A lot of the work in the above may not be immediately apparent (which is good I think). Specifically, there are actually two shapes of butterflies in the above not just one. They look like this:

and are arrayed as in a checkerboard with a 90 degree rotation applied to cells of opposite parity — the geometry might be clearer here in which I create these butterfly tiles unadorned with my tool EscherDraw (before I had beziers working in EscherDraw) Thus any time a butterfly moves to a cell with opposite parity I need to not only handle the scale and rotation tweening imposed by the spiral lattice, I also have to simultaneously do the 90 degree rotation imposed by the butterfly tiles and simultaneously morph the actual sprites above from one to the other.

I created the above sprites as vector art using tile outlines exported from Escher draw as SVG and wrote a Python script that morphs SVG as the publically available software for morphing vector graphics is surprisingly non-existent. I am going to post about my vector morphing code here soon, I just need to clean it up for use by people other than me.

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My code is here: link.

The first function is splitting a cubic bezier into two curves at a given

... public Tuple<Bezier,Bezier> Split(double t) { PointD E = Interpolate(t, CtrlPoint1, CtrlPoint2); PointD F = Interpolate(t, CtrlPoint2, CtrlPoint3); PointD G = Interpolate(t, CtrlPoint3, CtrlPoint4); PointD H = Interpolate(t, E, F); PointD J = Interpolate(t, F, G); PointD K = Interpolate(t, H, J); return new Tuple<Bezier, Bezier>( new Bezier(CtrlPoint1, E, H, K), new Bezier(K, J, G, CtrlPoint4) ); } static private PointD Interpolate(double t, PointD pt1, PointD pt2) { double x = (1.0 - t) * pt1.X + t * pt2.X; double y = (1.0 - t) * pt1.Y + t * pt2.Y; return new PointD(x, y); } ...The other function is considerably harder: finding the nearest point on a bezier curve to a given point. If you scan the literature (i.e. perform the google search) you will find that what most people end up using is some code from an article in the book

But more relevantly I just don’t get this code on a fundamental level. The nearest point on a bezier problem is difficult to

The formulation I am talking about is as follows, if

public Tuple<PointD, double> FindNearestPoint(PointD pt) { var polyQuintic = GetNearestBezierPtQuintic( _points[0].X, _points[0].Y, _points[1].X, _points[1].Y, _points[2].X, _points[2].Y, _points[3].X, _points[3].Y, pt.X, pt.Y ); List<Complex> roots = FindAllRoots(polyQuintic); // Filter out roots with nonzero imaginary parts and roots // with real parts that are not between 0 and 1. List<double> candidates = roots.FindAll( root => root.Real > 0 && root.Real < 1.0 && Math.Abs(root.Imaginary) < ROOT_EPS ).Select( root => root.Real ).ToList(); // add t=0 and t=1 ... the edge cases. candidates.Add(0.0); candidates.Add(1.0); // find the candidate that yields the closest point on the bezier to the given point. double t = double.NaN; PointD output = new PointD(double.NaN,double.NaN); double minDistance = double.MaxValue; foreach (double candidate in candidates) { var ptAtCandidate = GetPoint(candidate); double distance = DistSqu(ptAtCandidate, pt); if (distance < minDistance) { minDistance = distance; t = candidate; output = ptAtCandidate; } } return new Tuple<PointD, double>(output, t); }which reduces the problem to implementing GetNearestBezierPtQuintic() and a numeric root finding function for polynomials, given that quintic equations cannot be solved via a closed-form formula like the quadratic equation.

