## ICN5D's Notation System

Higher-dimensional geometry (previously "Polyshapes").

### ICN5D's Notation System

So, I decided to unearth an old post, in relation to one that quickfur pointed me to. It was brought to my attention that he made another notation system, but it didn't use a lathe operator. Not only have I been using the lathe operator, but I've also been using toratope notations as well, that expand on the crosscut types. This part was necessary to describe certain surtopes on the surface of certain shapes. The toratope notation I use is a linear step by step construction plus cartesian products. Once again, this notation evolved along with a computation system to derive the lacing elements through linear operators, cartesian products, or projective edge lacing. It's probably the only one that includes all toratopes, tapertopes, rotopes, and (m,n)-duoprisms.

Here are some things I have collected together:

Using the acronym " STEMP " that stands for Spin, Taper, Extrude, Manifold, Product, we have:
-------------------------------------------------------------------------------------------------------------------------
• mO# - rotate shape M around N-1 plane into N+1, where axis '' # '' is the rotating axis
• m> -taper shape M into N+1
• m| - extrude shape M into N+1
• m(qa) - extrude shape M along surface of shape Q into N+a
• m[q] - cartesian product with shape M and Q, the (m,q)-prism

* Where "a" is number of operators in parentheses

General Shape Families:
-----------------------------
• |a = N-Cube
• |>a = N-Simplex
• |Oa = N-Sphere
• |Oa>b = N-Sphone
• ||a>b = N-Pyramid**
• |>a|b = N-Pyramid**
• |>a|b>c = N-Pyramid**
• ||aOb = N-Cylinder
• ||aOb>c = N-Cylindrone
• |...[|O] = N-Cylinder
• |...[|>] = N-Trianglinder

* Where "a,b,c" is ≥ 1
** These are just a few of the endless examples as | and > don't commute

0-D
n
* - POINT

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

1-D
X
| - LINE
(O) - HOLLOW CIRCLE / GLOMOLATRIX

----------------------------------
2-D
XY
|O - CIRCLE
|> - TRIANGLE
|| - SQUARE
|(O) - LINE TORUS
|^(O) - ANCHORED LINE TORUS
(OO) - HOLLOW SPHERE / GLOMOHEDRIX
(O)(O) - HOLLOW TORUS / TORIHEDRIX
[(O)(O)] - DUOCYLINDER MARGIN

Complex manifold examples:

(O>) - CONIHEDRIX / HOLLOW CONE
(>>) - TETRAHEDRIX / HOLLOW TETRAHEDRON

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

3-D
XYZ
|OO - SPHERE
|O> - CONE
||O - CYLINDER
|>> - TETRAHEDRON
|>| - TRIANGLE PRISM
||> - SQUARE PYRAMID
||| - CUBE
|O(O) - TORUS
||(O) - SQUARE TORUS
|>(O) - TRIANGLE TORUS
(OOO) - HOLLOW GLOME / GLOMOCHORIX
(OO)(O) - HOLLOW SPHERITORUS
(O)(OO) - HOLLOW TORISPHERE
(O)(O)(O) - HOLLOW DITORUS
|(OO) - LINE TORISPHERE
|(O)(O) - LINE DITORUS
|[(O)(O)] - LINE TIGER
[(O)(O)(O)] - TRIOCYLINDER MARGIN
[(OO)(O)] - CYLSPHERINDER MARGIN
[[(O)(O)](O)] - CYLTORINDER MARGIN

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

4-D
XYZW
|OOO - GLOME
|OO> - SPHONE
|OO| - SPHERINDER
|O>> - DICONE
|O>| - CONINDER
|O|O - DUOCYLINDER
||O> - CYLINDRONE
|>>> - PENTACHORON
|>>| - TETRAHEDRINDER
|>|O - CYLTRIANGLINDER
|>|> - TRIANGLE PRISM PYRAMID
|>|| - TRIANGLE DIPRISM
||>> - DIPYRAMID
||>| - PYRAMID PRISM
|||O - CUBINDER
|||> - HEMDODECACHORON / CUBE PYRAMID
|||| - TESSERACT
|>[|>] - DUOTRIANGLINDER
|OO(O) - SPHERITORUS
|O(OO) - TORISPHERE
|O(O)(O) - DITORUS
|O[(O)(O)] - TIGER
|O>(O) - CONE TORUS
||O(O) - CYLINDER TORUS / TORINDER
|>>(O) - TETRAHEDRON TORUS
|>|(O) - TRIANGLE PRISM TORUS
||>(O) - SQUARE PYRAMID TORUS
|||(O) - CUBE TORUS
||(O)(O) - SQUARE DITORUS
|>(O)(O) - TRIANGLE DITORUS
||(OO) - SQUARE TORISPHERE
|>(OO) - TRIANGLE TORISPHERE
(OOOO) - HOLLOW PENTASPHERE / GLOMOTERIX
(OOO)(O) - HOLLOW GLOMITORUS
(O)(OOO) - HOLLOW TORIGLOME
(OO)(OO) - HOLLOW SPHERITORISPHERE
(OO)(O)(O) - HOLLOW SPHERIC DITORUS
(O)(OO)(O) - HOLLOW TORISPHERIC TORUS
(O)(O)(OO) - HOLLOW TORIC TORISPHERE
(O)(O)(O)(O) - HOLLOW TRITORUS
|(OO)(O) - LINE TORISPHERIC TORUS
|(O)(OO) - LINE TORIC TORISPHERE
|(O)(O)(O) - LINE TRITORUS
|(OOO) - LINE TORIGLOME

