Alternative Methods for the Same Goal

Higher-dimensional geometry (previously "Polyshapes").

Alternative Methods for the Same Goal

Postby ICN5D » Sun Nov 17, 2013 6:37 pm

Wendy, I can tell that you have been refining your terminology for quite some time. It's easy to get caught up in it when you've become so familiar over the years, I know! I suppose we all come to understand these higher dimensions in our own, unique way. I certainly have with my development of this dimensional transformation mathematics. I've been working on it for over 5 and a half years.

Ultimately we take different paths to enlightenment, but I believe we understand many of the same things. I also have come to see the duocylinder as having two sides from a completely different method. Interesting, huh? I like my funny names, too. They are etymologically sound and tap into my linguistic side. I am also interested in educating the masses about higher dimensional spaces, as I have found that they aren't so hard to see, once the right tricks are applied.

So you created the " spherate" motion, huh? I have seen that word all over the place with any reference to higher dimensions. Funny how far a word can travel through the internet. I myself prefer the "spin" motion. It is my specific way to describe rotopes and torii. Of course there is no right or wrong method as long as it is congruent to the end result.

It does create some ambiguity, though. If a spin motion is paired with a taper or extrude motion, it seems to take on a quantum superposition nature, having multiple rearrangements create the same thing. Like the cone and cylinder: both can be created two different ways. I found this interesting: a cylinder can be made by extruding a circle or spinning a square. A cone can be a tapering of a circle or spin of a triangle. Using an old syntax, it looks like this:

CONE
|0> = taper of circle
|>0 = spin of triangle

CONE - [ |0> ]

[ |0 ] ---z---> [ * ] == [ |0> ]
-----------------------------------------
[ *(0):xy ] --> [ * ] == [ |z(0):xy ]
-------------------------- [ |0-*:z ]


CONE-B - [ |>0 ]

[ |> ] --z--> [ 0x ] == [ |>0 ]
----------------------------------------
[ |2:x ] --> [ (0) ] == [ |y(0):xz ]
[ |-*:y ] ---> [ 0 ] == [ |0-*:y ]


CYLINDER
|0| = extrude of circle
||0 = spin of square

CYLINDER - [ |0| ]

[ |0 ] ----z---> [ |0 ] == [ |0| ]
-------------------------------------------
[ *(0):xy ] --> [ *(0) ] == [ |z(0):xy ]
----------------------------- [ |02:z ]


CYLINDER-B - [ ||0 ]

[ || ] --z--> [ 0x ] == [ ||0 ]
---------------------------------------
[ |2:x ] --> [ (0) ] == [ |y(0):xz ]
[ |2:y ] ---> [ 0 ] == [ |02:y ]



I wondered if there were any versions in the higher dimensions, and there most certainly are! Through much conceptual evolution and familiarizing myself with the nuances, I have found MANY:

CYLINDRAMID ( or cylinder-pyramid )
||>0 = spin of square-pyramid
||0> = taper of cylinder

CUBINDER ( cylinder-prism )
|||0 = spin of cube
||0| = extrude of cylinder

TESSERINDER ( circle-triprism, cubinder-prism )
||||0 = spin of tesseract
|||0| = extrude of cubinder

CYLTRIANGLINDRAMID ( cyltrianglinder-pyramid )
|>|>0 = spin of trianglindramid ( triangle-prism-pyramid )
|>|0> = taper of cyltrianglinder

just to name a few.

I found that in 6-D a new kind of "prime shape" arises. Primes are shapes with only one arrangement to it's sequence. This effect can be seen as early as 4-D, and arise again in 6-D with the operations of the |>|0, from its triangular nature during the motions. In fact, any triangular-like shape be it ||>|0, |||>|0, ||||>|0, etc has this effect:

|>|00| = DUOCYLTRIANGLINDRINDER - cartesian product of a spherinder and a triangle

|>||00 = DUOCUBTRIANGLINDER, or |>|0|0 - cartesian product of a duocylinder and a triangle

After deriving their 5-D surface panels, I found that they are not the same. When it came to a triangular-like shape, spin then extrude creates something different from extrude then spin. Similar to the difference between a cyltrianglinder and a coninder:

|>|0 = Has 3 cylinders + 1 triangle-torus on the surface
|>0| = Has 1 cylinder + 2 cones + 1 square-torus on the surface


CYLTRIANGLINDER - [ |>|0 ]

[ |>| ] ---w--> [ 0z ] == [ |>|0 ]
----------------------------------------
[ ||2:x ] -----> [ 0 ] == [ ||02:x ]
[ ||-|:y ] ----> [ 0 ] == [ ||0-|0:y ]
[ |>2:z ] ---> [ (0) ] == [ |>xy(0):zw ]


CONINDER - [ |0>| ]

[ |0> ] ---w---> [ |0> ] == [ |0>| ]
------------------------------------------------
[ |z(0):xy ] -> [ |(0) ] == [ ||zw(0):xy ]
[ |0-*:z ] ---> [ |0-* ] == [ ||0-|:z ]
------------------------------ [ |0>2:w ]


Talk about funny names, here's a list up to 5-D of all non-manifold shapes as described by this Dimensionometry of mine. Of course, there are an infinite number of shapes, below is a simplified version focused the basics. Once again, this is just one way to the path of enlightenment, (my specific way):

multiform: shapes with multiple sequences, denoted with asterix
duplicate: additional multiforms not listed
prime: having one sequence
| = extrude
> = taper
0 = spin

