Cool tricks for deriving surface elements

Higher-dimensional geometry (previously "Polyshapes").

Cool tricks for deriving surface elements

Postby ICN5D » Tue Nov 26, 2013 7:19 am

Using a series of linear operations on a particular axis(es), this is a way to derive the surface elements of a shape in any dimension. Based on a simplified version of Calculus, this algorithm works directly with the surfaces of certain polytopes. Cartesian products, Tegum Products, Linear Operations, Bi-linear Operations, and Perspective Products can be used in this systematic way.

An example of the Cylconinder derived three different ways:

The Cartesian Product is a cross multiplication of the elements between the two shapes. It follows manifold embedding rules.

C[|0(|0>)]

C{ |0xy , (|0>)zwv } == |0>|0xyzwv
===================================
C{ (0)xy , |0>zwv } == [|0>zwv(0)xy]
C{ |0xy , (|v(0)zw) } == [|0|xyv(0)zw]
C{ |0xy , (|0zw--*) } == [|0|0xyzw--|0xy]


C[circle(cone)]

C{ circle-xy , (cone-zwv manifold) } == cylconinder-xyzwv
=========================================================
C{ [dot]-torus-xy , cone-zwv } == [cone-zwv]-torus-xy
C{ circle-xy , ([line-v]-torus-zw) }== [cyl-xyv]-torus-zw
C{ circle-xy , (circle-zw--dot) } == duocyl-xyzw--circle-xy



The Linear Operation is a transforming motion applied to the surface elements of a base shape into the N+1 dimension, deriving the N+1 surface elements. The Bi-linear is an N+2 transforming motion that uses N+2 rules in a cartesian product layout.

Linear Operation

L[0w,v]:[|>0|]

L{ |>0|xyzw , [0w] } == [ |>0|0xyzwv ]
======================================
L{ ||zw(0)-xy , 0 }XY == [ ||0zwv(0)xy ]
L{ |0|xyw--|w , 0 }Z == [ |0|0xywv--|0wv ]
L{ |0>--2xyz , (0) }W == [ |0>xyz(0)wv ]


[lathing along W , intoV]: of the [coninder]

L{ coninder-xyzw , [lathe W as EQ] } == cylconinder-xyzwv
==========================================================
L{ [sqr-zw]-torus-xy , [lathe] }XY == [cyl-zwv]-torus-xy
L{ cyl-xyw--line-w , [lathe] }Z == duocyl-xywv--circle-wv
L{ cone--2-xyz , [torus] }W == [cone-xyv]-torus-wv



The Perspective Product relies on the coloring notation. It shows how to rotate the shape around and view it from different surface element pairs, deriving the connecting elements between them. It follows element joining rules.

[|0|0--|0]:v

P{ |0|0xyzw , |0xy } == [ |>0|0xyzwv ]
=====================================
P{ |0zw(0)xy , (0)xy } == [ |0>zwv(0)xy ]
P{ |0xy(0)zw , |0xy } === [ |0|xyv(0)zw ]
________________________[ |0|0xyzw--|0xy ]:v


[duocyl tapers=down=to circle : along V]

P{ duocyl-xyzw , circle-xy } =============== cylconinder-xyzwv
=================================================================
P{ [circle-zw]-torus-xy , [dot]-torus-xy } == [cone-zwv]-torus-xy
P{ [circle-xy]-torus-zw , circle-xy } ======= [cyl-xyv]-torus-zw
________________________________________ [ duocyl-xyzw--circle-xy ]: paired along v


I'll detail the rest when I get a chance...

-Philip
Last edited by ICN5D on Wed Nov 27, 2013 4:30 am, edited 1 time in total.
in search of combinatorial objects of finite extent
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Re: Cool tricks for deriving surface elements

Postby wendy » Tue Nov 26, 2013 10:23 am

The surface elements of the prism and comb products, are the prism-products of the elements. For the tegum and pyramid product, they're the pyramid product. Consider this table for the pentagon-line products.

