Augmented polyhedral prisms

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Re: Augmented polyhedral prisms

Postby Klitzing » Wed May 10, 2017 3:22 pm

First of all, these values of student91 would reflect the required angles of the trips attached to the 4 polar triangles. The other ones would ask for the third dihedral angle, not mentioned in his post.

The required value of about 73 degrees at a square can be matched easily by several small attaching cells. But the required value of 58 cannot be matched by any cell!

This is simply because the "sharpest" segmentochoron with one trip base and without a retrograde top base clearly is the trippy = pt || trip. And that one already has dihedral angles of
• at {4} between squippy and trip: arccos(sqrt[1/6]) = 65.905157°

At least for straight segmentochora. If you look at sheered segmentochora (thus asking for a dimensional degenerate top base!) you might be more lucky.

But the case of a point for top base already is done by trippy. The case of a line segment there is done either by bidrap or by tepe (depending on relative orientation). And the case of a triangle for opposing base is done by triddip resp. traf = K-4.6 (depending on relative gyration). - Therin only bidrap implements some shift of the degenerate base. But none of these would bow to the above requirements.

This then concludes the proof that a snadow prism cannot be augmented.

--- rk
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Re: Augmented polyhedral prisms

Postby Klitzing » Sun May 14, 2017 10:05 am

But then what about 3 snadows adjoined mutually around their sharp angles (more are impossible)?
Could that be continued somehow? (Possibly with tets at the neighbouring triangles and perhaps further snadows at the opposite sharp angle as well?)

(Okay, that won't be an augmented prism, so should be placed within a different thread, I suppose...)
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Re: Augmented polyhedral prisms

Postby student91 » Sun May 14, 2017 1:15 pm

That won't work either I suppose, the distance between the outer vertices will become very weird, making it impossible for one of the known Johnson solids to fit inbetween. Setting more than one solid inbetween will result in non-convexity
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Re: Augmented polyhedral prisms

Postby Klitzing » Sun May 14, 2017 6:00 pm

Well, my original idea here was, kind of something similar to the great antiprism or some weird expansion of a duoprism, using the snadows as the separating elements (along with possibly some other stuff) ...
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Re: Augmented polyhedral prisms

Postby quickfur » Mon May 15, 2017 4:27 am

It may be hard for a single Johnson to fit between the "weird distances" of the outer vertices, but what if we have several smaller pieces that confer more flexibility? Or do we still run into the same problem eventually because 4D vertices must be rigid?

Also, if there's not enough space between, say, a floret of 3 snadows around an edge to insert bridging cells, what if we Stott-expand the configuration by inserting triangular prisms between the snadows? Though that does have the wrinkle of introducing square faces that must join to something else, which may limit the possibilities. But OTOH, it may expand the possibilities because then we'd be able to bring, say, squippies into play where tetrahedral dihedral angles may prove insufficient.
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Re: Augmented polyhedral prisms

Postby Klitzing » Mon May 15, 2017 9:30 pm

Klitzing wrote:Putting it all together, ...

The second type of variation is limited by ... P=2 for the second, where the polychora become dimensionally degenerate (becoming J51 and J57, resp. their prisms). Accordingly we have here the following bunch of valide CRFs:

...

oa3xo3oo xo&#zx = ope + 4 (alternate) trippies
oa3xo3xx xo&#zx = tuttip + 4 tripufs
xb3xo3oo xo&#zx = tuttip + 4 trippies
xb3xo3xx xo&#zx = tope + 4 (alternate) tripufs
(where a = (2+sqrt(10))/3 = 1.720759, b = a+x = 2.720759)

oa3xo4oo xo&#zx = cope + 6 cubpies
oa3xo4xx xo&#zx = ticcup + 6 (ortho) squipufs
xb3xo4oo xo&#zx = tope + 6 cubpies
xb3xo4xx xo&#zx = gircope + 6 (ortho) squipufs
(where a = w/q = 1.707107, b = a+x = 2.707107)

of3xo5oo xo&#zx = iddip + 12 peppies
of3xo5xx xo&#zx = tiddip + 12 (ortho) pepufs
xF3xo5oo xo&#zx = tipe + 12 peppies
xF3xo5xx xo&#zx = griddip + 12 (ortho) pepufs
(where F = ff = f+x)

...


Just like to mention that the highlighted 3 together with the above mentioned J51-prism, which then is nothing but tisdip + 3 squippyp (or: line || cube, i.e. di-puf), already was looming up way back in November of 2012, cf. that post (within the 2nd half of it). I just re-found that one by hazard this weekend! ;)

In those days I still was not aware of the full shape, not knowing all their dihedrals, even was asking for coords. So, by means of those student91-style tegum sum notations provided in this recent post (i.e. oa3xoPxx xo&#zx), together with the also provided respective measures "a", we are a big step ahead now. 8)

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