## ursa rings!

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

### ursa rings!

ursa-rings: the evolved form of ursatopes (sorry couldn't resist saying that )

if take an ursatope a semi/anti cupola (xoo||oxo) of the dual(of the shape for which the ursatope is based) and an antiprism (based on that dual pair) and put them end to end in a ring.

the only CRF Polychoron is K4.33 (aka the trigonal tridimished icosahedral wedge) this was the inspiration behind them,
generally the 4D cases have

1| n-prism
1| n-antiprism
n| 5-pyramids
n| 4-pyramids
1| n-ursahedron

n is the number of sides of the polygon for which they are based

I'm still working on the 5D cases atm
wintersolstice
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Posts: 91
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### Re: ursa rings!

wintersolstice wrote:ursa-rings: the evolved form of ursatopes (sorry couldn't resist saying that )

Suppose you mean here the sequence ofx&#xt (= {5}), ofx3xoo&#xt (= teddi), ofx3xooNooo&#xt (N = 3, 4, 5) - right?

if take an ursatope a semi/anti cupola (xoo||oxo) of the dual(of the shape for which the ursatope is based) and an antiprism (based on that dual pair) and put them end to end in a ring.

Rather unclear, sorry. What is a "semi/anti cupola"? Do you mean xo3oxNoo&#x by that? Which is your base of reference of the ursatope? Are you sure to mean and antiprism and not a prism (K-4.33 uses a trip)?

the only CRF Polychoron is K4.33 (aka the trigonal tridimished icosahedral wedge) this was the inspiration behind them,
generally the 4D cases have

1| n-prism
1| n-antiprism
n| 5-pyramids
n| 4-pyramids
1| n-ursahedron

n is the number of sides of the polygon for which they are based

First of all, you are missing the tetrahedra in your list. Next, what would be an n-usehedron?
Do you have perhaps ofx4qoo&#xt and ofx5foo&#xt in mind?

I'm still working on the 5D cases atm

--- rk
Klitzing
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### Re: ursa rings!

Klitzing wrote:What is a "semi/anti cupola"?

CRF_polychora_discovery_project#Cupolae_of_regular_polyhedra

semicupola: xoo‖oxo

Keiji

Posts: 1962
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: ursa rings!

Finally got an idea of what wintersolstice was after.

So, I wouldn't speak of a ring here!
Just as a triangle is no circle.

In fact you rather describe ursachoron based wedges.

For the 4D one you'd have a lace tower of 3 stacks and one more triangle symmetric vertex layer (in fact a triangle) in an orthogonal direction. Thus it would be described by the following lace city:
Code: Select all
`o3xf3o  o3x      : K-4.33x3o`

Its facets then are:
Code: Select all
`o3xf3o  . .      : teddix3oo3x. .  o3x      : K-3.4 = trip. .. .. .  o3x      : K-3.2 = octx3oo .f .  o .      : K-3.8 = peppyx .. x. o  . x      : K-3.3 = squippy. .. .. o  . x      : K-3.1 = tet. o`

But sure, this one was known so far (K-4.33).

Now let's turn towards 5D.

A similar lace tower description again describes the 3 ursachora. Again you want to place a single polyhedron (each) in an orthogonal direction. As far as I could dechiffer your writings - please tell, if I've done it right, or not - those would be the 3 polytera according to the following lace city description (N=3,4,5):
Code: Select all
`o3xNo             f3oNo  o3oNx      x3oNo             `

The facets of those thingies then would be:
Code: Select all
`o3xNo               tet,oct-ursachoronf3oNo  . . .      : oct,co-ursachoron x3oNo               ike,id-ursachorono3xNo               K-4.5 = oct,tet-cupola = rap. . .  o3oNx      : K-4.35 = co,cube-cupola. . .               K-4.77 = id,doe-cupola. . .               K-4.2 = tet,dual-tet-antiprism = hex. . .  o3oNx      : K-4.15 = oct,cube-antiprismx3oNo               K-4.78 = ike,doe-antiprismo3x .               teddi-pyramid = teddipyf3o .  o3o .      : teddi-pyramid = teddipyx3o .               teddi-pyramid = teddipyo . o               K-4.86 = peppypyf . o  o . x      : K-4.86 = peppypyx . o               K-4.86 = peppypy. xNo               K-4.3 = oct-pyramid = octpy. oNo  . oNx      : K-4.17 = squap-pyramid = squappy. . .               K-4.80 = pap-pyramid = pappy. . .               K-4.1 = pen. oNo  . oNx      : K-4.4 = squippypy. oNo               K-4.86 = peppypy`

--- rk
Klitzing
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### Re: ursa rings!

