And each time you run the script, you get something different. Another useful feature of the turtle drawing class is being able to draw not just lines but polygonal surfaces. To do this, just do t. Once you're in face mode, each time you draw a line, the code actually draws a triangle: one vertex is the point the turtle was when t.
This draws a pentagon tilted on its side by 45 degrees:. Draw a triangle. Now take each line of the triangle and replace it by a line with a triangular bump on it. Like on the animated image public domain, based on Wikipedia , you get a Koch snowflake. This can be modeled by a super-simple turtle-graphics program. Then the initial triangle can be drawn by:.
So here's how we can generate a snowflake. Each F in it represents a line. If we keep on going, we generate the Koch snowflake. This is a simple example of an L-system. We apply this a bunch of times and we get a rather complicated string.
For instance, after two iterations in the snowflake case, we get:. After four, we get the image above. To really get a fractal, the lines should shrink with each iteration, but that won't work in Minecraft.
There is a lot of really good information in this free PDF book. I wrote a very simple lsystem. To implement the snowflake, start with boilerplate:. Finally we need to tell the system what each of the symbols in the strings mean. For different L-systems, we will assign different meanings to them rotations by different angles, for instance. So we need a second python dictionary specifying what is done for each symbol.
If you don't specify an action for a symbol, the symbol is ignored when it is time for the turtle to draw the output but it might be important for the generation. The meanings are given by a dictionary that specifies a function to call for each symbol. One-line functions can be specified with the lambda operator. In this case, all the functions are one-liners:. There is one special trick for some L-systems. These L-systems are grid-aligned, with all the rotations being 90 degrees.
The square curve squarecurve. The trick is to call, somewhere near the beginning of your code:. This moves your turtle to an integer grid location, and aligns its heading to a grid direction.
After this, the turtle will remain exactly grid aligned provided that you move it only by integer amounts e. The dragon curve code also gives an example of a forward function that is a bit more complicated--instead of just a line, it draws a wall with an opening in it.
Actually, calling gridalign can sometimes be a good idea even if not all of your angles are right angles. You will probably get some round-off problems in a large image, but it can still look better. See the spacefilling curve example rendered in purple stained glass! L-systems don't have to be two-dimensional. You can include symbols that do yaw, pitch and roll rotations.
For designing trees, a useful trick is to have stack commands: '[' to save the current drawing state to a stack and ']' to restore it. This will use the turtle module's push and pop methods.
For instance, here's a snippet of code for drawing a simple tree ltree. Think of the L as a leaf though this simple code doesn't actually draw the leaf--that would need to be added to the dictionary.
We start with FL , which is a trunk plus a leaf. This is a set of three branches, each tilted by 20 degrees from the trunk, degrees apart. This repeats, so that the leaves on the branches are replaced by triples of branches, and so on. A more realistic tree could have more complex code for '['. It could make the succeeding branches shorter and thinner, and change their material as we get closer to the leaves and then restore on ']'. I include such a tree as the demo code in lsystem. You can also do grid-aligned 3D things.
For instance, hilbert. Finally, you might want to introduce some randomization into the L-system rules. So far our rules were deterministic: a single string was given that replaces a symbol, e. But instead of a simple replacement string, one can give a Python list of pairs p,string , where p is a probability from 0 to 1 and string is the string to use with that probability. The probabilities for a given source symbol had better not add up to more than 1, but they can add up to less than one--in that case, there is a chance that there will be no replacement.
For instance, here's a slightly randomized version of geeky. This isn't very random--more randomness would make things even more lifelike. I attach a screenshot of a fairly sparse forest randomly generated using this rule and random placement of trees, subject to a minimum distance.
This initializes the Minecraft connection and brings math and block names into the namespace. Then you have several convenience methods:. For instance, you can do a rectangle sloped at 45 degrees made of glass at level 0 at the spawn point with:.
For fun, I adapted Simon Tatham's really neat polyhedron generation code to use the Minecraft drawing class. His code starts with a number of random points on a sphere, and then simulates them repelling themselves until they stabilize into a configuration that is often regular. Then he has two neat methods of making a polyhedron out of these points, a face method and a vertex method.
I put this in polyhedron. This script takes commandline arguments. For instance, to draw an icosahedron use 12 points and the vertex construction. Directly from Minecraft:. For an dodecahedron, change vertex to face. You can also add one more argument for size. The polyhedron will be made of glass with stone edges and will be centered around the player. In either case, you can access the Minecraft object via d. Minecraft normally pauses the game and goes to the Game menu when you alt-tab out of it or otherwise lose focus.
This makes it harder to see what your python scripts are doing if you're launching them from outside of Minecraft. To do that, you can edit the options. This is particularly nice for running python scripts interactively. There are basically two different techniques for drawing mathematically defined objects with a python script in Minecraft.
One way is to define a solid object by an inequality. For instance, a sphere centered on x0,y0,z0 with radius r can be defined by the inequality:. Start with the standard header and init Minecraft connection code:. This draws a sphere of the specified radius above the player, and a little offset in the z-direction. I use the same technique, but with a more complicated formula, in my donut.
While the inequality technique works best for solid shapes, you can use it for hollow shapes in two ways. One way is to use two inequalities, for instance in the case of the sphere one to make sure that we're within the outer radius of the center and another to make sure we're not closer than the inner radius. The other way is just to draw another object with smaller dimensions made out of air inside the larger solid object, much as in my donut.
One can also draw a surface by parametrizing it with two parameters, say a and b, and then looping over a range of these parameters, setting blocks where needed. For instance, the Mobius strip see my mobius.
You can think of b as defining the angle around the circuit, and a moving one from one edge to the other. Using scripts like this, you need to ensure that in your loops over a and b, the steps are sufficiently small that there are no gaps in the surface. Unless that's the effect you're after. For details and examples see mobius. You can find parametric equation for knots on the net. This time, we're going to do things slightly different from before.
Before, we had loops driving calls to mc. But in our surface plots, such as the Klein bottle, often the same block would get drawn multiple times, which is slow and inefficient. A better way is to keep track of the set of exact blocks that were already drawn to avoid redoing the same thing. Let me go through an example like that in knot.
Start with a standard header like:. We now need to generate our knot. I used the cinquefoil formulas from here. I used steps. Since this is done in-memory, and computers are fast, and overlapping blocks are only sent once to Minecraft, it's easier to easier to do more steps than to think how many is enough. It's important for the coordinates that go in the dictionary to be integers so we can tell that the same block is being drawing a block at 1. We first initialize and set the position of the knot relative to the player.
Note that the player position need not be an integer you might not be standing exactly aligned with a block and should be turned into an integer.
Now we make an empty set named done to store the coordinates we've already drawn:. This only draws data that isn't already drawn. Note that we need to round off the x, y and z coordinates with the int function. That's the magic behind the overlap removal: when the rounded coordinates are the same, only one block is drawn.
Note: the double parenthesis in the done. The knot would look better if the rope were thicker. There are many ways one can do that. An inefficient way, but easy since computers are fast these days, is just to draw a little ball instead of a point at each pixel.
To do that, first I make a little utility function to draw a ball while checking in the done set to ensure there are no duplicate blocks:.
Then just modify our knot-making while loop to make a ball instead of just a point:. Other sample knots are in trefoil. If you draw with multiple materials, you can use a dictionary in place of a set, or just go sequentially through the different materials and clear the set before each that's what I do in trefoil2. Using python and a modified version of the NeuroPy package , you can now control Minecraft with your brain. Here's how to do it, either with the Mindflex or with a full Mindwave Mobile set.
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