x3o4o3x regiment facts

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

x3o4o3x regiment facts

Postby polychoronlover » Sun May 14, 2017 4:41 am

When testing a method to exhaustively search regiments, I decided to try it on this regiment, as well as other already known ones. It turn out that there are several interesting things about it.

First of all, the polychoron ipi is the blend of three srits (o3ox3o4x), blending two to a vertex figure. The octs blend out. In fact, this is what happens when gico, the compound of three tesseracts, gets cantellated and the 2octs get removed This is already known, but it turns out that 6 out of 7 srit regiment members can do the same thing. Rawvatoth blends to ridi, spript blends to sarnipadi, garpit becomes sirc when the ops blend to srohs, and pattinoth and pinpith also become nadupti and dupapdi respectively.

The remaining member of the srit regiment is sirdo, which is interesting because instead of forming a blend it forms a compound, which also has coinciding edges I just looked up this compound of 3 sirdoes and it appears to be called sepdi, for small prismatodisicositetrachoron.

There is another way to construct sirc. Since garpit is a blend of three odips (octagonal duoprisms), sirc is a blend of nine odips, blending 4 to a vertex figure. I wonder if there is a similar compound of nine square-octagonal duoprisms or even tesseracts.

There is also a way to take a compound of ridi and ipi without faces or cells coinciding. As this is very clearly an "inter-regimental" compound, I would like to know if there is any important reason why it was not mentioned here while a similar compound in the o3o3o3x3o3o regiment was. In fact, this kind of compound seems to be pretty common. I've seen a few of them when I searched the gacoca regiment.

Finally, half the cells of sirc can be removed and all the original faces will still be in place, each incident to one cell instead of two. This may be known as well, but I thought it was interesting how it shared this property with hex, dittady, chon, etc. Unfortunately this leads to no other company members.
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Re: x3o4o3x regiment facts

Postby username5243 » Sun May 14, 2017 10:48 am

Nice one! I think I had noted this once - I was on Hedrondude's category 13 page (spic and giddic regiments) and noticed ipi's verf was described as a "2-wedge blend" and eventually noted the wedges were srit verfs.

Would I be correct that the conjugates of all of these have corresponding blends in the giddic regiment (from wavitoth members)?

I also found something somewhat similar in some other regiments by blending tessic omnitruncates at the toes:

blend of 3 gidpiths = sirpdi (in prico regiment)

blend of 3 gaquidpoths = girpdi (in gipti regiment)

blend of 3 thatpaths = girti (in ditdi regiment)

blend of 3 thaquitpaths = sirti (in suti regiment)

And even a few mor (blending these at coes/ohoes/choes):

blend of 3 prohs = sirdi (in srico regiment)
blend of 3 siphadoes = spirdy (in srico regiment)
blend of 3 sirpiths = gardi (in srico regiment)

blend of 3 quiprohs = girdi (in wavaty regiment)
blend of 3 giphadoes = gapirdy (in wavaty regiment)
blend of 3 girpiths = gaquerdy (in wavaty regiment)
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Re: x3o4o3x regiment facts

Postby polychoronlover » Sun May 14, 2017 8:24 pm

username5243 wrote:Would I be correct that the conjugates of all of these have corresponding blends in the giddic regiment (from wavitoth members)?


Yes. In the conjugate shape, the coordinates of each point became their algebraic conjugates; if the lowest degree polynomial (w/ integer coefficients) that contains the coordinate as a root is quadratic, the other root of the polynomial is taken. On the other hand, if the coordinate is rational and is therefore the root of a linear equation w/ integer coefficients, it doesn't get changed.

For example, the following coordinates form a unit-edged srit centered on the origin:
perms(±0(1/2 + √(1/2)), ±1(1/2 + √(1/2)), ±21/2, ±31/2)
Here, the subscripts on the ± symbols mean that they are independent of each other. "perms" means that all 6 permutations of these numbers can be taken. This makes 96 points.

The important thing to realize is that 1/2 + √(1/2) and -1/2 - √(1/2) can both be achieved but not 1/2 - √(1/2) or -1/2 + √(1/2). 1/2 ± √(1/2) are the roots to 4x^2 + 4x - 1 and -1/2 ± √(1/2) are the roots to 4x^2 - 4x - 1. As you can see, one root was chosen from each polynomial.

Qrit, the conjugate of srit, has coordinates like this:
perms(±0(1/2 - √(1/2)), ±1(1/2 - √(1/2)), ±21/2, ±31/2)
Now in each polynomial, the other root was chosen! Here, 1/2 - √(1/2) and -1/2 + √(1/2) are the coordinates gotten.

username5243 wrote:I also found something somewhat similar in some other regiments by blending tessic omnitruncates at the toes:


Cool! Similar things can be constructed in the tessic regiments by blending prisms. For example:

blend of 4 ticcups = sirdo (srit regiment)

blend of 4 sircopes = shafipto (sidpith regiment)
blend of 4 soccopes = stefacoth
blend of 4 srohps completely blends out, leaving nothing behind.

blend of 4 gircopes = sirpdo (prit regiment)

These are similar to the construction of sroh as a blend of three ops (octagonal prisms). Similar things exist in 5 dimensions:

blend of 5 tattips = slawn (sirn regiment)

blend of 5 srittips = shippin (span regiment)

blend of 5 grittips = slapirn (prin regiment)

blend of 5 sidpithips = scotan (scant regiment)

...

And of course, these all have conjugates found by replacing octagons or pseudo-octagons with octagrams.
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Re: x3o4o3x regiment facts

Postby username5243 » Sun May 14, 2017 11:00 pm

Nice. I suspect this kind of thing works in all dimensions, but of course the higher ones remain unnamed.

I also wondered about the blend of 4 cotcopes - at least from the list of cells, it would seem to be kaviptit from the skiviphado regiment (wih srohs, grohs, cotcoes,and hips). It seems many of the cubic/tessic symmetric polychora have blended prisms that work like this (though not all of them). And indeed, I would think this works generally for any higher dimension.
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