Multicupolic screw-gauges

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

Multicupolic screw-gauges

Postby wintersolstice » Sat Jun 30, 2012 7:39 pm

these are a generalisation of the cupola and bicupolic rings (which are lever 1 and 2 respectively:D)

I've only just started studying these, so I can't say much atm

the simplex and hypercube cases can be CRF polytopes but only the simplex, square, and cube forms can be gyrated (with the latter the cube becomes a octahedron. (btw I haven't yet ennumerated gyrations and dimishes of the general case case)

this because a regular polytope needs to have a ditope angle (angle between the facets) of less than 120 degrees in order to form a cupola (the 16 cell is exactly 120 so doesn't work._

but I'm not going to restrict my self to CRF, this is a seperate investigation:D

to start with there's an interesting property of the hypercube cases (which is what I'm using as a foundation to actually define these shapes) (the simplex also have similar properties that I haven't looked at properly yet)

3D

Level one: if you take a rhombicuboctahedron it can be cut into 3 pieces:

square cupola + 8-prism + square cupola

4D

Level one: if you take a runcinated tesseract it can be cut into 3 pieces

cube cupola + rhombicuboctahedron prism + cube cupola

Level two: if you take a cube cupola it can be cut into 3 pieces

square bicupolic ring + square cupola prism + square bicupolic ring

(btw it's "orthobi":D)

5D

Level one: if you take a stericated penteract it can be cut into 3 pieces

tesseract cupola + runcinated tesseract prism + tesseract cupola

Level two: if you take a tesseract cupola it can be cut into 3 pieces

cube bicupolic ring + cube cupola prism + cube bicupolic ring

(btw neither cube has been changed to an octahedron)

Level three: if you take a cube bicupolic ring it can be cut into 3 pieces

square tricupolic screw-guage + square bicupolic ring prism + square tricupolic screw-guage

6D

Level one: if you take a pentalated hexeract it can be cut into 3 pieces

penteract cupola + stericated penteract prism + penteract cupola

Level two: if you take a penteract cupola it can be cut into 3 pieces

tesseract bicupolic ring + tesseract cupola prism + tesseract bicupolic ring

Level three: if you take a tesseract bicupolic ring it can be cut into 3 pieces

cube tricupolic screw-guage + cube bicupolic ring prism + cube tricupolic screw-guage

Level four: if you take a cube tricupolic screw-guage it can be cut into 3 pieces

square quadricupolic screw-guage + square tricupolic screw-guage prism + square quadricupolic screw-guage
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Re: Multicupolic screw-gauges

Postby wintersolstice » Thu Jul 12, 2012 1:38 pm

I've figured out more about these shapes and have a new definition:

Definition:

if you take a prismatoid (not a duoprismatoid, just one that has vertices in two planes) and define the two planes using two polytopes P&Q

EP||P X Q

where EP means "expanded P" (shrink the facets and insert new ones between them)

P X Q means the cartesian product P and Q

now if Q = "Point" then the final shape is a "P cupola"

if Q = "line segment" then the final shape is a "P orthobicupolic ring"

now if you take the duoprism base there are two types of facet (arranged in two girdles/rings)

one girdle has a number of facets equal to the number of facets in P.

take this girdle and indentify all the elements within it, these join to the "expanded P" in the same way as they would in a cupola (when Q is a point this is the only girdle)

the facets in the other girdle join directly to the whole of EP the facets that join it are lower screw guages

e.g. in a bicupolic ring the bases of the prism (the product of a line with anything) they join to the expanded P with a cupola (the previous level)

if you take a triangle as Q then there will be 3 prisms of P as facets in the duoprism these will join to EP with a orthobicupolic ring

etc...

Now for Gyro and Magna forms:

for a magna form you just change it to

P||EP X Q

the elements in P (not P's girdle in the duoprism) join to the facets of the EP girdle of the duoprism in the same way as before, while the whole of P is joined to the facets in the other girdle by lower Magna forms

for gyro forms:

you need to "gyrate the duoprism" *
I haven't figured out what this does to the duoprism yet so I'm not sure how to identify the facets, however you only gyrate the facets in the Q girdle* so the facets in THIS girdle are now "Gyrated Duoprisms" * these join the whole of EP with Gyro forms but it may not be all the facets that have been gyrated* so some will still be ortho forms.

*look at my other thread for "Gyrated antiprisms"
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Re: Multicupolic screw-gauges

Postby quickfur » Thu Jul 12, 2012 4:31 pm

wintersolstice wrote:I've figured out more about these shapes and have a new definition:

Definition:

if you take a prismatoid (not a duoprismatoid, just one that has vertices in two planes) and define the two planes using two polytopes P&Q

EP||P X Q

where EP means "expanded P" (shrink the facets and insert new ones between them)

P X Q means the cartesian product P and Q

now if Q = "Point" then the final shape is a "P cupola"

if Q = "line segment" then the final shape is a "P orthobicupolic ring"

That is cool!! So there is a nice systematic way to derive these things. I like it!

[...]Now for Gyro and Magna forms:

for a magna form you just change it to

P||EP X Q

the elements in P (not P's girdle in the duoprism) join to the facets of the EP girdle of the duoprism in the same way as before, while the whole of P is joined to the facets in the other girdle by lower Magna forms

This is cool.

for gyro forms:

you need to "gyrate the duoprism" *
I haven't figured out what this does to the duoprism yet so I'm not sure how to identify the facets, however you only gyrate the facets in the Q girdle* so the facets in THIS girdle are now "Gyrated Duoprisms" * these join the whole of EP with Gyro forms but it may not be all the facets that have been gyrated* so some will still be ortho forms.[...]

Seems complicated, need to look more into this.
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Re: Multicupolic screw-gauges

Postby wintersolstice » Tue Jul 17, 2012 7:27 pm

some closer investigating indicates that these shapes get even more complicated!

In 4D the cube cupola has "square orthobicupolic rings" as a sub part,

The octahedron cupola has a "square bipyramidal bicupoilc ring" (I made that name up myself) as a sub part

the "level 3" ortho-cupolic screw guages (I'm thinking of using tri quadri penta etc for the cases when Q is a polygon) exist as a sub part of the orthobicuplic rings the magna forms have their own fragments and I'm not convinced that there lower magna forms (though I can't be absolutly sure)

both of these could have generalisations that could go even more complex:D
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