GetNearestBezierPtQuintic() –which returns the coefficients of ⋅

static private List<Complex> GetNearestBezierPtQuintic(double x_0, double y_0, double x_1, double y_1, double x_2, double y_2, double x_3, double y_3, double x, double y) { double t5 = 3 * x_0 * x_0 - 18 * x_0 * x_1 + 27 * x_1 * x_1 + 18 * x_0 * x_2 - 54 * x_1 * x_2 + 27 * x_2 * x_2 - 6 * x_0 * x_3 + 18 * x_1 * x_3 - 18 * x_2 * x_3 + 3 * x_3 * x_3 + 3 * y_0 * y_0 - 18 * y_0 * y_1 + 27 * y_1 * y_1 + 18 * y_0 * y_2 - 54 * y_1 * y_2 + 27 * y_2 * y_2 - 6 * y_0 * y_3 + 18 * y_1 * y_3 - 18 * y_2 * y_3 + 3 * y_3 * y_3; double t4 = -15 * x_0 * x_0 + 75 * x_0 * x_1 - 90 * x_1 * x_1 - 60 * x_0 * x_2 + 135 * x_1 * x_2 - 45 * x_2 * x_2 + 15 * x_0 * x_3 - 30 * x_1 * x_3 + 15 * x_2 * x_3 - 15 * y_0 * y_0 + 75 * y_0 * y_1 - 90 * y_1 * y_1 - 60 * y_0 * y_2 + 135 * y_1 * y_2 - 45 * y_2 * y_2 + 15 * y_0 * y_3 - 30 * y_1 * y_3 + 15 * y_2 * y_3; double t3 = 30 * x_0 * x_0 - 120 * x_0 * x_1 + 108 * x_1 * x_1 + 72 * x_0 * x_2 - 108 * x_1 * x_2 + 18 * x_2 * x_2 - 12 * x_0 * x_3 + 12 * x_1 * x_3 + 30 * y_0 * y_0 - 120 * y_0 * y_1 + 108 * y_1 * y_1 + 72 * y_0 * y_2 - 108 * y_1 * y_2 + 18 * y_2 * y_2 - 12 * y_0 * y_3 + 12 * y_1 * y_3; double t2 = 3 * x * x_0 - 30 * x_0 * x_0 - 9 * x * x_1 + 90 * x_0 * x_1 - 54 * x_1 * x_1 + 9 * x * x_2 - 36 * x_0 * x_2 + 27 * x_1 * x_2 - 3 * x * x_3 + 3 * x_0 * x_3 + 3 * y * y_0 - 30 * y_0 * y_0 - 9 * y * y_1 + 90 * y_0 * y_1 - 54 * y_1 * y_1 + 9 * y * y_2 - 36 * y_0 * y_2 + 27 * y_1 * y_2 - 3 * y * y_3 + 3 * y_0 * y_3; double t1 = -6 * x * x_0 + 15 * x_0 * x_0 + 12 * x * x_1 - 30 * x_0 * x_1 + 9 * x_1 * x_1 - 6 * x * x_2 + 6 * x_0 * x_2 - 6 * y * y_0 + 15 * y_0 * y_0 + 12 * y * y_1 - 30 * y_0 * y_1 + 9 * y_1 * y_1 - 6 * y * y_2 + 6 * y_0 * y_2; double t0 = 3 * x * x_0 - 3 * x_0 * x_0 - 3 * x * x_1 + 3 * x_0 * x_1 + 3 * y * y_0 - 3 * y_0 * y_0 - 3 * y * y_1 + 3 * y_0 * y_1; return new List<Complex> { (Complex)t0/t5, (Complex)t1/t5, (Complex)t2/t5, (Complex)t3/t5, (Complex)t4/t5, (Complex)1.0 }; }and decided to use Laguerre’s Method to solve the equation.

I chose Laguerre’s Method because it is optimized for polynomials, is less flaky than Newton-Raphson, is relatively concise, and is easy to port from C++ given Microsoft's implementation of complex numbers found in the System.Numerics namespace. I ported the implementation of Laguerre’s Method found in

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Notes on a novel tiling of the plane

Rendered in Escher draw

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These applications are “too limited” in the sense that none encompass everything possible that one would regard as a 2D tessellation. You have some applications that simplistically shoehorn you into a few kinds of geometries ignoring the richness of the whole domain of tiled patterns. You have other applications that try to take the domain seriously and follow the mathematical literature on the subject. What this means, generally, is that they allow the user to construct a tessellation in terms of its “wallpaper group” which is the mathematical classification of a plane pattern based on the symmetries the pattern admits.

