Archive for the Category Mathematics

 
 

Rhombo

Over the past couple of weeks I wrote some code in C# to generate dissections of the rhombic triacontahedron into golden rhombohedrons. George Hart discusses these types of dissections here  and also talks about the problem of enumerating them in an appendix here — briefly, all this material by Hart and others is about how the fact that the rhombic triacontahedron and the rhombic enneacontahedron are zonohedra lead to both having interesting combinatoric properties which can be explored by coloring their dissections.

I was, however, more interested in how such dissections could be turned into an interlocking puzzle, akin to a traditional burr puzzle. amd as such needed code to generate 3D models of the dissections. My generation code is a dumb, constructive, brute force approach in which I just traverse the search space adding rhombohedrons to a candidate dissection in progress and backtracking when reaching a state in which it is impossible to add a rhombohedron without intersecting the one that was already added or the containing triacontahedron, keeping track of configurations that have already been explored.

Dissections of the rhombic triacontahedron into golden rhombohedrons (hereafter “blocks”) turns out to always need 10 and 10 of the two types of blocks that Hart refers to in the above as the “pointy” and “flat” varieties (and which I refer to as yellow and blue). Further it turns out that in all of these dissections there are four blocks that are completely internal, i.e. sharing no face with the triacontahedron; I also believe that the four internal blocks are always three blue and one yellow, but I’m not sure about that.

My strategy for finding an interlocking puzzle was the following:

  • Generate a bunch of raw dissections into blocks
  • For each dissection, search the adjacency graph for four pieces, the union of sets of five blocks, such that
    • Each piece forms a simple path in the dissection; that is, each block in the piece
      • is either an end block that is face adjacent to a next or previous block in the piece or is a non-end block that is face adjacent to a next block and a previous block.
      • and does not share any edges with other blocks in the piece except for the edges of the face adjacencies.
    • Each piece contains at least one fully internal block.
    • Each piece is “single axis disentangle-able” from each other piece, where we mean by that that there exists some edge e in the complete construction such that if given piece p1 and piece p2, if you offset p1 in the direction of  e by a small amount p1 does not intersect p2.
    • Each piece is not single axis disentangle-able from the union of the other three pieces.

I never managed to succeed in doing a complete enumeration, generating all of the dissections for reasons that I don’t feel like going into. (As I said above, I did not do anything fancy and it would be easier to just be smarter about how I do the generation than to make what I have more efficient; i.e. could have done the George Hart algorithm if I had known abouyt that or there are ways of transforming one dissection into another that I don’t do — I do an exhaustive search, period — but I never did the smarter stuff because I found what I was looking for, see below)

But from about 10 dissections I found one set of pieces that uniquely satisfies all of the above:

Here’s some video. (The individual blocks were 3D printed and super glued together)

I’m calling the above “rhombo”. Those pieces are rough because I only 3D printed the individual rhombohedrons and then superglued them together into the pieces, which is imprecise. I had to sand them heavily to get them to behave  nicely. I’ll eventually put full piece models up on Shapeways.

In the course of doing this work, it became apparent that there is no good computational geometry library for C# to use for something like this. There is one called Math.Net Numerics along with Math.Net Spatial that will get you vectors and matrices but not with all the convenience routines you’d expect to treat vectors like 3D points and so forth. What I ended up doing was extracting the vectors and matrices out of monogame and search-and-replacing “float” to “double” to get double precision. Here is that code on github. I also included in there 3D line segment/line segment intersection code and 3D triangle/triangle intersection code which I transliterated to C#. The line segment intersection code came from Paul Bourke’s web site. And the triangle intersection code came from running Tomas Moller’s C code through just a C preprocessor to resolve all the macros and then transliterating the result to C#.

RIP Magnus Wenninger

Father Magnus Wenninger died last month.

Is there hexagonal analog of Conway’s Game of Life?

The short answer is that in the hexagonal case the best analog of Conway’s Game of Life — in my opinion as someone who has been a CA hobbyist for 30 years or so — is an original creation which I will describe for the first time in detail in this blog post. (Note: The links to cellular automata in this post go to a custom web-based player which may not run correctly on mobile devices.)

Regarding a hexagonal Game of Life, a key thing to understand is that Conway Life isn’t just the rule; it is the rule (Life 2333) running on a square grid with an 8-cell neighborhood i.e. the neighborhood of four squares directly adjacent to a square plus those diagonally adjacent. You can, of course, apply the same rule on a hexagonal grid using the natural six hexagon neighborhood but what you will get won’t look anything like Life. It will look like this: Life 2333 on a hexagon grid.

So if the above does not qualify as a hexagonal Game of Life then what would? Well, at the very least we need a glider. Carter Bays, a professor at the University of South Carolina, presented a Conway-like rule that admits a glider on the hexagonal lattice in 2005, Life 3,5/2 in his notation. However, by Bays’ own admission “this rule is not as rich as Conway’s Life” and indeed when we run Bays’ Hex Life on random input we do not see gliders: Life 3,5/2 on random input. The problem is that its glider is too big to occur randomly. Its glider is in a sense artificial. Part of the beauty of Conway Life is that gliders are frequently spontaneously generated. The other thing you’ll notice about Bays’ Life 3,5/2 is that it descends into still-lifes and oscillators too quickly. Conway Life is dynamic. It sprawls and grows, descends into bounded chaos, before finally decaying into still-lifes and oscillators.

To summarize, we want a hexagonal cellular automaton that

  1. Has a glider that is frequently generated by random initial input.
  2. Frequently exhibits bounded growth from random initial input.

It is my contention that there is no simple totalistic rule over two states and the hexagonal grid using the natural 6-cell neighborhood that exhibits both 1. and 2.

In order to achieve what we want, we need to drop one of the constraints. Using my cellular automata breeding application Lifelike, I explored dropping the constraint that the rule must be a simple totalistic rule over two states. What I have come up with is a cellular automaton that uses a simple totalistic rule over three states,  states 0 to 2. One way to view this move is to view the live cells in Conway Life as counters — beans, pennies, whatever — and imagine dropping the constraint that a cell can only contain one counter. Anyway, my rule is as follows:

  • Take the sum S of the 6-cell neighborhood where death=0, yellow=1, and red=2.
    • If the cell is currently dead it comes to life as yellow if S is exactly 4.
    • If the cell is yellow it goes to red if S is 1 to 4 inclusive or is exactly 6.
    • If the cell is red it stays red if S is 1 or 2 and goes to yellow if S is 4.
    • Otherwise it is dead in the next generation.

The above, which I call “Joe Life”, is as rich as Conway life: click this link to see it run. It exhibits bounded growth with about the same burn rate as Conway Life and features two gliders, the little fish and the big fish, which are frequently generated spontaneously.

The little fish

The big fish

I concede Conway’s rule is more elegant than Joe Life’s rule but if one thinks about it, Conway Life’s neighborhood is less natural than the six hexagon neighborhood in that it is kind of weird to include the the diagonally adjacent squares. So in my opinion, elegance that Joe Life lacks in its state table it makes up for in its neighborhood.

A Woven Icosahedron

You can weave an icosahedron from 10 strips of paper — or anyway you can weave a construction that has icosahedral geometry; it is actually more of a snub icosahedron.

From about 11″ strips of printer paper folded lengthwise to have two layers with a little overlap to lock into rings, as pictured below, locks well and is rigid:

Construction follows the pattern implied by the following: