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.

G4G9, Day 4: Lasers, Sculptures, and Balloon Polyhedra

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

We woke up the next day, and soon realized that the first talk had already started, but only by around a minute. Luckily, the conference was in the hotel I was staying in, so I only arrived a few minutes late. The first talk was by Jean Pedersen, about the extended face planes of various polyhedra. The next few talks were rather interesting:  Zdravko Zivkovic introduced a puzzle called “MemorIQ” where you have to make various shapes out of octagonal pieces which are colored on the sides. The sides of the pieces touching must also be the same, so it is a bit of a challenge to make a square with the pieces. Al Seckel then did a talk on “The Nature of Belief”, talking about various ambiguous optical illusions which change completely when you add a simple line to them, as well as a music track reversed which originally sounds like gibberish, but when words are added, comes out very clear. Greg Federickson did a talk on “Symmetry vs. Economy in Dissections of Squares and Cubes”.  In it, he showed many demonstrations of  dissecting squares and cubes into many smaller squares and cubes, in very symmetrical ways and also in the minimum number of pieces. He also showed examples for hinged dissections, some of which were very ingenious, especially for the cubes.  Lastly, Robert Crease talked about his new book about some of the most important equations in mathematics and science.

After a short break, the 2nd session began. Pablos Holman stated out with a great talk about “Hackers and Invention” in which he demonstrated how to kill mosquitoes by shooting lasers, changed the voicemail sound on Al Seckel’s phone by spoofing his caller ID, displayed a robot that wheels up to people and shows them their passwords, and showed how to pick a lock very quickly using a filed-down key and a hammer. After this talk, I went out with Bill Gosper, who was going to show John Conway the Universal Game Of Life Computer which Calcyman had made computing Pi. Bill also showed Conway some other Game of Life patterns, such as the same universal computer computing the digits of the Golden Ratio, and a Python script for going to a particular step in a Life simulation faster than the normal algorithm, which he demonstrated by simulating a pattern to a googol-1 steps. Because of this, I was a bit late for the last talk of the day, the overview of the math sculptures that were to be made later that day at Tom Rodger’s house, which ranged from a button knot to a huge zonohedral pavilion.

I had a quick lunch (i.e, none) and boarded the bus that would be going to Tom’s house. On the way there, I tried to figure out some particularly hard puzzles which had little or no instructions, and also talked with some of the other attendees. When we arrived, they had a lot of Japanese-style lunches set out on a table for us to eat before building the various sculptures and seeing some of the things that were already set up. Some of the most interesting things there were a metal polyhedral-ish sculpture that George Hart was making, an impossible box that you could stand in, and a huge black hyperbola that towered over everything else.

After eating my lunch, I helped build the base for the zonohedral pavilion by soaping the pieces and then placing them into place on the supports. When that was done, they started on the roof of the pavillion, and I showed a few puzzles to other attendees, inlcuding a version of the Enigma puzzle as well as a “chopstick” puzzle using some of the left-over chopsticks from lunch.

Afterwards , I helped out on another sculpture, this time a metal sculpture of a three-dimensional Peano curve, which had to be put together using  near-identical pieces and screws. The pieces were very rusty, so my hands got very dirty. Eventually it was almost done and I wandered off somewhere else. Back near the house, Vi Hart had been showing people how to make various polyhedra out of  balloons, such as simple octahedra and cubes.

I went with Gareth Conway and Max to explore a section of the landscape which Max said was an entrance to a gold or a silver mine, and which was almost completely covered with leaves from the surrounding trees. At some point, Max said that we’ll get famous for discovering this gold mine, to which Gareth responded that he was already famous for that he knew 130 digits of pi. I promptly responded with all of the digits of Pi I knew (only 30), and Gareth corrected me when I added on a few extra digits. It’s good that Michael Keith, the author of a book entirely written in Pilish wasn’t there at that point, because then I’d have to listen to quite a lot of digits of Pi. Eventually, however, it turned out that the “gold mine” was actually just a well.

Meanwhile, the polyhedral balloon-making had gotten completely out of control:

I went back to the main area, where I saw that a lot of the sculptures had been finished, such as the Chinese Button Knot and George Hart’s sculpture. I got to talk with Clifford Pickover about various things, such as the non-paradox that 100% of all integers have a 9 in them, and about some of the artwork in The Math Book, Pickover’s new book. Nearby was Ivan Moscovich, whom I talked with as well about various puzzles, such as his Mirrorkal series of sliding block puzzles in which you have to make a certain image with the pieces, which have mirrors on them so that the first puzzle is figuring out what configuration the blocks should be in afterwards. Soon, nearly all of the sculptures had been finished except for the pavilion which was almost finished and it was getting dark.