----------------------------------------
1st Dimension: 1 total, 1 prime
X
| - LINE
----------------------------------------
2nd Dimension: 3 total, 3 prime
XY
|0 - CIRCLE
|> - TRIANGLE
|| - SQUARE
---------------------------------------------------------------------------
3rd Dimension: 9 total, 5 prime, 2 multiform*, 2 duplicates
XYZ
|00 - SPHERE
|0> - CONE*
||0 - CYLINDER*
|>> - TETRAHEDRON
|>| - TRIANGLINDER / TRIANGLE PRISM
||> - PYRAMID / HEMOCTAHEDRON
||| - CUBE / ORTHOHEXAHEDRON
--------------------------------------------------------------------------------
4th Dimension: 27 total, 12 prime, 5 multiform*, 10 duplicates
XYZW
|000 - GLOME
|00> - SPHERACONE*
|00| - SPHERINDER
|0>> - DICONE*
|0>| - CONINDER
||00 - DUOCYLINDER*
||0> - CYLINDRAMID*
|>>> - PENTACHORON
|>>| - TETRAHEDRINDER
|>|0 - CYLTRIANGLINDER
|>|> - TRIANGLINDRAMID
|>|| - TRIANGDILINDER
||>> - DIPYRAMID
||>| - PYRAMINDER
|||0 - CUBINDER*
|||> - HEMDODECACHORON
|||| - TESSERACT / ORTHOCTACHORON
---------------------------------------------------------------------------------
5th Dimension: 81 total, 27 prime, 16 multiform*, 38 duplicates
XYZWV
|0000 - PENTASPHERE
|000> - GLOMACONE*
|000| - GLOMINDER
|00>> - SPHERADICONE*
|00>| - SPHERACONINDER
|00|0 - CYLSPHERINDER
|00|> - SPHERINDRAMID
|00|| - SPHERADILINDER
|0>>0 - CYLDICONE? / SPHERADICONE?*
|0>>> - TRICONE*
|0>>| - DICONINDER*
|0>|0 - CYLCONINDER*
|0>|> - CONINDRAMID*
|0>|| - CONDILINDER*
||000 - UNKNOWN ??*
||00> - DUOCYLINDRAMID*
||0>> - CYLINDRADIMID*
||0>| - CYLINDRAMINDER*
|>>>> - HEXATERON
|>>>| - PENTACHORINDER
|>>|0 - CYLTETRAHEDRINDER
|>>|> - TETRAHEDRINAMID
|>>|| - TETRAHEDRADILINDER
|>|00 - DUOCYLTRIANGLINDER
|>|0> - CYLTRIANGLINDRAMID
|>|0| - CUBTRIANGLINDER
|>|>> - TRIANGLINDRADIMID
|>|>| - TRIANGLINDRAMINDER
|>||> - TRIANGDILINDRAMID
|>||| - TRIANGTRILINDER
||>>0 - CYLINDRADIMID*
||>>> - TRIPYRAMID
||>>| - DIPYRAMINDER
||>|0 - CYLHEMOCTAHEDRINDER
||>|> - PYRAMINDRAMID / HEMOCTAHEDRINAMID
||>|| - PYRADIMID / HEMOCTAHEDRADILINDER
|||00 - DUOCUBINDER*
|||0> - CUBINDRAMID / CYLHEMDODECACHORON*
|||>> - CUBE DIPYRAMID / HEMDODECACHORAMID
|||>| - CUBE PYRAMINDER / HEMDODECACHORINDER
||||0 - TESSERINDER*
||||> - TESSERACT PYRAMID / HEMHEXADECATERON
||||| - PENTERACT / ORTHODECATERON / 5-D CUBE
-------------------------------------------------------------------
6-D: 243 total, 54 prime, 59 multiform, 130 duplicates
7-D: 729 total, 81 prime, 226 multiform, 422 duplicates
etc.....

TOTAL POSSIBLE NON-MANIFOLD SHAPES IN N-TH DIMENSION
----------------------------------------------------------------------------

Tn = 3^(n-1)

(Tn-Pn) = (Mn+Dn)

Mn = Tn-Dn-Pn

Pn = Tn-(Mn+Dn)

Dn = Tn-(Pn+Mn)

Tn = Pn+Mn+Dn

(M+D)n = (1+n)(3^(n-3))

(P+M)n = 3(P[n-1]+M[n-1])-2^(n-2)

--------------------------------------------------------
Tn = Total number of shapes in dimension n
Pn = Primes ( Monoforms )
Mn = Multiforms
Dn = Duplicates ( extra multiforms )
n = number of dimensions
[n-1] = previous dimension, n-1

-Philip
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Re: Alternative Methods for the Same Goal

Postby wendy » Mon Nov 18, 2013 9:35 am

Looking at the notation, i get this feeling:

| leading, is a line, following, makes a prism
> makes a pyramid,
0 makes something like a crind on some unidentified axis.

Ok, we have been dealing with five different products, rather than three here. I'm not sure on the spin cycle, since there are several different ways you can spin a figure, eg the hexagon prism has two different spins. We deal with tapers under 'lace prisms', where one can 'taper' a square into an octagon. Klitzing has prepared a list of all of the equalateral tapers inscrubable in two planes through a sphere (segmentotopes).

Not overly fussed with the word mangling. It's hardly etymology, and terribly offends people who are well versed in that subject.

The trouble with using words line 'bipyramid', is that the meaning falls apart rapidly in higher dimensions. In 3d, a bipyramid a figure made by joining two pyramids at their bases. In four dimensions, one can make a similar figure, or one can make an entirely different figure based on a pyramid whose base is a pyramid. And simply saying something like "dodecagon bipyramid" can be read as two different figures in two different dimensions. In my terminology, one can make these from two different products: a pair of pyramids joined at the base, is a tegum of the base and the axis. Tegum is a product that produces the duals of prisms. If you have a pyramid whose base is a pyramid, then its "'base' - line pyramid", which is an entirely different thing.

In four dimensions, there are even stranger things: there is the bi-circular tegum, which one might form by throwing a skin over two orthogonal circles.

What you find, is that for pretty much below eight dimensions, there are figures that serve different roles. It's only when you get out to six or eight dimensions, that each thread has a different reflex, or that the reflex is common throughout. The 24choron in 4d, is in 3d, both the cuboctahedron and the rhombic-dodecahedron, but these 3d figures have also two other figures in 4d (the runcinated pentachoron, and the rhombic icosachoron). In five dimensions, there are four different figures, which remain that way there-after.
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Tue Nov 19, 2013 5:28 am

Terribly offended, huh? Well, that's too bad. I try not to get too offended by someone continuing to misname parts of a bicycle. I just pass them off as " one of those people ". It is apparent that I still have much to learn, thank you for the insight.
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Re: Alternative Methods for the Same Goal

Postby Keiji » Wed Nov 20, 2013 10:32 pm

Just so we're not missing anything out here I'll go through and write the names I use for each of your objects:

XYZW
|000 - GLOME glome
|00> - SPHERACONE* sphone
|00| - SPHERINDER spherinder
|0>> - DICONE* dicone
|0>| - CONINDER coninder
||00 - DUOCYLINDER* duocylinder
||0> - CYLINDRAMID* cylindrone
|>>> - PENTACHORON pyrochoron (was previously pentachoron)
|>>| - TETRAHEDRINDER tetrahedral prism (but I may adopt tetrahedrinder, or perhaps pyrohedrinder, in future)
|>|0 - CYLTRIANGLINDER cyltrianglinder
|>|> - TRIANGLINDRAMID triangular prismic pyramid (but I may adopt trianglindrone in future)
|>|| - TRIANGDILINDER triangular diprism
||>> - DIPYRAMID square dipyramid
||>| - PYRAMINDER square pyramidal prism (but I may adopt hemoctahedrinder, or hemaerohedrinder, in future)
|||0 - CUBINDER* cubinder
|||> - HEMDODECACHORON cubic pyramid (but I may adopt hemaerochoron in future... cubone is also a nice nickname, and a fun reference ;) )
|||| - TESSERACT / ORTHOCTACHORON geochoron (was previously tesseract)
missing: duotrianglinder (Cartesian product of two triangles)