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

5-D
XYZWV
|OOOO - PENTASPHERE
|OOO> - GLONE
|OOO| - GLOMINDER
|OO>> - DISPHONE
|OO>| - SPHONINDER
|OO|O - CYLSPHERINDER
|OO|> - SPHERINDRONE
|OO|| - CUBSPHERINDER
|O>>> - TRICONE
|O>>| - DICONINDER
|O>|O - CYLCONINDER
|O>|> - CONINDER PYRAMID
|O>|| - CONE DIPRISM
||OO> - DUOCYLINDRONE
||O>> - DICYLINDRONE
||O>| - CYLINDRONE PRISM
|>>>> - HEXATERON
|>>>| - PENTACHORINDER
|>>|O - CYLTETRAHEDRINDER
|>>|> - TETRAHEDRINDER PYRAMID
|>>|| - TETRAHEDRON DIPRISM
|>|OO - DUOCYLTRIANGLINDER
|>|O> - CYLTRIANGLINDRONE
|>|O| - CYLTRIANDYINDER
|>|>> - TRIANGLE PRISM DIPYRAMID
|>|>| - TRIANGLIE PRISM PYRAMID PRISM
|>||> - TRIANGLE DIPRISM PYRAMID
|>||| - TRIANGLE TRIPRISM
||>>> - SQUARE-TRIPYRAMID
||>>| - SQUARE DIPYRAMID PRISM
||>|O - CYLHEMOCTAHEDRINDER
||>|> - SQUARE PYRAMID PRISM PYRAMID
||>|| - SQUARE PYRAMID DIPRISM
|O|O| - DUOCYLDYINDER
|||O> - CUBINDRONE
|||>> - CUBE DIPYRAMID
|||>| - CUBE PYRAMID PRISM
||||O - TESSERINDER
||||> - TESSERACT PYRAMID
||||| - PENTERACT
|OO[|>] - SPHENTRIANGLINDER
|O>[|>] - CONTRIANGLINDER
||>[|>] - HEMOCTAHEDROTRIANGLINDER
|>>[|>] - TETRAHEDROTRIANGLINDER
|>[|>]| - DUOTRIANGLINDYINDER
|>[|>]> - DUOTRIANGLINDRIC PYRAMID
|OOO(O) - GLOMITORUS
|O(OOO) - TORIGLOME
|OO(OO) - SPHERITORISPHERE
|OO(O)(O) - SPHERIC DITORUS
|O(OO)(O) - TORISPHERIC TORUS
|O(O)(OO) - TORIC TORISPHERE
|O(O)(O)(O) - TRITORUS
|O[(OO)(O)] - CYLSPHERINTIGROID
|O[[(O)(O)](O)] - CYLTORINTIGROID
|O[(O)(O)](O) - TIGRITORUS
|OO[(O)(O)] - SPHERIC TIGER
|O(O)|O - CYLTORINDER
|O|O(O) - DUOCYLINDRITORUS

Inflated complex manifold:

|O>(O>) - DUOCONTERIX

....and many more toratopes with crosscuts of any other 1,2,3,4D non-toratopes above.

So, I've been working on the construction tables, trying to slim it down a bit. They seem kind of distracting with all of the extra crap. This is my slimmed down version, going downwards now, with the flow of the operators. It's still a mess trying to keep the operators spaced out in the larger shape surtope computations. It's basically a linear progression of surtope lacing from a point.

Here's one of my favorite, of course:

Cylconinder |O>|O = |xOy>z|wOv

* == [ * ]
|x = [ * ]
------------
| == [ *-2 ]

| = [ *-2 ]
Oy = [ (O) ]
---------------
|O = [ *(O) ]

|O == [ *(O) ]
>z === [ * ]
--------------------------
|O> = [ |(O) , |O-* ]

|O> == [ |(O) , |O-* ]
|w === [ |(O) , |O-* ]
-----------------------------------------
|O>| = [ ||(O) , |O|-| , |O>-2 ]

|O>| == [ ||(O) , |O|-| , |O>-2 ]
Ov ===== [ O ....... O ....... (O) ]
-------------------------------------------------
|O>|O = [ ||O(O) , |O|O-|O , |O>(O) ]

-- Philip
Last edited by ICN5D on Fri Jul 25, 2014 12:30 am, edited 6 times in total.
It is by will alone, I set my donuts in motion
ICN5D
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### Re: Yet another improved notation for shapes

Sorry, but I'm splitting this post, as it has nothing to do with SSC2 (other than both being notations).

Keiji

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### Re: Yet another improved notation for shapes

ICN5D wrote:So, I decided to unearth an old post, in relation to one that quickfur pointed me to. It was brought to my attention that he made another notation system, but it didn't use a lathe operator. Not only have I been using the lathe operator, but I've also been using toratope notations as well, that expand on the crosscut types.

Well, the whole point of my notation was to produce the maximum variety of shapes from the minimal number of operators. It wasn't intended to be all-inclusive, but just a little experiment into what can be produced from what is essentially just a single underlying operation (extrusion with not necessarily linear scaling). Obviously, many shapes can't be produced this way, such as the duocylinder. Adding the Cartesian product does produce the duocylinder, and a bunch of other stuff, but there are of course some other shapes that can't be reached, like the toratopes. It was enlightening to see what can be reached just with those two minimal operations, though.

For the rest of your post, I haven't read it carefully yet, will do so later as I have to run now.
quickfur
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### Re: Yet another improved notation for shapes

Cool. Keiji, I just wanted it to be out there, materialized in some form. Someone may appreciate it someday! I'm not really looking to revolutionize anything
It is by will alone, I set my donuts in motion
ICN5D
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### Re: Yet another improved notation for shapes

Philip's "|" (bar) and Quickfur's "I" (capital i) are equivalent, the prismation (extrusion).

Philip's ">" and Quickfur's "A" are equivalent too, the pyramidization (tapering).

Philip's "O" (spin - of some axis of the "base") and Quickfur's "O" (roundish bi-tapering of the base) generally differ.

Quickfur further added in his much older setup also the operator "X" of bi-tapering.

To my opinion "X", if taken not combinatorically but rather metrically, so seeming like a good operator for a square base (providing the octahedron), looks quite odd. For soon beyond, the base could have a radius larger than sqrt(2) resp. smaller than that. Then the bi-tapering becomes rather flat, disc shaped resp. rather tall, needle shaped. Both usually tend to run out of interesting shapes in general.

Same holds true for Quickfur's roundish bi-tapering. Not only that those shapes tend to get any kind of elliptic cross-sections, you even get thingies like the intersection of 2 orthogonal cylinders instead of truely round things. - So I fear Philip's spin is much better suited to describe round things. Even if he still has to work out some consistent set of rules, how to select that axis of the base, which would perform the rotation...

For polytopists thus the most interesting operations are extrusion "|" / "I" and tapering ">" / "A". Note that both operations as such are defind combinatorically a priori. But one could consider those metrically as well. Then this imposes some restrictions onto the tapering however. Then the radius of the base surely has to be lower than unity: else a unit lacing edge could not connect the base vertices to the aimed for tip.

It should be mentioned here, when restricting to these 2 operations only, that both operational results will be monostratic only. If considered metrically, i.e. using unit edges throughout, you even would result in a subset of convex segmentotopes only. This tiny observation then implies that a circumradius of those shapes always would be well defined. - And thus could be observed for the above tapering restriction.

Extrusions /prismations surely are always possible. Even if the radius already exceeds unity by far. Any additional prismation increases the radius: r2D+1 = r2D + (1/2)2.

Tapering / pyramidization is possible only when the base radius is smaller unity. The height then is calculated according to: h2D+1 = 12 - r2D.

Even so several of the larger base polytopes cannot be tapered the higher the dimension goes, the smaller ones never would run out of count, as the mere simplex (only taperings being applied) gets lower and lower radius the higher the dimension climbs: r2D-simplex = D/[2(D+1)].

