by ICN5D » Tue Nov 19, 2013 4:38 am
@ Keiji: Hey, thanks for taking your time to respond. I like your input about new additions to my little creation. A side effect of self-teaching and development outside other academic sources can lead it to be radical and incomplete. I am a 30 year old professional bicycle mechanic and this is my side hobby. I am quite aware that there are infinite shapes in any dimension, many of which can be described by the processes that are discussed here. Judging by my joining date, you can tell I've been away for some time.
Although, I am unsure as to why you would ask me to stop. I am not disproving anything said on the forum, nor am I spreading disinformation. I bring new ideas and methods, to an appropriate forum where these things are being discussed. It's not off topic, not harassing, not vulgar or full of uneducated "text speak". Collaborating ideas is how ideas grow. Perhaps you will find some value to my outside perspective. Maybe you find my username " ICN5D " to have a dose of mild arrogance, thus ruining any beneficial first impression, I don't know. I do not claim to know everything, you know more than I do in this topic. I'm the guy who fixes your $10,000 racing bicycle, you are the admin for HDDB.
So, think of it as an an alternative path to "enlightenment" to aid in sharpening one's visual acuity above 3-D. It is a very basic form of what many of you are talking about. It definitely does not include powertopes and the like. It can be used as an introductory education tool to prepare for the other material on HDDB. The goal was to educate and familiarize people who have never thought about these things or wondered what the heck a tesseract really was. Being that it is a simplified linear construction language, shapes above 4-D can still be grasped even by the most non-mathematical types.
As for proving the transformation of a triangle prism into what I call the cyltrianglinder: I'm still kicking myself in the butt for not taking calculus in high school. I know that it is a very powerful tool for such things. If you are referring to the word "lathe" as my "spin" motion, then I can try. However, as stated above, I have developed my own proof system using the graphical matrix computation format. I designed it to be easily understood by any mathematically minded person with an eye for patterns. If you are asking me for help, then maybe you see something of value to it! It has helped me teach myself. Mathematics ( other than statistics) is an exact thing, no shades of gray. And because of that, there is always an exact solution, something to strive for in conceptual hurdles. Below is going to be a verbose, heavy read as I attempt to explain my thought process after 5 and a half years of familiarity.
In order to do so, I have to walk you through the some of the same steps I took. The way I calculate shapes is by reducing them into their N-1 surface panels and pairing them up on a particular axis. I felt there was no need to identify whether the pair was attached on a triangular-like shape, like the line torus on a cone as being attached at the vertex. The simplicity of Dimensionometry allows me to do this with every shape for consistency.
0 = spin ( lathe )
> = taper ( scale )
| = extrude ( translate )
(0) = extrude along path of line that has been spun, i.e: circle
* = vertex
Applying the extrude or taper is very straightforward, but the spin is tricky. When I visualize an object spinning around in place into the N+1, I see some of the N-1 surface panels undergo different motions. Some side panel pairs sit in place and apply the spin ( lathe ) motion, while only one pair joins together into a torus. I use the terms POLAR and EQUATORIAL to denote them. Much like how our planet spins around, at the equator an object is flung around in a great big circle, tracing out the path of a "circular extrusion " of a manifold. Whereas at the geometric poles, an object more or less sits in place and rotates ( lathes ). This is of course an oversimplified visual trick to identify which sides spin and which become a torus. After some time, I noticed that in this method, there are 3 types of spins ( and perhaps even more). I named them PRISMIC, TRIANGULAR, and SPHERICAL spins. The difference is how and where to apply polar vs equatorial.