We had quite a nice dinner, although the tables were full so I had to sit nearby, where Gosper was.  We talked for some time, and I mentioned a formula that can calculate Pi to 42 billion digits but then soon diverges. After the dinner, I went into Tom’s house which, as I have said before, is absolutely filled with puzzles. I played with a few puzzles, including  a 3-piece burr and a few Japanese puzzle boxes but then encountered a puzzle that fell apart and then was impossible to put back together. By that time, it was time to go back to the hotel. I boarded the bus in the back- right next to George Hart and a few other people who had made the sculptures at Tom’s house that day, who I talked with for the ride back.

It had been a great day, and there was only 1 day of the conference left.

G4G9, Day 3: Random(Blog), Crazy Detectives, and the Rubik’s Cube

This is the fourth post in a series of blog posts about Gathering For Gardner 9: 1 2 3

We started out the 3rd day by changing the hotel where we were from the Peachtrees to the Ritz-Carlton, where I missed the first talk, which was apparently about “The Odd One Out and Unrevealing Coin Weighings”

The very first talk that I saw, then, was by John Edmark about “Geometric Patterns of Change”. It was mostly about the sculptures that he has made, some based on the Fibonacci sequence and the Golden Angle, while others were on various spirals which could change direction by simply changing the angle at the top. Adrian Fisher also did a talk on that he was making Custom Designed Mazes, specifically hedge mazes for any people who had a castle somewhere and liked mazes. Last in the first session was a 15-minute talk by Ed Pegg, called “Meet the Attendees”, which was where he would bring up various attendees who weren’t doing talks and have them describe themselves in 20 seconds, as he would show a slide that he had made for them.  I thought that he would only bring up the attendees who wanted a slide in the presentation.

Turns out, I was wrong. He really had made 70 individual slides, one for each attendee who wasn’t giving a talk, including me.

I was around 5th, but because many of the attendees had decided not to come up, I was instead in 2nd place for a 20-second talk. Of course, I hadn’t expected this, and so I had around 30 seconds to figure out what I was going to say.  When my time came, I went up and gave a very short description of my website, this blog, and my Scratch Projects, somehow in less than the 20 second I had. Many other people came up and gave short descriptions of what they did, some seeming to go over 1 minute, but Ed’s talk still came in before the 15 minutes he had.

The next session started out with two Dr. Matrix (one of Martin Gardner’s characters, a numerologist) impersonators, Scot Morris and Bruce Oberg, talk about the number 9. Scot’s talk was about “Cosmic 9” which detailed how 9 lay at the center of the universe: He pointed out the methods of counting out nines, that 9 was a square number, and so on. Bruce Oberg’s talk was about “Nein to Nine”, in which he pointed out how bad 9 was. My favorite line in his talk: “First, I will show that 9 is lazy. What happened in 9 A.D.? (pause) ABSOLUTELY NOTHING!” After a few more talks, Stephen Wolfram did a talk on all the work he has been doing, such as Mathematica, Wolfram|Alpha, and A New Kind of Science, a rather large book weighing in at 1,200 pages.

We had a short lunch break, in which I skipped eating in order to buy a few puzzles, which included a combinatorial puzzle in which you have to rotate 3 controls in order to get 10 disks to line up, as well as an interesting packing set of polyhedra. After this, I went back downstairs for the 3rd session.

Steve Macknik and Susana Martinez-Conde started out with a talk on why we are fooled by magic. They pointed out that this was because of the magician’s skillful use of misdirection, and showed us a few videos on this effect, starting out with a card trick:

And then following up with a case of “Whodunit”, where there are 21 changes in the scene:

David Kaye also did a talk on how to perform magic for groups of  children, using a video as an example where he is dressed up as a clown and proceeds to do a trick with scarves, except that many things go wrong while he is doing the trick. Adam Rubin then did a talk on “Gravity Unmatched” which was a magic trick where a knife, attached to a string which goes over a pole and is tied to a pen, is falling towards him, yet it stops just before stabbing him. Kenichi Mura then did a talk on using Reulaux triangles for buckets in a chaos experiment.