1D:
Code: Select all
`| - edge`

2D:
Code: Select all
`|| - {4}|> - {3}`

3D:
Code: Select all
`||| - cube||> - squippy|>| - trip|>> - tet`

4D:
Code: Select all
`|||| - tes   (R = 1)|||> - cubpy (R = 1)||>| - squippyp||>> - squasc|>|| - tisdip|>|> - trippy|>>| - tepe|>>> - pen`

5D:
Code: Select all
`||||| - pent        (R > 1)||||> - tespy       (H = 0)|||>| - cubpyp      (R > 1)|||>> - cubasc      (H = 0)||>|| - squasquippy (R = 1)||>|> - squippyippy||>>| - squascop||>>> - squete|>||| - tracube     (R > 1)|>||> - tisdippy|>|>| - trippyp|>|>> - trippasc|>>|| - squatet|>>|> - tepepy|>>>| - penp|>>>> - hix`

--- rk
Klitzing
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### Re: Yet another improved notation for shapes

We could be even be a bit more explicit:

These values rD+1 and hD+1 are easily to be calculated by Iteration (for only unit edged figures), provided rD, the radius of the base, is known.
This surely is true for a single (unit) edge (i.e. r1 = 1/2).
Thus, considering any combination of prismations and pyramidizations only, we could derive the respective values quite easily for all of these!

Prismation / Extrusion:
rD+12 = rD2 + hD+12
hD+1 = 1

Pyramidization / Tapering:
rD+1 = 1 / (2 hD+1)
hD+12 = 12 - rD2

--- rk
Klitzing
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### Re: Yet another improved notation for shapes

Thanks for all those formulas! They're certainly important when considering unit edge rotopes. One good thing to add would also be how each lathe operator only "inflates" the shape in place, and holds off the degenerate height effect for another dimension. It allows a shape to add another dimension while apparently remaining unaffected. A unit edge cone is made from lathing a unit edge triangle. A unit edge cylinder is made from lathing a unit square, unit cubinder made from lathing a unit cube, etc.

Even if he still has to work out some consistent set of rules, how to select that axis of the base, which would perform the rotation...

Hint, hint, nudge, nudge.

Well, you're right, let's do that now I should start by adding that I sort of reverse derived the spin out of shapes that had a cartesian product with a circle. As I stated before, any (Q,circle)-duorpism can also be made using linear operators, the extrude | followed by the spin O. The extrusion of Q turns the shape into a Q-prism. The spin will then rotate this Q-prism into N+1, turning into a kind of cylinder, with a rolling side. This is what the lathe does, it adds a new rolling surface to the end shape. The number of lathe operators equals the number of rolling faces, and thus directions of freedom.

A well known property of the cartesian products is that it combines the attributes of the surfaces of both shapes together. When doing a cartesian product with shape Q and a circle, all faces on shape Q have the appended |O operators (mult by entire disk), while the entire shape Q will be multiplied with the edge of the circle, the 1D glomolatrix, (O). This will always turn into the Q-torus, the new rolling face with a crosscut of shape Q. This effect is directly reflected in rotations around an N-1 bisecting plane. Assuming this N-1 plane also bisects the faces themselves, rotations around it will always leave one axis left over. This one axis undergoes a non-bisecting rotation. All others have a bisecting rotation, around this N-1 plane. The number of dimensions a shape has is equal to the number of available bisecting N-1 planes.

For Rotating into N+1 around an N-1 Bisecting Plane:

1D- x - one 0-plane: origin (0,0)

2D - xy - two 1-planes: x, y

3D - xyz - three 2-planes: xy, xz, yz

4D - xyzw - four 3-planes: xyz, xyw, xzw, yzw

5D - xyzwv - five 4-planes: xyzw, xyzv, xzwv, xywv, yzwv

6D - xyzwvu - six 5-planes: xyzwv, xyzwu, xyzvu, xywvu, xzwvu, yzwvu

We can establish the stationary plane of rotation by picking any one of the available n-1 planes. By pairing equally opposed faces, single torii, or attached faces along an axis, we can then establish which faces have the additional O appended, and which single pair become joined together into a torus, (O). Another, even easier way to do it would be to address only the moving axis, since there will always be one. Establishing which is the axis in motion means all the others are the stationary.

Using the O operator, by adding the axis in motion in a subscript, we can establish the stationary axes and make a specific shape. One can make a different shape by the same operator when switching the axis in motion. I'm used to using letters for the axes, but quickfur thinks I should use numbers. I'm still not sure what I want to do about that

For example, the cylinder:

|O| = |xOy|z = [ |z(O) ]xy , [ |Oxy-2 ]z, we have two parallel circles, separated along z [|Oxy-2]z , and a line torus extending into z, occupying the xy plane [|z(O)]xy.

Rotation around N-1 planes of the Cylinder:

Bisecting stationary plane: YZ
|O|Ox = [ |z(OO) ]xyw , [ |OOxyw-2 ]z == |OO| spherinder
The major radius of line torus has moving axis X, becomes line toprisphere; circles are on the bisecting plane, becoming spheres

Bisecting stationary plane: XZ
|O|Oy = [ |z(OO) ]xyw , [ |OOxyw-2 ]z == |OO| spherinder
The major radius of line torus has moving axis Y, becomes line toprisphere; circles are on the bisecting plane, becoming spheres

Bisecting stationary plane: XY
|O|Oz = [ |Ozw(O) ]xy , [ |Oxy(O) ]zw == |O|O duocylinder
The minor radius of line torus has the moving axis Z, becomes circle torus; circles are on moving axis, becoming circle torus

Another simpler way to assign the axis in motion is to simply look at which operator is previous to the appending spin O. If there's an extrude, assign the moving axis as last constructed axis. If there's a taper, we can assign the second-last constructed. However, if there is another spin, simply assigning last constructed will place the moving axis on a surcell's toratope major radius. In the commutative property, we get a sticky spin. Adding a spin next to a spin will join the two and become inseparable. This way, we only need to do something different with a taper+spin >O, because extrude+spin and spin+spin will be done the same way. This is an attempt at standardizing the spin, while trying to clean up all of the little intricate symbology. But, of course, higher-D shapes are more complex and can have many different bisecting planes to make many interesting new things. This is probably where the subscript for the moving axis should be used, in 4D and higher. It's sort of along the lines of " Want to apply a different spin? Simply rearrange the construction sequence! ". But, the cyltrianglinder |>|O cannot commute, while having three unique bisecting rotations. So that sort of rules out dropping the subscript, for specific spins that is....

Back to the cylinder lathing:

|O| is another orientation of ||O. If using xyz as axis of reference, |O|O is different than ||OO.

* Arrangement |O| has the circle end caps separated along Z. Last constructed axis rule will rotate the circle ends and make a duocylinder |O|O.

