polytwisters

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

polytwisters

Postby alkaline » Mon Dec 08, 2003 6:28 pm

so what exactly is a polytwister?
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Postby Polyhedron Dude » Mon Dec 08, 2003 9:07 pm

A polytwister is like a polychoron except that it's sides are not flat, but they curve the way a cylinder does. I'll start with the dodecatwister:

To construct the dodecatwister, we'll need twelve identical sides, the sides look like the following: take a rod which has a pentagonal cross section - some sort of pentagonal pole - bulge out the rod so that the pentagon sections look bloated a little - now take this rod and twist it full circle - you now have something that looks like a Twizzler - now curve this into tetraspace to form a ring - That is one of the sides - to show where the next 5 sides would be - imagine that the ring was unwound back into a twisted rod, now place 5 more of these twisted rods - so they touch each of the 5 sides of the rod - but slant them so they seem to be swirling about the first rod - now fold each of them into rings - the sides would now wrap around each other in a swirling fashion - complete this with the 6 remaining sides which are orthogonal to the first sides and you have the dodecatwister. This swirligig has 30 ridges, each ridge joins two of the sides - the ridges look like a long rectangular strip that has been twisted full circle and then curved into tetraspace to form some twisted ring. Finally there are 20 circles which act like peaks, three sides meet at these circles - these circles are actually equators - but now there are 20 - they are arranged so that they seem to spiral aound each other.

If a tetronian had a dodecatwister and rolled it like a die, it could land on any of the twelve sides and be rolling on that side the way a cylinder rolls - if it turns over on an adjacent side - it would start rolling off at a different angle. The dodecatwister can also be formed by the intersection of 6 duocylinders placed in such a way that the result is a regular figure with twisted symmetries. It is related to Hopf Fibration - Hopf Fibration is the division of the glome into circles - to picture it, take a glome and make it swirl (double rotation where both rotations are at the same speed) - now imagine holding a pencil to it so that the swirling glome would cause a circle to be drawn on it - no matter where you place the pencil - a circle will be drawn - and not just any circle - but an equator - you could keep doing this until the whole glome has been covered in circles, the circles dont intersect unless they are exactly the same - the circles swirl around each other. 20 of these circles would be the circles of the dodecatwister.

Alkaline - if you're interested, I could send you cross section pictures of a couple of polytwisters, and you can post them to your site. This would be extremely helpful to seeing what they are.

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Postby alkaline » Mon Dec 08, 2003 9:26 pm

yes send me some pictures, this is quite some mind-boggling stuff :)
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Postby Keiji » Mon Dec 08, 2003 10:40 pm

rather, maybe you can upload them? if you can't find anywhere to upload them, you could upload them on my forums ( [/me does not want to advertise, tell me if neccacery] ). It would be better to upload them so that everyone can see the pics.
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Postby alkaline » Mon Dec 08, 2003 11:16 pm

i meant for him to send them to me so that i could upload them to the tetraspace website.
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Postby Polyhedron Dude » Tue Dec 09, 2003 6:00 am

alkaline wrote:yes send me some pictures, this is quite some mind-boggling stuff :)


Sho can do!
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Postby Aale de Winkel » Tue Dec 09, 2003 6:10 am

Yes pictures please, also to the hyperoctagons.
shaving off the hypercubes corners I fully understand, but blowing out faces can be done ad infinitum even to the point where one ends up with an hypersphere.

numbernotatation, if I now understand it:
1: line
2: circle
3: sphere
so:
12: line-circle so cylinder (but how to denote a cone)
13: line-sphere so spherinder (but what about the sphone)

22: circle-circle so duocircle (why this was confused with "duocylinder" 1212(?))
though using this notation 122 would be the first pentaspace object that might be called a duocylinder (which means that 1212 contracts somehow to 122(?), though I do think 1212 is a hexa-space object, which is different from 1122 (the duocubinder(?)))
Is there a way to describe the donut in this manner, which is thus 12 but the 1 is circling the "but" onto the "face" circle.

A notationally confused trionian :lol:
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Postby Polyhedron Dude » Tue Dec 09, 2003 7:50 am

Aale de Winkel wrote:Yes pictures please, also to the hyperoctagons.
shaving off the hypercubes corners I fully understand, but blowing out faces can be done ad infinitum even to the point where one ends up with an hypersphere.