The latter seems like what one would want but in practice it isn’t, at least it is not what I want. It is too heavy of a constraint to work only in terms of wallpaper group because there are all kinds of patterns one might be interested in that fall outside of this formalism: Escher’s tessellations of the Poincare disk model of the hyperbolic plane, for example, or, say, any kind of aperiodic tiling or any tessellations that admit similarity as a symmetry, i.e. scaling, such as Escher’s square limit etchings, or my case: various spiral patterns.

It is instructive that Escher himself did not work in terms of symmetry groups. He created his own classification system which Doris Schattschneider refers to as Escher’s “layman’s theory” of tilings in her book

What I like about Escher’s approach is that it is geared towards construction. When you try to make one of these things you start with a tile or some a small set of simple tiles that tile the plane. You try to modify them until you get something you like. I decided to implement software that captures that approach. I didn’t want to have to be too concerned with “wallpaper group” or other abstractions which I don’t personally find to be intuitive anyway.

What I came up with is the following application that is written in C#. Below is video in which I use the application to construct something like Escher’s butterfly tessellation — system IX-D in his layman’s theory — but on a double logarithmic spiral (This is going to be the basis for the art of the first level of my game; the butterflies will be animated, flapping their wings, etc.)

The way the application works is that there are two kinds of tiles: normal tiles and tile references. Normal tiles are composed of “sides”. There are two kinds of sides: normal sides and side references. Tile references and side references point to another tile or side and say basically, “I am just like that tile or side, except apply this transformation to me”. The transformations that tile references can apply to a tile are defined in terms of affine transformation matrices. The transformations that side references can apply are just two kinds of flipping relative to the sides’ end points, either flipping “horizontally” or flipping “vertically” or both. The application then allows the user to add and move vertices on any sides that are on tiles that are marked as editable and then resolves all the references in real-time.

Right now, all of the information about tiles and sides and which is a reference to which has to be hand coded as an XML file, which is a pain (for the above I wrote a separate Python program that generated the XML defining a double logarithmic spiral of tiles that interact with each other according to Escher’s IX-D system), but it is an extremely flexible formulation (it could, for example, handle hyperbolic tessellations in theory … if you added a kind of side that is a circular arc and moebius transformations to the set of transformations the applications knows how to apply to referenced tiles).

Eventually I’d like to release this application as open source but I am not sure it will be useful to anyone but me in its current form. I need to incorporate the part that one has to hand code in XML into the actual GUI … fix bugs and bezier curves and so forth, but please feel free to contact me if you are interested in the application in its current state or otherwise.

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The final app will be styled similar to one of M. C. Escher’s tessellations of the plane with the tessellation changing at level breaks. I don't have a name yet...

Gameplay will be like the following, which is video of a prototype I wrote in C# (just WinForms to the GDI, nothing fancy). I’m going to write the real thing in Swift using Sprite Kit, unless Swift turns out to be too immature in which case I’ll use C++ and Cocos2d-x again.

*: The identity of which I leave as an exercise for the reader.

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By "discrete distibution" we just mean the roll you get from something like an unfair die, e.g. you want a random number from 0 to 5 but you want 4 and 5 to be twice as likely as 0, 1, 2, or 3. If we think of each possible random value as having a weight, in this case 0, 1, 2, and 3 would have a weight of 1 and 4 and 5 would have a weight of 2.