* Arrangement ||O has the major radius of the line torus on Z, making the "last constructed" rule assign the moving axis to rotate the main circle of the line torus, making a line torisphere and sphere endcaps, the spherinder ||OO = |OO| .

xyz
||O : axis Z is line torus' major radius
|O| axis Z is separating circles

||OO = |OO|, only the second | can commute, the 2 spins are stuck together once attached
|O|O = |O|O, none commute, except for an entire |O reflecting cartesian product property

So, in summary,

[...]|O - last constructed axis in motion, extruded end-cap faces [...] becomes [...](O) rolling face, all others stationary becoming [...]O

[...]OO - last constructed axis in motion, toratope face [...](O) becomes [...](OO) torisphere with [...] crosscut, all others stationary becoming [...]O

[...]>O - second-last constructed axis in motion, surtope pair here becomes [...](O) rolling side, all others stationary becoming [...]O

[...]Oq - axis Q in motion, surtope pair here become [...](O) rolling side, all others stationary becoming [...]O

--Philip
It is by will alone, I set my donuts in motion
ICN5D
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### Re: Yet another improved notation for shapes

Klitzing wrote:Philip's "|" (bar) and Quickfur's "I" (capital i) are equivalent, the prismation (extrusion).

Philip's ">" and Quickfur's "A" are equivalent too, the pyramidization (tapering).

Philip's "O" (spin - of some axis of the "base") and Quickfur's "O" (roundish bi-tapering of the base) generally differ.

Quickfur further added in his much older setup also the operator "X" of bi-tapering.

To my opinion "X", if taken not combinatorically but rather metrically, so seeming like a good operator for a square base (providing the octahedron), looks quite odd. For soon beyond, the base could have a radius larger than sqrt(2) resp. smaller than that. Then the bi-tapering becomes rather flat, disc shaped resp. rather tall, needle shaped. Both usually tend to run out of interesting shapes in general.

Yeah, I think my X and O operators are really only interesting if treated topologically.

Same holds true for Quickfur's roundish bi-tapering. Not only that those shapes tend to get any kind of elliptic cross-sections, you even get thingies like the intersection of 2 orthogonal cylinders instead of truely round things. - So I fear Philip's spin is much better suited to describe round things. Even if he still has to work out some consistent set of rules, how to select that axis of the base, which would perform the rotation...

I would argue that cylindrical intersections (crinds) represent an under-considered class of shapes. Keiji actually liked the fact that my O operators start producing crind-like shapes in higher dimensions. They do have interesting topological properties too, such as in things in 4D that have surfaces that look like polygonal lunes or circular lunes. But I agree that it's not very good at describing roundish shapes. For that, some kind of true rotation like Philip's O would probably be better suited.
quickfur
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### Re: Yet another improved notation for shapes

So, is that a sufficient definition of the lathe operator?
It is by will alone, I set my donuts in motion
ICN5D
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### Re: Yet another improved notation for shapes

ICN5D wrote:[...][...]
For Rotating into N+1 around an N-1 Bisecting Plane:

1D- x - one 0-plane: origin (0,0)

2D - xy - two 1-planes: x, y

3D - xyz - three 2-planes: xy, xz, yz

4D - xyzw - four 3-planes: xyz, xyw, xzw, yzw

5D - xyzwv - five 4-planes: xyzw, xyzv, xzwv, xywv, yzwv

6D - xyzwvu - six 5-planes: xyzwv, xyzwu, xyzvu, xywvu, xzwvu, yzwvu
[...]

I don't like this at all. The ordering of the letters is completely counterintuitive: you start with x, then go to y and z, and then for no good reason jump back to w and start counting backwards! And then what happens when you reach past 'a'? Well OK, I understand the reason, but I don't like it. A general notation should not encode what was essentially a historical accident (x, y, and z were chosen because at the time people didn't know geometry existed beyond 3D). Using numerical indices will solve both the ordering problem and the extension problem (you don't have to start reaching for Greek letters once you go past 26D), and it also allows you to easily tell which axis you're talking about without having to count your fingers every time, and it corresponds directly with the dimension it's introduced in: 1 = X axis = the dimension introduced in 1D, 2 = Y axis = the dimension introduced in 2D, 3 = Z axis = the dimension introduced in 3D, etc.. Then when you see a pair of indices like "7,9", you know immediately the rotation involves the two axes that were introduced, respectively, in 7D and 9D. (As opposed to "rotation around the f-r plane": quick, without counting your fingers, which dimensions were the f and r axes introduced in?)

Now I realize that familiarity can be a big barrier sometimes -- we all grew up with X, Y, and Z, and for the most part, popular culture has associated W with 4 for no good reason, and we are all resistant to change. But this kind of notation is really inconsistent, and once you get used to numerical indices, you'll realize it's a far superior approach.

Having said all that, though, this is just my opinion, and you don't have to follow it.
quickfur
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### Re: Yet another improved notation for shapes

As opposed to "rotation around the f-r plane": quick, without counting your fingers, which dimensions were the f and r axes introduced in?)

Hahaha, you crack me up! Alright, we'll use numbers then. I guess I just needed you real, unobstructed opinion. Remember, now, I don't want you holding back! I must apologize for my photographic memory, I can recite the alphabet backwards from any letter at will. It comes from learning the alphabet backwards in the form of a single word. Not really much of an achievement, though. Just a fluke of the nature. And, before you say I have too much time on my hands, trust me, I already know. It was about 15 years ago.

Okay, so, numbers then:

1D- 1 - one 0-plane: origin (0,0)

2D - 12 - two 1-planes: 1, 2

3D - 123 - three 2-planes: 12, 13, 23

4D - 1234 - four 3-planes: 123, 124, 134, 234

5D - 12345 - five 4-planes: 1234, 1235, 1245, 1345, 2345

6D - 123456 - six 5-planes: 12345, 12346, 12356, 12456, 13456, 23456

[...]Oq = rotate shape [...] into N+1, assign axis q as the axis in motion, turning the faces here into a torus. All other axes are stationary.

What do you think about the |O, >O, OO rules? Unnecessary now with the subscript rotating axis?
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### Re: Yet another improved notation for shapes

Seeing those neat incidence trees for the rotopes inspired me to use something similar in my computation system. In these trees, or more like matrices, all elements are represented without the use of incidence markers. ASCII art cannot show the complex lacing very well, and it's not the focus in this algorithm.

In the first matrix, we list all n-cells of the cone |O>, then the circle |O. Now, the next step is to multiply these two together to get their cartesian product. By multiply, all we're doing is sticking terms together. When we do this with two shapes with N and M dimensions, make n*m-prism in n+m dimensions. So, we place the new product on the appropriate dimension level. For better part of six years I have mastered the n-1 cell computation, by use of my notation and algorithms. Now, with this new system, I can enumerate all n-cells at once.

