Tiling and tessellation with a Context Free Grammar - Part 2

In my previous blog post we looked at how to use a Context Free Grammar to make simple grids of circles and then use colour variation to produce a rainbow effect. Now let's look at making more interesting patterns of shapes.

Simple spirals

First let's draw a series of ever decreasing, spiralling triangles...

#lang s-exp stamps/lang

(define-shape spiral
  (triangle)
  (spiral [r 22]
          [x .8]
          [s .95]))
          
(maximum-render-cycles 1000)
(start-shape spiral)

Play with those numbers in square brackets to produce something you like. Remember, r=rotation, x=x translation, s=scale.

Recursive loops

This example draws a hexagon with a circle of hexagons around it, and for each of those hexagons, it draws a circle of hexagons around it, recursively building up a pattern that fills the artwork.

#lang s-exp stamps/lang

(define-shape hex-circle
  (hexagon)
  ((loop ([i 6])
         (hex-circle 
           [r (* 360 (/ i 6))]
           [y .9])))
  )

(define-shape scene
  (hex-circle [alpha -.7]
              [b .2])
  )

(maximum-render-cycles 1000)
(start-shape scene)

There's a few new concepts in this code:
  • A loop construct to draw a circle of 6 hexagons. There's a lot of brackets in there! These denote the loop; the binding, in this case i takes the values from 1 to 6; then the loop body, in this case a call to hex-circle with attributes to draw each at the right position around the central hexagon.
  • A bit of lisp-style maths: (* 360 (/ i 6)) means 360*(i/6), in other words, rotate by a sixth of a circle for each loop iteration. 
  • We use a scene function to set up the basic attributes, for everything drawn: reduced alpha (transparency) and reduced brightness.
As with previous examples, play with the numbers, or add in extra adjustments to create something you like.

Recursive branching

You may have noticed in the previous example that we drew many hexagons on top of other hexagons, resulting in dark hexagons in the centre of the pattern. As an alternative we can instead draw just two outer shapes, positioned at alternative angles (in this case -60˚ and 60˚), the effect of recursion then fills the plane.

#lang s-exp stamps/lang 

(define-shape C
  (circle)
  
  (C [r -60] [y .5])
  (C [r 60] [y .5])
)

(define-shape scene
  (C 
     [alpha -.9]
     [b .2]))

(maximum-render-cycles 1000)
(start-shape scene)

Let's introduce a few variables to make playing with the numbers easier. This code is the pretty much the same as the above, but with the angle and space set at the start of the program (and red dots):

#lang s-exp stamps/lang 

(define angle (/ 360 8))
(define space 3)

(define-shape C
  (circle)
  
  (C [r (- angle)] [y space])
  (C [r angle] [y space])
)

(define-shape scene
  (C 
     [alpha -.9]
     [b 1]
     [sat 1]
     [hue 0]))

(maximum-render-cycles 50000)
(start-shape scene)

What next? Let's explore tessellation with regular shapes such as pentagons, hexagons and heptagons... (coming soon)

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