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Geometry. Three-Spoke Dovetailing Tile Tessellation. "trispokedovetile"?




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#1 Peter Dow

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Posted November 19 2016 - 01:06 AM

I've posted this image which I've described as a
"Three-spoke dovetailing tile tessellation".
 
30936980531_67a6d41ab0_o.jpg
 
which is a tessellation of this tile shape,
 
30938091111_5a347b71fa_o.png
 
Check my Flickr page for the preceding design iterations and inspiration.
 
I've named the shape
Trispokedovetile
which is a contraction of "tri-spoke dovetailing tile".
  • "tri-spoke" because the shape is similar to a 3-spoke motorcycle wheel with three bites taken out of it.
  • "dovetailing" because the tiles interlock like a dovetail joint
 
But it is quite possible this shape is already known to geometry and already has been named.
 
Do you know if this shape has already been named? If so please do reply to tell me what the name is.

Attached Files


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#2 ʈuɱɓɭiɳɠ.ƌičɛ

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Posted November 19 2016 - 05:05 AM

Klein Bottle.

 

Klein_bottle.png


348r02f.jpg


#3 Peter Dow

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Posted November 19 2016 - 03:01 PM

 
I've programmed a webpage using Javascript to display an animation which shows a range of different trispokedovetiles, each of which can be specified by a "CIRCLE" percentage, which is the ratio as a percent of two parameters -
 
1. A "HEXAGON" parameter length - always nominally "100%"
2. A "CIRCLE" parameter length - the animation varies this between 100% and 135%, though up to 150% is possible in theory.
 
So you can specify the "CIRCLE" percentage to specify a particular shape of trispokedovetile.
 
Here is a still image from the animation - showing the trispokedovetiles with "CIRCLE" = 125%.
 
Attached File  trispokedovetiles_1200.png   95.98KB   0 downloads
 
I've tested the animation in Chrome and Internet Explorer browsers and it works fine for me.
 
But it is not working in Firefox for some as yet unknown reason.
 
Let me know by replying here if you have any other problems with the animation not displaying for you.

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#4 guerillabedlam

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Posted November 19 2016 - 03:10 PM

I've tested the animation in Chrome and Internet Explorer browsers and it works fine for me.
 
But it is not working in Firefox for some as yet unknown reason.
 
Let me know by replying here if you have any other problems with the animation not displaying for you.


For your firefox issue, if you used the -webkit- prefix to run it, maybe try copying those lines replacing that prefix with -moz- and leaving the rest of the code the same.

3i6eiPI.png?

 


#5 Peter Dow

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Posted November 23 2016 - 04:39 AM

Attached File  trispokedovetile_1x3D.png   49.52KB   0 downloads
 
Tutorial on how to produce 3D images from my trispokedovetiles webpage
 
Step 1.
 
Create a line image using my webpage, this is for CIRCLE = 120. Right click save image as.
 
Attached File  trispokedovetile_line_C120_600px.png   17.75KB   0 downloads
 
Step 2
 
Use an image editing program such as paint.net
 
Crop the image to suit and use the paint bucket tool to fill the tiles you want to make 3D with another colour, I chose red.
 
 
Step 3
 
Use the magic wand tool with option flood mode global to select all the red area and then copy and paste the red area into a new image
 
Attached File  trispokedovetile_C120_3_cut_paste_red.png   7.07KB   0 downloads
 
Step 4
 
Use paint bucket tool to colour the tiles black
 
Attached File  3-trispokedovetiles.png   10.01KB   0 downloads
 
Step 5
 
Upload the black and white image to selva3d.com with option "Logo/text", adjust the height and rotate the image to suit and right click to save the image.
 
Attached File  trispokedovetiles_3x3D.png   42.37KB   0 downloads

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#6 Peter Dow

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Posted December 06 2016 - 10:47 AM

Now to consider the important issue of interlocking trispokedovetiles against movement in the direction normal to the tiled plane, which for the application of tiled armour would be the normal to the armour surface, in the direction of a bullet's path.
BILAYER TRISPOKEDOVETILES
I propose that the unit armour tile be comprised of 2 joined trispokedovetiles with matching HEXAGON parameters but each with a different CIRCLE percentage.

For example, suppose we choose trispokedovetiles with CIRCLE = 100% and 121%.

Attached File  C100.jpg   12.08KB   0 downloads Attached File  C121.jpg   14.32KB   0 downloads

The reason for choosing C100 for the outer layer of the armour is because its 120 degree angle corners would be more robust.

The reason for choosing C121 for the inner layer of the armour is because CIRCLE = 121% offers the largest percentage where the neck attaching the outer part rings is at least twice the thickness of the ring, attempting to balance the robustness of the ring parts to the robustness of the neck versus tensile stresses.

Attached File  why121.jpg   16.36KB   0 downloads

Stacking and joining those together forms a bilayer trispokedovetile, "C100+C121".

Attached File  C100C121.jpg   18.31KB   0 downloads Attached File  C100+C121_3D.jpg   74.34KB   0 downloads

Drawing the 2 layers semi-transparently we can see how the bilayer trispokedovetiles would interlock in the normal to the plane.

Attached File  100+121_crop_low.jpg   84.53KB   0 downloads

2/3rds of the tiles can be slotted together, either the yellows and the blues or the yellows and the purples or the blues and the purples.

However the final 1/3rd of the tiles would not simply slot in and would have to be inserted by joining the two halves of the bilayer trispokedovetile in situ.
 


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#7 Peter Dow

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Posted December 06 2016 - 10:55 AM

For Computer-Aided Manufacturing of trispokedovetiles, I'm publishing my new Javascript web-page -

Trispokedovetiles: CNC code to cut tiles web-page G-code generator [LINK]

- which generates Computer Numerical Control (CNC) code which I've tested with CNC Simulator Pro [LINK] but not on a real CNC cutter as yet.


Edited by Peter Dow, December 06 2016 - 10:59 AM.

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#8 Wu Li Heron

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Posted December 06 2016 - 11:33 AM

I'm a Rubic's Cube fanatic and you could possibly make a sphere with that shape, just a thought.



#9 Peter Dow

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Posted December 18 2016 - 09:00 AM

 
My trispokedovetile CAD-CAM webpage now offers an (X,Y,Z) option for router cutting using rotary tool paths between spaced out tiles and options to cut holes in the tiles too - 
 
Attached File  trispokedovetiles_spaced.png   131.13KB   0 downloads
 
I've simulated the CNC router code - 
 
Attached File  routersimulation_50.jpg   86.77KB   0 downloads

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#10 Logan 5

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Posted December 22 2016 - 09:36 PM

Too bad you don't have it in .stl or .obj....


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