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 Post subject: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 3:03 am 
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Could it be made?

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It seems that the 3 fold pentagon corners could be left unfudged, but how much fudging would be needed to turn the hexagon-pentagon-pentagon vertexes into 3 fold symmetric corners?

Also, what is the dual of this solid? Wikipedia mentions 'hexakis truncated tetrahedron', but what does this look like? I cant visualise it, and I cant find it.

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 Post subject: Re: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 4:32 am 
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I suspect that is indeed possible, but I have yet to identify the best way to create the geometry for fudged puzzles, Oskar seems to be the champ at that! He made the Illegal Cube work, and it seems like it should require more fudging than this.

It is worth mentioning that a few puzzles have used this shape before.

First was Timur's Skyglobe, which turns about the hexagonal faces - although fudged, this puzzle does not jumble
Then came David Pitcher's Jumball, which turns about the irregular pentagonal faces, making it a jumbling puzzle

The dual, by the way, looks like this:
Attachment:
TruncatedTriakisTetrahedronDual.png
TruncatedTriakisTetrahedronDual.png [ 23.3 KiB | Viewed 931 times ]


It looks a lot like an icosahedron at first, but notice that there is one point in view where 6 triangles meet at a corner, instead of the usual 5. There are 28 faces and 16 points. 4 of those points are surrounded by 6 triangles, the rest by 5 triangles (exaclty one of which is regular). There are 4 regular triangles and the other 24 are all the same isosceles triangle: approx 64.55, 64.55, 52.89 degrees. Looking at it after being so used to seeing icosahedrons is a little off-putting, you can feel that the geometry is slightly off but it takes a little effort to conciously spot the illegal 6-fold symmetries (local only, the global symmetry is only 3-fold). It is a bit reminiscent of a Geodesic dome, though I am not sure if it has a specific name - I was unable to find it on Wikipedia.

Using the Conway Polyhedron Notation, I believe it should be called a Hexakis Truncated Tetrahedron

You should ask Oskar to make it! 8-)

Peace,
Matt Galla


Last edited by Allagem on Thu Jul 18, 2013 5:05 am, edited 2 times in total.

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 Post subject: Re: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 4:54 am 
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Since the pentagons on the Truncated triakis tetrahedron are irregular, when you constructed the points for the dual, how did you find the "centre" of these pentagonal faces?

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 Post subject: Re: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 4:58 am 
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Gus wrote:
Since the pentagons on the Truncated triakis tetrahedron are irregular, when you constructed the points for the dual, how did you find the "centre" of these pentagonal faces?


I don't know what program Allagem uses, but for me, solidworks has a 'point' feature which creates a point bases on references. You can choose the face and it (somehow) calculated the center.

More on topic, the mechanism for the puzzle probably will not be incredibly hard to do. It is the fudging that is the tricky part. Just guessing.

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 Post subject: Re: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 5:10 am 
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Gus wrote:
Since the pentagons on the Truncated triakis tetrahedron are irregular, when you constructed the points for the dual, how did you find the "centre" of these pentagonal faces?


That's not the only way of creating a dual :wink:
The dual is also the surface bounded by the planes perpendicular to the lines from the center of the puzzle to each vertex, aligned to intersect that vertex

By the way, I am using the program Cabri 3D. It is essentially a 3-D version of using a straight edge and compass on a computer, as it can only perform basic geometric manipulations, but if you can identify midpoints, rotations, circles, perpendicular planes, spheres, or other simple geometry and their intersections to define the shape you wish to make it can generate some nice sketches with relatively little effort. It cannot handle any sort of cutting or extruding though - it isn't a real CAD program, just geometry!

Peace
Matt Galla


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 Post subject: Re: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 5:36 am 
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Interesting software Matt. I knew there would be a sneaky (meaning: correct) mathematical way of making the dual. By the way, according to this site there is more than one way of creating the TTT - which one is best for our purposes?

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 Post subject: Re: Dino truncated triakis tetrahedron? (idea)
PostPosted: Thu Jul 18, 2013 6:10 am 
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Gus wrote:
Interesting software Matt. I knew there would be a sneaky (meaning: correct) mathematical way of making the dual. By the way, according to this site there is more than one way of creating the TTT - which one is best for our purposes?



The one that produces the closest to 3 fold hexagon-pentagon-pentagon vertexes.

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