And, boy did I find some surprises with the cylconinder, one I thought I knew intricately well. It was the duocylinder margin [(O)(O)], and the line torus |(O)(O) that I didn't expect. The duocyl margin I should have expected as a 2-cell, but I'd never seen it in a computation other than use with the tiger |O[(O)(O)]. I knew a duocylinder was a 4-cell, but I never derived these elements in this way before. It was more the line ditorus |(O)(O) that surprised me. But, it makes perfect sense. As a 3-cell, it is the single binding surface between both of the curved 4-cells, torinder and cone torus. This kind of line ditorus is better represented by |^(O)(O), where the ^ means " attached face". It's the cartesian product with the curved 2-cell of the cone |^(O) and the disk-edge (O). The |^(O) is the special case of the line torus, where one 0-cell of the line is anchored in place during the torus operation, making the triangular rolling side of a cone.

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`5                                  [  |O>|O                  ]  cylconinder teron                                   [                         ]4                                  [  |O|O , |O|(O) , |O>(O) ]  duocylinder , torinder , cone-torus                                    [                         ]3  [ |O>       ]                   [  2-|O(O) , |(O)(O)      ]  2x ortho torii , line-ditorus   [           ]                   [                         ]2  [ |O , |(O) ]     [ |O  ]       [  |O , [(O)(O)]          ]  disk , duocylinder margin   [           ]     [     ]       [                         ]1  [ (O)       ]     [ (O) ]       [  (O) , [∞]              ]disk edge, infinite edges along duocyl margin   [           ]  X  [     ]   ==  [                         ]0  [  *        ]     [  ∞  ]       [   ∞                     ]  infinite vertices along disk edge   [           ]     [     ]       [                         ]-1 [  n        ]     [  n  ]       [   n                     ]  namon`

So, by starting with all cells, products are very easy to do. I haven't played with the linear operators yet, but I suspect | and > are just as equally intuitive. The lathe will be more complex, as the axis of orientation will need to be addressed, in order to establish the bisecting plane. A nice, complex challenge.

--Philip
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### Re: Yet another improved notation for shapes

This one could also be used to represent the lacing.

Along the columns and rows:
"---" , "|" , "\" , "/" is a direct connection
" : " is a separation

Along the rows:
"2+" is an ortho attached pair of 2
"2-" is a separate pair of 2
"2^" is an attached pair of 2
"n[...]q" is n shapes of q in [...] arrangement

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`5                              [           |O>|O         ]5]             cylconinder                                [        /    |     \     ]             /      |      \4                              [ |O|^(O)---|O>(O)---|O|O ]4]  torinder---cone-torus---duocylinder                                [    \    /         /     ]         |      /           /3  [ |O>        ]              [ |^(O)(O)---2+|O(O)      ]3]  line-ditorus---2x ortho torii   [  |  \      ]              [  /   \     /            ]       /         \      /2  [ |O---|^(O) ]   [ |O  ]    [ |O : [(O)(O)]           ]2]  disk : duocylinder margin   [  |  /      ]   [  |  ]    [  |  :  |                ]       |   :         |1  [ (O)        ]   [ (O) ]    [ (O) : [∞]               ]1]  disk edge : inf edges on duocyl margin   [  :         ] X [  |  ] == [  |  :  |                ]       |          :       \0  [  *         ]   [ (∞) ]    [ (∞) : [∞]               ]0] inf vert on disk edge : inf V on duocyl margin   [  |         ]   [  |  ]    [   \  /                  ]                    \      /-1 [  n         ]   [  n  ]    [    n                    ]-1]                  namon`

Actually, I like this better. It has more information in it, without destroying the simplicity. I feel it's important to notate what the incidences are. Everything there is to know about the shape is in matrices. The notation tells us what the products are when we stick them together. It's awesome, and there will be more.

--Philip
Last edited by ICN5D on Mon Feb 24, 2014 6:52 am, edited 2 times in total.
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### Re: Yet another improved notation for shapes

|O|O : Duocylinder
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`4                           [   |O|O   ]4]  duocylinder                            [    |     ]      |3                           [ 2+|O(O)  ]3]  2x ortho torii                             [    |     ]      |2  [ |O  ]     [ |O  ]      [ [(O)(O)] ]2]  duocylinder margin    [  |  ]     [  |  ]      [    |     ]      |1  [ (O) ]     [ (O) ]      [   [∞]    ]1]  infinite edges on margin   [  |  ]  X  [  |  ]  ==  [    |     ]      |0  [ (∞) ]     [ (∞) ]      [   [∞]    ]0]  infinite vertices on margin   [  |  ]     [  |  ]      [    |     ]      |-1 [  n  ]     [  n  ]      [    n     ]-1] namon`

|>|O : Cyltrianglinder
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`4                           [     |>|O      ]4]             cyltrianglinder                            [     /   \     ]                 /           \3                           [ |>(O)---3^||O ]3]     triangle-torus----3x attached cylinders                            [    |     |    ]                |             |2  [ |>  ]     [ |O  ]      [ 3^|(O)---3-|O ]2]  3x attached line-torii----3x circles   [  |  ]     [  |  ]      [      \  /     ]                 \           /1  [ 3^| ]     [ (O) ]      [      3-(O)    ]1]               3x disk edges   [  |  ]  X  [  |  ]  ==  [       |       ]                       |0  [ 3-* ]     [ (∞) ]      [      3-(∞)    ]0]            3x inf vert disk edges   [  |  ]     [  |  ]      [       |       ]                       |-1 [  n  ]     [  n  ]      [       n       ]-1]                  namon`

|||O : Cubinder
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`4                           [     |||O      ]4]                 cubinder                            [     /   \     ]                  /        \3                           [ ||(O)---4^||O ]3]      square-torus----4x attached cylinders                            [    |     |    ]                 |           |2  [ ||  ]     [ |O  ]      [ 4^|(O)---4-|O ]2]  4x attached line-torii----4x circles   [  |  ]     [  |  ]      [      \  /     ]                  \         /1  [ 4^| ]     [ (O) ]      [      4-(O)    ]1]               4x disk edges   [  |  ]  X  [  |  ]  ==  [       |       ]                       |0  [ 4-* ]     [ (∞) ]      [      4-(∞)    ]0]            4x inf vert disk edges   [  |  ]     [  |  ]      [       |       ]                       |-1 [  n  ]     [  n  ]      [       n       ]-1]                  namon`

|>[|>] : Duotrianglinder
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`4                           [   |>[|>]    ]4]         duotrianglinder                            [     |       ]                 |         3                           [    6^|>|    ]3]   6x attached triangle prisms                            [    /   \    ]              /      \             2  [ |>  ]     [ |>  ]      [ 6-|>---9^|| ]2]  6x triangles----9x attached squares   [  |  ]     [  |  ]      [    \   /    ]              \      /          1  [ 3^| ]     [ 3^| ]      [     18^|    ]1]        18x attached edges   [  |  ]  X  [  |  ]  ==  [      |      ]                 |0  [ 3-* ]     [ 3-* ]      [     9-*     ]0]           9x vertices   [  |  ]     [  |  ]      [      |      ]                 |-1 [  n  ]     [  n  ]      [      n      ]-1]            namon`

