Platonic Solids

[DrJB]

I tried, but no success so far. I'm still not sure how you can tile this ...

... into this

The only one I could do is shown below

 

[Didumos69]

The same way of stacking is visible in this 'cloud':

When using the same pattern with longer axles (32L) for the slanted connections inside the primitive (not between primitives), then you get this. The primitive has progressed towards a octahedron and the small squares have become long rectangles. Inbetween the octahedra you can see the (red) tetrahedra with paired edges (LXF-file).

Edited April 12, 2017 by Didumos69

[Didumos69]

59 minutes ago, DrJB said:

The only one I could do is shown below

Can you share the LXF? I have trouble seeing what it actually is. It certainly looks interesting.

[aeh5040]

30 minutes ago, Didumos69 said:

The same way of stacking is visible in this 'cloud':

When using the same pattern with longer axles (32L) for the slanted connections inside the primitive (not between primitives), then you get this. The primitive has progressed towards a octahedron and the small squares have become long rectangles. Inbetween the octahedra you can see the (red) tetrahedra with paired edges (LXF-file).

That's right, although it also works if you replace precisely all the 2L axles between 'primitives' (the ones that are sticking out in the first picture) with 32. (This is what I meant originally).

In both versions, all the edges of the resulting lattice are 'paired'.  It's harder to see this for the octahedra above because of where you chose to stop...

The one with all 2L axles lengthened is cool too btw!

I like the way these produce 45 degree angles without using any 45 degree part, and without any approximations (like a 7-7-10 triangle). A priori I would have thought this might not be possible.

Edited April 12, 2017 by aeh5040

[DrJB]

Certainly, here it is.

tetrahedron.lxf

3 minutes ago, aeh5040 said:

That's right, although it also works if you replace precisely all the 2L axles between 'primitives' (the ones that are sticking out in the first picture) with 32. Then all edges of the resulting lattice are paired. (This is what I meant originally).

Ah, ok, found my mistake. I replaced ALL the 2L axles with longer ones

Edited April 12, 2017 by DrJB

[Didumos69]

1 minute ago, aeh5040 said:

That's right, although it also works if you replace precisely all the 2L axles between 'primitives' (the ones that are sticking out in the first yellow picture) with 32. Then all edges of the resulting lattice are paired. (This is what I meant originally).

I get it now. It would give a nicer picture, but it would yield the same infinite pattern if I'm correct.

[aeh5040]

8 minutes ago, Didumos69 said:

I get it now. It would give a nicer picture, but it would yield the same infinite pattern if I'm correct.

I see, I actually had not realized that! I think you are right. In the original picture you can see another copy of the primitive nestled between the others, in multiple colors...

Btw, it would be nice to see the original with all 2L axles replaced with 3L. Then all squares would be the same size. Then the question arises, can we put a connector in the middle of each of those 3Ls and connect them somehow...

Edited April 12, 2017 by aeh5040

[Didumos69]

15 minutes ago, DrJB said:

Certainly, here it is.

tetrahedron.lxf

Ah, ok, found my mistake. I replaced ALL the 2L axles with longer ones

Thanks! If I'm not mistaken, the gaps enclosed by your octahedra are in fact cuboctahedra, which would mean you can fill space with octahedra and cuboctahedra. I haven't red about that anywhere...

Edited April 12, 2017 by Didumos69

[aeh5040]

Just now, Didumos69 said:

Thanks! If I'm not mistaken, the gaps enclosed by your octahedra are in fact cuboctahedra, which would mean you can fill space with octahedra and cuboctahedra.

Cool! I can at least see why such a space filling is possible (independently of the Lego!). If you make the cuboctahedra meet at the square faces they form a cubic lattice, and the gaps between them are octahedra.

[Didumos69]

24 minutes ago, aeh5040 said:

Cool! I can at least see why such a space filling is possible (independently of the Lego!). If you make the cuboctahedra meet at the square faces they form a cubic lattice, and the gaps between them are octahedra.

That makes sense. Here's the cuboctahedron excerpt from @DrJB's honeycomb.

The difference with the tetrahedron-octahedron honeycomb is that @DrJB stacked the octahedra as

44 minutes ago, DrJB said:

Ah, ok, found my mistake. I replaced ALL the 2L axles with longer ones

Btw, great to see that we have come to a shared understanding in this matter . I like to think that if all these regular solids and there relationships weren't discovered already, they would have been now .

Edited April 12, 2017 by Didumos69

[aeh5040]

21 minutes ago, Didumos69 said:

That makes sense. Here's the cuboctahedron excerpt from @DrJB's honeycomb.

Cool. For some reason that picture especially helps clarify what is going on for me.

One more little observation: you can replace half the connectors in one type of square (the big one in that last picture) with 10197 for yet another variation.

[mocbuild101]

It all seems interesting, cool, confusing and amazing all at the same time ! the only one I have ever made is a cube .

Edited April 13, 2017 by mocbuild101

[aeh5040]

Hi folks, I managed to make a tensegrity icosahedron:

It is a tensegrity structure: the axles are all in compression, and do not touch each other, the chains are all in tension, and it holds its shape.

 

[Didumos69]

4 hours ago, aeh5040 said:

Hi folks, I managed to make a tensegrity icosahedron:

It is a tensegrity structure: the axles are all in compression, and do not touch each other, the chains are all in tension, and it holds its shape.

Brilliant! There is one close to my work. Built by students in a single night as a joke in 1974. Made from old telphone poles. It's still standing.

Edited May 8, 2017 by Didumos69

[aeh5040]

7 minutes ago, Didumos69 said:

Brilliant! There is one close to my work. Built by students in a single night as a joke in 1974. Made from  old telphones poles. It's still standing.

Nice!  Dutch students must have a very sophisticated sense of humour...

Edited May 8, 2017 by aeh5040

[ColletArrow]

Impressive, although I completely failed to understand or even pronounce it's name...
Are you saying that it can't be deformed in any way as all the chains act against each other? Or would it just go splat if you picked it up and threw it?

[aeh5040]

20 hours ago, ColletArrow said:

Impressive, although I completely failed to understand or even pronounce it's name...
Are you saying that it can't be deformed in any way as all the chains act against each other? Or would it just go splat if you picked it up and threw it?

That's correct - it basically cannot be deformed.  It is fairly amazing to me that tensegrity structures exist.  Of course, if you throw it hard enough it will break, like any Lego model.  And there is a small amount of flex, because of stretching of the chains and bending of the axles - it actually has quite a nice feeling of "bounce" to it.

[lcvisser]

On 5/8/2017 at 7:34 PM, aeh5040 said:

Nice!  Dutch students must have a very sophisticated sense of humour...

The structure @Didumos69 mentions is on the campus of a technical university famous for its geek students (I can know, I studied there ).