A Holiday Star Bucks the Trend

Happy Holidays, dear readers!  Today I offer you a Holiday Star I just created, along with the story of its creation. First of all, the “Star” of the show:

Holiday Star - copyright Phil Webster 2011

 

Pretty, don’t you think?  So, where did it come from?  Why don’t you study it for a moment before reading on.  See what you you can notice about it. For example:

  • What different kinds of shapes can you see?
  • Do they have anything in common?
  • Is there any symmetry going on? Any asymmetry?

Inspiration Number One – The Starbucks Pattern

So, as my wife, brother, and sister-in-law will attest, around Thanksgiving time I became rather obsessed with a pattern that Starbucks is using pervasively in its holiday advertising. Now would be the time to go re-read the post title, groan a little, and then resume reading.

I mentioned this pattern in my last newsletter, but didn’t say much about it. That’s about to change. :-) First of all, here is the pattern, which I copied off of a tumbler I purchased:

Starbucks pattern (dotted lines show repeat)

 

As the dotted lines show, the pattern does repeat, although with a long enough cycle that it’s not immediately obvious. For those who are interested, every other place I’ve checked (mugs, displays, even my Starbucks gold card!) the pattern is the same.

Deconstructing the Pattern – The Pieces

So, the first and most obvious interesting thing about the pattern is the pieces. It doesn’t take long to realize that they are all combinations of the sub-pieces you get if you take a pentagon and draw a five-sided star (pentagram) within it. My next step was to figure out how many different such pieces there are (remember, I said I was obsessed!). Here is my summary of every piece I could find, not counting reflections and rotations of the same piece:

Summary of possible pieces from subdivided pentagon

 

It’s worth noting that there are a couple of “filler pieces” in the pattern that are not from this set, though they’re closely related.  Can you find them??

Deconstructing the Pattern – The Grid

So, knowing that (most) of the pieces fit inside pentagons, it seemed a natural next step to lay out the underlying pentagons to see if any pattern emerged:

Starbucks pattern with underlying pentagons (blue for upright, purple for inverted)

 

Lo and behold, when you remove the pattern itself, suddenly more underlying order is revelaed than is immediately apparent:

Grid of pentagons underneath the Starbucks pattern

 

Inspiration Number Two – My College Doodle

So, having set that all aside a few weeks ago, the other day an old college friend reminded me of an elaborate doodle I had once created during a mind-numbing lecture we were in.

The Doodle

I haven’t found the original doodle but remembered it well enough to recreate it quickly:

My college doodle - "Pentagon Rosette" copyright 1986 Phil Webster

 

OK, I’m sure you see where I’m headed with this now…

The Star Variation

Knowing I wanted to create something holiday related, I first modified my original doodle into a star shape, and added the shading I used for the Starbucks grid, in order to create my foundation:

Pentagon grid for my Holiday Star

Putting It All Together

From here, it was just a matter of laying in some of the possible tiles onto the grid, while trying to maintain a nice sense of overall balance, make sure the gaps between tiles were nice shapes, and make the outside border conform to something nice. The result is my Holiday Star for 2011, shown at the top of the post.

How It “Bucks the Trend”

While it’s true that I titled the post mostly just to make the horrible pun, it is also true that my star “bucks the trend” set by the Starbucks pattern in a few ways. Namely:

  • I used a few tiles from the pentagon-pentagram set that Starbucks did not. Can you find them?
  • My pattern has rotational symmetry, while theirs has translational (“glide”) symmetry
  • Due to the nature of my grid, I used a different “filler piece” (piece not derived from the pentagon-pentagram system) than they did. Again., can you spot it?

Special Thanks

That’s about it, except that I would be remiss without offering some special thanks to:

  • My wife Jen, brother Brian, and sister-in-law Katy for putting up with my ranting about “this awesome pattern… look, it repeats!… check out the grid!…”
  • My buddy Barak for reminding my of that college doodle
  • The staff of my local Starbucks (where I sat to write this post), not only for their awesome friendly service, but for letting my take some of their holiday-patterned display pieces now that they’re done with them!
Most of all, thanks to YOU, my readers, for your interest in Geometric Arts. Happy Holidays and a Happy New Year!

 

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About Phil Webster

Phil is the creator of GeometricArts.com. You can reach him on the Contact page.
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