Beyond the Fifth Dimension

Higher-dimensional geometry (previously "Polyshapes").

Beyond the Fifth Dimension

Postby ICN5D » Wed Nov 27, 2013 6:33 am

So, yeah, 5D is cool and all, but I'm interested in some of the bigger monsters out there. I don't expect the wiki to detail ALL the shapes, that would be a very long process. At least, not until regular patterns can be extrapolated and plugged into a computer program. Then the sky's the limit!


The bi-cyltrianglinder prism , square{cyltrianglinder,cyltrianglinder}, an eight dimensional monster unveiled

Also made by:

* cube{triangle-prism,cone,circle}


Cartesian Product: |>|O(|>|O) , |>|(|O>)(|O)


C{ |>|Oxyzw , (|>|Ovuts) } == |>|O(|>|O)
-----------------------------------------
C{ |>(O) , |>|O }xy == [ |>||>O(O) ]xy
C{ ||O-2 , |>|O }z == [ |>|||OO-2 ]z
C{ ||O-|O , |>|O }w == [ |>|||OO-|>||OO ]w
C{ |>|O , |>(O) }vu == [ |>||>O(O) ]vu
C{ |>|O , ||O-2 }t == [ |>|||OO-2 ]t
C{ |>|O , ||O-|O }s == [ |>|||OO-|>||OO ]s



The features that we can derive out of this are:

* 6 flat |>|||OO side panels
--- In 2 groups of 3, along perpendicular planes zw and ts

* 2 |>||>0 torii connecting curved surfaces of flat panels

* 6 triangular attachments, at a |>||OO for a vertex

* 2 curved rolling surfaces, bisected by planes xy and vu

* Rolls along 2 perpendicular directions

* A |>||>O for a contact patch resting on a 7-plane

* Net of 6 |>|||OO , bound to 2 |>||>O prisms

* Duocylinder-like rolling capabilities combined with 2 bisecting perpendicular triangles. So, it could be called, in a ways, a duocylindric duotrianglinder.


Cross sections are:

* A |>|||OO scaling down to a |>||OO through a flat side

* A |>||>O expanding into a |>||>O-prism, then collapsing back into a |>||>O through either rolling surface


Hypervolumes:

* 7-surface Bulk: 6(|>|||OO ) + 2(|>||>O(O)) 7-volumes

* 8-surface Bulk: yeah right, that way beyond me!
Last edited by ICN5D on Tue Dec 03, 2013 12:16 am, edited 1 time in total.
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The Conindric-Trianglinder of the 6th Dimension

Postby ICN5D » Sat Nov 30, 2013 1:20 am

The Conindric-Trianglinder: triangle-prism*cone, cone*triangle-prism, coninder*triangle

* Surface elements of a triangle-prism, |>| : [ ||^2x , ||--|y , |>-2z ] and [ ||^2x , |>-2y , ||--|z ], 3 squares, 2 triangles

* Surface elements of a Cone, |O> , |>O : [ |z(O)xy , |O--*z ], 1 circle, 1 line-torus ( glomolatrix-prism)

Cartesian Product: |>|(|O>) , cross multiply the two shapes and their surface elements together. Symbols in parentheses are the surface element manifolds, making them the final linear operations added to other shape:

C{ |>|xyz , |O>wvu } == [ |>|(|O>) ]xyzwvu
--------------------------------------------------------------------------
C{ (||^2) , |O> }X == [ |O>||^2 ]:X
C{ (||--|) , |O> }Y == [ |O>||--|O>| ]:Y
C{ (|>-2) , |O> } Z == [ |O>(|>)-2 ]:Z
C{ |>| , (|(O)) }WV == [ |>||(O) ]:WV
C{ |>| , (|O--*) }U == [ |>||O--|>| ]:U


So, we get [ |>|(|O>) ] == [ |O>||^2x , |O>||--|O>|y , |O>(|>)-2z , |>||(O)wv , |>||O--|>|u ]

* 3 conic diprisms, 2 contrianglinders, 1 cyltriandyinder, and 1 triangle diprism torus.

The |>|(|O>) has 5 perspectives to view the shape from. Of those 5, two are unique and three are identical.