Code: Select all
                        Prism                                    Tegum

                    (line)      2 vertex                       2 vertices      (1 null)

  (pentagon)        (pent prism )  2 pentagons          5 edg   2 vert^5 edg     null^ 5 edg
                                                               = 10 triang     =  5  edg

   5 edges        5 line*line    5 line*2 point      5 vert   2 vert^5vert    nul ^ 5 vert
                   = 5 squares    = 10 lines                    = 10 edges      = 5 vert

  5 vertex       5 vertex*line   5 point*2 point    (1 null)  2 vert^ null    null ^ null
                   = 5 lines      = 10 vertices                 = 2 vert       = 1 (null)



The other two products contain a similar process, but include different elements. The effect of including the null element in the tegum, is that the original pentagon and line surfaces show through, but the bases are entirely inside the tegum. Likewise, the effect of including the body means that the body of the factors is part of the surface.
The dream you dream alone is only a dream
the dream we dream together is reality.
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Re: Cool tricks for deriving surface elements

Postby ICN5D » Thu Jan 30, 2014 3:35 am

X
| - LINE
--------------------------------
XY
|0 - CIRCLE
|> - TRIANGLE
|| - SQUARE
-----------------------------------------------------------
XYZ
|00 - SPHERE
|0> - CONE
||0 - CYLINDER
|>> - TETRAHEDRON
|>| - TRIANGLE PRISM
||> - SQUARE PYRAMID / HEMOCTAHEDRON
||| - CUBE
---------------------------------------------------------------
XYZW
|000 - GLOME
|00> - SPHONE
|00| - SPHERINDER
|0>> - DICONE
|0>| - CONINDER
||00 - DUOCYLINDER
||0> - CYLINDRONE
|>>> - PENTACHORON
|>>| - TETRAHEDRINDER
|>|0 - CYLTRIANGLINDER
|>|> - TRIANGLE PRISM PYRAMID
|>|| - TRIANGLE DIPRISM
||>> - DIPYRAMID
||>| - PYRAMID PRISM
|||0 - CUBINDER
|||> - HEMDODECACHORON
|||| - TESSERACT
---------------------------------------------------------------
XYZWV
|0000 - PENTASPHERE
|000> - GLONE
|000| - GLOMINDER
|00>> - DISPHONE
|00>| - SPHONINDER
|00|0 - CYLSPHERINDER
|00|> - SPHERINDRONE
|00|| - CUBSPHERINDER
|0>>> - TRICONE
|0>>| - DICONINDER
|0>|0 - CYLCONINDER
|0>|> - CONINDER PYRAMID
|0>|| - CONE DIPRISM
||00> - DUOCYLINDRONE
||0>> - DICYLINDRONE
||0>| - CYLINDRONE PRISM
|>>>> - HEXATERON
|>>>| - PENTACHORINDER
|>>|0 - CYLTETRAHEDRINDER
|>>|> - TETRAHEDRINDER PYRAMID
|>>|| - TETRAHEDRON DIPRISM
|>|00 - DUOCYLTRIANGLINDER
|>|0> - CYLTRIANGLINDRONE
|>|0| - CYLTRIANDYINDER
|>|>> - TRIANGLE PRISM DIPYRAMID
|>|>| - TRIANGLIE PRISM PYRAMID PRISM
|>||> - TRIANGLE DIPRISM PYRAMID
|>||| - TRIANGLE TRIPRISM
||>>> - SQUARE-TRIPYRAMID
||>>| - SQUARE DIPYRAMID PRISM
||>|0 - CYLHEMOCTAHEDRINDER
||>|> - SQUARE PYRAMID PRISM PYRAMID
||>|| - SQUARE PYRAMID DIPRISM
|0|0| - DUOCYLDYINDER
|||0> - CUBINDRONE
|||>> - CUBE DIPYRAMID
|||>| - CUBE PYRAMID PRISM
||||0 - TESSERINDER
||||> - TESSERACT PYRAMID
||||| - PENTERACT


NAME - [ SEQUENCE ]

[ N-BASE ] ---Q+1 AXIS---> [ OPERATE ] = [ PRODUCT ]
-----------------------------------------------------------------
[ N-1 PANEL:Q ] ----------> [ OPERATE ] = [ N+1 SURFACE PANEL:Q ]
[ N-1 PANEL:Q ] ----------> [ OPERATE ] = [ N+1 SURFACE PANEL:Q ]
[ N-1 PANEL:Q ] ----------> [ OPERATE ] = [ N+1 SURFACE PANEL:Q ]
----------------------------------------- [ BASE-OPERATE :Q+1 ]