XYZWV
|0000 - PENTASPHERE pentasphere
|000> - GLOMACONE* glone
|000| - GLOMINDER glominder
|00>> - SPHERADICONE* disphone
|00>| - SPHERACONINDER sphoninder
|00|0 - CYLSPHERINDER cylspherinder
|00|> - SPHERINDRAMID spherindrone
|00|| - SPHERADILINDER cubspherinder
|0>>0 - CYLDICONE? / SPHERADICONE?* unknown
|0>>> - TRICONE* tricone
|0>>| - DICONINDER* diconic prism
|0>|0 - CYLCONINDER* cylconinder
|0>|> - CONINDRAMID* conindric pyramid
|0>|| - CONDILINDER* conic diprism
||000 - UNKNOWN ??* glominder
||00> - DUOCYLINDRAMID* duocylindrone
||0>> - CYLINDRADIMID* dicylindrone
||0>| - CYLINDRAMINDER* cylindronic prism
|>>>> - HEXATERON pyroteron (was previously hexateron)
|>>>| - PENTACHORINDER pyrochoric prism
|>>|0 - CYLTETRAHEDRINDER cyltetrahedrinder
|>>|> - TETRAHEDRINAMID tetrahedral prismic pyramid
|>>|| - TETRAHEDRADILINDER tetrahedral diprism
|>|00 - DUOCYLTRIANGLINDER unknown
|>|0> - CYLTRIANGLINDRAMID cyltrianglindrone
|>|0| - CUBTRIANGLINDER cyltriandyinder
|>|>> - TRIANGLINDRADIMID triangular prismic dipyramid
|>|>| - TRIANGLINDRAMINDER triangular prismic pyramidal prism
|>||> - TRIANGDILINDRAMID triangular diprismic pyramid
|>||| - TRIANGTRILINDER triangular triprism
||>>0 - CYLINDRADIMID* unknown
||>>> - TRIPYRAMID square tripyramid
||>>| - DIPYRAMINDER square dipyramidal prism
||>|0 - CYLHEMOCTAHEDRINDER cylhemoctahedrinder
||>|> - PYRAMINDRAMID / HEMOCTAHEDRINAMID square pyramidal prismic pyramod (may adopt hemaerohedrindrone)
||>|| - PYRADIMID / HEMOCTAHEDRADILINDER square pyramidal diprism (may adopt hemaerohedral diprism)
|||00 - DUOCUBINDER* cubspherinder
|||0> - CUBINDRAMID / CYLHEMDODECACHORON* cubindrone
|||>> - CUBE DIPYRAMID / HEMDODECACHORAMID cubic dipyramid
|||>| - CUBE PYRAMINDER / HEMDODECACHORINDER cubic pyramidal prism (may adopt hemaerochorinder)
||||0 - TESSERINDER* tesserinder
||||> - TESSERACT PYRAMID / HEMHEXADECATERON tesseric pyramid (may adopt hemaeroteron)
||||| - PENTERACT / ORTHODECATERON / 5-D CUBE geoteron (was previously penteract)
missing: duocyldyinder sphentrianglinder duotrianglindyinder contrianglinder hemoctahedrotrianglinder tetrahedrotrianglinder duotrianglindric pyramid
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Re: Alternative Methods for the Same Goal

Postby wendy » Thu Nov 21, 2013 7:16 am

Would |O|O become the duocylinder? It seems it to me.
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Thu Nov 21, 2013 10:51 pm

@ Wendy: Yes, |0|0 is the duocylinder as well as ||00.

@ Keiji: Thanks for translating those for me. It will be useful when I mention them in the future. I didn't realize that this was such a well developed field! I thought it was an unknown branch of math that was a free for all for others rushing to name the shape first. I am interested in how you interpret some of the shapes:

|>|> : wouldn't that be the duotrianglinder? One of the powertopes as a triangle squared?

|>|00 : lathing a cyltrianglinder into 5D. This creates a cone-like shape out of the spin of the triangular nature of |>|0. Much like lathing a triangle creates the cone, |>|0 is like a flat triangle of cylinders according to 5D. Deriving cylinder-lathe as duocylinder, adding the duo- to cyltrianglinder making duocyltrianglinder.

||>>0 : dicylindrone, lathe of square dipyramid is equal to dipyramid of cylinder [ ||>>0 ] == [ ||0>> ] == [ |0|>> ]. In a strictly worded sense, it has the pattern of "lathe of (square dipyramid)" == "(lathe of square) dipyramid"

|||00 : lathe of cubinder is equal to extrude of duocylinder [ ||00| ], deriving cylinder prism as cubinder, using cub- in place of cyl- for duocubinder, duocyl-prism. I associate |00||0 or ||0|0| as cubspherinder, extrude of cylspherinder |00|0

|>|>> : duotrianglindric pyramid? Going by |>|> as duotrianglinder?

||000 : lathe of duocylinder, If |000| is glominder, pairing the extrude at beginning makes a square as starting shape for duocylinder. It could be the glominder, not sure how to spin the darn thing around that has two curved surfaces into 5D. It can also be written |0|00. But, I know that |00| is not equal to ||00, so it seems that ||000 not equal to |000|.

|00|| : sphere diprism? If |0| is cylinder, |0|| is cubinder, wouldn't |00|0| or |00||0 be cubspherinder of 6D?

|>|0| : also |>||0, cyltrianglinder prism, how do you derive the name cyltriandyinder? Deriving the extrude of cylinder as cubinder, using cub- in place of cyl- for cubtrianglinder

|0>>0 : Disphone, lathe of dicone equal to dipyramid of sphere. Similar to lathe of cone [ |0>0 ] = [ |00> ], [ |0>>0 ] == [ |00>> ] == [ |>0>0 ] == [ |>>00 ], also double lathe of tetrahedron, I'm pretty sure at this point

I'm curious about these, they seem to be 6D if I understand the names correctly. It's the -dyinder suffix, looks like the extrude of something? If |>|0 is cyltrianglinder and you interpret |>|0| as cyltriandyinder, then -linder becomes -dyinder for diprisms?

duocyldyinder : ?

sphentrianglinder : ?

duotrianglindyinder : ?

contrianglinder: ?

hemoctahedrotrianglinder : ||>|>|, hemoctahedron prismic pyramidal prism?

tetrahedrotrianglinder : |>>|>|, tetrahedron prismic pyramidal prism?