Klitzing wrote:
So, I wouldn't speak of a ring here!
Just as a triangle is no circle.

In fact you rather describe ursachoron based wedges.

I know it's a triangle not a circle it's just that, the shapes composed of two cupola and a prism looped in 4d and the shapes made as two antiprisms and a prism looped in 4d (both involve a triangle) were called "Bicupolic Rings" and "Antiprismatic Rings" respectively (by people on this forum) so I borrowed the name from them
wintersolstice
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Posts: 91
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### Re: ursa rings!

Here is the incidence matrix for that single polychoron, K-4.33:
Code: Select all
`3 * * * | 2 2 1 0 0 0 0 0 | 1 2 2 1 2 0 0 0 0 0 0 | 1 1 2 1 0 0* 3 * * | 0 2 0 1 2 0 0 0 | 0 1 0 2 2 2 1 0 0 0 0 | 1 0 1 2 1 0* * 3 * | 0 0 0 1 0 2 2 0 | 0 0 0 2 0 2 0 1 2 1 0 | 1 0 0 2 1 1* * * 3 | 0 0 1 0 2 0 2 2 | 0 0 2 0 2 2 2 0 1 2 1 | 0 1 2 1 2 1--------+-----------------+-----------------------+------------2 0 0 0 | 3 * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 | 1 1 1 0 0 01 1 0 0 | * 6 * * * * * * | 0 1 0 1 1 0 0 0 0 0 0 | 1 0 1 1 0 01 0 0 1 | * * 3 * * * * * | 0 0 2 0 2 0 0 0 0 0 0 | 0 1 2 1 0 00 1 1 0 | * * * 3 * * * * | 0 0 0 2 0 2 0 0 0 0 0 | 1 0 0 2 1 00 1 0 1 | * * * * 6 * * * | 0 0 0 0 1 1 1 0 0 0 0 | 0 0 1 1 1 00 0 2 0 | * * * * * 3 * * | 0 0 0 1 0 0 0 1 1 0 0 | 1 0 0 1 0 10 0 1 1 | * * * * * * 6 * | 0 0 0 0 0 1 0 0 1 1 0 | 0 0 0 1 1 10 0 0 2 | * * * * * * * 3 | 0 0 1 0 0 0 1 0 0 1 1 | 0 1 1 0 1 1--------+-----------------+-----------------------+------------3 0 0 0 | 3 0 0 0 0 0 0 0 | 1 * * * * * * * * * * | 1 1 0 0 0 02 1 0 0 | 1 2 0 0 0 0 0 0 | * 3 * * * * * * * * * | 1 0 1 0 0 02 0 0 2 | 1 0 2 0 0 0 0 1 | * * 3 * * * * * * * * | 0 1 1 0 0 01 2 2 0 | 0 2 0 2 0 1 0 0 | * * * 3 * * * * * * * | 1 0 0 1 0 01 1 0 1 | 0 1 1 0 1 0 0 0 | * * * * 6 * * * * * * | 0 0 1 1 0 00 1 1 1 | 0 0 0 1 1 0 1 0 | * * * * * 6 * * * * * | 0 0 0 1 1 00 1 0 2 | 0 0 0 0 2 0 0 1 | * * * * * * 3 * * * * | 0 0 1 0 1 00 0 3 0 | 0 0 0 0 0 3 0 0 | * * * * * * * 1 * * * | 1 0 0 0 0 10 0 2 1 | 0 0 0 0 0 1 2 0 | * * * * * * * * 3 * * | 0 0 0 1 0 10 0 1 2 | 0 0 0 0 0 0 2 1 | * * * * * * * * * 3 * | 0 0 0 0 1 10 0 0 3 | 0 0 0 0 0 0 0 3 | * * * * * * * * * * 1 | 0 1 0 0 0 1--------+-----------------+-----------------------+------------3 3 3 0 | 3 6 0 3 0 3 0 0 | 1 3 0 3 0 0 0 1 0 0 0 | 1 * * * * * teddi3 0 0 3 | 3 0 3 0 0 0 0 3 | 1 0 3 0 0 0 0 0 0 0 1 | * 1 * * * * trip2 1 0 2 | 1 2 2 0 2 0 0 1 | 0 1 1 0 2 0 1 0 0 0 0 | * * 3 * * * squippy1 2 2 1 | 0 2 1 2 2 1 2 0 | 0 0 0 1 2 2 0 0 1 0 0 | * * * 3 * * peppy0 1 1 2 | 0 0 0 1 2 0 2 1 | 0 0 0 0 0 2 1 0 0 1 0 | * * * * 3 * tet0 0 3 3 | 0 0 0 0 0 3 6 3 | 0 0 0 0 0 0 0 1 3 3 1 | * * * * * 1 oct`

