Hat types from equilateral Cairo pentagons

-

I created these shapes in the grid below, using the Polyform Puzzle Solver by Jaap Scherphuis.  I then reconstructed them in VectorEngineer Pro-Tools using angles of 69.295, 90 and 114.295 that were suggested to me by Jaap.


The DXF files were converted using Bob Nungester's small but effective DXF to TXT utility, for use in Arnaud Chéritat's dedicated tiling applet (links at the foot).



The Sombrero: 


This tile can simulate the chair tiling if reflections are allowed:



Crumpled hat:



A group of four tiles can produce an attractive periodic tiling:



A six-tile motif combined with its reflection, which is also rotated, produces a striped pattern:





Beret:


This tile when used with its reflection, produces strips that can be glued together in different ways.  However, there's only one configuration that the beret can tile if reflections are not used.




Novelty hat:



This polygon can produce misshapen mesh-like structures, if allowing reflections.


Links:

Bob Nungester's DXF to TXT convertor:
https://www.mediafire.com/file/0ifl4o7mnv6fi7q/DXFtoTXT.exe/file

Arnauld Chéritat's tiling app:
https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Monotile/generic/v4.5.1/









Comments

Post a Comment

Popular posts from this blog

Using cellular automata for retro maps

-
Image
There are several cellular automata map examples online but very few resemble mine.  I use fewer generations,  initial seeds commonly of between  1 - 5% and   a variety of alternative life rules that I have experimented with over the years. Programs used: MCell http://www.mirekw.com/ca/ Mirek Wojtowicz (Mirosław Wójtowicz) Irfanview (for batch conversion of files) https://www.irfanview.com/ Irfan Skiljan Jasc Paintshop Pro 7 (for flood fill and borders) Image to Ascii Converter https://www.softpedia.com/get/Multimedia/Graphic/Image-Convertors/Bitmap-to-Ascii-converter.shtml Peter Bone The larger images of 1280x960 pixels (each pixel representing a 5x5 or 8x8 user defined graphic block) are scaled down versions of entire maps.  Those below them are a 3x zoom section to show the scale.  I purposely chose a grid size of 160x96 pixels as games (known as homebrews) are still being produced for the Mattel Intellivision games console (now owned by Atari). Leg...
Read more

Scottie dog, rabbit and donkey

-
Image
I made up these shapes by overlapping and creating voids from other Spectre related polygons.  From left to right, the Scottie dog,   rabbit   and donkey.  They either don't need or won't work with their reflection, so are essentially single tiles: Later research revealed that all of these polygons can be constructed with Bram Cohen's ducks and rotary blades. The Scottie dog is cute but the most versatile of the three is the  rabbit .  It can do everything and more!  Putting only two together back to back creates a small hexagon-like figure: Each ring can be constructed in three different configurations.  These motifs can be filled in a variety of ways and are periodic, even if one was the size of a planet: Other central cores include feet in, heads in and chaotic: It seems that spirals can have any number of arms.  Those shown are 2, 3, 5, 11 and 12: Anthills.  The surrounding area can be filled in all manner of ways: Start of...
Read more

Fractal eXtreme

-
Image
I've been using Fractal eXtreme of late, creating many zoomed images of Mandelbrot variants.  These include Perpendicular Mandelbrot (top right),  Burning Ship, Perpendicular Burning Ship,  Buffalo and Perpendicular Buffalo (bottom right) . The program was last updated in January 2016  but is amazingly fast and has antialiasing options when generating/previewing fractal zooms.  More information here: https://www.cygnus-software.com/ The preset palettes in Fractal eXtreme are mainly made up of primary colours (garish) which I didn't particularly like so I opted for greyscale.  I later modified it to produce a dark sepia-like duotone. Mandelbrot  (we all know and love). Perpendicular Mandelbrot Burning Ship Perpendicular Burning ship Buffalo Perpendicular Buffalo More Perpendicular Mandelbrot and Perpendicular Burning Ship images (in slideshow format) on my website: https://the-orangery.weebly.com/perpendicular-mandelbrot.html https://the-orangery.weebly...
Read more