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Re: Alternative Methods for the Same Goal

Postby ICN5D » Thu Nov 21, 2013 11:15 pm

@ wendy, what's this bi-circular tegum you mentioned? You mentioned the dual pyramids as tegums, as in the octahedron being the dual pyramid of the square pyramid. Is the bi-circular tegum like a duocylinder version of this? Or is it more like two circles intersecting at a right angle with the outside envelope smoothing the surface together? I think that's what you mean by skin.

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Re: Alternative Methods for the Same Goal

Postby wendy » Fri Nov 22, 2013 7:53 am

The bi-circular tegum is |O+O in your notation, where + creates a pyramid on each side of the base. In practice, it's actually the product of the line between the apexes and the surface of the base figure.
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Fri Nov 22, 2013 4:45 pm

So, if |O+ is like a double cone, that is a circle attached to two vertices, each one above or below the xy plane, then |O+O is the lathing of this double cone?
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Re: Alternative Methods for the Same Goal

Postby Keiji » Fri Nov 22, 2013 8:14 pm

ICN5D wrote:|>|> : wouldn't that be the duotrianglinder? One of the powertopes as a triangle squared?


No - linear processes cannot produce the duotrianglinder.

| = line, |> = triangle, |>| = triangular prism, |>|> is a pyramid where the base is a triangular prism.

The duotrianglinder is not a pyramid. Here is a picture of the duotrianglinder in fact: https://en.wikipedia.org/wiki/File:3-3_duoprism.png You can't pick one vertex on there such that all other vertices lie in the same realm (3D spaaace), hence it is not a pyramid.

And while yes the duotrianglinder is technically a powertope, we say it is a trivial powertope since the exponent is one of the fundamental bricks (square - the others are circle and "diamond", or square rotated 45 degrees). Usually when we are talking about powertopes, we are concerned with more elaborate ones like octagoltriates, where the exponent is an octagon.

|>|00 : lathing a cyltrianglinder into 5D. This creates a cone-like shape out of the spin of the triangular nature of |>|0. Much like lathing a triangle creates the cone, |>|0 is like a flat triangle of cylinders according to 5D. Deriving cylinder-lathe as duocylinder, adding the duo- to cyltrianglinder making duocyltrianglinder.


If that final lathe creates one (and only one) duocylinder and creates a cone-like shape, it could be the cylconinder, which is an object we already have as |0>|0.

So the final lathe turns each of the 3 cylinders into duocylinders? I do not see how that would work, as remember the squares in the triangular prism have two different types of edges - edges on one triangular face, and edges connecting opposite triangular faces. Those between opposite triangular faces can easily turn into circles, and thus the squares between them into cylinders, however those on the same triangular face cannot turn into circles, because then the triangle in the middle would turn into a cone in 3 directions at once, which makes no sense.

||>>0 : dicylindrone, lathe of square dipyramid is equal to dipyramid of cylinder [ ||>>0 ] == [ ||0>> ] == [ |0|>> ]. In a strictly worded sense, it has the pattern of "lathe of (square dipyramid)" == "(lathe of square) dipyramid"


Oh, I see. I didn't notice you had it in your list twice, was that intentional?

|||00 : lathe of cubinder is equal to extrude of duocylinder [ ||00| ], deriving cylinder prism as cubinder, using cub- in place of cyl- for duocubinder, duocyl-prism. I associate |00||0 or ||0|0| as cubspherinder, extrude of cylspherinder |00|0


Then this is the duocyldyinder.

|>|>> : duotrianglindric pyramid? Going by |>|> as duotrianglinder?


No, see above why |>|> is not the duotrianglinder.

||000 : lathe of duocylinder, If |000| is glominder, pairing the extrude at beginning makes a square as starting shape for duocylinder. It could be the glominder, not sure how to spin the darn thing around that has two curved surfaces into 5D. It can also be written |0|00. But, I know that |00| is not equal to ||00, so it seems that ||000 not equal to |000|.


Okay, then ||000 is the cylspherinder, same as |00|0. Again, was this intentionally listed twice?

|00|| : sphere diprism? If |0| is cylinder, |0|| is cubinder, wouldn't |00|0| or |00||0 be cubspherinder of 6D?


sphere diprism is the cubspherinder.

|>|0| : also |>||0, cyltrianglinder prism, how do you derive the name cyltriandyinder? Deriving the extrude of cylinder as cubinder, using cub- in place of cyl- for cubtrianglinder


cyltrianglinder prism could be thought of as "cyltrianglinder-inder", the "inder" is repeated, so di-"inder", but diinder doesn't read properly, hence i changes to y. "gl" also disappears to make it readable.

|0>>0 : Disphone, lathe of dicone equal to dipyramid of sphere. Similar to lathe of cone [ |0>0 ] = [ |00> ], [ |0>>0 ] == [ |00>> ] == [ |>0>0 ] == [ |>>00 ], also double lathe of tetrahedron, I'm pretty sure at this point


Then, again, why would you list the same object twice?

I'm curious about these, they seem to be 6D if I understand the names correctly. It's the -dyinder suffix, looks like the extrude of something? If |>|0 is cyltrianglinder and you interpret |>|0| as cyltriandyinder, then -linder becomes -dyinder for diprisms?


duocyldyinder - duocylinder is 4D, so duocyldyinder is 5D prism of duocylinder

sphentrianglinder - sphere x triangle. "re" changes to "n" for readability

duotrianglindyinder - duotrianglinder is 4D, so duotrianglindyinder is 5D prism of duotrianglinder.

contrianglinder - cone x triangle, "e" omitted for readability

hemoctahedrotrianglinder : ||>|>|, hemoctahedron prismic pyramidal prism? - No - your suggestion is 6D, the hemoctahedrotrianglinder is 5D, it is hemoctahedron x triangle, which cannot be represented in your notation

tetrahedrotrianglinder : |>>|>|, tetrahedron prismic pyramidal prism? - No - same as above, your suggestion is 6D, the tetrahedrotrianglinder is 5D, it is tetrahedron x triangle, which cannot be represented in your notation

As before, this is why you need to include products, not just linear processes if you are to enumerate tapertopes :)
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Sat Nov 23, 2013 1:32 am

I read up on similar methods that I have developed and what I use is called a Generating Set of operations. Think of the series of symbols as operations along a specific axis, starting with a point. The sequence for the cyltrianglinder being |>|0, reads as a point that has been extruded along X makes A, shape A tapered along Y makes B, shape B extruded along Z makes C, and shape C lathed into W makes D.