2-D SHAPES

----------------------------

CIRCLE - [ |0 ]

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



TRIANGLE - [ |> ]

[ | ] --y--> [ * ] == [ |> ]
-------------------------------
[ *2:x ] --> [ * ] == [ |2:x ]
----------------------[ |-*:y ]




SQUARE - [ || ]

[ | ] ---y--> [ | ] == [ || ]
--------------------------------
[ *2:x ] --> [ *2 ] == [ |2:x ]
---------------------- [ |2:y ]







3-D


SPHERE - [ |00 ]

[ |0 ] --z---> [ 0xy ] == [ |00 ]
--------------------------------------
[ *(0):xy ] --> [ 0 ] == [ *(00):xyz ]



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| ]

[ |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 ]



TETRAHEDRON - [ |>> ]

[ |> ] --z--> [ * ] == [ |>> ]
----------------------------------
[ |2:x ] ---> [ * ] == [ |>2:x ]
[ |-*:y ] --> [ * ] == [ |>-*:y ]
---------------------- [ |>-*:z ]



TRIANGLE PRISM -i - [ |>| ] , [ |>2:z ]

[ |> ] --z--> [ |> ] == [ |>| ]
-----------------------------------
[ |2:x ] ---> [ |2 ] == [ ||2:x ]
[ |-*:y ] -> [ |-* ] == [ ||-|:y ]
----------------------- [ |>2:z ]



TRIANGLE PRISM -ii - [ |>| ] , [ ||-|:z ]

[ || ] --z--> [ | ] == [ |>| ]
---------------------------------
[ |2:x ] ---> [ * ] == [ |>2:x ]
[ |2:y ] ---> [ | ] == [ ||2:y ]
---------------------- [ ||-|:z ]



PYRAMID-i - [ ||> ] , [ ||-*:z ]

[ || ] --z--> [ * ] == [ ||> ]
--------------------------------
[ |2:x ] ---> [ * ] == [ |>2:x ]
[ |2:y ] ---> [ * ] == [ |>2:y ]
---------------------- [ ||-*:z ]


PYRAMID-ii - [ ||> ] , [|>-|:z]

[ |> ] --z--> [ | ] == [ ||> ]
----------------------------------
[ |2:x ] ---> [ * ] == [ |>2:x ]
[ |-*:y ] --> [ | ] == [ ||-|>:y ]
---------------------- [ |>-|:z ]



CUBE - [ ||| ]

[ || ] --z-> [ || ] == [ ||| ]
---------------------------------
[ |2:x ] --> [ |2 ] == [ ||2:x ]
[ |2:y ] --> [ |2 ] == [ ||2:y ]
---------------------- [ ||2:z ]



CIRCLE-TORUS - [ |0(0) ]

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



TRIANGLE-TORUS - [ |>(0) ]

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



SQUARE-TORUS - [ ||(0) ]

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



LINE-TORISPHERE - [ |(00) ]

[ |(0) ] --z--> [ 0xy ] == [ |(00) ]
----------------------------------------
[ |x(0)xy ] ----> [ 0 ] == [ |x(00):xyz ]








4-D


GLOME - [ |000 ]

[ |00 ] --w--> [ 0xyz ] == [ |000 ]
-------------------------------------------
[ *(00):xyz ] --> [ 0 ] == [ *(000):xyzw ]



SPHONE - [ |00> ]

[ |00 ] ----w---> [ * ] == [ |00> ]
------------------------------------------
[ *(00):xyz ] --> [ * ] == [ |w(00):xyz ]
-------------------------- [ |00-*:w ]


SPHONE-B - [ |0>0 ]

[ |0> ] ---W---> [ 0xy ] == [ |0>0 ]
-------------------------------------------
[ |z(0):xy ] ----> [ 0 ] == [ |w(00):xyz ]
[ |0-*:z ] ------> [ 0 ] == [ |00-*:w ]



SPHERINDER - [ |00| ]

[ |00 ] ----w----> [ |00 ] == [ |00| ]
----------------------------------------------
[ *(00):xyz ] --> [ *(00) ] == [ |w(00):xyz ]
------------------------------ [ |002:w ]



DICONE - [ |0>> ]

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



CONINDER-i - [ |0>| ] , [ |0>2:w ]

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


CONINDER-ii - [ |0>| ] , [ ||0-|:w ]