There was a short break, in which I went to the Thinkfun exhibit showing nearly all of the games and puzzles that Thinkfun has made, from its first puzzles based on the Chinese Rings to the classic Pentominoes to the new Tipover. I talked with some of the creators, such as Bill Ritchie and Tanya Tompson, and said that many of their old puzzles were really neat, and that perhaps they should do sort of a “2nd edition” of some of them.

The last session of the day was themed around the Rubik’s Cube, and started out with Jerry Slocum doing a talk on the history of the Rubik’s Cube which was very interesting especially in the part where he talked about various Rubik’s Cube variants, such as the Void Cube or some of Bram Cohen and Oskar Van Deventer’s twisty puzzles. Lucas Garron followed up by talking about speedcubing and other types of Rubik’s cube. My favorite talk of the session, though, was Bram Cohen’s demonstration of the twisty puzzles that he has been making, in which the cubes can have very strange forms once twisted in certain ways (They no longer in any way resemble cubes) and also where the cube is distorted and so will not permit certain moves once twisted. Many of the cubes he and Oskar have invented can be seen at Oskar’s Youtube page:

Rik Van Grol, editor of Cubism For Fun, did a talk on “The Quest for God’s Algorithm”  which is the algorithm which solves the Rubik’s Cube in the minimum number of moves. He detailed on how the number has gone down from a high 60 to a lower bound of 20 and an upper bound of 22. (News Flash! Tomas Rokicki has found an algorithm which solves the Cube in 21 steps. Could this be God’s Algorithm?) Roice Nelson, creator of many wonderful programs, then did a talk on his program for displaying 3-d Rubik’s Cubes as 2-dimensional stereographic projections which you can rotate.  Julian and Corey then went up and gave a talk entitles “Fun with the Minsky Circle Algorithm”. It summarized nearly all of their research with the Minsky Circle Algorithm, which is supposed to make circles, but they managed to tweak the variables so that it makes crazy fractal-like structures. For some reason, the plots of the periods often have symmetry, often based around a central point:

Click to view full size

After the last session, we waited while the room in which the talks were held was being converted into a dinner/magic room. While we were waiting in line to get food, a person managed to find me and said “Stephen Wolfram wants to see you.” I was absolutely amazed by this, so I followed her to where, in fact, Stephen Wolfram was. I talked with him for a bit about various cellular automata and his book, and then went back in line to get food.

The magic show was amazing. It started out with Mark Mitton bringing Gareth Conway (he must be getting awfully tired of these magic shows) up to demonstrate an optical illusion with a rotating spiral. Then a dancer came up and performed an act in which she would produce seemingly endless flowers and cards from a single flower. Mark went back up for an act in which he would get a (very confused) audience member to perform a magic trick, without him speaking any words. A few other magicians came up for acts, and Gary Foshee presented a gift to Tom Rodgers. Lennart Green did an amazing card trick where he would blindfold himself, duct-tape his entire face,  cover it with aluminum foil, and then perform a magic trick, sometimes spilling cards, but performing the trick flawlessly. I was actually called up for a trick by Derek Hughes, in which he would perform a card trick in which supposedly, whatever answers I gave to his questions, he would show that I did not have free will by showing that I chose one particular card.

Apparently I do have free will, because I managed to somehow mess him up by not cutting the cards.

There were many other acts, and the show in general was great. In the above video, there’s a multicolored blob to the left, which was because the first act was of Caspar Schwabe blowing up a giant inflatable model of the 59th stellation of the icosahedron.

After the magic show, we went back upstairs and went to bed.

Gathering For Gardner, Day 2: Fractals, Puzzles, and Magic

Continued from a previous post… and the one before that

The next day was Thursday, marking the start of the talks, where various mathematicians, optical illusionists, computer programmers and magicians would give short 10-30 minute talks about various subjects. The talks started at 8:30, but we got there a bit early, so my mom dropped off my exchange gift (a puzzle where you have to put together 9 nonahedral shapes together to make a nonahedron), while I watched the start of the talks. The very first talk was by Erez Lieberman-Aiden, who talked about how the human genome might fold itself into spacefilling curves, rather than in a big tangle. The talk was supposed to be 30 minutes long, but he finished 3 minutes early, so (due to a rule/tradition that any speaker who finished before his time limit was given 1 dollar for each minute that he was under time) he received 3 dollars. Vladmir Bulatov did a talk on models of hyperbolic geometry, starting with Escher’s Circle Limits and moving on to computer models and animations. Jason Rosenhouse also did a talk on “The Monty Hall Problem, Revisited” in which he described various variations on the Monty Hall problem, such as a Monty who completely chooses random doors, and sometimes shows the car before he allows you to make a decision. Gary Foshee did a 1-minute talk on the Tuseday Birthday Problem, based on the original birthday problem, except that one of the children is born on a Tuesday.