Here is a picture of the duotrianglinder in fact: https://en.wikipedia.org/wiki/File:3-3_duoprism.png


Ahh yes! I have thought of such things! And I do in fact have a way for describing them. The duotrianglinder is simply the triangle-prism version of the cyltrianglinder. Rather than cylinders joined into a 4-D triangle, this one has triangle-prisms joined into a 4-D triangle. Another way I see it is as a triangle prism tapering down to a triangle along the w-axis. I will learn your methods of describing shapes, but I have to translate them for now into the self-taught terms I have come to learn. In my notation it would look like this: [ |>|-|>:w ] as the perspective. I cannot describe it now with the current notation system, but it can be adapted.

It's definitely not a triangle embedded into a triangular manifold. I describe torii and other manifolds as a(b), where "a" is the sub-shape, and "b" is a modifier of the default extrusion. If "0" means to spin(lathe), then (0) means to extrude along the path of a line that has been spun, in this case a circle. I can totally see how it really is the cartesian product of two triangles. It's like the second triangle represents the shape of the prism, with a triangle embedded into this triangle. If the triangles were replaced with squares, it would be a cube tapering down to a square. A perspective of the triangle diprism [ |>|| ], looking through one of its cube sides. If you loosen the definition of vertex to include shapes higher than a point, then the duotrianglinder is a kind of pyramid. It has a triangle-prism at the base connecting to a triangle at the "vertex", so to speak. I understand that the little triangle inside is merely one of the N-2 edges, connecting the N-1 surface panels of triangle-prisms. I'm pretty sure that |>(|>) or |>(>|) wouldn't do the trick, but I'll have to investigate.


|>|00 : If that final lathe creates one (and only one) duocylinder and creates a cone-like shape, it could be the cylconinder, which is an object we already have as |0>|0. So the final lathe turns each of the 3 cylinders into duocylinders? I do not see how that would work, as remember the squares in the triangular prism have two different types of edges - edges on one triangular face, and edges connecting opposite triangular faces. Those between opposite triangular faces can easily turn into circles, and thus the squares between them into cylinders, however those on the same triangular face cannot turn into circles, because then the triangle in the middle would turn into a cone in 3 directions at once, which makes no sense.


My bad, I didn't explain that one too well. When I ascend my mind into 5-D and look "down" on a cyltrianglinder, I perceive it as a flat triangle made of cylinders. The 5-D lathing of this shape looks very identical to the 3-D lathing of a triangle into a cone. Through the same process that the 3 lines of a triangle create one line-spin (circle) and one line-torus, I imagine the 3 cylinders of |>|0 doing the same thing. So, one cylinder lathes to become a duocylinder, and the two other cyls extrude along the path of a circle and join into a torus, a cylinder torus. Lying along the same axis of spin, |>|0 has a "circle-vertex" which is paired with the lathing cylinder in its motion, and thus becomes lathed into a sphere. This is the direct result of lathing the perspective of a cylinder tapering down to a circle along W, connected by two cylinders and a triangle-torus. I believe that the |>|00 is a duocylinder tapering down to a sphere along the V-axis connected by a cylinder-torus and a cone-torus. The additional cone-torus comes from lathing the sub-shape of the initial triangle-torus in |>|0 , triangle becomes cone.

I feel that it is different from the cylconinder, where a cone was extruded into a cone-prism, then lathed into 5-D. Another perspective of the cone-prism is a cylinder tapering down to a line along W, connected by two cones and a square-torus. Lathing this shape will create a duocylinder tapering down to a circle, connected by a cylinder-torus and cone-torus. It is very similar, though! One is a 5D cone made from lathing a triangle of cylinders. The other is a 5D cylinder made from lathing a cone-prism. Does that make any sense? I can show it in my surface calculation tables if you want.


Okay, then ||000 is the cylspherinder, same as |00|0. Again, was this intentionally listed twice?


I am still not sure about this one. That's why it's listed there and potentially a duplicate. I still haven't grasped how to lathe a duocylinder yet. I lathe things differently depending on its configuration of surface panels. Whether or not it is prism, triangle, or sphere-like changes which sides join into a torus and which lathe in place. If it is sphere-like, then yes it may also be the cylspherinder, good connection. I have seen that the duocylinder has two perpendicular circle-torii on its surface. Just not sure how to spin it around.


cyltrianglinder prism could be thought of as "cyltrianglinder-inder", the "inder" is repeated, so di-"inder", but diinder doesn't read properly, hence i changes to y. "gl" also disappears to make it readable.:


Totally get the naming now. You use -dyinder in place of my -dilinder. Got it! For triprisms, would you use -tryinder? It would follow along the same lines. I notice you also use cub- to represent diprisms of n-spheres. Got it. So for triprisms of n-spheres, would you use tesser-, as in |00||| is the tesserspherinder ?

I think that the only things I listed twice were the Disphone: |00>> and the Dicylindrone: ||0>>. The other sequence of the disphone |0>>0 I wasn't sure about at the time of compiling the list. I believe they're equal now. As for the dicylindrone, a minor oversight late at night. I goofed on that one :)

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Re: Alternative Methods for the Same Goal

Postby wendy » Sat Nov 23, 2013 7:12 am

I have some trouble with the lathe/spin process, as it is not clearly defined.

When one supposes it to be a real lathe, then the axis of spin needs to have some sort of symmetry about it. You can't spin a triangle onto an inverse triangle.

You can't simply spin a cylinder. Rather one spins the cylinder against the height (|O|O) against the spin across the circle (which gives ||OO). If you suppose that the 'O' operator works on the previous constucted axis, then |O|O gives a duocylinder (where the grain runs perpendicular to the circles), and ||OO gives a spherical cylinder (as the circles are spun into spheres). This makes sense if one reads |, |O, |OO etc as line, circle, sphere, and the concatenation of these gives a prism product.
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Re: Alternative Methods for the Same Goal

Postby Keiji » Sat Nov 23, 2013 8:43 am

ICN5D wrote:cylconinder


Perhaps the problem is that what you think is the cylconinder actually isn't the cylconinder (remember the definition of cylconinder is circle x cone, regardless of how you construct that), and what you are saying is not the cylconinder actually is. That way they are still two different shapes. I haven't yet been able to figure out all the elements of a cylconinder (note the question marks on the wiki page), which doesn't help.

ICN5D wrote:I still haven't grasped how to lathe a duocylinder yet.


Okay, there are two ways to lathe a duocylinder, and they have the exact same result - one of the circles turns into a sphere (which way you lathe it only determines which circle is which). Thus lathing a duocylinder ALWAYS gives you a cylspherinder.