--- rk
Klitzing
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Posts: 1377
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### Re: ursa rings!

And here follows the incidence matrix for the ursachoron based wedge for the case N=5, i.e. the one according to the following lace city:
Code: Select all
`o3x5o       f3o5o  o3o5xx3o5o       `

Code: Select all
`30  *  *  * |  4  2  2  0  0  0  0  0 |  2  2  4  4  1   4  1  0  0  0  0  0  0 | 1  2  2  2  2  4  0  2  0  0 0  0  0  0 0 | 1 1  2  2  1  0 0 * 12  *  * |  0  5  0  1  5  0  0  0 |  0  0  5  0  5  10  0  5  5  0  0  0  0 | 0  1  5  0  0  5 10  5  5  1 0  0  0  0 0 | 1 0  1  5  5  1 0 *  * 12  * |  0  0  0  1  0  5  5  0 |  0  0  0  0  5   0  0  5  0  5 10  5  0 | 0  0  5  0  0  0 10  0  5  0 1  5  5  1 0 | 1 0  0  5  5  1 1 *  *  * 20 |  0  0  3  0  3  0  3  3 |  0  0  0  3  0   6  3  3  6  0  3  6  3 | 0  0  0  3  1  3  3  6  6  3 0  1  3  3 1 | 0 1  3  1  3  3 1------------+-------------------------+-----------------------------------------+-------------------------------------------+------------------ 2  0  0  0 | 60  *  *  *  *  *  *  * |  1  1  1  1  0   0  0  0  0  0  0  0  0 | 1  1  1  1  1  1  0  0  0  0 0  0  0  0 0 | 1 1  1  1  0  0 0 1  1  0  0 |  * 60  *  *  *  *  *  * |  0  0  2  0  1   2  0  0  0  0  0  0  0 | 0  1  2  0  0  2  2  1  0  0 0  0  0  0 0 | 1 0  1  2  1  0 0 1  0  0  1 |  *  * 60  *  *  *  *  * |  0  0  0  2  0   2  1  0  0  0  0  0  0 | 0  0  0  2  1  2  1  2  0  0 0  0  0  0 0 | 0 1  2  1  1  0 0 0  1  1  0 |  *  *  * 12  *  *  *  * |  0  0  0  0  5   0  0  5  0  0  0  0  0 | 0  0  5  0  0  0 10  0  5  0 0  0  0  0 0 | 1 0  0  5  5  1 0 0  1  0  1 |  *  *  *  * 60  *  *  * |  0  0  0  0  0   2  0  1  2  0  0  0  0 | 0  0  0  0  0  1  2  2  2  1 0  0  0  0 0 | 0 0  1  1  2  1 0 0  0  2  0 |  *  *  *  *  * 30  *  * |  0  0  0  0  1   0  0  0  0  2  2  0  0 | 0  0  2  0  0  0  2  0  0  0 1  2  1  0 0 | 1 0  0  2  1  0 1 0  0  1  1 |  *  *  *  *  *  * 60  * |  0  0  0  0  0   0  0  1  0  0  2  2  0 | 0  0  0  0  0  0  2  0  2  0 0  1  2  1 0 | 0 0  0  1  2  1 1 0  0  0  2 |  *  *  *  *  *  *  * 30 |  0  0  0  0  0   0  1  0  2  0  0  2  2 | 0  0  0  2  0  0  0  2  2  2 0  0  1  2 1 | 0 1  2  0  1  2 1------------+-------------------------+-----------------------------------------+-------------------------------------------+------------------ 5  0  0  0 |  5  0  0  0  0  0  0  0 | 12  *  *  *  *   *  *  *  *  *  *  *  * | 1  1  0  1  0  0  0  0  0  0 0  0  0  0 0 | 1 1  1  0  0  0 0 3  0  0  0 |  3  0  0  0  0  0  0  0 |  * 20  *  *  *   *  *  *  *  *  *  *  * | 1  0  