[ ||0 ] ---w--> [ |x ] == [ |0>| ]
-----------------------------------------
[ |02:x ] ----> [ *2 ] == [ |0>2:x ]
[ |x(0):yz ] -> [ | ] == [ ||xw(0):yz ]
------------------------- [ ||0-|:w ]


DUOCYLINDER - [ |0|0 ]

[ ||0 ] ---w---> [ 0z ] == [ ||00 ]
------------------------------------------
[ |z(0):xy ] --> [ 0 ] == [ |0zw(0):xy ]
[ |02:z ] -----> [ (0) ] == [ |0xy(0):zw ]



CYLINDRONE-i - [ ||0> ] , [ ||0-*:w ]

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


CYLINDRONE-ii - [ ||0> ] , [ |0>-|0:w ]

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


CYLINDRONE-B - [ ||>0 ]

[ ||> ] ----w---> [ 0x ] == [ ||>0 ]
-------------------------------------------
[ |>2:x ] -----> [ (0) ] == [ |>yz(0):xw ]
[ |>2:y ] -------> [ 0 ] == [ |>02:y ]
[ ||-*:z ] ------> [ 0 ] == [ ||0-*:z ]



PENTACHORON - [ |>>> ]

[ |>> ] --w-> [ * ] == [ |>>> ]
-----------------------------------
[ |>2:x ] --> [ * ] == [ |>>2:x ]
[ |>-*:y ]--> [ * ] == [ |>>-*:y ]
[ |>-*:z ]--> [ * ] == [ |>>-*:z ]
---------------------- [ |>>-*:w ]



TETRAHEDRINDER-i - [ |>>| ] , [ |>>2:w ]

[ |>> ] ---w--> [ |>> ] == [ |>>| ]
--------------------------------------
[ |>2:x ] ---> [ |>2 ] == [ |>|2:x ]
[ |>-*:y ] --> [ |>-* ] == [ |>|-|:y ]
[ |>-*:z ] --> [ |>-* ] == [ |>|-|:z ]
-------------------------- [ |>>2:w ]


TETRAHEDRINDER-ii - [ |>>| ] , [ |>|-|:w ]

[ |>| ] ---w--> [ |x ] == [ |>>| ]
--------------------------------------
[ |>2:x ] ----> [ *2 ] == [ |>>2:x ]
[ ||2:y ] ----> [ | ] == [ |>|2:y ]
[ ||-|:z ] ---> [ | ] == [ |>|-||:z ]
------------------------- [ |>|-|:w ]


CYLTRIANGLINDER - [ |>|0 ]

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


CYLTRIANGLINDER-ii - [ |>|0 ] , [ ||0-|0:w ]

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



TRIANGLE PRISM PYRAMID-i - [ |>|> ] , [ |>|-*:w ]

[ |>| ] ---w--> [ * ] == [ |>|> ]
------------------------------------
[ ||2:x ] ----> [ * ] == [ ||>2:x ]
[ ||-|:y ] ---> [ * ] == [ ||>-|:y ]
[ |>2:z ] ----> [ * ] == [ |>>2:z ]
------------------------ [ |>|-*:w ]


TRIANGLE PRISM PYRAMID-ii - [ |>|> ] , [ ||>-|:w ]

[ ||> ] --w--> [ |x ] == [ |>|> ]
------------------------------------
[ |>2:x ] ---> [ *2 ] == [ |>>2:x ]
[ |>2:y ] ----> [ | ] == [ ||>2:y ]
[ ||-*:z ] ---> [ | ] == [ |>|-|>:z ]
------------------------ [ ||>-|:w ]


TRIANGLE DIPRISM-i - [ |>|| ] , [ |>|2:w ]

[ |>| ] --w--> [ |>| ] == [ |>|| ]
--------------------------------------
[ ||2:x ] ---> [ ||2 ] == [ |||2:x ]
[ ||-|:y ]--> [ ||-| ] == [ |||-||:y ]
[ |>2:z ] ---> [ |>2 ] == [ |>|2:z ]
------------------------- [ |>|2:w ]


TRIANGLE DIPRISM-ii - [ |>|| ] , [ |||-||:w ]