This is where it helps to know Garrett Jones' rotatopic notation, and the rules for extruding and lathing:

*The notation is a series of numbers, each number is a hypersphere, with 1 being a line, 2 being a circle, 3 being a sphere, 4 being a glome etc.
*Multiple numbers represent a Cartesian product of those hyperspheres.
*When you extrude an object, you insert a '1' into the series of numbers.
*When you lathe an object, you pick one of the numbers and add one to that number.
*It does not matter which order you write the numbers in the series.

So you get:

1 - line

11 - square (line extruded - inserts '1' into the series)
2 - circle (line lathed - adds one to the only number in "1")

111 - cube (square extruded - inserts '1' into the series)
12 or 21 - cylinder (line lathed - adds one to one of the '1's - or circle extruded - inserts '1' into the series)
3 - sphere (circle lathed - adds one to the only number in "2")

1111 - tesseract (cube extruded)
112 or 121 or 211 - cubinder (cube lathed, or cylinder extruded)
22 - duocylinder (cylinder lathed by adding one to the '1')
31 or 13 - spherinder (sphere extruded, or cylinder lathed by adding one to the '2')
4 - glome (sphere lathed)

It therefore follows that if you lathe a duocylinder, you get either 23 or 32, which are the same thing, and are circle x sphere (or sphere x circle, which means the same), which is the definition of a cylspherinder.
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Sat Nov 23, 2013 6:39 pm

A cylconinder is the Cartesian product of a circle and a cone.

One of its tera is a duocylinder and the other is some curved 4-surface. It also includes one circular face. However, its other elements are currently unknown.


Taken right off the Wiki. Well, what you have derived is very identical to what I have found. You see the duocylinder and the circle, just like I have. So, let's just say that I'm proposing a theory that the surface elements of a cylconinder are:

[ |>0|0 ] == [ ||0(0)xy , |>0(0)zw , ||00-|0v ]

[ CYLCONINDER ] == [ CYLINDER TORUSxy , CONE TORUSzw , DUOCYLINDER--CIRCLEv ]

Now, I don't know calculus, but I'll bet some of you do! I believe that by adding the hypervolume bulk of a cone torus, cylinder torus, and a duocylinder together, one can derive the surteron hypervolume of a cylconinder this way. Another is the long way by lathing a cone-prism with calculus, by establishing the lathe as the grain being perpendicular to the cone ends of the cone-prism. This way will be the true proof of concept and is outside my academic capabilities. It would be awesome if someone could do that! I'm willing to bet that both methods will be equal.

I can't seem to find the cylinder-torus on the wiki, it may be the toracubinder but I'm not sure. Also can't find the cone-torus. I wanted to add the formulas for the bulk of these to the duocylinder.

If you suppose that the 'O' operator works on the previous constucted axis, then |O|O gives a duocylinder (where the grain runs perpendicular to the circles), and ||OO gives a spherical cylinder (as the circles are spun into spheres).


I did notice this some years ago with the cylinder. One of the projections is a circle within a circle connected by the line-torus ( what is the real name for this? ). By keeping this projection "flat" and lathing it like a circle, then yes it will indeed produce the spherinder. The two circles become spheres and the line-torus becomes the glomohedrix-prism connecting the two sphere ends. At the time, I didn't think this was a valid way to lathe the cylinder and dismissed it. However, if you are mentioning it Wendy, then perhaps I need to include it from now on.

The difference between |0|0 and ||00 should now be associated by the pairing of the two lathe operators, and how this ultimately determines the outcome. If they are paired, then it should follow along the same lines as the n-spheres |0, |00, |000, etc. This also agrees with the lathe of the duocylinder, the two lathe operators paired up lock it in to being associated with a sphere. ||000 , |0|00 , and |00|0 would all be the same. If ||00 and |00| are the spherinder, then lathing perpendicular to the sphere ends will produce the cylspherinder.

****I've been doing some thinking about the |>|00, and it may be the triangular configuration of spherinders, just like how |>|0 is a triangular config of cylinders. |>|00 could be square{triangle,sphere}. What do you think? That means this is the sphentrianglinder. A triangle of three spherinders mutually attached by a glomohedrix prism pyramid.

What are the real names for these?
*line-torus on cylinder?
*line-torus on cone?
*1-D surface of a circle? glomogonix ?
*3-d surface of a glome? glomochorix / glomotopix ?
*4-D surface of a pentasphere? glomoterix ?
|0|0|0, (circle x circle x circle), triocylinder?

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Definition of the Lathe

Postby ICN5D » Sat Nov 23, 2013 9:54 pm

Go figure, not more than an hour away from the keyboard and I come up with something new. Wendy, you asked me to give you a good definition of the lathe and here it is:

In reality I suppose there are two ways to lathe:

- Hold a shape in place and rotate into N+1 around an axis that divides the shape in half : "lathe"

- Hold a surface element in place ( most likely one on the previous constructed axis ) and rotate into N+1 : "flip"


The "flip" can create some tegum products, as in the flip of a triangle creates a bicone, the tegum product of a circle and digon.

The part I want to focus on is the lathe around the center. During an N+1 lathe, an N-dimensional shape can be cut in half by an N-1 dimensional shape ( or surface element for that matter ). The dimensionality of this dividing shape is what determines the stationary axis/axes. The stationary axes I call POLAR, the moving axes are EQUATORIAL.

- 1D cut by point, none stationary
- 2D cut by line, one stationary
- 3D cut by square/circle/triangle, two stationary
- 4D cut by cube/sphere/cyl/etc, three stationary
- 5D cut by geochoron/glome/etc, four stationary
- 6D cut by geoteron/pentasphere/etc, five stationary


Note the pattern: Lathing an N-D shape allows N-1 axes to remain stationary. This means in each case, only one axis is moving around, tracing out the disk of a circle. The pair of surface elements on this moving axis join together into a torus, while the stationaries lathe in place, in the typical manner.

Consider the lathings of the hypercubes:

| : has 1 point pair, 1 axis
|0 : has one dot-torus // zero stationary, one moving axis

|| : 2 line pairs, 2 axes
||0 : 1 circle pair, 1 line-torus // one stationary, one moving

||| : 3 square pairs, 3 axes
|||0 : 2 cylinder pairs, 1 square-torus // two stationary, one moving

|||| : 4 cube pairs, 4 axes
||||0 : 3 cubinder pairs, 1 cube-torus // three stationary, one moving

||||| : 5 geochoron pairs, 5 axes
|||||0 : 4 tesserinder pairs, 1 geochoron-torus // four stationary, one moving

|||||| : 6 geoteron pairs, 6 axes
||||||0 : 5 penterinder pairs, 1 geoteron-torus // five stationary, one moving


Within the lathe operation are four sub-types: prismic, triangular, spherical, and toric. They determine which axis is the moving axis based on the general configuration of surface panel pairs. The toric lathe is a special case of the spherical, where a spherical lathe is applied to the manifold of the torus as well as the n-sphere ends.