PRISMIC SPIN : equatorial applied to last extruded pair in the sequence
TRIANGULAR SPIN : eqtrl applied to second last created pair, one of the connecting sides between base and vertex
SPHERICAL SPIN : eqtrl applied evenly to entire surface, no polar pairs
Consider how the square, " || " spins into 3-D to trace out the shape of a cylinder. The square has four lines on its surface, as two pairs separated by the x and y axes. When I say "on the x-axis", I really mean "separated by/along the x-axis". My symbols for these are:
" |2:x " = line pair on X
" |2:y " = line pair on Y
So, || = [ |2:x , |2:y ]. Since a square has only two pairs, applying the spin operation means one line pair becomes POLAR and the other EQUATORIAL. (Above 3-D I found that only one pair can be equatorial, but many can be polar.) The polar line-pair sits in place and spins to trace out a circle-pair, " |02 ". The equatorial line-pair is flung around in a circle, and therefore extruded along the path of a circle, joining to create a line-torus " |(0) ". This symbol " |(0) " means " line extruded along the path of a circle". The spin operation in parentheses is the modifier of the default extrude, meaning extruded along the path of a line that has been spun. This gives us two circles attached by a line torus, which agrees with what we see as the surface of a cylinder. The graphical representation is like this:
[ || ] --z--> [ 0y ] == [ ||0 ] --- " square ---> spins into Z == cylinder " , where the y-axis pair is equatorial
--------------------------------------
[ |2:x ] ---> [ 0 ] == [ |02:x ] --- " line pair of x ---> spins == circle pair "
[ |2:y ] --> [ (0) ] == [ |(0):yz ] --- " line pair of y ---> extrudes along path of circle == line torus"
Extruding a circle also creates the cylinder:
[ |0 ] ----z---> [ |0 ] == [ |0| ] --- " circle ---> circle = cylinder "
----------------------------------------
[ *(0):xy ] --> [ *(0) ] == [ |(0):xy ] --- " dot-torus ---> dot-torus = line-torus " , the dot-torus is the n-1 wireframe surface of a circle **
----------------------------- [ |02:z ] --- final pair is circle to circle along z, which creates the cylinder
** The torus transformations have a partial distributive law, where only the sub-shape is modified and the manifold left alone. However, during a spherical spin the manifold is modified and the sub-shape is left alone.
Spinning a cube into 4-D traces out the shape of a cubinder, or cylinder-prism. Since a cube has 6 square sides, there are 3 pairs of squares on their respective axes. During the spin, only one pair is equatorial, the rest as polar. So, CUBE : [ ||| ] = [ ||2:x , ||2:y , ||2:z ]
Graph detailing the spin of the cube:
[ ||| ] --w--> [ 0z ] == [ |||0 ] --- " cube ---> spin into W == cubinder, where z-pair is equatorial"
----------------------------------------------
[ ||2:x ] -----> [ 0 ] == [ ||02:x ] --- " square pair of x ---> spin == cylinder pair of x "
[ ||2:y ] -----> [ 0 ] == [ ||02:y ] --- " square pair of y ---> spin == cylinder pair of y "
[ ||2:z ] ---> [ (0) ] == [ ||(0):zw ] --- " square pair of z ---> extruded along path of circle zw == square torus "
When looking at the projection of a cubinder, we see a cylinder within a cylinder, attached by two more cylinders and a square torus. Another calculation of the cubinder is the extrusion of the cylinder: [ |||0 ] = [ ||0| ] = [ |0|| ]
[ ||0 ] ---w---> [ ||0 ] == [ ||0| ] --- " cylinder ---> extruded along w to cylinder == cubinder "
---------------------------------------------------
[ |(0):xy ] ---> [ |(0) ] == [ ||(0):xy ] --- " line torus --> line torus == square torus " , line --> line = square, the manifold isn't modified
[ |02:z ] ------> [ |02 ] == [ ||02:z ] --- " circle pair ---> circle pair = cylinder pair "
-------------------------------- [ ||02:w ] --- final pair is how shape was made, cyl-pair joined along w creates cubinder
In both graphs, the end result is that a cubinder has four cylinders attached by a square torus. Both the spin of the cube and extrude of the cylinder agree in the same n-1 surface panels. Another description is the Cartesian product of a square and a circle.
Here we apply the spin to a triangle-prism. Another way to create the cyltrianglinder ( god I love that word ), is to taper a cylinder down to a circle along the W-axis. This method is identical to tapering a line to a point, it is simply the same version where a circle is embedded into every point.
The triangle-prism : [ |>| ] , extrude of triangle along Z, or taper of square to line along Z.