* The 3 conic-diprisms, |O>|| are attached into a triangle parallel to the XY plane
---- Each attach by a cone-prism |O>|
------Tapering a conic-diprism down to a cone-prism also makes a Conindric-Trianglinder

* The 2 contrianglinders, |O>(|>) are attached into a perfect prism along Z, and is another way to make the Conindric-Trianglinder
---- This extrusion making the contrianglinder prism, |O>(|>|) is the cartesian dual of the original sequence |>|(|O>)

* The 1 cyltriandyinder, |>||O tapers down to a triangle prism, |>| along U, and is another way to make the Conindric-Trianglinder
---- This is the coninder-like perspective, a curved shape tapers to an uncurved hyper-vertex, in this case a triangle-prism

* All 6 flat side panels are joined by one triangle-diprism-torus, |>||(O)

* Makes a triangle-diprism |>|| contact patch when placed on its rolling surface


CROSS SECTIONS along a 5-PLANE:

* A conic-diprism shrinking down to a cone-prism, |O>||---> |O>|

* A contrianglinder of unchanging size, |O>|(>) ---> |O>(|>)

* A cyltriandyinder shrinking down to a triangle-prism, |>||O ---> |>|

* A triangle-diprism expands into a triangle-triprism, then collapses back into a triangle-diprism, |>|| ---> |>||| ---> |>||
Last edited by ICN5D on Tue Dec 03, 2013 2:10 am, edited 2 times in total.
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Perspectives of the Conindric Trianglinder

Postby ICN5D » Tue Dec 03, 2013 1:59 am

|>|(|O>) == [ |O>||^2 , |O>||--|O>| , |O>(|>)--2 , |>||(O) , |>||O--|>| ]

The 3 perspectives of |>|(|O>) :

* [ |O>||--|O>| ]:U , Cone-Diprism scales down to Cone-Prism

* [ |O>(|>)--2 ]:U , Contrianglinder extrudes to Contrianglinder

* [ |>||O--|>| ]:U , Cyltriandyinder scales down to Triangle-Prism



[ |O>||--|O>| ]:U

P[ |O>|| ]--U--[ |O>| ] ===== [ |>|(|O>) ]
--------------------------------------------------------
P[ |||(O) ]-----[ ||(O) ]:XY == [ |>||(O) ]:xy
P[ |||O--|| ]--[ ||O--| ]:Z == [ |>||O--|>| ]:z
P[ |O>|--2 ]--[ |O>--2 ]:W == [ |O>(|>)--2 ]:w
P[ |O>|--2 ]-----[ |O>| ]:V == [ |O>||^2 ]:v
-------------------------------------[ |O>||--|O>| ]:u





[ |O>(|>)--2 ]:U

P[ |O>(|>) ]--U---[ |O>(|>) ] ==== [ |O>(|>|) ] , [ |O>|(|>) ]
---------------------------------------------------------------------
P[ |>|(O) ]---------[ |>|(O) ]:XY == [ |>||(O) ]:xy
P[ |>|O--|> ]-----[ |>|O--|> ]:Z == [ |>||O--|>| ]:z
P[ |O>|^2 ]--------[ |O>|^2 ]:W == [ |O>||^2 ]:w
P[ |O>|--|O> ]--[ |O>|--|O> ]:V ==[ |O>||--|O>| ]:v
-------------------------------------------[ |O>(|>)--2 ]:u





[ |>||O--|>| ]:U

P[ |>||O ]----U----[ |>| ] ==== [ |>|(|O>) ]
--------------------------------------------------------
P[ |||O^2 ]-------[ ||^2 ]:X == [ |O>||^2 ]:x
P[ |||O--||O ]---[ ||--| ]:Y == [ |O>||--|O>| ]:y
P[ |>|O--2 ]------[ |>--2 ]:Z == [ |O>(|>)--2 ]:z
P[ |>|(O) ]-------[ |>| ]:WV == [ |>||(O) ]:wv
--------------------------------------[ |>||O--|>| ]:u



In all 3 viewing angles, we see how each surface element of the near shape connects to those of the far shape, producing all of the same elements found on the surface of the Conindric Trianglinder.

-Philip
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