For lathing an N-D shape:

Prismic Lathe 0|: Last constructed axis in motion / Equatorial

Triangular Lathe 0>: Second-last constructed axis is in motion / Equatorial

Spherical Lathe 00: One axis stationary / Polar

Toric Lathe 0m: Axes parallel to the torus manifold in motion / Equatorial



I consider hypercubes and hyperspheres to have the maximum amount of symmetry. Changing which axes are moving or stationary won't affect the outcome. However, if there is any kind of break in the symmetry, the outcome is different. Just like alternating the moving/stationary axes during a cylinder lathe will alternate between a duocylinder and a spherinder.

-Philip
Last edited by ICN5D on Sun Nov 24, 2013 5:18 am, edited 1 time in total.
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Sat Nov 23, 2013 11:29 pm

Okay, so I mentioned how I didn't think that |>(|>) would create the duotrianglinder. But it actually does in fact. If Q(0) means to embed shape Q into the 1-manifold of a circle, then Q(|>) means to embed shape Q into the 2-manifold of a triangle. (|>) has two symbols for the 2D manifold, |>(|>) has four symbols for the 4 dimensions of a duotrianglinder.

sphentrianglinder : |00(|>)

duotrianglindyinder : |>(|>|) or |>(|>)|
, I'm not sure here could be equal

contrianglinder: |>0(|>)

hemoctahedrotrianglinder : ||>(|>)

tetrahedrotrianglinder : |>>(|>)


It should be noted that:
||(|>) == |>||, triangle diprism
|0(|>) == |>|0, cyltrianglinder
|0(|0) == |0|0, duocylinder
|>(|>) =/= |>|>, two are different
|(|>) == |>|, triangle prism
|(|0) == ||0, cylinder


If the |>|0 has a triangular lathe applied, it produces the strange shape with the duocyl tapering to a sphere. However, the toric lathe of |>|0 will create the sphentrianglinder. That is, |>|00m == |00(|>)

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Re: Alternative Methods for the Same Goal

Postby wendy » Sun Nov 24, 2013 8:38 am

Let's look at toruses and kindred products.

The actual description of the 'tiger' as a product is a Comb{ circle ø circle ø circle}. The torus in 3d is comb{circle ø circle}.

There is a way of generating toruses in a linear way like Philip does, but that does not lead to the Comb product. It does understand linkages, so we shall explore it.

A one-dimensional torus consists of two disjoint lines, like ---AAA----AAA----. If one supposes that the parts are held at the same distance, one can make a chain, like --AA-------BB--AA-------BB--. These are held to the line and so can't be uncoupled: they're linked.

One expands a link into the next dimension in two different ways. A link AA---aa is made into two circles, of diameter AA and of aa (say lx). Alternately, you can make an arch, so the circle-diameters are AA and aa still, but the link is Aa---aA (say ld). You get nested circles, or an anulus.

In three dimensions, the the 1d link can make lxx and ldd (which forms the familiar link-and-ball chain one sees holding pens, the link implements two small spheres which are at fixed distance, and the ball has a little hole that allows the link, but not the knob to pass through. The other kind of chain is lxd + ldx, which is a link of hollow circles.

In 4d, you have lxxx + lddd and lxxd + lddx, only the second chain is made of solid links: the first is a chain of bars and glomes as the pen-chain in 3d. And so it goes.

The thing to notice here is that when you add the l's and d's together, for any given link, you get the dimension of hollow link. This is the dimension of the cloth needed to span the hole. So an ordinary hollow circle (ld) is 2d, a hollow sphere (ldd) is 3d and so forth.

THE COMB PRODUCT

The comb product is the 'repetition of surface'. Its regular figure is the series of 'number-line', 'square tiling', 'cubic tiling', etc, which, while are usually seen as tilings, can be seen as a surface of an infinite polytope, covering half-space. When seen this way, the 'number-line' (horogon), comes to be the x-axis and all values below it. It's a very big polygon. We multiply it by an other big polygon, to get a polyhedron.

Unlike the brick products (tegum, prism, crind), the comb product is a pondering one: it reduces the product by one dimension on each use. So our expression for the tiger as (circle ø circle ø circle), a brick product gives a 6-d figure here, the comb-product reduces it by two dimensions to four.

"Repetition"of "surface", means "cartesian product" of "net". Here, 'net' is what you might cut out of a stiff peice of cardboard to fold up to make a cube or tetrahedron. It's the unfolded surface of the figure. The net of a circle is a straight line, which you join end to end.

In 3d, there is a product of CB{circle ø circle} = torus. You see that the net or unfolded surface of a circle is a line. The cartesian product of two lines is a rectangle. Let's suppose it's lying in the z=0 plane. You now roll the x-axis up to make it a circle. This means that it crosses the x=0,y=0 at two points (the diameter of the circle). You have now a cylinder, which is the cartesian product of the maded-up surface of the circle xz, and the net of the circle yz. When we close the ends of the cylinder yz, we get a torus surface, standing so that the x=0,y=0 line crosses it four times. The surface is now the 'cartesian product' of the two elements, and the volume is ready to be filled. Note here that we used the z axis for all figures, this is the axis of pondering. They do not all have to align, just that an axis has to be shared between the two elements.

You can replace 'circle' by 'polygon' at any point here.

In 4d, you have the 'tiger' as the comb product of {circle ø circle ø circle} Let's make the tiger from the cartesian product of the nets. These are three lines, we can set them however long we need. Make one 1 inch, and the other two 1 foot. As before, the thing lies in the plane z=0, with the one-inch in w=0. The tiger is the 'spherated bi-glomohedric prism'.

The first step is to make a thin roll in the w-z plane. We roll the thing up so we get a 1ft * 1 ft thing like a roll-up ciggie. (This little thin cylinder-square thing is rather like a 'spherated square', where we replace each point of a square with an orthogonal circle.)

The second step is to make one of the glomohedrixes (or hollow circles). This is done in the x-z space, involves bending the previous circle so that the x-ends of the square meet, and we get a cylinder, three times longer than it is round. The figure here is a cartesian product of the little w-z circle, and the x-z glomohedrix, and y-z is still the net of the other glomohedrix.