Then there was a 20-minute break, in which I went up to the exhibit room to help and watch the exhibitors set up.  Hans Schepker was setting up a large staircase which appeared to defy gravity, even though wires were attached to each of the cubes that made it up. He also made a type of flexagon based on seven tetrahedra taped along their edges in which the shape folded out progressively around the circle instead of all at once. John Edmark was also there, with many sculptures based on the Fibonacci sequence, the Golden Ratio, and the Golden Angle, such as a whirligig which, when spun one way, made a smooth spiral, and when spun the other way, made a shape that looked like a pine cone.

The next session started out with John Conway doing a talk on the Lexicode Theorem Non-Theorem Puzzle, which led to the system of Nimbers, in which 8+8=0, and where 8*11=9. Uri Levi was next, with a demonstration of a new puzzle he had found called the “Magnetic Tower of Hanoi”  which normally needs 3^n moves to solve, but variations on it can have rather complicated formulas for the minimum moves required. Neil Sloane also announced that the OEIS was going into a wiki format, and Benjamin Chaffin did a talk on computing the curling number conjecture and the Recaman Sequence.

By then it was time for lunch, and I skipped lunch to have a look at the sales rooms, where various puzzle creators were selling their puzzles for various prices. The first booth that I recognized when I first came in was that of Pavel Curtis, creator of  insanely hard puzzles, who was selling nearly all of the puzzles he had on his website. I also noticed that the people who made ZomeTool had set up a booth selling the product. Inside the other room was even more puzzles, including various combinatoric puzzles, mathematical books, puzzle boxes and suitcases, and much more. Sandro Del-Prete, who I had met before before the Bar Bets session, was there and my mom bought one of his books for me, provided that he would sign the book in German, and that I would have to read what he had written.  Nearby was Clifford Pickover, one of my favorite writers of math and computer science books, who I talked with shortly and then – something that would only ever happen in Gathering For Gardner- Ivan Moscovich, another one of my favorite authors of math and puzzle books, turned out to be right beside us. Of course, I talked with him for a while, and then went back to the other room, where I noticed that Kadon Enterprises, makers of tons of polyomino-based puzzles, were there, and quickly solved one of their easier puzzles, a set of pentominoes which could be stacked to make 3D shapes. By that time I went back down for the next set of talks, as an hour had already passed.

The next set of talks started out with a set of puzzle fonts by Erik Demaine, where you have to solve a puzzle to even figure out what the letter was, and then repeat that for each letter in the text. Kenneth Brecher did a talk on ambiguous figures, in 2D and also in 3D, and proposed a problem about 4 or more perspectives of an ambiguous object that I quickly solved by placing the Rubin Vase on a type of striped disk which produces either 4 or 6 perspectives, depending on what you consider it to be. Clifford Pickover did a talk on the making of his newest book, called The Math Book, and Glen Whitney finished off the session with a talk on The Museum of Mathematics, which is to be built very soon. Another short break, and then the last session for the day began.

First, there was a 30-minute talk on “The Art of Throwing Up” which is not what you may think it is. It was actually about juggling, and by the end of the talk I could actually juggle three scarves without grabbing everywhere. Tomas Rokicki, one of the programmers of Golly and a searcher for God’s Number on the Rubik’s Cube, then did a talk on ‘Modern Life” which was about recent developments in Conway’s Game of Life patterns. David Spies introduced GamesCrafters, a service where you can play around  70 games against a perfect opponent, and Robert Bosch talked about using the Traveling Salesman Problem to generate artworks. Sandro Del-Prete also did a talk about some of his illusions, a few of which were animated. Alex Bellos, author of a new book, Here’s Looking at Euclid ( Alex’s Adventures in Numberland in Britain) talked about why they still use abacuses in Japan (those kids are scary fast), and Eve Torrence, lastly, gave an improvement to Lewis Carroll’s Condensation Method.

