Mixing Hexagons and Pentagons

Here's a pattern that alternates a loop of six pentagons with the green hexagons. The loops of pentagons have two orientations. The three holes surrounded by the pentagons and hexagons are the most interesting structures to look at. I'll try to make a few variations of this pattern where I replace the hexagons with other polygons.

Unraveling a Pentagon Loop

See if you can construct the process of unraveling a loop of six pentagons. This is a similar process to Jan. 3rd's pattern. For me, I enjoy looking at the holes in this pattern: the s-shapes and a shape similar to the "Cross of Lorraine" (the double cross). Let me know if you can construct this.

Unwound Heptagon Loop

In this busy time of year, I've had a tough time attending to the blog. I get the feeling that the first part of this semester will be a bit crazy as well. So that is warning in advance. This pattern comes from a simple loop of six heptagons that are unwound by six hexagon-square pairs. As a visual explanation of this process, the picture below shows a rule that is applied to a pattern. The rule is placed the box; it shows a pair of red heptagons being replaced by a pair of heptagons with an orange square and an olive hexagon being inserted between the pair. When this rule is applied to the six instances of heptagons pairs in the adjacent loop of heptagons, the pattern on the right of the picture results. I call this process unwinding because of the angles introduced by the hexagons makes the loop go in the opposite direction. For another example of this, see July 10th's pattern.

To create the original pattern, a number of resulting unwound loop is connected in a checker board fashion and the opposite square are slightly elaborated with squares and heptagons.

Patch Quilt

This quilt has three sizes of squares: the small white square, the medium olive square, and the rest. The pattern can take a while to understand but the colours help. Since this quilt doesn't have a four-way rotational symmetry, the program written isn't as short as I would like. When I get a little more free time I'll experiment with some more unravelling of loops. This pattern like the last is from "Tilings and Patterns".

Tri-triangles

Three different triangles fitting nicely together. This one came directly out of the wonderful book Tilings and Patterns (p. 74). There are a few ones similar to this one as well that I may put up there. I'm more a fan of the patterns that come from polygon loops that are combined rather than ones like this. Still these ones are easily described since they have only one local neighbourhood. The PolygonR&D program has only one subprogram that describes the path from a purple triangle it's neighbour. The smallest triangles are not drawn since they are white.

tri {
  right
  scale 0.625
  left
  sides 3
  paint orange
  scale 1.6
  sides 3
  ifon none {
    paint purple
    3 { create tri
        left
    }
  } {}
}

A New Checker Board

This one has a lot of patterns in it. At some vertices you can see three differently coloured teardrop shapes that form a propeller. In fact, each teardrop shape belongs to only one propeller. The pattern can be thought of as a particular colouring of a hexagonal grid where certain hexagons are broken into six triangles. Let me know if you want to see more patterns like this one.

Eggplant

Well this one is simply labeled eggplant because of the colours. The pattern demonstrates some local three-way rotational symmetry and some local three-way reflectional symmetry. The program that created it only has two sub-programs: one that describes the neighbourhood of nonagon with four triangles around it and another one that describes the neighbourhood of the nonagon with only three triangles around it. Those are the only local neighbourhoods in the pattern.