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Showing posts from January, 2007

Mixing Hexagons and Pentagons

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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

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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

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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 check…