Yes I could send pictures of the 3-D and 4-D ones.

Aale de Winkel wrote:numbernotatation, if I now understand it:
1: line
2: circle
3: sphere
so:
12: line-circle so cylinder (but how to denote a cone)
13: line-sphere so spherinder (but what about the sphone)


I do have a notation for cones, and relatives - I use a notation that I informally call the "Sony Playstation Notation" - I call it this due to the symbols I use (X A O D) - D supposed to be a square, and A a triangle.
X-represents cross product
A-represents pyramid-extenstion
O-represents spinning into next dimension (lathing)
D-represents prism extention
assumption is that you start with a line segment.

a triangle is A, a square is D, a circle is O
AA is a tet, DD is a cube, AD is a trip (triangle prism), DA is a square pyramid. OA=AO would be the cone, OD=DO is a cylinder, OO is a sphere.
AXA is the triangle duoprism (4-D), OXO is the duocylinder, the sphone is OOA. DXD=DDD=tesseract, MXD (M can be any sequence) = MDD.
sometimes you may need parentheses, as in this example:
(AAXO)AX(AXAXA)A - the (AAXO) part leads to the cross product of a tet and a circle (we'll call it a "tepinder" (I call the tet prism a tepe)) - then you take the pyramid of that to get (AAXO)A - now you need to cross product this thing with (AXAXA)A - which is the pyramid of the cross product of three triangles (ie. the triangle triprism pyramid) - so the result is a tepinder pyramid - triangle triprism pyramid duoprism - nice name for a 13 dimensional shape!

Aale de Winkel wrote:22: circle-circle so duocircle (why this was confused with "duocylinder" 1212(?))
though using this notation 122 would be the first pentaspace object that might be called a duocylinder (which means that 1212 contracts somehow to 122(?), though I do think 1212 is a hexa-space object, which is different from 1122 (the duocubinder(?)))


I called the 22 a duocylinder since it can be rolled like a cylinder in two ways, calling the 22 a duocircle is not a bad idea - since 122 would roll like the cylinder in two ways and can stand up like a cylinder - it could be dubbed the duocylinder, so lets call the 22 a duocircle and 122 a duocylinder (uh oh more changes in the glossary). 33 can be called a duosphere, 133 can be called a duospherinder. 1122 is actually the same as 1212 - the reason is because 1122 is the cross product of a line, a line, a circle, and a circle - 1212 is the cross product of a line, a circle, a line, and a circle. Since cross producting is communative, 1122=1212 - this is the cylinder duoprism - the name duocubinder is fitting. Here's one - lets call the 23 a "sphercle" and the 123 a sphercylinder.

Aale de Winkel wrote:Is there a way to describe the donut in this manner, which is thus 12 but the 1 is circling the "but" onto the "face" circle.

A notationally confused trionian :lol:


Haven't came up with a notation for donuts yet, but here are a couple 4-D donuts:

doball - looks like a ball with a ball hollow inside the way a donut looks like a circle with a circle hollow - the connection of the inner and outer ball is a circle.
sphonut - looks like a circle with circle hollow, but has a spherical connection (also the result of warping a spherinder into a ring)

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Postby Aale de Winkel » Tue Dec 09, 2003 9:16 am

quite illuminating, combining things I gather:

2A: the cone
3A: the sphone
11A: the cubone
23A: the spherone (?)

must work your cross product out some day to understand fully. but I get the idea.

yet more confusing objects, the Mobius-belt and the Kleins-flask etc. Might not be dealt with in this matter.

Note: my formulae, worked out in the (general fourth dimension) glossary posting might also come in handy in this context! It showed me that the duocircle had no place in the hypercylinder family, but was the front-end of the hyperduocylinders!

Your "doball" looks to me belonging to the "hyper-ring"-family { x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], r = R[sub]1[/sub] .. R[sub]2[/sub] } which serves as a basis for the donut, which is the ring cross product(?) with a disc (circle(?)).
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post- and pre-lathing(?)