|>|| : Triangle Diprism
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`4                           [     |>||      ]4]            triangle diprism                            [     /  \      ]                   /      \     3                           [ 4^|>|---3^||| ]3]  4x triangle prisms---3x cubes                            [    |    |     ]                  |        |             2  [ |>  ]     [ ||  ]      [ 4-|>---15^||  ]2]       4x triangles---15x squares   [  |  ]     [  |  ]      [     \  /      ]                   \       /          1  [ 3^| ]     [ 4^| ]      [     24^|      ]1]            24x attached edges      [  |  ]  X  [  |  ]  ==  [      |        ]                      |0  [ 3-* ]     [ 4-* ]      [     12-*      ]0]               12x vertices      [  |  ]     [  |  ]      [      |        ]                       |-1 [  n  ]     [  n  ]      [      n        ]-1]                  namon`

|||| : Tesseract
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`4                           [  ||||   ]4]    tesseract                            [   |     ]          |          3                           [  8^|||  ]3]    8x cubes                            [    |    ]          |                    2  [ ||  ]     [ ||  ]      [  24^||  ]2]  24x squares   [  |  ]     [  |  ]      [    |    ]          |                1  [ 4^| ]     [ 4^| ]      [  32^|   ]1]   32x edges    [  |  ]  X  [  |  ]  ==  [    |    ]          |         0  [ 4-* ]     [ 4-* ]      [  16-*   ]0]   16x vertices      [  |  ]     [  |  ]      [    |    ]          |          -1 [  n  ]     [  n  ]      [    n    ]-1]     namon`
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### Re: Yet another improved notation for shapes

Still refining the notation. Just played around with the trees, and I noticed that it's very polynomial-like. The two shapes mult together is binomial expansion, and the product builds the higher shape up with combining terms. So, I rearranged them into these:

Triangle diprism
{|> • 3^| • 3:* } x {|| • 4^| • 4:* } == {|>|| • {4^|>| + 3^|||} • {4:|> + 15^||} • 24^| • 12:* }

Duocylinder
{|O • (O) • (∞) } x {|O • (O) • (∞) } == {|O|O • 2+|O(O) • [(O)(O)] • [∞] • [∞] }

Cyltrianglinder
{|> • 3^| • 3:* } x {|O • (O) • (∞) } == {|>|O • {3^|O| + |>(O)} • {3:|O + 3^|(O)} • 3:(O) • 3:(∞) }

Cubinder
{|| • 4^| • 4:* } x {|O • (O) • (∞) } == {|||O • {4^|O| + ||(O)} • {4:|O + 4^|(O)} • 4:(O) • 4:(∞) }

Duotrianglinder
{|> • 3^| • 3:* } x {|> • 3^| • 3:* } == {|>[|>] • 6^|>| • {6:|> + 9^||} • 18^| • 9:* }

Tesseract
{|| • 4^| • 4:* } x {|| • 4^| • 4:* } == {|||| • 8^||| • 24^|| • 32^| • 16:* }

Cylconinder
{|O> • {|O + |^(O)} • (O) : {(∞) : * } } x {|O • (O) • (∞) } ==

{|O>|O • {|O|O + |O>(O) + |O|^(O)} • {2+|O(O) + |^(O)(O)} • {|O : [(O)(O)]} • {(O) : [∞]} • {(∞) : [∞]} }

This is rapidly evolving into something easier to work with and express, and reflects actual equations. Next step is to further develop the linear operators and figure out how to apply the spin. And 6D cartesian products, of course! Probably that conindric trianglinder |O>[|>|] I detailed before.
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### Re: Yet another improved notation for shapes

By using the symbols as a numbering system, I have created several algorithms that compute all of the cells of the listed shapes. It's very powerful because it does a lot of math all at once. That's the beauty of it.

Organizing the cells by dimension level has the advantage of turning the sequence into a polynomial. We can then apply well known techniques to this system, and do amazing math that condenses calculus big time. All symbols can be translated into equations, which when added up, produce the same outcome as calculus. The next step would be to prove this of course! I call it ' Dimensionometry ', a new topological math language that focuses on surfaces. There is a lot more work to be done, of course!

I'm almost done creating the 5D poly equations in that huge list, all others are complete. I'll post it when I'm done.

-- Philip
Last edited by ICN5D on Mon Mar 03, 2014 6:25 am, edited 1 time in total.
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### Re: Yet another improved notation for shapes

Well, it took some time and some work, and this is the first time I did all of 5D on the list above. There may be some errors.

POLYNOMIAL EQUATIONS
----------------------
Below are the topology formulas of all shapes up to 5D. Organizing n-cells by dimension level has a similarity with the powers of ' x ', as in x² + 2x + 1, which resembles the sequence of the line {| • 2:* • n} . Cartesian Products are done with binomial expansion of both surfaces. Tapering is done with +[*], lace to point and add base n-cells to final product. Extrusion is done with the line product. Lathe is done in a lathe table, where all n-cells are lined up on axes to establish the stationary bisecting plane. This is still in development, so the original version is used [()] . I have omitted the last " • n ", in all below.

GENERAL TERMINOLOGY
---------------------
Q = general shape variable
n,m = number of shapes
Q^(...) = anchored Q-torus, where one N-1 cell of Q was held in place during torus (...) operation
margin = n-2 edge of (n-sphere,n-sphere)-prism
" • " == the dimension level separator

ARRANGEMENTS
--------------
2+Q == an ortho attached pair of 2 shape Q's , only found on (n-sphere,n-sphere)-prism

2:Q == a separate pair of 2 shape Q's , plays the role of vertices

2^Q == an attached pair of 2 shape Q's , bound by n-2 cells

n[...]Q == n shapes of Q in [...] arrangement

n.Q == n mixed arrangements of shapes Q , n>3

N-CELL COMPUTATIONS
--------------------
Q+[*] = {B + Q>} , taper shape Q to a point, adding base n-cells B to final product

2:Q+[*] = 2^Q> , taper 2 separate Q's to one point, making attached pair of Q>

Q[| • 2:* • n] = Q| , extrude shape Q into N+1

nQ+[O] = nQO , lathe shape Q into N+1

nQ+[2()] = {Q(O) + (n-2)QO} , only 2 n^Q-cells get Q(O) / (n-2)^QO for n>4 / (n-2):QO for n=4

__________________0D___________________

0D

Point - *

[ * ]

___________________1D__________________

1D • 0D

Line - |

*
+[*]
---------
| • 2:*

__________________2D___________________

2D • 1D • 0D

Circle - |O

| • 2:*
[O]-[2()]
---------
|O • (O)