1  0  1  0  0  0  0  0 0  0  0  0 0 | 1 1  0  1  0  0 0 2  1  0  0 |  1  2  0  0  0  0  0  0 |  *  * 60  *  *   *  *  *  *  *  *  *  * | 0  1  1  0  0  1  0  0  0  0 0  0  0  0 0 | 1 0  1  1  0  0 0 2  0  0  1 |  1  0  2  0  0  0  0  0 |  *  *  * 60  *   *  *  *  *  *  *  *  * | 0  0  0  1  1  1  0  0  0  0 0  0  0  0 0 | 0 1  1  1  0  0 0 1  2  2  0 |  0  2  0  2  0  1  0  0 |  *  *  *  * 30   *  *  *  *  *  *  *  * | 0  0  2  0  0  0  2  0  0  0 0  0  0  0 0 | 1 0  0  2  1  0 0 1  1  0  1 |  0  1  1  0  1  0  0  0 |  *  *  *  *  * 120  *  *  *  *  *  *  * | 0  0  0  0  0  1  1  1  0  0 0  0  0  0 0 | 0 0  1  1  1  0 0 1  0  0  2 |  0  0  2  0  0  0  0  1 |  *  *  *  *  *   * 30  *  *  *  *  *  * | 0  0  0  2  0  0  0  2  0  0 0  0  0  0 0 | 0 1  2  0  1  0 0 0  1  1  1 |  0  0  0  1  1  0  1  0 |  *  *  *  *  *   *  * 60  *  *  *  *  * | 0  0  0  0  0  0  2  0  2  0 0  0  0  0 0 | 0 0  0  1  2  1 0 0  1  0  2 |  0  0  0  0  2  0  0  1 |  *  *  *  *  *   *  *  * 60  *  *  *  * | 0  0  0  0  0  0  0  1  1  1 0  0  0  0 0 | 0 0  1  0  1  1 0 0  0  3  0 |  0  0  0  0  0  3  0  0 |  *  *  *  *  *   *  *  *  * 20  *  *  * | 0  0  1  0  0  0  0  0  0  0 1  1  0  0 0 | 1 0  0  1  0  0 1 0  0  2  1 |  0  0  0  0  0  1  2  0 |  *  *  *  *  *   *  *  *  *  * 60  *  * | 0  0  0  0  0  0  1  0  0  0 0  1  1  0 0 | 0 0  0  1  1  0 1 0  0  1  2 |  0  0  0  0  0  0  2  1 |  *  *  *  *  *   *  *  *  *  *  * 60  * | 0  0  0  0  0  0  0  0  1  0 0  0  1  1 0 | 0 0  0  0  1  1 1 0  0  0  5 |  0  0  0  0  0  0  0  5 |  *  *  *  *  *   *  *  *  *  *  *  * 12 | 0  0  0  0  0  0  0  0  0  1 0  0  0  1 1 | 0 1  1  0  0  1 1------------+-------------------------+-----------------------------------------+-------------------------------------------+------------------30  0  0  0 | 60  0  0  0  0  0  0  0 | 12 20  0  0  0   0  0  0  0  0  0  0  0 | 1  *  *  *  *  *  *  *  *  * *  *  *  * * | 1 1  0  0  0  0 0 id 5  1  0  0 |  5  5  0  0  0  0  0  0 |  1  0  5  0  0   0  0  0  0  0  0  0  0 | * 12  *  *  *  *  *  *  *  * *  *  *  * * | 1 0  1  0  0  0 0 peppy 3  3  3  0 |  3  6  0  3  0  3  0  0 |  0  1  3  0  3   0  0  0  0  1  0  0  0 | *  * 20  *  *  *  *  *  *  * *  *  *  * * | 1 0  0  1  0  0 0 teddi 5  0  0  5 |  5  0 10  0  0  0  0  5 |  1  0  0  5  0   0  5  0  0  0  0  0  0 | *  *  * 12  *  *  *  *  *  * *  *  *  * * | 0 1  1  0  0  0 0 pap 3  0  0  1 |  3  0  3  0  0  0  0  0 |  0  1  0  3  0   0  0  0  0  0  0  0  0 | *  *  *  * 20  *  *  *  *  * *  *  *  * * | 0 1  0  1  0  0 0 tet 2  1  0  1 |  1  2  2  0  1  0  0  0 |  0  0  1  1  0   2  0  0  0  0  0  0  0 | *  *  *  *  * 60  *  *  *  * *  *  *  * * | 0 0  1  1  0  0 0 tet 1  2  2  1 |  0  2  1  2  2  1  2  0 |  0  0  0  0  1   2  0  2  0  0  1  0  0 | *  *  *  *  *  * 60  *  *  * *  *  *  * * | 0 0  0  1  1  0 0 peppy 1  1  0  2 |  0  1  2  0  2  0  0  1 |  0  0  0  0  0   2  1  0  1  0  0  0  0 | *  *  *  *  *  *  * 60  *  * *  *  *  * * | 0 0  1  0  1  0 0 tet 0  1  1  2 |  0  0  0  1  2  0  2  1 |  0  0  0  0  0   0  0  2  1  0  0  1  0 | *  *  *  *  *  *  *  * 60  * *  *  *  * * | 0 0  0  0  1  1 0 tet 0  1  0  5 |  0  0  0  0  5  0  0  5 |  0  0  0  0  0   0  0  0  5  0  0  0  1 | *  *  *  *  *  *  *  *  * 12 *  *  *  * * | 0 0  1  0  0  1 0 peppy 0  0 12  0 |  0  0  0  0  0 30  0  0 |  0  0  0  0  0   0  0  0  0 20  0  0  0 | *  *  *  *  *  *  *  *  *  * 1  *  *  * * | 1 0  0  0  0  0 1 ike 0  0  3  1 |  0  0  0  0  0  3  3  0 |  0  0  0  0  0   0  0  0  0  1  3  0  0 | *  *  *  *  *  *  *  *  *  * * 20  *  * * | 0 0  0  1  0  0 1 tet 0  0  2  2 |  0  0  0  0  0  1  4  1 |  0  0  0  0  0   0  0  0  0  0  2  2  0 | *  *  *  *  *  *  *  *  *  * *  * 30  * * | 0 0  0  0  1  0 1 tet 0  0  1  5 |  0  0  0  0  0  0  5  5 |  0  0  0  0  0   0  0  0  0  0  0  5  1 | *  *  *  *  *  *  *  *  *  * *  *  * 12 * | 0 0  0  0  0  1 1 peppy 0  0  0 20 |  0  0  0  0  0  0  0 30 |  0  0  0  0  0   0  0  0  0  0  0  0 12 | *  *  *  *  *  *  *  *  *  * *  *  *  * 1 | 0 1  0  0  0  0 1 doe------------+-------------------------+-----------------------------------------+-------------------------------------------+------------------30 12 12  0 | 60 60  0 12  0 30  0  0 | 12 20 60  0 30   0  0  0  0 20  0  0  0 | 1 12 20  0  0  0  0  0  0  0 1  0  0  0 0 | 1 *  *  *  *  * * id,ike-urs.30  0  0 20 | 60  0 60  0  0  0  0 30 | 12 20  0 60  0   0 30  0  0  0  0  0 12 | 1  0  0 12 20  0  0  0  0  0 0  0  0  0 1 | * 1  *  *  *  * * id,doe-cup. 5  1  0  5 |  5  5 10  0  5  0  0  5 |  1  0  5  5  0  10  5  0  5  0  0  0  1 | 0  1  0  1  0  5  0  5  0  1 0  0  0  0 0 | * * 12  *  *  * * pappy 3  3  3  1 |  3  6  3  3  3  3  3  0 |  0  1  3  3  3   6  0  3  0  1  3  0  0 | 0  0  1  0  1  3  3  0  0  0 0  1  0  0 0 | * *  * 20  *  * * teddipy 1  2  2  2 |  0  2  2  2  4  1  4  1 |  0  0  0  0  1   4  1  4  2  0  2  2  0 | 0  0  0  0  0  0  2  2  2  0 0  0  1  0 0 | * *  *  * 30  * * peppypy 0  1  1  5 |  0  0  0  1  5  0  5  5 |  0  0  0  0  0   0  0  5  5  0  0  5  1 | 0  0  0  0  0  0  0  0  5  1 0  0  0  1 0 | * *  *  *  * 12 * peppypy 0  0 12 20 |  0  0  0  0  0 30 60 30 |  0  0  0  0  0   0  0  0  0 20 60 60 12 | 0  0  0  0  0  0  0  0  0  0 1 20 30 12 1 | * *  *  *  *  * 1 ike,doe-ap.`