[ ||| ] --w-> [ ||xy ] == [ |>|| ]
--------------------------------------
[ ||2:x ] ---> [ |2 ] == [ |>|2:x ]
[ ||2:y ] ---> [ |2 ] == [ |>|2:y ]
[ ||2:z ] ---> [ || ] == [ |||2:z ]
------------------------ [ |||-||:w ]


DIPYRAMID-i - [ ||>> ] , [ ||>-*:w ]

[ ||> ] --w--> [ * ] == [ ||>> ]
-----------------------------------
[ |>2:x ] ---> [ * ] == [ |>>2:x ]
[ |>2:y ] ---> [ * ] == [ |>>2:y ]
[ ||-*:z ] --> [ * ] == [ ||>-*:z ]
----------------------- [ ||>-*:w ]


DIPYRAMID-ii - [ ||>> ] , [ |>>-|:w ]

[ |>> ] --w---> [ |x ] == [ ||>> ]
------------------------------------
[ |>2:x ] ----> [ *2 ] == [ |>>2:x ]
[ |>-*:y ] ---> [ |x ] == [ ||>-|>:y ]
[ |>-*:z ] ---> [ |x ] == [ ||>-|>:z ]
[ |:yz ] -----> [ |x ] == [ |>>xyz ] <--new addition expl missing |>>
------------------------- [ |>>-|:w ]


PYRAMID PRISM-i - [ ||>| ] , [ ||>2:w ]

[ ||> ] --w--> [ ||> ] == [ ||>| ]
-------------------------------------
[ |>2:x ] ---> [ |>2 ] == [ |>|2:x ]
[ |>2:y ] ---> [ |>2 ] == [ |>|2:y ]
[ ||-*:z ] -> [ ||-* ] == [ |||-|:z ]
------------------------- [ ||>2:w ]


PYRAMID PRISM-ii - [ ||>| ] , [ |||-|:w ]

[ ||| ] --w--> [ |x ] == [ ||>| ]
-------------------------------------
[ ||2:x ] ---> [ *2 ] == [ ||>2:x ]
[ ||2:y ] ----> [ | ] == [ |>|2:y ]
[ ||2:z ] ----> [ | ] == [ |>|2:z ]
------------------------ [ |||-|:w ]


PYRAMID PRISM-iii - [ ||>| ] , [ |>|-||:w ]

[ |>| ] -w-> [ ||xy ] == [ ||>| ]
-------------------------------------
[ ||2:x ] ---> [ |2 ] == [ |>|2:x ]
[ |>2:y ] ---> [ |2 ] == [ ||>2:y ]
[ ||-|:z ] --> [ || ] == [ |||-|>|:z ]
------------------------ [ |>|-||:w ]


CUBINDER - [ |||0 ]

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



CUBINDER-B - [ ||0| ]

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



HEMDODECACHORON-i - [ |||> ] , [ |||-*:w ]

[ ||| ] --w--> [ * ] == [ |||> ]
-----------------------------------
[ ||2:x ] ---> [ * ] == [ ||>2:x ]
[ ||2:y ] ---> [ * ] == [ ||>2:y ]
[ ||2:z ] ---> [ * ] == [ ||>2:z ]
----------------------- [ |||-*:w ]


HEMDODECACHORON-ii - [ |||> ] , [ ||>-||:w ]

[ ||> ] --w--> [ ||xy ] == [ |||> ]
--------------------------------------
[ |>2:x ] -----> [ |2 ] == [ ||>2:x ]
[ |>2:y ] -----> [ |2 ] == [ ||>2:y ]
[ ||-*:z ] ----> [ || ] == [ |||-||>:z ]
-------------------------- [ ||>-||:w ]


TESSERACT - [ |||| ]

[ ||| ] --w-> [ ||| ] == [ |||| ]
------------------------------------
[ ||2:x ] --> [ ||2 ] == [ |||2:x ]
[ ||2:y ] --> [ ||2 ] == [ |||2:y ]
[ ||2:z ] --> [ ||2 ] == [ |||2:z ]
------------------------ [ |||2:w ]
in search of combinatorial objects of finite extent
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The 4 Bisecting Rotations of the Duotrianglinder

Postby ICN5D » Wed Feb 12, 2014 10:28 pm

I wanted to experiment with the lathe operator, and apply them to |>[|>]. Using the 4 bisecting planes, we should get 4 shapes in 5D.