[ |> ] --z--> [ |> ] == [ |>| ]
-----------------------------------
[ |2:x ] ---> [ |2 ] == [ ||2:x ]
[ |-*:y ] --> [ |-* ] == [ ||-|:y ]
-------------------------- [ |>2:z ]
[ || ] ---z--> [ | ] == [ |>| ] --- " square --> line along Z = triangle-prism"
---------------------------------
[ |2:x ] ---> [ *2 ] == [ |>2:x ] --- " line-pair --> point-pair = triangle-pair" , viewing a line from the x-axis we see it from it's natural 1-D perspective, two points
[ |2:y ] ----> [ | ] == [ ||2:y ] --- " line-pair --> line = square-pair" , viewing the line from above & below in y-space, seeing it it from a higher perspective, a line
------------------------- [ ||-|:z ] -- " final pair is the square-line pair, to account for the last square on surface unaccounted for
Spinning, or lathing a triangle-prism is the process of rotating |>| as if it were "flat", which it is according to 4-D. Since the two triangle ends are considered the final extruded pair, I assign them as equatorial. Associating any other side panel pair as eqt will not create the same thing. Whipping the triangle ends around causes them to become extruded along the path of a circle and join together, just like how the square spins around into 3-D. This creates the triangle-torus. Since the three squares are polar, they sit in place and have the spin motion applied. In the same way that the three squares are attached in a triangular fashion, these three cylinders also attach the same way. And because they have curved rolling surfaces themselves ( line-torii ), when they attach the end result is the triangle-torus.
Something I noticed with the spin operation: rearranging the symbols of n-cylinders allows me to conclude that they are also spins of prisms. By moving the spin around in the sequence for a cubinder, [ ||0| ] to [ |||0 ], I was able to derive the spin of a cube [ |||0 ] as being equal to the extrude of a cylinder [ ||0| ] . This is where I took notice of the sequence for a cyltrianglinder as [ |>|0 ]. It also can be described as the spin of a triangle prism [ |>| ]. I set out to understand this little trick and it works for every single one of them in all dimensions. Every calculation of every sequence and alternate perspective equals the same thing. I like symmetry and in this case, it all works out perfectly. That's how I know I'm on the right track.
Spin of triangle-prism:
[ |>| ] ---w--> [ 0z ] == [ |>|0 ] -- " triangle-prism --> spin into W = cyltriang" , where z-pair is EQT
----------------------------------------
[ ||2:x ] -----> [ 0 ] == [ ||02:x ] -- " square-pair --> spin = cyl-pair"
[ ||-|:y ] ----> [ 0 ] == [ ||0-|0:y ] -- " square-line pair --> spin = cyl-circle pair"
[ |>2:z ] ---> [ (0) ] == [ |>(0):zw ] -- " triangle-pair --> extrude along path of circle = triangle-torus"
Taper of cylinder to circle:
[ ||0 ] --w--> [ |0xy ] == [ |>|0 ] --- " cylinder ---> circle in xy plane == cyltrianglinder"
-------------------------------------------
[ |(0):xy ] ---> [ *(0) ] == [ |>(0):xy ] -- " line-torus --> dot torus == triangle-torus" , viewing a circle from its natural 2-D perspective, we see a dot-torus
[ |02:z ] ------> [ |0 ] == [ ||02:z ] -- " circle-pair --> circle = cylinder-pair" , viewing a circle from above & below in Z-space, we see an entire circle
----------------------------- [ ||0-|0:w ] -- final pair from method of creation, cyl to circle along w
So, in both graphs, we see the |>|0 as having three cylinders attached by a triangle torus. By examining the projection of a |>|0, we see a circle inside a cylinder attached by two more cylinders and a triangle-torus.
Applying this method of spinning prisms to create cylinders in 5-D will shed some light on the difficult cylhemoctahedrinder [ ||>|0 ] and cyltetrahedrinder [ |>>|0 ]. These shapes were hideous and impossible for me to understand and see for a while. Then one day, I discovered how to reduce shapes to their surface panels and apply operations to them.
Let's examine the [ ||>|0 ]: It is the spin of the pyramid-prism
[ ||> ] is the square pyramid, has a square at the base tapering to a vertex connected by four triangles.
PYRAMID-i - [ ||> ] , [ ||-*:z ]
[ || ] --z--> [ * ] == [ ||> ]
-----------------------------------
[ |2:x ] ---> [ * ] == [ |>2:x ]
[ |2:y ] ---> [ * ] == [ |>2:y ]
------------------------ [ ||-*:z ]
Another perspective is viewing it through one of the triangle sides, where we see a triangle tapering to a line:
PYRAMID-ii - [ ||> ] , [|>-|:z]
[ |> ] --z--> [ | ] == [ ||> ]
------------------------------------
[ |2:x ] --> [ *2 ] == [ |>2:x ]
[ |-*:y ] --> [ | ] == [ ||-|>:y ]
----------------------- [ |>-|:z ]
In both graphs we see a square and four triangles, derived two different ways. The pyramid prism that I call the pyraminder, is made by extruding a pyramid to a pyramid along the w-axis. This extrusion is also applied to every surface panel. When the two pyramids connect, each panel connects to their respective partner. Two squares connect to form a cube and eight triangles connect forming four triangle-prisms. It is also important to include the vertex as both are connected into a line. The resulting shape has three different surface panels allowing three perspectives to view the object from.