--- rk
Klitzing
Pentonian

Posts: 1377
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: ursa rings!

Oops, there is a problem - at least with the N=5 ursachoron based wedge ofx3xoo5ooo&#xt || o3o5x!

That ursachoron itself, i.e. ofx3xoo5ooo&#xt, is nothing but the 600-cell (= ex) rotunda with the single vertex of the first and all the 20 vertices of the third vertex layer being diminished (i.e. a 21-diminishing of that rotunda).

Btw., that rotunda could be considered a tetra-stratic stack of 5 vertex layers:
o3o5o || x3o5o || o3o5x || f3o5o || o3x5o
(where f would mark a tau scaled edge size).

Hence the dodecahedron, which is to be used as subdimensional opposite base of that wedge of consideration, would just be the 600-cell section at this very third layer. That is, the height (measured in the fifth direction orthogonal to the 4D base ursachoron) of that wedge is just zero! It thus becomes a degenerate flat 4D object only.

And true, all the lacing edges, being needed for the x3o5o || o3o5x antiprism, are used as edges between the second and third vertex layer of the 600-cell (resp. its rotunda). And too, all the lacing edges, used in the o3o5x || o3x5o (semi)cupola, are used in the 600-cell as edges connecting the third and the fifth vertex layer.

Given in a different way, here we'd have the lacing city not in the wedge-type way shown recently:
Code: Select all
`x3o5o        f3o5o   o3o5xo3x5o         `

but more metrically correct rather in a single (flat) stack:
Code: Select all
`x3o5oo3o5xf3o5oo3x5o`

(So far I have not looked into cases N=3 and N=4, so.)

--- rk
Klitzing
Pentonian

Posts: 1377
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### Re: ursa rings!

And here it comes, the remaining analysis:

I just calculated the total height of the ursachora as a stack of 2 lace prisms. That one would have to serve for he wedge base, kind as the hypothenuse of a triangle. - On the other hand there are the heights of the wedge sides to be calculated. Those here too are lace prisms. - And finally, the sum of those sides should be strictly larger than the hypothenuse, if that wedge wants to be non-degenerate!

Code: Select all
`hypothenuse: h(o3x3o || f3o3o) + h(f3o3o || x3o3o) = 0.572061 + 0.925615 = 1.497676sum of legs: h(o3x3o || o3o3x) + h(x3o3o || o3o3x) = 0.790569 + 0.707107 = 1.497676hypothenuse: h(o3x4o || f3o4o) + h(f3o4o || x3o4o) = 0.555893 + 0.899454 = 1.455347sum of legs: h(o3x4o || o3o4x) + h(x3o4o || o3o4x) = 0.814993 + 0.676097 = 1.491090hypothenuse: h(o3x5o || f3o5o) + h(f3o5o || x3o5o) = 0.500000 + 0.809017 = 1.309017sum of legs: h(o3x5o || o3o5x) + h(x3o5o || o3o5x) = 0.809017 + 0.500000 = 1.309017`

Thus it turns out after all, that the cases N=3 and N=5 both are the limiting ones, where the questioned height becomes zero. Therefore only the single case N=4 would remain, which provides a strictly positive height (if N is restricted to be integral).

--- rk
Klitzing
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### Re: ursa rings!

... and here are finally the respective heights (in edge units):

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`H([o3x3o || f3o3o || x3o3o] || o3o3x) = 0H([o3x4o || f3o4o || x3o4o] || o3o4x) = 0.161520 = sqrt[sqrt(2)-2*sqrt(5)+sqrt(10)]/2H([o3x5o || f3o5o || x3o5o] || o3o5x) = 0`

--- rk
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### Re: ursa rings!

