Corresponding Tessellations

(maybe someone has already done this)

proposition:

for every convex polygon plane tessellation there is a corresponding convex polygon tessellation such that each polygon is [can be represented as] a vertex at it's center that connects through an edge to each other polygon's vertex-center once through each exterior mutually adjacent segment.

With two sets of parallel lines (right angles/rectangles/squares) its corresponding tessellation is itself.

With a tiling with triangles its corresponding tessellation is offset hexagons
conversely the tiling with offset hexagons is triangles.

If you take two perpindicular parallel line sets and then add two perpindicular parallel line sets at a 45 degree angle [squares with an 'X' crossing it] its corresponding tessellation is offset octagons with small squares at each quadrant.
Again the corresponding tessellation is the 'X'ed squares.
Interestingly in this case it means the offset octagons have a better [more linear] flow pattern than its converse the 'X'ed squares.

If you think of the vertexes as the 'structure' and the polygon faces as the flow-through faces the corresponding tessellation is the flow pattern of the original. (and visa-versa)

I believe this is also true for non-regular tessellations as well as regular.
It may be true of concave polygons also, but would probably require using curved connection/edges. Multiple sequential segments adjacent to the same polygon would be represented as a set of multiple edges side-by-side between the two vertexes.

I suspect this is also true of space filling polyhedra (3D). ie. if you find a center point of each polyhedra and connect each pair of centers[polyhedra] with an edge it will create a wireframe of its corresponding space filling polyhedra.

This is pretty obvious with cubes [three sets of perpindicular parallel plane sets].
I am quite sure this is true for perpindicular intersections in any number of dimensions. (Hypercubes, etc.)

If it is true It would mean that any polygon/polyhedra/4D(?) filling would be completely representable as a graph structure.
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Originally posted by slank


If it is true It would mean that any polygon/polyhedra/4D(?) filling would be completely representable as a graph structure.
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Yes, wouldn't it simply be the integral of the function representing the polyhedra segment used to connect each centerpoint and in the range from the initial polyhedra centerpoint to the ending polyhedra centerpoint - and for a tessellation I assume the integral would be to infinity.

And I might be wrong - that's why I'm pitching that out to see if it makes sense to you.

I think there has been work done on this, but I'm going to be the first to admit that I absolutly don't understand it.
scholar.google.com...

In fact, this may address your question:
www.turpion.org...

I have DONE tessellations. I love drawing them. But my eyes kind of cross when people mention math to me. It has to be introduced slowly so it doesn't panic my neurons.

Wow, you just gave me a mental image of a 3D tessellation. I am not a math wiz but I have long been a fan of MC Escher and the mathematics involved in creating tessellations. I too enjoyed creating them but I had always considered them in 2D. I am thinking of a cube with his fish tessellation (hey it's simple ok) and how these fish would look in that cube.

Just a possibility I have never before thought of, thanks.

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I was thinking that tetrahedrons can be used to fill space. Then i did some angle calculations and found problems.

Here is a great page of space filling solids.
mathworld.wolfram.com...
They are in java so you can rotate them onscreen.
A wireframe version might be more helpful in thinking about my proposition, though.

It is so difficult to think about space filling solids that i think i would have to have a wire frame model in front of me just to think about it.

I guess with many space filling systems it can/might work by thinking about sets of parallel planes and how they can intersect in a regular fashion.
(but it is still way over my head)
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Originally posted by slank


It is so difficult to think about space filling solids that i think i would have to have a wire frame model in front of me just to think about it.


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Actually, you don't need the visualization at all if you do it mathematically. If you take the function that describes the geometric shape of the solids being used, and you establish the boundary conditions that must be met in order to "place" each solid the way you want relative to each other, you can establish your tessellation via satisfying those boundary conditions.