Looking through the pyramid side:
PYRAMINDER-i - [ ||>| ] , [ ||>2:w ]
[ ||> ] --w--> [ ||> ] == [ ||>| ]
-----------------------------------------
[ |>2:x ] ---> [ |>2 ] == [ |>|2:x ]
[ |>2:y ] ---> [ |>2 ] == [ |>|2:y ]
[ ||-*:z ] --> [ ||-* ] == [ |||-|:z ]
---------------------------- [ ||>2:w ]
Looking through the cube side:
PYRAMINDER-ii - [ ||>| ] , [ |||-|:w ]
[ ||| ] --w--> [ |x ] == [ ||>| ]
----------------------------------------
[ ||2:x ] ---> [ *2 ] == [ ||>2:x ]
[ ||2:y ] ----> [ | ] == [ |>|2:y ]
[ ||2:z ] ----> [ | ] == [ |>|2:z ]
-------------------------- [ |||-|:w ]
Looking through the triangle-prism side
PYRAMINDER-iii - [ ||>| ] , [ |>|-||:w ]
[ |>| ] -w-> [ ||xy ] == [ ||>| ]
----------------------------------------
[ ||2:x ] ---> [ |2 ] == [ |>|2:x ]
[ |>2:y ] ---> [ |2 ] == [ ||>2:y ]
[ ||-|:z ] --> [ || ] == [ |||-|>|:z ]
-------------------------- [ |>|-||:w ]
All three graphs end up with the same number and type of side panels: 1 x cube [ ||| ], 2 x pyramids [ ||> ], 4 x triangle-prisms [ |>| ]. Note how the perspectives of cube->line pair and trianglinder->square pair are the direct extrusions of the square->dot pair and triangle->line pair. More lovely symmetry!
Applying the spin operation to this shape is pretty straightforward now. The last extruded pair were the pyramid ends, and so become equatorial. The rest are polar. Do not worry, no pyramid prisms were injured during this operation ( get it? )
[ ||>| ] --v--> [ 0w ] == [ ||>|0 ]
---------------------------------------------
[ |>|2:x ] ------> [ 0 ] == [ |>|02:x ]
[ |>|2:y ] ------> [ 0 ] == [ |>|02:y ]
[ |||-|:z ] -----> [ 0 ] == [ |||0-|0:z ]
[ ||>2:w ] ---> [ (0) ] == [ ||>(0):wv ]
Being that this shape is closely related to all of the intricacies of a pyramid, it also has the same 3 perspectives. The pyramid to pyramid was rounded out during the spin into a torus ( above ):
Looking through cubinder side:
[ |||0 ] --v--> [ |0xy ] == [ ||>|0 ]
-----------------------------------------------
[ ||(0):xy ] ---> [ *(0) ] == [ ||>(0):xy ]
[ ||02:z ] ------> [ |0 ] == [ |>|02:z ]
[ ||02:w ] -----> [ |0 ] == [ |>|02:w ]
------------------------------ [ |||0-|0:v ]
Looking through cyltrianglinder side:
[ |>|0 ] ---v--> [ ||0xyz ] == [ ||>|0 ]
------------------------------------------------------
[ |>(0):xy ] -------> [ |(0) ] == [ ||>(0):xy ]
[ ||02:z ] ---------> [ |02 ] == [ |>|02:z ]
[ ||0-|0:w ] ------> [ ||0 ] == [ |||0-|>|0:w ]
----------------------------------- [ |>|0-||0:v ]
In all 3 graphs we derive the same n-1 surface panels: 1 x cubinder, 4 x cyltrianglinders, 1 x pyramid-torus. Once again, the cubinder->circle and cyltrianglinder->cylinder perspectives are directly related to the square->dot and triangle->line of the pyramid.
By now, you can probably derive the 5-D surface panels of the [ |||>|0 ]. I call it the cylhemdodecachorinder ( sil' hem' doe' dessah' kor' in' dur ). The spin of the prism of the hemdodecachoron, or cube-pyramid. Also the cartesian product of a circle and a cube-pyramid.