The final step is the same as the previous one, except that y-z axis is rolled up. We have made all three circles on the z axis, which means that the otherwise six dimensions of three circles, is reduced to four, by having three share the one z-axis.
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Help with Cylconinder

Postby ICN5D » Sun Nov 24, 2013 8:51 am

I haven't yet been able to figure out all the elements of a cylconinder (note the question marks on the wiki page), which doesn't help.


I can help you with that. It's not just one curved 4-surface you're looking for, it's 2 of them. The cylconinder has two torii connecting the duocylinder to the circle-vertex. This is because the duocylinder at the base has two perpendicular circle torii on its surface. The torus-cell perpendicular to the circle connects by a cone-torus. The torus-cell parallel to the circle connects by a cylinder torus.

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Re: Alternative Methods for the Same Goal

Postby Keiji » Sun Nov 24, 2013 12:16 pm

Thanks for that :)

One advantage of your colouring notation is that it's helped me to look at the cone analogously to the triangle. Knowing what the elements of the cyltrianglinder are and how they arise, I've been able to enumerate the elements of the cylconinder in the same way.

There are two 3D torii, which are cells of the duocylinder at the base. One of those torii connects to the opposite circle by means of a curved cone-torus teron, and the other connects connects by the Cartesian product of a disc and the curved surface of a 3D cone. This is basically what you said, but the way you phrased it was confusing to me until I worked it out myself.
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Re: Alternative Methods for the Same Goal

Postby ICN5D » Sun Nov 24, 2013 8:31 pm

@ Keiji:

Awesome! I'm glad I could help. I see that you've made some new additions to the wiki, most notably the cylconinder, cylhemoctahedrinder, and the cyltetrahedrinder. I love this forum and it's really cool that that came from me! The other curved tera for the ||>|0 and |>>|0 are:

||>|0 : 1x sq-pyr torus, bounding the 1x cubinder and 4x cyltrianglinders
Flunic cross sections are: cubinder tapering to circle, cyltrianglinder tapering to cylinder, and sq-pyr expanding into sq-pyr prism, then collapsing back into sq-pyr

|>>|0 : 1x tetrahedron torus, bounding the 4x cyltrianglinders
Flunic cross sections are: cyltrianglinder tapering to circle, tetrahedron expanding into tetrahedrinder, then collapsing back into tetrahedron.

A curious effect I've seen in cartesian products with circles:

{Shape-N x circle} == lathe of N-Prism: This always creates a torus of N bound by the cartesian products of a circle and the surface elements of shape-N.

Cross sections through a surface element: Always the cartesian products of a circle and the cross sections of shape-N

Cross sections through the torus(es): Always Shape-N expands into N-Prism, then collapses back into Shape-N

Contact patch of curved surface: Always Shape-N

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Re: Alternative Methods for the Same Goal

Postby wendy » Mon Nov 25, 2013 10:24 am

I answered Keiji and Philip's questions elsewhere: viewtopic.php?f=31&t=1823&p=19289#p19289

In essence, you can use a comb product with a solid element, then that solid element is going to be interior to the resulting comb. The order is then irrelevant, apparently.

For Philip's question. Glomo-hedr-ix is my name, as described in the polygloss http://os2fan2.com/glossn/ptbridge.html as being a 'round 2d cloth'. Cloths are usually things by themselves, while patches (glomohedron = round 2d patch) do something (here hold volume). So the first is a sphere, the second a ball.
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Re: Alternative Methods for the Same Goal

Postby quickfur » Wed Dec 18, 2013 4:44 pm

ICN5D wrote:@ wendy, what's this bi-circular tegum you mentioned? You mentioned the dual pyramids as tegums, as in the octahedron being the dual pyramid of the square pyramid. Is the bi-circular tegum like a duocylinder version of this? Or is it more like two circles intersecting at a right angle with the outside envelope smoothing the surface together? I think that's what you mean by skin.

-Philip

The bi-circular tegum is sorta like a "dual" of the duocylinder (in a crude sense of the word). Basically, where you have two toroidal 3-manifolds bounding the duocylinder, you replace with two circles, so you have two circles in orthogonal planes (e.g., in the XY and ZW planes, both circles centered on the origin), then you take the convex hull of that. So where you have the toroidal 2-manifold of the duocylinder's ridge (i.e., where the two bounding surfaces meet), you now have a kind of triangular-torus type 3-manifold that now forms the single surface of the bicircular tegum.

Just as the duocylinder may be thought of as the limiting shape of n,n-duoprisms as n approaches infinity, so the bicircular tegum can be thought of as the limiting shape of the n-gonal-n-gonal tegums as n approaches infinity. Just as the n,n-duoprisms are dual to the n,n-tegums, so also the duocylinder may be thought of as being "dual" to the bicircular tegum.

Another way to think about it is, the duocylinder is the Cartesian product of two circles, whereas the bicircular tegum is the convex hull of two circles (lying in orthogonal planes).

I understand 4D mainly via projections into 3D, so I like to think of 4D objects that way. You can see some renderings I made for the duocylinder here. The bicircular tegum I haven't had a chance to model and render yet, but one possible projection has an envelope in the shape of two cones joined base-to-base, with the circle around the base being the image of one of the tegum's circles, and the other circle projecting to a single line connecting the tips of the two cones, passing through the center of the projection. In this 4D viewpoint, you can quite easily see the triangular-torus shaped manifold that forms the tegum's surface. It's something like a torus with triangular cross-sections and a "hole" of radius 0. (Of course, the thing is curved into 4D, so should not be misconstrued as a 3D construct.)
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Re: Alternative Methods for the Same Goal

Postby wendy » Mon Dec 23, 2013 9:10 am

It's (the tegum) not just 'sort of the dual of' the duocylinder, it is the dual. Sections along a circle, would show, a circle-tegum (cone on each side of a circle), grow and contract in the sine wave.

The equation for the surface of the canonical coordinates are: 1 = sqrt(w^2+x^2)+sqrt(y^2+z^2).

Sections in other directions present elude me. I imagine the diagonal through xy, might run from a line, to a crind-form rhomb.

It's volume and surface area are easy to find from coherent products. L^2 P2 = square L, L^3 P3 = cubic L, L^4 P4 = tesseractic or biquadrate L)

Area of circle = (pi/4). d^2 C2, or (pi/2) d^2. T2. Volume of tegum is (pi^2/4) a^2 b^2 T4, or (pi^2/96) a^2 b^2 P4 or 1/3 a^2 b^2 C4. It's as big as 1/3 of the enclosing sphere. The surface area comes from the in-diameter is sqrt(2), if the diameter is 2. So you get

(pi^2/4) a^4 T4 divide by a/sqrt(2), gives (pi^2 sqrt(2)/4) a^3 T3, or (pi^2 sqrt(2)/24) a^3 O3
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