Afterwards, we went to the 50th floor of a nearby tower for a large dinner with other attendees of G4G9. After the dinner, we were led into one of a few rooms where we were shown a number of short magic shows. I was in the room with Gareth Conway and John Conway, who I talked with about the Game of Life (it was originally simulated on Go boards), the talk about Nimbers he gave, and the Century Sliding Block Puzzle, which he apparently found by modifying the L’Ane Rouge puzzle. The magic shows were great, and I noticed that for some reason, Gareth, my mom, and I were chosen very frequently. Some of my favorite acts were a trick by Victoria Skye, who had 3 cards which would correspond to any answer to one of the questions she asked you; A trick by Mark Mitton in which he would place a card on the table, stand on top of a chair in a corner, and the card would turn out to be whatever the person named; and nearly all of Lennart Green’s card tricks. I was especially amazed by a trick by John Railing in which he turned a pack of cards into a sheet of plexiglass. This was especially amazing to me because I was holding the pack of cards at the time, and my hand was small enough that I could see in from the outside, and I still couldn’t tell when the switch happened.

Afterwards, we went back to our hotel and went to sleep, amazed by what had happened today.

G4G9, Day 1: Pencils, Optical Illusions, and Bar Bets

Continued from the previous post

The next day, Wednesday, was the first official day of Gathering For Gardner. The only session that day was the Bar Bets section, which was where the magicians and some mathematicians would show various tricks and trick bets which were mathematically related or interesting. However, the session was in the afternoon, so in the morning we had some time to do whatever we wanted to.

Julian Ziegler Hunts and his family had arrived overnight, so we got to have breakfast with them, in which he showed me some interesting Minsky Circle maps based on varying ξpsilon and zeta in the Minsky circle algorithm and plotting the period. After this, my mom and I, as well as the Zieglers and Gosper slept in until 11:00,at which point we decided to head back over to Tom Rodger’s house to play with puzzles while we waited for the session to begin.

As I have mentioned before, Tom has a huge collection of puzzles and sculptures. Since Julian had never been here before, and Tom was on a quick errand, I quickly gave him a tour of the house. Inside the puzzle rooms, Bill noticed that there were many impossible objects made by Gary Foshee, who makes sculptures where the puzzle is to determine how the object got into the current state, not to get it out. A classic example is of the “arrow through the coke bottle”:

Of course, Tom had many others, such as multiple coke bottles, strung together in impossible ways:

Later, Tom Rodgers came back from his errand and showed us some secret closets filled with puzzles. He placed out a few of his favorites on the table for us, and we attempted to solve all of them:

Many of the puzzles I knew the solution to, such as the nails puzzle and the ring puzzle, others I was able to solve, but the majority of them completely stumped me and everyone else. Akio Hizume showed us two interesting programs he wrote, called Real Number Music and Real Kekak System. They were both based on using the coefficients of the continued fraction of the number to generate music, and often made music which I think I’ve heard in some songs. At around 2:00 P.M., we went to the Ritz-Carlton for the before-conference meet.

There at the meet were lots of people who were going to G4G9, such as Lucas Garron, a speedcuber who has some very interesting modded cubes, such as one which transforms the edges to the corners and the corners to the edges, and is equivalent to a Shepard’s cube. There were many puzzles there, including Oskar’s Gears and a set of 9 3×3 paper-folding puzzles which varied from easy to AAUUGGHH! I also got to meet Sandro Del-Prete, one of my favorite optical illusion artists and talk to him about his optical illusions and what he was inspired by to make some of his drawings. He didn’t have perfect English, and my German is terrible, so my mom had to act as a translator at some parts. I was still able to understand what he was saying, even in German, though.

by Sandro Del-Prete

At around 6:30, we were led into an adjoining room for the Bar Bets session, in which various people demonstrated interesting and amazing magic tricks and bar bets. One person attempted unsucessfuly to drop a cork so that it would balance on its edge, another was successful at the same thing with matchboxes. The Great Jordini showed how to solve a certain puzzle by blowing on it, and I even got to solve a simple matchstick puzzle, shown below:

Get the dime out of the glass by moving only two matchsticks

Many of the tricks originated from Martin Gardner, such as a trick where a person moves a ring from a lower upperhand knot to a higher one. This went on until around 11:30, at which point we went back to our hotel and slept.

Naturally, I was excited for tomorrow.

Gathering For Gardner 9: Prelude

I’ll have to break up a single long post about Gathering For Gardner 9 into several medium-sized posts, as WordPress could crash before the post was completed.

