Saturday, November 27, 2010

More estimates

Last time, I talked about the height of a hexagon; the strategy I used works with any shape with a pair of parallel sides.  The parallel sides allow you to stack the polygons and add up the heights without having to worry about angles. For instance, it will work with a regular octagon.  The picture to the left shows 5 stacked octagons that are slightly more than 12 square (otherwise in TileLand the top purple square would be wiped out).  That means that 5 octagons > 12 squares or that  1 octagon > 12/5 =2.4 (since they are unit squares). 
For those who like to exact lengths, you can break the height of an octagon up into three pieces.  This diagram below demonstrates the breakdown: two half squares of length 1/ root 2  (since the diagonal is 1) and a rectangle with the long side of length 1.  Root 2 plus 1 is 1.41421+1 = 2.41421.   

Look forward to when I tackle odd sided polygons that don't have parallel sides.

Tuesday, November 16, 2010

Root 3

Who said approximating can't be fun?  This is an activity that I designed to get geometry students to come up with estimates for the height of a hexagon.  The tool I asked them to use was Tileland.  Tileland is a world of polygon paths (well known to those who follow this blog regularly) that doesn't allow for polygons to overlap (the top green squares replaced vertical squares that were there previously).  The following polygon path demonstrates an estimate for the height of a hexagon with unit length sides that is pretty good.  Here the height of four hexagons is slightly less than seven squares; this means that the height of one hexagon is slight less than 7/4 = 1.75.  Using the diagram of the hexagon composed of equilateral triangles and the Pythagorean Theorem, it is easy to show that the hexagon's height is actually root 3. Thus root 3 must be slightly under 1.75.  It is pretty close since root 3 is about 1.732--so within 0.02.

The reason I prefer to use types of approximations like this has to do with having a concrete physical focus to talk about.   It allows for variations and better discussions.  These two diagrams are more of the conclusions to the discussions and experiments.  Also I like reducing and/or delaying  using the magical calculator.  To me, it seems to be an ender rather than a promoter of thought.  In fact, I prefer to have students multiply 1.75 by itself to show how close it is to 3.
For better estimates, you can construct paths corresponding to these fractions 12/7, 19/11, or 26/15 fairly easily in Tileland.  To obtain 45/26, you have to use a more wiggly approach....

Thursday, October 28, 2010

Flower of Crowns

This one has an interesting centre. As it expands out it becomes clear that there are only 15 orientations of the crown pattern.  Given the nature of the centre it was clear that there it had to be a multiple of 5.    

I am still searching for a true spiral pattern that uses these crowns.  Currently I have only found things that look spiral but really have rings instead.  Perhaps, the shape is too complex.  

I guess I'll have to give it some more thought.

Wednesday, October 20, 2010

So many choices...

After playing around with the sun burst in the previous entry, I remembered these 30-gons with interiors that are made from nice loops of pentagons and triangles that meet exactly at the centre of the 30-gon.  I'm sure I have already made a bunch of patterns that already have this 30-gon but I haven't been able to find a blog entry with one.

My goal for patterns is to have an interesting and symmetric way of making neighbourhoods for tiling.  These 30-gons turn out to have a fair number of choices on how they can relate to their neighbours.  Essentially, coming up with easy to follow rules for describing the pattern is a way of completely understanding the pattern.  These rules embody what I like about patterns: access to the infinite.

My rule I choose to explore here, has to do with having the internal structures back-to-back.  I'll label the structures: loops and arrows.  The loops refer to the three positive-space structures composed of six pentagons and 5 triangles.  The arrows refer to the three negative space structures (white space) that are pointy and all touch the center of each 30-gon.

To see my rule in affect, imagine the pattern without the pink-centred 30-gons.  All the purle-centred 30-gons are connected with back-to-back arrow structures.  Paths that follow these arrow structures form a hexagonal grid with the purple centres being at the vertices.  A similar story occurs for the loops structures but there is a hitch--there are two types of purple-centred 30-gons: ones to the right of a pink-centred 30-gon and ones to the left.  If we ignore all the purple-centred 30-gons on the right, we end up with similar hexagonal grid formed by paths following the loops.

Seeing the above structures makes my rule seem complete.  However, pairing the pink-centred 30-gons with the purple-centred 30-gons on the right leaves us with only back-to-back pairs of loops with arrows.   Still I am fairly content with this patterns symmetry and it's overall appeal.

I have included this blurred patch that can be used as a background.

Tuesday, October 19, 2010

Sun burst tiling

Here is the pattern I mentioned last entry.  In this pattern, the E-shaped white spaces radiate out from the central circle.  I guess a better label for these E's are described as 3-point crowns since here there are also 4-point crowns.  Should the pattern be extended indefinitely there would many more 4-point crowns than 3-point crowns. The spiral effect seems to be produced by all the points pointing in the clockwise direction around the centre of the the circle.  Then again, maybe it has more to do with the curved lines that drawing the eye in a clockwise direction.  

Although the repetitive aspect of the outer rings is rigidly defined, the inner part of the circle has many options.  I have explored a few possibilities in a different context.  But I have one that incorporates the 4-point crown.  I'm not sure if I should include the middle 3 pentagons.  The three humpback whale shapes in white space can be replaced by a shape that has nice rotational symmetry. Here's an alternative centre.

Maybe this is a better colouring....

Monday, October 18, 2010

Playing with pentagons and triangles

On a very long train ride from Buffalo to Albuquerque, I ended up chatting with a rapper on his way back to LA.  We both had a lot a hours to fill so I showed him some of my polygon programs.  Nearly every time I play with TileLand I end up creating something new (at least to me).  This time was no different as you can see from the picture on the left.

This pattern with pentagons and triangles started out as a big 30-sided polygon (constructed by alternating pentagons and triangles) and slowly turned into a triangular grid of green triangles that has been augmented with zig-zag spacers that resemble purple and red dresses. 

I'm not yet finished with playing with the fruits of this play.  I'm going to attempt to make something different with the E shaped white space.  The interesting thing about this E is that it can be completely surrounded by polygons by inserting another pentagon where you have the two kissing red triangles. Check back in a couple of days and I should have the new pattern completed.

 Here is a blurred version to use as a background.  One of these days I'll have to eventually address the idea of colours--but that day is not here yet. :)

Friday, January 29, 2010

Playing with old program

I was about to progam a PolygonR&D program to compute with Fibonacci codings but I got distracted with an old pattern. I get so sucked in with all things pentagon. I wonder if there is some occult connection... pentagram etc. Kidding.

OK, here's my procrastination pattern.  It's related to leaves.  I thought I had a similar pattern posted but I couldn't find it (I probably should have a better index of images).   Each square in leaves is replaced with a pair of triangles.  The colouring adds a little bit of complexity as well but nothing too crazy.

With a small modification I'll post the pattern that I thought I had previously posted.... If you are the one person following this blog and happen across this  pattern, like me know.

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