A simple way to generate these kinds of random values is the following. Given some n such that we want random values ranging from 0 to n-1 where for each 0 <= i < n we have a weight w(i):

- Build a data structure mapping cumulative weight to each value i. By cumulative weight we mean for each i the sum of w(0), ... , w(i-1).
- Generate a random number r from 0 to W-1 inclusive where W is the total weight i.e. the sum of w(i) for all i.
- If r is a cumulative weight in our data structure return the value associated with it; otherwise, find the value v that is the first item in the data structure that has a cumulative weight greater than r and return v-1.

However, in C# you can't just use a SortedDictionary, which is a binary search tree under the hood. You can't use a SortedDictionary because it does not expose the equivalent of C++'s std::lower_bound and std::upper_bound or the equivalient of Java's TreeMap.floorEntry(...) and TreeMap.ceilingEntry(...). In order to perform the "otherwise" part of 3. above you need to efficiently be able to find the spot in the data structure where a key would go if it was in the data structure when it is in fact not in the data structure. There is no efficient way to do this with a SortedDictionary.

However, C#'s List<T> does support a BinarySearch method that will return the bitwise complement of the index of the next element that is larger than the item you searched for so you can use that.

The downside of this whole approach is that there will be no way to efficiently add or remove items to the discrete distribution, but often you don't need this functionality anyway and the code to do the whole algorithm is very concise:

class DiscreteDistributionRnd { private List<int> m_accumulatedWeights; private int m_totalWeight; private Random m_rnd; public DiscreteDistributionRnd(IEnumerable<int> weights, Random rnd = null) { int accumulator = 0; m_accumulatedWeights = weights.Select( (int prob) => { int output = accumulator; accumulator += prob; return output; } ).ToList(); m_totalWeight = accumulator; m_rnd = (rnd != null) ? rnd : new Random(); } public DiscreteDistributionRnd(Random rnd, params int[] weights) : this(weights, rnd) { } public DiscreteDistributionRnd(params int[] weights) : this(weights, null) { } public int Next() { int index = m_accumulatedWeights.BinarySearch(m_rnd.Next(m_totalWeight)); return (index >= 0) ? index : ~index - 1; } }where usage is like

DiscreteDistributionRnd rnd = new DiscreteDistributionRnd(3,1,2,6); int[] ary = new int[4] {0,0,0,0}; for (int i = 0; i < 100000; i++) ary[rnd.Next()]++; System.Diagnostics.Debug.WriteLine( "0 => {0}, 1 => {1}, 2 => {2}, 3 => {3}", (float)(ary[0] / 100000.0), (float)(ary[1] / 100000.0), (float)(ary[2] / 100000.0), (float)(ary[3] / 100000.0) );Source]]>

If you want to use physics in a Cocos2d-x game the current standard way to do this is to use the integrated physics classes, which are Chipmunk-based by default and can use Box2d too in some hybrid way the details of which are not at all clear. However, in my case I want to use Box2d, period, Box2d in a non-integrated manner for transparency and in order to leverage the vast amount of code that you get to peruse and possibly use by writing to vanilla Box2d. When doing something like this it can be hard to know where to start given that any sample code you find will be broken.

For setting up a Box2d/Cocos2d-x project there was always this BreakOut implementation to Cocos2d-iphone by Ray Wenderlich, link, which is transliterated into Cocos2d-x here but to relatively ancient versions of both Box2D and Cocos2d-x. I’ve taken that code and updated it to Cocos2d-x version 3.2 and Box2d version 2.3.

Here is my updated version.

To use do the following:

- Setup a cocos2d-x v3.2 project via the python script. This will give you Box2d v2.3 set up in your project without you having to do anything else.
- Copy the source code in the above zip file into your project’s Classes directory.
- Copy the image files the Ball.jpg, Block.jpg, and Paddle.jpg from here into your project’s Resources directory.
- Build.

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I will be updating the page I have for it here this week with more detailed instructions than what the app itself provides (There is a tutorial mode but it really just covers the basics), but basically it is an action puzzle game in which you lay tiles on a board to form interlocking words, but you’re building words against the clock where

- forming large grids of words puts time back on the clock
- not using tiles eventually leads to tile death which takes time off the clock.