Triangle - |>

| • 2:*
+[*]
---------------
|> • 3^| • 3:*

Square - ||

| • 2:*
[| • 2:*]
--------------
|| • 4^| • 4:*

_________________3D__________________

3D • 2D • 1D • 0D

|OO - SPHERE

|O • (O)
+[O]+[()]
----------
|OO • (OO)

|O> - CONE

|O • (O)
+[*]
----------------------------
|O> • {|O + |^(O)} • (O) • *

|O| - CYLINDER

|O • (O)
[| • 2:*]
---------------------------
|O| • {2:|O + |(O)} • 2:(O)

||O - CYLINDER-ii

|| • 4^| • 4:*
+[O]+[2()]+[2()]
---------------------------
||O • {2:|O + |(O)} • 2:(O)

|>> - TETRAHEDRON

|> • 3^| • 3:*
+[*]
----------------------
|>> • 4^|> • 6^| • 4:*

|>| - TRIANGLE PRISM

|> • 3^| • 3:*
[ | • 2:*]
-------------------------------
|>| • {2:|> + 3^||} • 9^| • 6:*

|>O - CONE-ii

|> • 3^| • 3:*
+[O]+[2()]+[2()]
----------------------------
|>O • {|^(O) + |O} • (O) • *

||> - SQUARE PYRAMID

|| • 4^| • 4:*
+[*]
-----------------------------
||> • {|| + 4^|>} • 8^| • 5:*

||| - CUBE

|| • 4^| • 4:*
[ | • 2:*]
-----------------------
||| • 6^|| • 12^| • 8:*

___________________4D______________________

4D • 3D • 2D • 1D • 0D

GLOME - |OOO

|OO • (OO)
[O] • [()]
------------
|OOO • (OOO)

|OO> - SPHONE

|OO • (OO)
+[*]
--------------------------------
|OO> • {|OO + |^(OO)} • (OO) • *

|OO| - SPHERINDER

|OO • (OO)
[| • 2:*]
------------------------------
|OO| • {2:|OO + |(OO)} • 2:(OO)

|O>> - DICONE

|O> • {|^(O) + |O} • (O) • *
+[*]
---------------------------------------------------------
|O>> • {2^|O> + |>^(O)} • {2^|^(O) + |O} • {(O) : |} • 2:*

|O>| - CONINDER

|O> • {|^(O) + |O} • (O) • *
[| • 2:*]
---------------------------------------------------------------------------
|O>| • {2:|O> + |O| + ||^(O)} • {2:|O + 2:|^(O) + |(O)} • {2:(O) + |} • 2:*

|O|O - DUOCYLINDER

|O • (O)
[|O • (O)]
------------------------
|O|O • 2+|O(O) • [(O)(O)]

||O> - CYLINDRONE

||O • {2:|O + |(O)} • 2:(O)
+[*]
-----------------------------------------------------------------
||O> • {||O + 2^|O> + |>^(O)} • {2:|O + |(O) + 2:|^(O)} • 2:(O) • *

|>>> - PENTACHORON

|>> • 4^|> • 6^| • 4:*
+[*]
---------------------------------
|>>> • 5^|>> • 10^|> • 10^| • 5:*

|>>| - TETRAHEDRINDER

|>> • 4^|> • 6^| • 4:*
[| • 2:*]
---------------------------------------------------
|>>| • {2:|>> + 4^|>|} • {8.|> + 6^||} • 16^| • 8:*

|>|O - CYLTRIANGLINDER

|> • 3^| • 3:*
[|O • (O)]
------------------------------------------------
|>|O • {3^|O| + |>(O)} • {3:|O + 3^|(O)} • 3:(O)

|>|> - TRIANGL PRISM PYRAMID

|>| • {2:|> + 3^||} • 9^| • 6:*
+[*]
----------------------------------------------------------
|>|> • {|>| + 2^|>> + 3^||>} • {11^|> + 3^||} • 15^| • 7:*

|>|| - TRIANGLE DIPRISM

|> • 3^| • 3:*
[|| • 4^| • 4:*]
-----------------------------------------------------
|>|| • {4^|>| + 3^|||} • {4:|> + 15^||} • 24^| • 12:*

||>> - DIPYRAMID

||> • {|| + 4^|>} • 8^| • 5:*
+[*]
--------------------------------------------------
||>> • {2^||> + 4^|>>} • {12^|> + ||} • 13^| • 6:*

||>| - PYRAMID PRISM

||> • {|| + 4^|>} • 8^| • 5:*
[| • 2:*]
----------------------------------------------------------
||>| • {||| + 2:||> + 4^|>|} • {10^|| + 8.|>} • 21^| • 10:*

|||O - CUBINDER

|| • 4^| • 4:*
[|O • (O)]
------------------------------------------------
|||O • {4^|O| + ||(O)} • {4:|O + 4^|(O)} • 4:(O)

|||> - HEMDODECACHORON / CUBE PYRAMID

||| • 6^|| • 12^| • 8:*
+[*]
--------------------------------------------------
|||> • {||| + 6^||>} • {6^|| + 12^|>} • 20^| • 9:*

|||| - TESSERACT

|| • 4^| • 4:*
[|| • 4^| • 4:*]
----------------------------------
|||| • 8^||| • 24^|| • 32^| • 16:*

|>[|>] - DUOTRIANGLINDER

|> • 3^| • 3:*
[|> • 3^| • 3:*]
-------------------------------------------
|>[|>] • 6^|>| • {6:|> + 9^||} • 18^| • 9:*

___________________________5D___________________________

5D • 4D • 3D • 2D • 1D • 0D

|OOOO - PENTASPHERE

|OOO • (OOO)
+[O] • +[()]
--------------
|OOOO • (OOOO)

|OOO> - GLONE

|OOO • (OOO)
+[*]
-----------------------------------
|OOO> • {|OOO + |^(OOO)} • (OOO) • *

|OOO| - GLOMINDER

|OOO • (OOO)
[| • 2:*]
-----------------------------------
|OOO| • {2:|OOO + |(OOO)} • 2:(OOO)

|OO>> - DISPHONE

|OO> • {|OO + |^(OO)} • (OO) • *
+[*]
--------------------------------------------------------
|OO>> • {2^|OO> + |>^(OO)} • {|OO + 2^|^(OO)} • (OO) • *

|OO>| - SPHONINDER

|OO> • {|OO + |^(OO)} • (OO) • *
[| • 2:*]
-----------------------------------------------------------------------------
|OO>| • {2:|OO> + |OO| + ||^(OO)} • {2:|OO + |(OO) + 2:|^(OO)} • (OO) • | • 2:*

|OO|O - CYLSPHERINDER

|OO • (OO)
[|O • (O)]
------------------------------------
|OO|O • {|OO(O) + |O(OO)} • [(OO)(O)]

|OO|> - SPHERINDRONE

|OO| • {2:|OO + |(OO)} • 2:(OO)
+[*]
--------------------------------------------------------------------------
|OO|> • {|OO| + 2^|OO> + |>^(OO)} • {2:|OO + |(OO) + 2^|^(OO)} • 2:(OO) • *