Construction of the duotrianglinder can be done with cartesian products or projection edge lacing. One can multiply two triangles together or lace a triangle atop triangle prism to get the |>[|>].

Cartesian product: [ |> , |> ]-duorpism. Surtope pairings in brackets are final operators appended to non-bracketed whole shape.


[ |> , |> ] === |>[|>]
------------------------------
[ |^2 ] , |> == [ |>|^2 ]1
[ |-* ] , |> == [ |>|-|> ]2
|> , [ |^2 ] == [ |>|^2 ]3
|> , [ |-* ] == [ |>|-|> ]4

|>[|>] = [ |>|^2 ]1 , [ |>|^2 ]2 , [ |>|-|> ]3 , [ |>|-|> ]4 , rearranged by rotating around in 4D, placing the pairings on different axes.

Two orthogonal triangles lace together their three edges, to make six triangle prisms arranged into two separate triangular attachments, the [ n|^2 ] , [ n|-n ] pattern.


Since |>[|>] is 4D, we have four separate bisecting 3-planes to rotate it around into 5D. They are the 1-2-3, 1-2-4, 1-3-4, and 2-3-4. The surchora pairs on the axes of these planes will remain stationary during the rotation, and become the new flat sutera containing round sides. The remaining axis will have a non-bisecting rotation and carve out a new rolling surface, becoming the new toratope surteron. Its crosscut will be one of the surchora from the moving axis. Using the sequence arrangement of the surchora pairs above, we can apply these rotations respectively. Because there are two identical surchora pairings on the surface of |>[|>], there are only two unique lathing figures that come out of it. According to the above arrangement, axis 1 and 2 are the same as well as 3 and 4.


• |>[|>]O1 : This rotation will hold plane 2-3-4 still and rotate axis 1 around into 5D. Identical to |>[|>]O2

|>[|>] ---5---> [ O1 ] == |>[|>]O1
------------------------------------------
[ |>|^2 ]1 ---> [ (O) ] == [ |>|(O) ]1-5
[ |>|^2 ]2 ----> [ O ] == [ |>|O^2 ]2
[ |>|-|> ]3 ---> [ O ] == [ |>|O-|>O ]3
[ |>|-|> ]4 ---> [ O ] == [ |>|O-|>O ]4

|>[|>]O1 = [ |>|(O) ]1-5 , [ |>|O^2 ]2 , [ |>|O-|>O ]3 , [ |>|O-|>O ]4

And what we get is two cone atop cyltrianglinder lacings [ |>|O-|>O ], and an attached pairing of two cyltrianglinders [ |>|O^2 ], all mutually connected by a triangle prism torus [ |>|(O) ].


• |>[|>]O3 : This rotation will hold plane 1-2-4 still and rotate axis 3 around into 5D. Identical to |>[|>]O4

|>[|>] ---5---> [ O3 ] == |>[|>]O3
------------------------------------------
[ |>|^2 ]1 ----> [ O ] == [ |>|O^2 ]1
[ |>|^2 ]2 ----> [ O ] == [ |>|O^2 ]2
[ |>|-|> ]3 -> [ (O) ] == [ |>|(O)-|>(O) ]3-5
[ |>|-|> ]4 ---> [ O ] == [ |>|O-|>O ]4

|>[|>]O3 = [ |>|O^2 ]1 , [ |>|O^2 ]2 , [ |>|(O)-|>(O) ]3-5 , [ |>|O-|>O ]4

And what we get is one cone atop cyltrianglinder lacing [ |>|O-|>O ], two attached pairings of cyltrianglinders [ |>|O^2 ], all mutually attached by a pairing of two toratope surtera, a triangle prism torus lacing to a triangle torus within its interior [ |>|(O)-|>(O) ], having a cocircular arrangement in an N-1 dimensional context.

-- Philip
in search of combinatorial objects of finite extent
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Re: Cool tricks for deriving surface elements

Postby ICN5D » Thu Feb 13, 2014 1:02 am

I've been playing around with different ways to illustrate the entire linear construction sequence from a point.