That's a class-4 type number, akin to what one finds in o3o4o3o5z.
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### Re: ursa rings!

yep, "sqrt(2)" relates to Dynkin symbol links numbered by "4", and "sqrt(5)" relates to such numbered by "5".
And "sqrt(10)" is nothing but the mixed term: sqrt(10) = sqrt(2 x 5) = sqrt(2) x sqrt(5).

--- rk
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### Re: ursa rings!

wintersolstice wrote:
Klitzing wrote:
So, I wouldn't speak of a ring here!
Just as a triangle is no circle.

In fact you rather describe ursachoron based wedges.

I know it's a triangle not a circle it's just that, the shapes composed of two cupola and a prism looped in 4d and the shapes made as two antiprisms and a prism looped in 4d (both involve a triangle) were called "Bicupolic Rings" and "Antiprismatic Rings" respectively (by people on this forum) so I borrowed the name from them

Yeah, about those "rings", in retrospect calling them rings was perhaps a bit of hyperbole; they are more like something between a wedge and a ring; there's lateral component to them (e.g. 8-gon||cube or 8-prism||square) but when viewed from the side, they are just 4D wedges. At the same time, though, they resemble irregular versions of the orthogonal rings in duoprism symmetries, so one could argue for the "ring" description too. So I say that they're ring-like wedges (or wedge-like rings, whatever you prefer). This latter argument, though, is a bit weakened by the fact that the members of the 3-member "ring" component are not transitive, which would suggest "wedge" is probably a better description.

In 3D, wedges have a linear leading edge; in 4D, this leading "edge" has become a (regular, in this case) polygon, so there is a circular component to it, even though the overall shape argues for the "wedge" description.
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### Re: ursa rings!

Yes, the "ring" naming was taken from analogy with the duocylinder. I was thinking a wedge would have a leading edge, rather than leading polygon, but that would be a bit more like a 3D spike.

It's up to you I guess - I'm happy to rename them wedges, if that's the consensus.

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### Re: ursa rings!

Keiji wrote:Yes, the "ring" naming was taken from analogy with the duocylinder. I was thinking a wedge would have a leading edge, rather than leading polygon, but that would be a bit more like a 3D spike.

It's up to you I guess - I'm happy to rename them wedges, if that's the consensus.

I don't know if there is consensus, but Klitzing did name them wedges long before we (re)discovered these shapes, so there's at least precedent for it.

As for how to generalize spikes and wedges to higher dimensions, this reminds me of my 4D sword-fighting thread from a while back. While a 4D sword with a linear cutting edge would be more penetrating, it also doesn't cover enough area to do maneuvres like fencing (blocking, etc.); in practice, it would be handled more like a spear than a sword. Such a sword also would be unable to cut things in half, so you couldn't cut off a limb, for example. OTOH, a 4D sword with a 2D cutting "edge" (ridge), while more amenable to blocking maneuvres as well as being able to cut things in half, are also much bulkier and therefore more cumbersome to maneuvre. It'd be more like swinging a big fat meat cleaver around than a true sleek blade.

So the former (with a leading edge) is more spike-like, and the latter (with a leading ridge) is more wedge-like. Funnily enough, 4D also has "true" spikes in the sense of tapering to a point, not just an edge. So there's a lot more variety going on here. Things get even more fun in 5D, where wedges would have a leading realm, and there's a progression of increasingly narrower leading "edges" down to a point.

As for why it makes sense to associate "wedge" with a leading 2D ridge, consider that in 3D, wedge-shaped doorstops have the property that they maintain their orientation in spite of pressure from the door on their top face. In 4D, an object with only a leading (1D) edge would not be able to do this; you need two (n-1)-hyperplanes meeting at an angle in order to be able to maintain orientation against pressure on the top hyperplane. And (n-1)-hyperplanes in 4D meet at a 2D area. A 4D object with a leading 1D edge has a ≥3 hyperplanes meeting at the edge, so they are more akin to a sharpened pencil tip (sharpened in the old traditional way with a knife, that is, which makes a number of planar cuts around the tip) than a true wedge.
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