Screenshot below:

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That year after college I was kind of shiftless. I chose to just not participate in the whole interview circus that normally accompanies one’s senior year of MIT and let myself graduate without having plans of any kind for the future. I think that I was tired mostly and was kind of sick of the universe of engineering and all of its hurdles and dog and pony shows, and I think I kind of half-understood that this would be the last time in my life in which it would be possible for me to be totally free of that universe — until, I guess, retirement.

Anyway, I lived in the slums of Boston — that is, Roxbury — with a friend of mine from high school and some roommates we found, one of whom ended going to jail (but that is another story), worked temp jobs, and didn’t do much of anything … except for some reason I became obsessed with aperiodic tilings of the plane and substitution tilings generally and put a lot of effort into coming up with one of my own. I wanted to find, for reasons that aren’t clear to me now, an aperiodic analog of the normal regular hexagon tiling.

It’s kind of a blur — I don’t really remember where the above came from as a sequence of steps — but the above is a patch of the best tiling I discovered during this period. It is generated via the following substitutions. The lengths of the sides of triangles and trapezoids are multiples of ϕ, the golden ratio.

As far as I know, the above is original and has not been discussed in the literature but I never was able to come up with local matching rules on the hexagons to enforce aperiodicity.

At that time I did develop software for working on these sorts of structures but what I came up with in retrospect wasn’t The Right Thing. This wasn’t all together my fault. Those were different times: no GitHub, no free code. If I wanted to output into a vector format it would have to be my own vector format. If I wanted to render that format to the screen I would have to write code to render that vector format to the screen, and so on. Also GUI applications were all the rage and were still new and shiny, so I was biased in that direction. I never really liked what I came up with then, and it wasn’t portable anyway; it was a black-and-white Macintosh application in C.

Having the negative example of that project all those years ago made it easy to see what I actually needed: not a GUI application but a programming language. So last week, I wrote one: a little language for specifying recursive structures like the above and rendering them in SVG. I’m calling it Snurtle because it is basically a combination of Python (a snake) and the turtle-based graphics of Logo. I chose Python syntax because I wrote the Snurtle interpreter in Python and thus got tokenization for free using Python’s “tokenize” module.

So, for example, the following simple substitution

is represented by the following Snurtle script:

sub square(n): terminal: poly("bisque", "orangered", 1 ): forward(n) turn(PI/2) forward(n) turn(PI/2) forward(n) nonterminal: rectangle(n) branch: turn(PI) forward(n/2) turn(-PI/2) forward(n/2) square(n/2) square(n/2) sub rectangle(n): terminal: poly("lightskyblue", "orangered", 1 ): forward(n/2) turn(PI/2) forward(n) turn(PI/2) forward(n/2) nonterminal: square(n/2) turn(PI) square(n/2)yielding

which, I think, is self-explanatory except for the “branch” block. Branch blocks tell Snurtle to push the state of the turtle on to a stack and then pop it when exiting the branch i.e. branch works like parentheses operators in L-systems. Also the following Snurtle constructs are not illustrated in the above:

- flip blocks: similar in syntax to branch above. Tell Snurtle to multiply the angle arguments passed to turn statements by -1. (i.e. flipping them)
- stroke and fill blocks: similar to “poly” above.

wheresnurtle.py -w -s square -k 500 -m 8 -o squares.html -c “10,10″ “snurtle_scriptssquares.snu

- -w : wrap the generated SVG in HTML (so it can be viewed in browser)
- -s : initital substitution used to kick off the recursion
- -k : scale factor in SVG output
- -m : max stack depth before substituting in terminal blocks rather than nonterminal and ending the recursion
- -o : output filename
- -c : starting coordinate when generating SVG. (There is also a -d which similarly specifies the width and height attributes of the SVG but I am using the default in the above

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