Geometry. Three-spoke dovetailing tile tessellation. "Trispokedovetile"?

I've posted this image which I've described as a
"Three-spoke dovetailing tile tessellation".


Trispokedovetile tessellation by Peter Dow, on Flickr

which is a tessellation of this tile shape,


Trispokedovetile by Peter Dow, on Flickr

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.

Trispokedovetiles Animation Webpage
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 trispokedovetiles with "CIRCLE" = 125%.


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.

I've absolute zero knowledge about mechanical engineering so it might be a dumb question, but can you tell what exactly is the significance of this? I am clueless.

[Prototype]

I've absolute zero knowledge about mechanical engineering so it might be a dumb question, but can you tell what exactly is the significance of this? I am clueless.

Well one issue with hexagonal ceramic armour tiles or square tiles is that such simple shapes don't interlock naturally, as would tiles that would slot together like jigsaw puzzle pieces.

A little bit of tile movement in and out of the plane is useful in absorbing a ballistic impact, such as from a bullet and better than the armour tile absorbing the same kinetic energy by shattering.
So armour tiles shaped like jigsaw puzzle shapes (or some hexagonal variation such as these trispokedovetiles) could perform well in this engineering application.

Peter Dow

Well one issue with hexagonal ceramic armour tiles or square tiles is that such simple shapes don't interlock naturally, as would tiles that would slot together like jigsaw puzzle pieces.

A little bit of tile movement in and out of the plane is useful in absorbing a ballistic impact, such as from a bullet and better than the armour tile absorbing the same kinetic energy by shattering.
So armour tiles shaped like jigsaw puzzle shapes (or some hexagonal variation such as these trispokedovetiles) could perform well in this engineering application.

Interesting.
What exactly is the above animation trying to illustrate?

[Prototype]

Interesting.

Thanks.

[Prototype]

What exactly is the above animation trying to illustrate?

What part of -

Peter Dow

Trispokedovetiles Animation Webpage
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 trispokedovetiles with "CIRCLE" = 125%.

- didn't you understand?

To make a familiar comparison - the rectangle shape is well known and the range rectangular shapes can be specified by the ratio of a rectangle's length to width.

Your PC monitor will have a particular ratio of length to width, perhaps 16:9. Different ratios are possible and work, such as 4:3 but some shapes of rectangle would not work for a PC monitor, but might for a pocket calculator display - 7:1, say.

But everyone understands the established convention of how to specify a rectangle shape in terms of ratio of length to width.

The trispokedovetile shapes would appear to be completely unfamiliar, not only to engineers, but also to scientists and maybe to most mathematicians too and so there is before now no established convention of how to specify different trispokedovetile shapes.

So a new convention of specifying trispokedovetile shapes has to be suggested by someone, presumably by me, for the purpose.

My animation helpfully does (it's not merely "trying") show how to specify a trispokedovetile shape as a ratio of two dimensions, (not of "length" and "width" as for the rectangle) but as a ratio of "circle" diameter to "hexagon" maximal diameter, which I have named the "CIRCLE" percentage with the hexagon maximal diameter normalised to "HEXAGON = 100%".

For the armour tile application there may be an optimal trispokedovetile shape. The best performing shape might be "CIRCLE = 120%", say, or it might be "CIRCLE = 110%". The best performing shape might depend on the material - ceramic armour trispokedovetiles may perform best with a shape of say "CIRCLE = 106%" but titanium armour trispokedovetiles may perform better at "CIRCLE = 112%".

Since I've not done that engineering research I don't know what CIRCLE percentage value would be best for trispokedovetile armour tiles.

I do know what length to width works well for a PC monitor - 16:9. But I don't know what CIRCLE percentage is best for armour trispokedovetiles.

But having a way to specify trispokedovetiles means that at least I can ask the question.

What CIRCLE percentage is best for trispokedovetile-shaped armour tiles?

Maybe one day, some engineers will do some practical research to find out the answer to that question?

Peter Dow

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.

The only people "trying" here would appear to be the company Mozilla who are trying to offer a browser called "Firefox" which is trying to run Javascript programs but Firefox may not be succeeding very well.

Peter Dow

The only people "trying" here would appear to be the company Mozilla who are trying to offer a browser called "Firefox" which is trying to run Javascript programs but Firefox may not be succeeding very well.

I uninstalled and reinstalled Firefox and my Javascript animation works fine now with Firefox too. I guess I must have had a bug in my Firefox installation somewhere?

Tetrapod Stock Photos, Royalty-Free Images & Vectors - Shutterstock

A.V.Ramani

Tetrapod Stock Photos, Royalty-Free Images & Vectors - Shutterstock

three-head arrows

A.V.Ramani

Tetrapod Stock Photos, Royalty-Free Images & Vectors - Shutterstock

Thanks for that "tetrapod" search term - it is indeed more useful for finding arty triangular tile tessellations than "tripod" which finds camera tripods more than anything else.

The term "tetrapod" is not really accurately descriptive for my 2D tile shapes though because "tetrapod" refers to a tetrahedral 3D-shape with triangular faces and to a 4-pod structure, which in silhouette admittedly only shows the 3-pods.


So whilst I have nothing against "tetrapods" per se, I am sticking with my "trispokedovetiles" name for now, if that's OK. 😎

As Shakespeare said,"What's ina name?'. Certainly the name "trispokedovetiles" should be fine. If you think it's a bit long see if 'Triloktile' will work.
This pattern in 2D is akin to yours in interlocking:
Seamless Geometric Pattern. Gray Abstract Geometrical Design. Flat Monochrome Design.Monochrome Striped Offset Tetrapods. Stock Vector Illustration 265176056 : Shutterstock

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.



Step 2

Use a 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



Step 4

Use paint bucket tool to colour the tiles black



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.

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%.

The reason for choosing C100 for the outer layer of the armour is because its 120 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 attacking 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.

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

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

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

For Computer-Aided Manufacturing of trispokedovetiles, I'm publishing today 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.

HOLES!
Now cut trispokedovetiles by CNC laser cutter with (or without) HOLES!

TRISPOKEDOVETILES ROUTER CUTTING
My trispokedovetile CAD-CAM webpage now offers an (X,Y,Z) option for router cutting using rotary tool paths between spaced out tiles -


I've simulated the CNC router code -

BEWARE BAD LINKS ABOVE!
I lost my website hosting in January. I was not "hacked" but the gcehosting administrator seems to have gone out of business and the hosting company he was using (godaddy) is serving various spurious adverts whenever anyone clicks one of my old links.
I have got the following pages back on-line with a different hosting company now so try these links. 😀
LINKS

• Trispokedovetiles: Three-spoke dovetailing tiles by Peter Dow
• Trispokedovetiles: CNC code to cut tiles by Peter Dow
• Trispokedovetiles Gallery

I can't edit my earlier posts to correct the old links, sorry.
I've not done anything with this project since January.