Postby Aale de Winkel » Wed Dec 10, 2003 5:59 am

Let R stand for the ring then

1R = R1 : the cylinderring and the ringcylinder (obviously the same thing)
2R : the doball (lathing the cylinderring around its center)
R2 : the donut (lathing the ringcylinder around the center of the ring)

(R3 : is then the sphonut you mentioned!
While 3R is a gongylring( dogongyl?) (this looks dimensionally a bit out of wack, but probably correct(!?) :lol: )

thic defines "prelathing" as lathing around the bodys center and "postlating" aound the bodys off center, thus going on:

12R = 2R1 : the cylinderdobal or dobalcylinder
22R : the duodoball (??)
2R2 : the doballdonut (??)

1R2 = R21 : the cylinderdonut or donutcylinder
2R2 : the doballdonut (?? but the same object as above)
R22 : the duodonut (??)

does this make some sense :?:

as a further possibiliy, one might "ring" the cylinder (ie 1 --> R)
12 = 21 : the cylinder
R2 : the donut
2R : the doball

though ringing the duocylinder can then be done in three different ways
221 -> 22R : the duodoball (??)
212 -> 2R2 : the dobaldonut (??)
122 -> R22 : the duodonut (??)
whether this makes some sense needs further study! :shock:
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twisting the moebius belt

Postby Aale de Winkel » Fri Dec 12, 2003 6:13 am

in order to obtain the moebius belt or strip:
http://mathworld.wolfram.com/MoebiusStrip.html
one could do the following:

let 1L = L1 : the cylinderline or linecylinder be alternative for the 11 rectangle.
Ringing them would make them bothe cylinders as
RL (ie R1) = LR (ie 1R) : ringcylinder or cylinderring (regularly called cylinder)

(note: there is another way of ringing the rectangle which transforms the rectangle into the ring defined 2 posts back, this might then be RL : ring as the analogon of the doball, however the ring is of the same dimension of the rectangle so this needs further analysis to define properly.)

twisting the cyllinder line gives the moebius strip so lets denote this twist by R[sub]180[/sub] then
LR[sub]180[/sub] : the moebius belt
as L is really '1' in disquise one can ring it into
RR[sub]180[/sub] : the Klein Bottle see: http://mathworld.wolfram.com/KleinBottle.html

R[sub]180[/sub]R[sub]180[/sub] : some twist of the Klein Bottle (???)

Having the twister denoted by R[sub]α[/sub] allows us to create several objects from say the octagoncylinder by twisting with angles 0,45,90,135,180,225,270,315,360,.......
with 0 it is the same as reglularly ringing O1 --> OR : the octagonring
angle 45 / 360 are singly twisted octagonrings
further angles are also possible introducing more then one twist though

LR[sub]360[/sub] already is doubly twisted (i believe)

(note: O stands here for octagon, not to be confused with Polyhedron Dude's lathing O (I simply adhockly needed a symbol))

Just trying to add some notation, there are some kinks still to straighten out. I think this is better left to someone more familiar with this kind of objects (you perhaps Polyhedron Dude :!: :?: ) :lol:


:idea: This and the previous two posting are NOT to be used elsewhere, these thing are yet to be formalized, and merely serves as the start of this discussion!! :idea:
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Postby Polyhedron Dude » Sat Dec 13, 2003 9:16 am

Those twisted ringed octagon prisms sounds like something that I've worked with - I would start with a duoprism (a cross product of two polytopes) and take a sub set of it's vertices (a sub set with identical vertices that is). An example - start with the odip (octagon duoprism) - it is 4-D with two orthogonal rings of 8 octagon prisms (16 8P's in all) - it has 64 vertices - the vertices can be labeled as MN (M and N goes from 1 to 8 representing the vertices of an octagon - M for the xy octagon, N for the zw octagon) - you can draw an 8x8 square grid to represent the vertices, to select all vertices would be symbolized by picking all the squares of this checkerboard, but you can pic a subset (how about the red squares) or how about the following 11, 23, 35, 47, 51, 63, 75, 87 - this would lead to a polychoron with 8 identical vertices. Another series would be 11, 24, 37, 42, 55, 68, 73, 86 - this is something fun to experiment with - you can also use these points (which are on a glome) to represent the location of tangent realms instead of vertices, to create polychora with only one type of cell (this would make nice 4-D dice) - you could even start with a heptagon duoprism and pick the following points from the 7x7 grid of points: 11, 23, 35, 47, 52, 64, 76 - use this as tangent points to create a strange 7 sided die in tetraspace, do this for the 13-gon and get 13 sided dice - finding polyhedral dice with odd number of identical sides happens in 4-space, but not here in 3-space - interesting stuff.