Cylconinder : |O>|O4
Code: Select all
[*] -1-> [*] │ [|] --2--> [O] │ [|O] --3--> [>] │ [|O>] --4-> [|O>] │ [|O>|] --5-> [O] == [|O>|O]
-------------│----------------│-----------------│-------------------│------------------------------
[*] ---> [*] │ [*-2] -> [(O)] │ [*(O)] ---> [*] │ [|(O)] --> [|(O)] │ [||(O)] ---> [O] == [||O(O)]
-------------│----------------│-----------------│ [|O-*] --> [|O-*] │ [|O|-|] ---> [O] == [|O|O-|O]
-------------│----------------│-----------------│-------------------│ [|O>-2] -> [(O)] == [|O>(O)]
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The 5 bisecting rotations of the Cylconinder

Postby ICN5D » Fri Feb 14, 2014 5:44 am

Bisecting rotations into 6D happen around a 4 dimensional plane. There are five such planes to rotate around in this manner. The cylconinder has two toratope faces lacing its base to the vertex. This makes two rotations identical out of the five possible 4-planes. Also, due to the pyramid-like nature of the cylconinder, rotating its height axis between base and vertex will carve out a crind-like shape.

5 Rotations of the Cylconinder : |O>|OO , |O>|O4On n = { 1=2 , 3 , 4=5 }


|O>|O4O1
Code: Select all
     |            O                >                |                   O4                  O1
     1            2                3                4                    5                   6
[*] -> [*]= [|] ----> [O] = [|O] -----> [>]=[|O>] -----> [|O>] = [|O>|] ---> [O] = [|O>|O] -----> [O1] = [|O>|OO]
----------│---------------│----------------│-------------------│-----------------│--------------------------------------
[*] -> [*]│[*-2]1 -> [(O)]│[*(O)]1-2 -> [*]│[|(O)]1-2 -> [|(O)]│[||(O)]1-2 -> [O]│[||O(O)]1-2 -> [(O)] = [||O(OO)]1-2-6
                                           │[|O-*]3 ---> [|O-*]│[|O|-|]3 ---> [O]│[|O|O-|O]3 ----> [O] = [|O|OO-|OO]3
                                                               │[|O>-2]4 -> [(O)]│[|O>(O)]4-5 ---> [O] = [|O>O(O)]4-5

|O>|O4O1 = [||O(OO)]1-2-6 , [|O|OO-|OO]3 , [|O>O(O)]4-5

* |O|OO is also the cylspherinder |OO|O
* |O>O(O) is also the sphone-torus |OO>(O), torus with sphone crosscut
* ||O(OO) is a torisphere with cylinder crosscut

Giving us ||O(OO) , |OO|O-|OO , |OO>(O) as the surtope elements.

In this rotation, we held plane 2-3-4-5 still and rotated axis 1 around into 6D. The resulting shape is a cylspherinder |O|OO lacing to a sphere |OO, joined by two new toratope faces. The original cylinder torus ||O(O) had its main circle on the moving axis, thus rotating the 2-spheric radius into a 3-spheric radius with a cylinder crosscut. The original cone torus |O>(O) had its minor crosscut shape on the moving axis, thus rotating the cone into a sphone, leaving the 2-sphere major radius unaffected. The base duocylinder was rotated into a cylspherinder, along with the opposing circle into a sphere. This shape is very identical to the cylsphoninder |OO>|O5, which is a cylspherinder lacing to a circle, instead of a sphere. Both shapes have the same exact lacing torii ||O(OO) and |OO>(O).



|O>|O4O3
Code: Select all
     |            O                >                |                   O4                  O3
     1            2                3                4                    5                   6
[*] -> [*]= [|] ----> [O] = [|O] -----> [>]=[|O>] -----> [|O>] = [|O>|] ---> [O] = [|O>|O] -----> [O3] = [|O>|OO]
----------│---------------│----------------│-------------------│-----------------│----------------------------------------
[*] -> [*]│[*-2]1 -> [(O)]│[*(O)]1-2 -> [*]│[|(O)]1-2 -> [|(O)]│[||(O)]1-2 -> [O]│[||O(O)]1-2 ---> [O] = [||OO(O)]1-2
                                           │[|O-*]3 ---> [|O-*]│[|O|-|]3 ---> [O]│[|O|O-|O]3 --> [(O)] = [|O|O(O)-|O(O)]3-6
                                                               │[|O>-2]4 -> [(O)]│[|O>(O)]4-5 ---> [O] = [|O>O(O)]4-5

|O>|O4O3 = [||OO(O)]1-2 , [|O|O(O)-|O(O)]3-6 , [|O>O(O)]4-5

* ||OO(O) is also the spherinder torus |OO|(O) , torus with spherinder crosscut
* |O>O(O) is also sphone torus |OO>(O)
* |O|O(O) is the duocylinder torus

Making |OO|(O) , |O|O(O)-|O(O) , |OO>(O) as the surtope elements.

