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More estimates

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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.

Root 3

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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 appr…

Flower of Crowns

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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.

So many choices...

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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.  …

Sun burst tiling

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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…

Playing with pentagons and triangles

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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 …

Playing with old program

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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.