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Postby Aale de Winkel » Wed Dec 17, 2003 9:10 am

It might be a too simple thought, but given the pictures, is a dodecatwister not simply a twisted dodecahedron, this allows simular notation as above, though twisting remains in the same dimension.

D: dodecahedron
DT[sub]α[/sub]: twisting the dodecahedron an angle α

(supposing here that a top and bottom pantagon is chosen)

Simularly one migth twist a cylinder.

11T[sub]α[/sub]

using a dodecatwister as a die, I don't see, since the twisted faces has other properties then the top and bottom pentagon. (but perhaps from a tetronian view it looks different)

Note: it might well be that this kind of twisting is something else, but it makes interesting objects :lol: 861T[sub]90[/sub]; the octahexaspherintwist(90[sup]o[/sup]) ( :?: :lol: :lol: :?: ).
(note: changed twister to twist, probably something needs be done to change a twist into a twister (ringing perhaps(?))

visiting your site (restriction at work where lifted) I realize that
DT[sub]α[/sub]: the twisted dodecahedron (dodecatwist(!?)) seems to be merely a trionian projection of the real dodecatwister!?
I guess I'll remain a trionian :lol:
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polygon (anti-) prisms ?

Postby Aale de Winkel » Wed Dec 17, 2003 5:25 pm

http://hometown.aol.com/Polycell/uniform.html
shows excellent picture of what might be call polygoncylinders, lets say P[sub]p[/sub] to stand for the p-agon (ie p-corners) then:

P[sub]p[/sub]1 : the p-agon cylinder (p-agon prism (!?))

this notation generates tip (p = 3) cube (p = 4), pip (p = 5), hip (p = 6) sip (p = 7) oip (op(??) p = 8 ) nip (p = 9) dip (p = 10) the same way as when using spherical based cylinder types

The shown pantagonal antiprism can simular be handled but needs some other "extrudor" lets say '>' then

P[sub]p[/sub]> : the p-agon anti-prism

thus generating tap / dap (the page shows only pap, don't know wheter p=4 would be called cap in which case cube = cip (hip from hexahedron won't do because of p=6))

Both types can be twisted or ringed as the prior postings suggests

trying to mak thing as simple as possible :lol: :lol:
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summary of the above transformations

Postby Aale de Winkel » Thu Dec 18, 2003 8:57 am

Here I try to summarize the transformations already posted

L: lathing, shape rotation into higher dimension; line (1) --> circle (2) --> sphere (3) etc.
E: extrusion, lineair progression into the next dimension at perpendicular angles of shape; line --> square --> cube etc.
>: anti extrusion, extrusion with adjacent points onto a single point; pe triangle --> tap
C: conic extrusion, extrusion onto a single point: line --> triangle --> tet etc.

T[sub]α[/sub]: twisting, tordation of object in itself around some body axis: 21T[sub]45[/sub] a cylindertwist(45[sup]o[/sup])
R: Rotation, rotating of front end onto backend within the same dimension around an outside boby point: square --> ring
R[sub]α[/sub]: Rotation through the next dimension front onto backend with possible twist: square (11) --> mobius strip (1R[sub]180[/sub])

(note the difference between R and R[sub]0[/sub]: 1R = ring; 1R[sub]0[/sub] = 21 = cylinder )

introducing in this post:
F: filling: fillng the object: circle --> disc, sphere --> ball etc.
-: subtracting: subtracting objects of different size: circle - circle --> docircle(? = ring! ?); ball - ball --> doball :?:
(one needs to add size-parameters to define subtraction more presicely.)
adding could also be defined for the unfilled types, however I consider this only for objects of same type, assuming by default the subtracted to be of say half size of the other, subtracting filled objects from filled objects is therby solidly defined, while adding here is quite redundant. With unfilled object one simply needs to ignore the technically present sign, or define addition also introducing the different objects T+T and T-T (with T=triangle)