In this rotation the 1-2-4-5 plane was held still, and rotated axis 3 around into 6D. Axis 3 is also the height axis between the duocylinder and circle. This makes two torii out of this pair, where the circle torus is laced into the interior of the duocylinder torus. The resulting lathing figure has all smooth, round sides, composed of complex lunes.



|O>|O4O4
Code: Select all
     |            O                >                |                   O4                  O4
     1            2                3                4                    5                   6
[*] -> [*]= [|] ----> [O] = [|O] -----> [>]=[|O>] -----> [|O>] = [|O>|] ---> [O] = [|O>|O] -----> [O4] = [|O>|OO]
----------│---------------│----------------│-------------------│-----------------│-----------------------------------
[*] -> [*]│[*-2]1 -> [(O)]│[*(O)]1-2 -> [*]│[|(O)]1-2 -> [|(O)]│[||(O)]1-2 -> [O]│[||O(O)]1-2 ---> [O] = [||OO(O)]1-2
                                           │[|O-*]3 ---> [|O-*]│[|O|-|]3 ---> [O]│[|O|O-|O]3 ----> [O] = [|O|OO-|OO]3
                                                               │[|O>-2]4 -> [(O)]│[|O>(O)]4-5 -> [(O)] = [|O>(OO)]4-5-6


This is the most important and interesting lathing shape. In this construction sequence, plane 1-2-3-5 is held stationary and rotated axis 4 around into 6D. Take notice that this is also the same axis as the last spin. A peculiar effect comes from rotating the same axis over and over again. Especially if you're rotating a shape that has a circle product in it. According to this last sequence:

|O>|O4O4 = [||OO(O)]1-2 , [|O|OO-|OO]3 , [|O>(OO)]4-5-6

* ||OO(O) is also the spherinder torus |OO|(O)
* |O|OO is also |OO|O cylspherinder
* |O>(OO) is the cone torisphere, or torisphere with cone crosscut

Which comes out to: |OO|(O) , |O|OO-|OO , |O>(OO) as the surtope elements. The cool thing about these are that they mean this shape is the (sphere,cone)-duoprism! How cool is that? This shape can be done with cartesian products and entirely through linear operations. So, |O>|O4O4 = |O>[|OO] = |OO[|O>] . This will add another rule to the commuting spin. Certain ones can go all over the place, in the proper context.

Derivation of the (sphere,cone)-prism done with cartesian product:

Code: Select all
[ |OO ]  x  [ |O> ]  │ |OO[|O>] = |O>[|OO]
---------------------│------------------
[ *(OO) ] --> |O>    │ [ |O>(OO) ]
  |OO ----> [ |(O) ] │ [ |OO|(O) ]
  |OO ----> [ |O-* ] │ [ |OO|O-|OO ]


can show the conveniently matching surtopes. It's the perfect analogy of a cylconinder, as the spherical version of it. Really, it's all in how the cylspherinder laces to its sphere-vertex. The cylspherinder has only two faces on it, just like the duocylinder. They are a torisphere bound orthogonally to a spheritorus. That is, a sphere innertube along with a spherical torus. So, when these two toratopes, both of which have a spherical attribute, lace to a single sphere, they collapse one of their diameters down to zero. Since a sphere is the endpoint of the lacing, the spherical radius of either toratope cell will be the one that remains unchanged. The other radius, which is 2-spheric for both, is the one that collapses, or tapers down to nothing.

Cylspherinder : |OO|O = [ |OO(O) , |O(OO) ]

|OO(O) --> |OO = |OO|(O) - spheritorus has spherical minor radius, circular major. Major shrinks to zero while maintaining spherical part, lacing both spheres like a prism into a spherinder torus


|O(OO) --> (OO) = |O>(OO) - torisphere has spherical major radius, circular minor. Minor shrinks to zero while maintaining spherical part, tapering a 2-plane of circles to a 2-plane of points on surface of sphere, making cone-torisphere.

-- Philip
in search of combinatorial objects of finite extent
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