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

Discussion in 'Physics & Math' started by Peter Dow, Nov 18, 2016.

1. ### Peter DowRegistered Senior Member

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I'm going to try for the "Physics & Math" forum for this topic but no worries if the mods decide to move this to "Art & Culture" or "Architecture & Engineering".

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.

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Last edited: Nov 18, 2016

3. ### spidergoatValued Senior Member

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It's a triangle grid based tessellation, there are literally an infinite number of shapes one could make. Cool though. I'm reading a book entitled "Escher on Escher", where he details his methods. Pretty amazing considering he didn't even have a computer.

5. ### Peter DowRegistered Senior Member

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But I've circled the triangle, too.

Well it was inspired by geometric art by Jay Friedenberg [link] but is my "trispokedovetile" shape just more cool art, or could there be engineering applications, such as armour tiles [link], on account of the dovetailed joint between tiles?

Well I should credit my computer javascript programming that helped me to produce this design.

7. ### spidergoatValued Senior Member

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We need to make a 3D version, and see if we can create interlocking armor.

8. ### Peter DowRegistered Senior Member

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Well what did you have in mind?

Meanwhile, I've been busy in 2 dimensions ...

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%.

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.

9. ### spidergoatValued Senior Member

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Very nice!

Peter Dow likes this.
10. ### Peter DowRegistered Senior Member

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185
Last edited: Nov 23, 2016
11. ### Peter DowRegistered Senior Member

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

12. ### Peter DowRegistered Senior Member

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Typos -

"its 120 degree angle corners"
"the neck attaching the outer part rings"
"would have to be inserted by joining"

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15. ### arfa branecall me arfValued Senior Member

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Apart from the dovetailing, your tiles have six sides but are not regular hexagons.

16. ### Peter DowRegistered Senior Member

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Indeed. The regular hexagon shapes are only partly drawn (see diagram) and the hexagon edges do not correspond to the edges of single tiles, but I find the term "HEXAGON" useful for a parameter length for trispokedovetiles, referring to the length of the maximal diameter of the partly drawn regular hexagons.

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19. ### Peter DowRegistered Senior Member

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I've collected a number of similar images on my "three-head arrows" board on pinterest. [LINK]

Er, thanks.

Oh I nearly forgot - I've now added a Trispokedovetiles Gallery page [LINK] because I've got some real tiles cut now.

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