Probably this is not the full range. The step from say triangle --> hexagon; square --> octagon, could also be formalized. I currently don't see pentagon and heptagon easily formulated, but the p-agon P[sub]p[/sub] is easily defined, as most likely other basic entities need be defined. :)
The lettering is adhockly defined and preliminairy at best, Polyhedron Dude please your comments on this monolog. :lol: I realize that my "polygon-cylinder" has some friction with your prism-sequence, but far easily to grasp then the schlaefli symbols etc. :wink:

a strange trionian wandering far from his range of expertise :D .

btw: introducing (same center) subtraction: subtracting a small circle from a larger circle thus this give the docircle :!: ; this way the doball is a filled dosphere to name but one of the possibilities of this notion.
All this gives a lot of possibilities to arrive at the same object though :lol:

btw http://mathworld.wolfram.com/NonorientableSurface.html showed me that R[sub]180[/sub]R[sub]180[/sub] is called "real projective plane".
.
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Postby Polyhedron Dude » Thu Dec 18, 2003 9:41 pm

Aale de Winkel wrote:It might be a too simple thought, but given the pictures, is a dodecatwister not simply a twisted dodecahedron, this allows simular notation as above, though twisting remains in the same dimension.



Looks kind of like a twisted dodecahedron, but it is quite different - it is 4-D, the pentagons in the picture are sections of the pentagonal ring, the top and bottom pentagon on each section are part of the same side!
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Re: polygon (anti-) prisms ?

Postby Polyhedron Dude » Thu Dec 18, 2003 9:50 pm

Aale de Winkel wrote:http://hometown.aol.com/Polycell/uniform.html
shows excellent picture of what might be call polygoncylinders, lets say P[sub]p[/sub] to stand for the p-agon (ie p-corners) then:

P[sub]p[/sub]1 : the p-agon cylinder (p-agon prism (!?))

this notation generates tip (p = 3) cube (p = 4), pip (p = 5), hip (p = 6) sip (p = 7) oip (op(??) p = 8 ) nip (p = 9) dip (p = 10) the same way as when using spherical based cylinder types

The shown pantagonal antiprism can simular be handled but needs some other "extrudor" lets say '>' then

P[sub]p[/sub]> : the p-agon anti-prism

thus generating tap / dap (the page shows only pap, don't know wheter p=4 would be called cap in which case cube = cip (hip from hexahedron won't do because of p=6))


the 4 AP and 4P are called squap and squip (sq for square), 8P is called op, 9P is called ep (for enneagon prism), 3P and 3AP are called trip and trap. You can ring any of these prismattic things to form a ring, polytwisters have various polygon type rings - the cubetwister for example has square "twisters" (a twisted ring formed from a square rod).

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Re: summary of the above transformations

Postby Polyhedron Dude » Thu Dec 18, 2003 10:01 pm

Aale de Winkel wrote:introducing in this post:
F: filling: fillng the object: circle --> disc, sphere --> ball etc.
-: subtracting: subtracting objects of different size: circle - circle --> docircle(? = ring! ?); ball - ball --> doball :?:
(one needs to add size-parameters to define subtraction more presicely.)
adding could also be defined for the unfilled types, however I consider this only for objects of same type, assuming by default the subtracted to be of say half size of the other, subtracting filled objects from filled objects is therby solidly defined, while adding here is quite redundant. With unfilled object one simply needs to ignore the technically present sign, or define addition also introducing the different objects T+T and T-T (with T=triangle)


circle - circle would be a wahser shape (a flat donut) - we need a filler as well - we could use circle, circle filled to represent circle - cirlce and bloated circularly.

Aale de Winkel wrote:Probably this is not the full range. The step from say triangle --> hexagon; square --> octagon, could also be formalized. I currently don't see pentagon and heptagon easily formulated, but the p-agon P[sub]p[/sub] is easily defined, as most likely other basic entities need be defined. :)
The lettering is adhockly defined and preliminairy at best, Polyhedron Dude please your comments on this monolog. :lol: I realize that my "polygon-cylinder" has some friction with your prism-sequence, but far easily to grasp then the schlaefli symbols etc. :wink:


the triangle --> hexagon is called truncation, the polygon cylinder can also be called polygon rod (if it is a tall cylinder), or polygon coin if it is flat, it is usually called polygon prism.