I like to think of cylinders as having a circle attached, or embedded into another shape. The common 3-D cylinder, I call the line-cylinder, is a circle embedded into every point of a line. The cubinder is a circle embedded into every point of a square, and takes on a square like appearance with the way its four cylinders attach. N-Cylinders can be created by the spin of n-prisms, extrude of lower n-cylinders, or extrusions of n-spheres. Spinning a square creates the line-cyl, spinning a cube creates the square-cyl ( cubinder ), and spinning a triangle prism creates the cyltrianglinder, or triangle-cylinder. What I am trying to do is show you the consistency of the graphs with what you already know to be true. After recognizing the patterns, you can see that it has an exact mathematical match, short of providing parametric equations or using the lingo you are familiar with. From there, I can apply the graphs to new concepts like the spin of the triangle-prism.
A few Cartesian Products and their effects on n-cyls ( contact patch is placing the rolling surface flat on our 3-D realmic plane for 4-D and above)
LINE x CIRCLE: line-cylinder ||0
--------------------------------------------
-CONTACT PATCH: line, rolls along 1 direction perpendicular to line
-CROSS-SECTIONS: line->square->line , circle->circle
-UNFOLDED: 2x circles, 1x square
SQUARE x CIRCLE: square-cylinder |||0
--------------------------------------------------
-CONTACT PATCH: square, rolls along 2 directions
-CROSS-SECTIONS: square->cube->square , cylinder->cylinder
-UNFOLDED: 4x cylinders, 1x cube
TRIANGLE x CIRCLE: triangle-cylinder |>|0
-------------------------------------------------------
-CONTACT PATCH: triangle, rolls along 3 directions
-CROSS-SECTIONS: triangle->triangle-prism->triangle , cylinder->circle
-UNFOLDED: 3x cylinders, 1x triangle-prism
CIRCLE x CIRCLE: circle-cylinder |0|0
------------------------------------------------
-CONTACT PATCH: circle, can roll along 2 simultaneous directions
-CROSS-SECTIONS: circle->cylinder->circle, circle->cylinder->circle
-UNFOLDED: 2x cylinders
PYRAMID x CIRCLE: pyramid-cylinder ||>|0 ( cylhemoctahedrinder )
----------------------------------------------------------------------------------------
-CONTACT PATCH: pyramid, rolls along 5 directions, each perpendicular to a side panel
-CROSS-SECTIONS: pyramid->pyramid prism->pyramid , cubinder->circle , cyltrianglinder->cylinder
-UNFOLDED: 1x cubinder, 4x cyltrianglinders, 1x pyramid-prism
TETRAHEDRON x CIRCLE: tetrahedron-cylinder |>>|0 ( cyltetrahedrinder )
----------------------------------------------------------------------------------------------
-CONTACT PATCH: tetrahedron, rolls along 4 directions, each perpendicular to a side panel
-CROSS-SECTIONS: tetrahedron->tetrahedron prism->tetrahedron , cyltrianglinder->circle
-UNFOLDED: 4x cyltrianglinders, 1x tetrahedron-prism
What I really need to find is a program to animate these things. Trying to describe using words may cause eye fatigue and eventual sleepiness. I can very clearly see the triangle prism going through a 4-D spin, causing the shapes I calculate to form. It was only in this very way that I knew what was right or wrong. Without a computer program to draw these projections, I drew them myself and imagined the heck out of them. But when I really examined the cool little gif animation of the rotation of a tesseract, I finally saw what the (N+1)-D spin looks like. That ah-ha moment came and I applied the very same looking motion (of turning inside-out) to all other prisms and shapes in my head. This was when I knew I was on the right track. One pair moves in an equatorial way, the others ( all others ) move in a polar way.
So, I'm not sure if this has been enough proof. Or if it makes any sense, being in a practically alien language. But, I tried to keep it simple and illustrate how spinning shapes produces certain patterns and effects, and how these can be applied in other ways. That's how I learned everything: experimenting, deriving principles and patterns, then testing to see if it's correct. After many a trial and error, so far, so good.
I can provide you with every graphical calculation of 2,3, and 4-D shapes with all perspectives and alternate sequences detailed, as described by dimensionometry, if you so choose. If you have to ask, yes, I spent 10 hours today typing this up for you. This is information that has been in my head and finally fleshed out on digital paper for the first time. Enjoy!
-Philip
in search of combinatorial objects of finite extent