), and turns left when it touches a cell of state 1. Using this generalization, there are some rather interesting cellular automata. For example, LLRR makes a cardiod shape:

Whilst one of the longer rules, LRRRRRLLR fills space around itself in a square.

Naturally, the infinity of 1-dimensional and 2-dimensional cellular automata wasn’t enough for some people, who proceeded on to 3-dimensional cellular automata. The notation for these is similar to the normal Life notation (i.e., B (something)/S (something)), except that the numbers go from 0 to 26 instead of from 0 to 8. There are some interesting analogs of 2d cellular automata, such as Brian’s Brain, which have been discovered (B4/S) :

As well as some new rules, such as the “Clouds” rule (B13,14,17,18,19 /S13,14,15,16,17,18,19,20,21,22,23,24) in which random patterns quickly form cloud-like blobs and bridges between the blobs. The “clouds” eventually shrink down, sometimes to nothing but sometimes forming rather simple oscillators:

There has even been a version of Life in 3D, however, it turns to simple oscillators very quickly. Supposedly, gliders can be formed, but I haven’t seen any.

The problem with 3D cellular automata, though, is that computer screens are 2-dimensional. When a computer screen displays a picture of a 3D cellular automata, the front (that we see) may be rather dull, while the other side may be very chaotic, but we wouldn’t know the difference. Also, there may be lots of action inside a blob, but we can’t see what is happening inside.

An interesting way to make a 3-dimensional shape out of  a cellular automata is to simply stack all the stages of  a 2-dimensional cellular automata on top of each other. This makes the cellular automaton seem quite a bit different. Patterns like the Gosper Glider Gun in Conway’s Game of Life turn into a tower with suspension cables on one side, Langton’s Ant into a Sears Tower-like skyscraper, and Brian’s Brain I don’t even want to think about. It’s rather fun to construct these out of blocks (specifically ones that can be joined together) , as the results are often surprising.

Part of Wolfram’s book was devoted to designing and finding certain cellular automata that can do anything– calculate what 2+2 is, emulate other cellular automata- even display letters- called Universal cellular automata. The simplest of these to show universal would be Conway’s Game of Life, by making AND gates, OR gates, a memory cell, a 90 degree reflector ,and a NOT gate. Many of these base on bashing gliders together to form certain outcomes, and the NOT gate is the hard one- it needs to use a glider gun, or something to send out gliders, in order to actually be a NOT gate. Once that’s made, the rest is simple.

A similar method can be used to show that WireWorld is universal- by making the necessary logic components, various computers can easily be made, such as Mark Owen’s massive prime calculator. There are even constructions made by putting logic gates together such that 1-dimensional cellular automata can be made!

Von Neumann also designed a 2-dimensional cellular automata, the sole purpose of which was to show that computers were possible in cellular automata. The rules are quite complex, mostly operate on signals passing through wires and writing cells, and the cellular automaton has a whopping 29 states. Replicators are possible, but they use humongous “tapes” to store how the structure should be built.

Now here’s the amazing part: Even 1-dimensional cellular automata can be universal. In particular, Wolfram showed a certain 19-state next-nearest neighbor cellular automaton which, given the right setup, will emulate any other 1-dimensional cellular automata on a huge basis (20 cells per cell). Some examples of it emulating cellular automata are below:

Rule 90 and Rule 30, emulated

In particular, although it is hard to see, the 19-state cellular automaton is emulating rule 90 and rule 30, respectively.

Most amazing, though is that, though it is anything but straightforward to prove, Rule 110 is a universal cellular automaton. This was done by showing how it could emulate another 1-dimensional cellular automata class, the cyclic tag system, and working from there. Eventually, Wolfram shows it emulating other elementary cellular automata, computing, and even emulating Turing machines.

Quite a lot of cellular automata programs exist (many of them are listed at http://cafaq.com/soft/index.php), so I’ll simply list some of the best ones that I have found.

One of my favorite programs is Mirek’s Cellebration (MCell), made by Mirek Wojtowicz, which has quite a lot of cellular automata rules (200+), and even more cellular automata patterns. It has a large Life pattern database, as well as allows you to make your own rules and save them easily. Probably the only problems with this are that the speed of the automaton may vary depending on the number of life cells on the board, and that the software is no longer developed. However, you can add on small extensions and even change the source code of the online Java version. You can either download it here, or see the Java implementation.

Another program for simulating cellular automata is Five Cellular Automata, which simulates exactly 5 types of cellular automata: A small generalization of Life, using 4 parameters and q states; The Belousov-Zhabotinsky reaction, as a cellular automaton;  a cellular automata in which blobs of colors try to meet with each other, and eventually take over the board; a probabilistic cellular automaton in which “viruses” break out among the population, kill everybody, and eventually die as the population regrows; and lastly, a DLA model.  The program simulates all 5 rather well, but it only does those 5, and there are no manual editing features. This makes it so that the program is good for watching, but not useful for any experimentation. You can download it at the Hermetic Systems website.

The best of these which is being developed on would easily be Golly, a cellular automata program that has infinite universes, uses Bill Gosper’s speedy Hashlife algorithm, has hundreds of patterns, including a few Life lexicons, and even is scriptable (with examples!) in both Python and Perl. And it reads practically every CA file ever made. The only problem is that completely new rules, such as making a rule table cellular automaton, isn’t very easy unless it’s a Life-like cellular automaton (B something/S something). You can download it at the project’s Sourceforge page.

Lastly, there’s CAPOW by Rudy Rucker, which is a program for generating continuous-valued cellular automata. It supports 1D and 2D rules, as well as a number of discrete-valued cellular automata. It also has a mode in which the 2D cellular automata is extruded, based on what state the cell is at, into a 3D grid. It has quite a lot of cellular automata, can make up new ones, and includes a screensaver which shows various cellular automata animating. The only bad part is that it’s a bit confusing to make different rules or make new CA classes. You can download it at Rudy Rucker’s website.

There are tons more cellular automata that have not been studied, so the field of Cellular Automata is still an interesting field to explore in and find new and interesting rules.