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Postby Aale de Winkel » Fri Dec 19, 2003 7:34 am

squap and squip I understand, 8P "op" I read on http://hometown.aol.com/Polycell/uniform.html , trip was simply not printed on my paper copy , trap also is logical then , 9P ep (it is simply the first time I read "enne" for 9)

As said I'm traveling far outside my realm of expertise, so i'm still learning.

With Truncation added to the list there is still no way to produce the "prime-agons" I think (so to be defined :!: :?: )

I simply used "do" in accordance with your "doball" (= ball - ball), you can then simply use do"shape"
circle - circle looks to me as a ring (which I redefined as not-filled), the earlier defined "ring" so is a filled ring.

rod as alternative for cylinder I knew, a circle is P[sub]infinit[/sup] (or infinitP as you seem to call it)

"op" as I understand your mnemonics, you take first letter of the regular name and trow in some letters to make them pronounceable, which thus makes "op" equivalent to "oip", for me though "oip" makes it more stand apart for "oap" (thus defining 'i' the normal prism, and 'a' the antiprism connector, with 'c' the "conicalizer" one could have a sloidly defined oc, but probably need some vocal to pronounce "trc" :lol: ).

But heh, who am I, it's your field of expertise, you tell me :cry: .
I suggest you decide upon the transformations, make a short list (like I did on my previous post), include rules like 1L = 2 (a lathed line is a circle) and more of these things :wink: .

A strange trionian ploughing his way into the higher dimension :lol:

(what you mean with "bloated circularly" I don't know, probably because I'm no a native english speaker :? )
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Postby Polyhedron Dude » Fri Dec 19, 2003 9:22 am

Aale de Winkel wrote:squap and squip I understand, 8P "op" I read on http://hometown.aol.com/Polycell/uniform.html , trip was simply not printed on my paper copy , trap also is logical then , 9P ep (it is simply the first time I read "enne" for 9)


the 9-gon can be called either the enneagon or the nonagon - many geometers seem to pick the former, however - between you and me, I like the later name better :wink: It may be better to call 9P nip instead of ep - enneagon is pronounced like N-eagon. 9AP can be called nap, just don't take 9APs while you're driving :P

Aale de Winkel wrote:"op" as I understand your mnemonics, you take first letter of the regular name and trow in some letters to make them pronounceable, which thus makes "op" equivalent to "oip", for me though "oip" makes it more stand apart for "oap" (thus defining 'i' the normal prism, and 'a' the antiprism connector, with 'c' the "conicalizer" one could have a sloidly defined oc, but probably need some vocal to pronounce "trc" :lol: ).


For my polytope short names, I usually add extra vowels when it's a necessity - for example for 5P, the abbr is pp - needs a vowel to pronounce it - so it becomes pip. For 8P - the abbr is op - this is already pronouncible, no need to add extra vowels. Other names include the cuboctahedron - abbr is CO - so I call it co. Another is the quasitruncated hexahedron - abbr is QTH - needs vowels - so I call it quith. I'll hafta start up a topic on polytope short names, this topic could be quite humorous, since some of the names get fairly silly sounding - I've actually had people laugh hystarically reading the names of various polychora. Here are some examples:

Polyhedra: gaquatid, quit sissid, gad, oho, did, snid, querco, gidditdid

Polychora: madtapchi hiccup, dod honho, frico, thatpath, gnappoth, picnut, gotto, frogfix, statupid hidxopchi, stupid pupthi, gixhinxhi, gixhi fohixhi, quippirgax, irp xanady, gaquerdy, sidhiquit paddy, gaquapac, coqroc, gidditdiddip, gaddiddip, sniccup, spiddit, spittid

Polytera: rat, cat, ratchet, hipbox, orpyax, orarpyax, oi ipyax, quabwabid, wiwabix, yarbawx, yackerd, dohbinhed, tonite, batatadit, titcat bacon, tetcanbocat, botatic dohunt, carntry hen
- I'm still in the process of naming the polytera, who knows how goofy sounding these names are gonna wind up. :lol:

Aale de Winkel wrote:(what you mean with "bloated circularly" I don't know, probably because I'm no a native english speaker :? )


I meant that the donut shape is bloated from a flat washer shape in a way that a vertical cross section of it looks like a circle.

Polyhedron Dude
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