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 Post subject: Pentagonal trapezohedra do not jumble!
PostPosted: Fri May 24, 2013 5:47 am 
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Title says it all. Basically, I was working on a face turning pentagonal trapezohedron design, then I realised that the shape was basically an extended bandaged face turning dodecahedron, which do not jumble.

Some time ago, David Pitcher posted a hypothesis saying that all trapezohedra whose faces are at equal dihedral angle will jumble, but this disproves it.

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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sat May 25, 2013 12:52 pm 
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I believe you are referring to my thread regarding convex solids and jumbling. One of the conditions stated was that the platonic solids are not jumbling (with the exception of the icosahedron). As you point out, the pentagonal trapezohedron can be viewed as a bandaged dodecahedron. So while it may not technically jumble, it still may be an interesting puzzle with blocked moves due to bandaging. Here's an image of the form:
Attachment:
face sliced pentagonal trapezohedron.jpg
face sliced pentagonal trapezohedron.jpg [ 61.65 KiB | Viewed 920 times ]

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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sun May 26, 2013 8:51 am 
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Face-turning Dodecahedron do not jumble, and extending two opposite faces with pentagonal pyramids consisting of golden triangles does form a pentagonal trapazohedron. Such a trapazohedrong does have geometry very similiar to a bandaged face-turning dodecahedron.

However, I do have a few questions:

1. If you are using strictly planar cuts, doesn't the extentsion from dodecahedron to pentagonal trapazohedron create new polar corner pieces with no analog on the dodecahedron? How does this effect the geometry?

2. Extending a dodecahedron is one way to generate a pentagonal dodecahedron, but is this the unique pentagonal trapazohedron, for are their kites other than 36-108-108-108(measures of angles in degrees) that can form a pentagonal trapazohedron? If there are others, wouldn't they jumble since they lack similiarity to the dodecahedron?

3. On a related note, the Trigonal trapazohedron is, in general, a jumbling geometry(correct me if I am wrong), but it has a specific case(the cube) which is doctraine. In general, are 3 and 5 the only n for which there exists a specific n-gonal trapazohedron such that its angles allow non-jumbling moves?

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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sun May 26, 2013 10:35 am 
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Jeffery Mewtamer wrote:
1. If you are using strictly planar cuts, doesn't the extentsion from dodecahedron to pentagonal trapazohedron create new polar corner pieces with no analog on the dodecahedron? How does this effect the geometry?

I believe those "corner" pieces and surrounding parts are actually equivalent to the centers and points on a face-turning Starminx, which would make sense.
Jeffery Mewtamer wrote:
2. Extending a dodecahedron is one way to generate a pentagonal dodecahedron, but is this the unique pentagonal trapazohedron, for are their kites other than 36-108-108-108(measures of angles in degrees) that can form a pentagonal trapazohedron? If there are others, wouldn't they jumble since they lack similiarity to the dodecahedron?

Yes, I believe there are other versions with different dihedral angles which do jumble.
Jeffery Mewtamer wrote:
3. On a related note, the Trigonal trapazohedron is, in general, a jumbling geometry(correct me if I am wrong), but it has a specific case(the cube) which is doctraine. In general, are 3 and 5 the only n for which there exists a specific n-gonal trapazohedron such that its angles allow non-jumbling moves?

Interesting question, and I think that they indeed are the only ones since none of the other regular solids that I've seen can be used to make regular trapezohedra. I may be wrong though.
This reminds me: what would a fully unbandaged face-turning pentagonal trapezohedron look like? It doesn't jumble, so there must be a way to fully unbandage it into a doctrinaire puzzle.

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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sun May 26, 2013 1:56 pm 
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What do you get if you extend all twelve faces of the dodecahedron like that? :o


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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sun May 26, 2013 3:04 pm 
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^The small stellated Dodecahedron, one of the kepler-poinstot solids, as also known as the regular star-polyhedra. The others in this family are the Great dodecahedron(Shape of Alexander's Star), Great stellated Dodecahedron(extend an Icosahedron with trigonal pyramids consisting of golden triangles), and the Great Icosahedron(not sure how to describe in text).

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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sun May 26, 2013 5:01 pm 
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The triangular trapezohedron doesn't jumble either!


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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Sun May 26, 2013 5:50 pm 
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Bram wrote:
The triangular trapezohedron doesn't jumble either!

Do you mean this type?
Attachment:
trap 1.jpg
trap 1.jpg [ 158.72 KiB | Viewed 721 times ]

Attachment:
trap 2.jpg
trap 2.jpg [ 151.94 KiB | Viewed 721 times ]

It is an extended face-turning octahedron, which does NOT jumble:
Attachment:
trap 3.jpg
trap 3.jpg [ 171.05 KiB | Viewed 721 times ]


EDIT: I take this back...the center of rotation is different than on a FTO...oops :oops:

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 Post subject: Re: Pentagonal trapezohedra do not jumble!
PostPosted: Mon May 27, 2013 5:30 am 
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I think Bram was referring to trigonal trapezohedra as in 'cube'

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