Gathering For Gardner is a large puzzle, mathematics, optical illusion, magic, and generally everything Martin Gardner-related conference held biannually in Atlanta, Georgia from  March 24-29. There are usually 300 people there, each specializing in a unique topic. Many famous people are there, such as Stephen Wolfram, who made Mathematica, John Conway, Scott Kim, Bill Gosper, Jerry Slocum, and many others. When I heard about this conference a few months ago from Bill Gosper, I was definitely very excited, and even more excited when Bill managed to get me an invitation.

Every participant is required to bring 300 copies of an exchange gift, which is simply defined as “something that you would want to give to Gardner”. This can be from puzzles to mathematical bookmarks to folding polyhedra.  The theme for Gathering for Gardner 9 was naturally 9, and many exchange gifts were based on this. My exchange gift for the conference was the 9-9-9 Puzzle, which is a packing puzzle where you have to combine nine nonahedral pieces to make a nonahedron.

We set out for Atlanta in an airplane on March 23, and arrived at around 5:00 P.M., so my mom and I had time to go to Tom Rodger’s, one of the organizers of the event and a puzzle collector’s, house. Lennart Green, Akio Hizume, and Caspar Schwabe were already there, so we weren’t the only people who were early for the conference.

Tom Rodger’s house shows how much of a puzzle collector he is. His front two gates have the Tangram and the Sei Shonagon, two very old pattern puzzles, embossed in the wood. In his yard, there are an amazing amount of great mathematical sculptures, such as this sculpture of five interlocking tetrahedra:

He also had a huge sculpture of a hyperbola made out of rare black bamboo which is conveniently right next to Tom’s house:

There was a smaller version of the hyperbola made with more common bamboo which can actually open and close.

Caspar Schwabe demonstrated this by asking me to step inside the hyperbola in its unfolded position and then closing it up around me. Obviously these people are not worried about damage, especially the “no touching” rule:

We soon went into Tom Rodger’s house after lots of amazing-sculpture-viewing.

Tom has a large and absolutely amazing Japanese-style house complete with an interior courtyard and a koi pond:

His house is also absolutely filled with puzzles, with a room and 3 closets, as well as a basement (supposedly) stuffed with amazing puzzles from around the world. Here he is demonstrating one of Oskar Van Deventer and Bram Cohen’s puzzles, the Caution Cube:

He has lots of drawers full of various puzzles, from puzzle boxes to locks to dexterity puzzles, and I was especially amazed when he showed us his large collection of sliding block puzzles, many from Minoru Abe, an excellent puzzlemaker from Japan. He even has 2 versions of the Panex puzzle, and all 4 of Minoru Abe’s Climb-24 series. He also allowed us to play with the puzzles, which of course made me very happy. Here are some pictures of the puzzles:

I also, after playing with the puzzles for some time, got to meet Lennart Green, probably the best card magician in the world. He showed me a few card tricks, involving taking any card I named out of the deck perfectly, even though I shuffled the deck terribly. He also separated 16 cards into 4 piles, each of which I saw the cards in, each pile containing a single ace. I took one of the piles which definitely had 1 ace, and then he picked another pile. It turned out that he somehow had all 4 aces, even though I had one of them. He then whacked his 4 cards against the 4 that I was holding in my hand, and then it turned out that I had all 4 aces and he had 4 normal cards. Of course how he did this is completely mysterious to me.

We then had a very tasty dinner at Tom’s house, and Akio Hizume showed me a certain structure he had made out of 6 interlocking stars, where the normal one could fold to a ring but the mirror image couldn’t. Many of the people there (Scott Hudson and Bruce Oberg had joined) couldn’t figure out why this was, and I pointed out to them that it was because of the places where 3 rods pass by each other. Aiko also gave me the set of 30 rods with which it can be constructed, but I have yet to assemble them in the correct way.Caspar Schwabe also gave me a kaleidoscope-like structure made out of mirrors that, when you looked inside, made a stellated dodecahedron and an unfolded circular net around the outside. It is a very convincing illusion, and the center stellated dodecahedron changes color depending on what you point the kaleidoscope at:

I also gave them one of the puzzles that I brought, specifically the one where you have to simply get the marble out. It only requires one move, but they almost didn’t solve it until my mom and I were getting ready to go.

All in all, it was a wonderful day, and a great sampler for what Gathering For Gardner would be.