|OO|| - CUBSPHERINDER

|OO • (OO)
[|| • 4^| • 4:*]
-----------------------------------------------------
|OO|| • {4^|OO| + ||(OO)} • {4:|OO + 4^|(OO)} • 4:(OO)

|O>>> - TRICONE

|O>> • {2^|O> + |>^(O)} • {2^|^(O) + |O} • {(O) : |} • 2:*
+[*]
----------------------------------------------------------------------------------------
|O>>> • {3^|O>> + |>>^(O)} • {3^|O> + 3^|>^(O)} • {3^|^(O) + |O : |>} • {(O) : 3^|} • 3:*

|O>>| - DICONINDER

|O>> • {2^|O> + |>^(O)} • {2^|^(O) + |O} • {(O) : |} • 2:*
[| • 2:*]
-----------------------------------------------------------------------------------------------------------------------------
|O>>| • {2:|O>> + 2^|O>| + |>|^(O)} • {||O + 4.|O> + 2:|>(O) + 2^||^(O)} • {2:|O + 4.|^(O) + |(O) : ||} • {2:(O) : 4^|} • 4:*

|O>|O - CYLCONINDER

|O> • {|O + |^(O)} • (O) • *
[|O • (O)]
--------------------------------------------------------------------------------
|O>|O • {|O|O + |O>(O) + |O|^(O)} • {|^(O)(O) + 2+|O(O)} • {[(O)(O)] : |O} • (O)

|O>|> - CONINDER PYRAMID

|O>| • {2:|O> + |O| + ||^(O)} • {2:|O + 2:|^(O) + |(O)} • {2:(O) + |} • 2:*
+[*]
-----------------------------------------------------------------------------------------------------------------------
|O>|> • {|O>| + |O>> + ||O> + ||>^(O)} • {4^|O> + ||O + ||^(O) + 3^|>^(O)} • {2:|O : |> + 4^|^(O)} • {2:(O) : 3^|} • 3:*

|O>|| - CONE DIPRISM

|O> • {|O + |^(O)} • (O) • *
[|| • 4^| • 4:*]
----------------------------------------------------------------------------------------------------------
|O>|| • {|||O + 4^|O>| + |||^(O)} • {4:|O> + 4^||O + 5^||(O)} • {|| : 4:|O + 8^|(O)} • {4^| : 4:(O)} • 4:*

||OO> - DUOCYLINDRONE

|O|O • 2+|O(O) • [(O)(O)]
+[*]
-----------------------------------------------------
|O|O> • {|O|O + 2+|O>(O)} • {2+|O(O) + |^[(O)(O)]} • *

||O>> - DICYLINDRONE

||O> • {||O + 2^|O> + |>^(O)} • {2:|O + |(O) + 2:|^(O)} • 2:(O) • *
+[*]
----------------------------------------------------------------------------------------------------
||O>> • {2^||O> + 2^|O>> + |>>^(O)} • {||O + 4^|O> + 4^|>^(O)} • {2:|O + 5^|^(O)} • {| : 2:(O)} • 2:*

||O>| - CYLINDRONE PRISM

||O> • {||O + 2^|O> + |>^(O)} • {2:|O + |(O) + 2:|^(O)} • 2:(O) • *
[| • 2:*]
--------------------------------------------------------------------------------------------------------------------------------
||O>| • {2:||O> + 2^|O>| + |||O + |>|^(O)} • {4:||O + ||(O) + 4.|O> + 2:||^(O) + 2:|>^(O)} • {4:|O + 8^|(O)} • {4:(O) : |} • 2:*

|>>>> - HEXATERON

|>>> • 5^|>> • 10^|> • 10^| • 5:*
+[*]
-------------------------------------------
|>>>> • 6^|>>> • 15^|>> • 20^|> • 15^| • 6:*

|>>>| - PENTACHORINDER

|>>> • 5^|>> • 10^|> • 10^| • 5:*
[| • 2:*]
----------------------------------------------------------------------------
|>>>| • {2:|>>> + 5^|>>|} • {10.|>> + 10^|>|} • {20.|> + 10^||} • 25^| • 10:*

|>>|O - CYLTETRAHEDRINDER

|>> • 4^|> • 6^| • 4:*
[|O • (O)]
----------------------------------------------------------------------
|>>|O • {4^|>|O + |>>(O)} • {6^||O + 4^|>(O)} • {4:|O + 6^|(O)} • 4:(O)

|>>|> - TETRAHEDRINDER PYRAMID

|>>| • {2:|>> + 4^|>|} • {2:4^|> + 6^||} • 16^| • 8:*
+[*]
----------------------------------------------------------------------------------------
|>>|> • {|>>| + 2^|>>> + 4^|>|>} • {10^|>> + 6^||> + 4^|>|} • {24^|> + 6^||} • 24^| • 9:*

|>>|| - TETRAHEDRON DIPRISM

|>> • 4^|> • 6^| • 4:*
[|| • 4^| • 4:*]
-----------------------------------------------------------------------------------
|>>|| • {4^|>>| + 4^|>||} • {4:|>> + 16^|>| + 6^|||} • {16.|> + 28^||} • 40^| • 16:*

|>|OO - DUOCYLTRIANGLINDER

|>|O • {3^||O + |>(O)} • {3:|O + 3^|(O)} • 3:(O)
+[O] • {[2()] + [O] } • {[2()] + [O] } • [2()]
--------------------------------------------------------------------------------
|>|OO • {|O|O + ||O^(O) + |>O(O)} • {|OO : 2+|O(O) + |(O)(O)} • {(OO) : [(O)(O)]}

|>|O> - CYLTRIANGLINDRONE

|>|O • {3^||O + |>(O)} • {3:|O + 3^|(O)} • 3:(O)
+[*]
-----------------------------------------------------------------------------------------------------
|>|O> • {|>|O + 3^||O> + |>>^(O)} • {3^||O + 3^|O> + 4^|>^(O)} • {3:|O + 3^|(O) + 3^|^(O)} • 3:(O) • *

|>|O| - CYLTRIANDYINDER

|>|O • {3^||O + |>(O)} • {3:|O + 3^|(O)} • 3:(O)
[| • 2:*]
---------------------------------------------------------------------------------------------------------
|>|O| • {2:|>|O + 3^|||O + |>|(O)} • {6.||O + 3:||O + 3^||(O) + 2:|>(O)} • {6:|O + 6.|(O) + 3:|(O)} • 6:(O)

|>|>> - TRIANGLE PRISM DIPYRAMID

|>|> • {|>| + 2^|>> + 3^||>} • {11^|> + 3^||} • 15^| • 7:*
+[*]
----------------------------------------------------------------------------------------
|>|>> • {2^|>|> + 2^|>>> + 3^||>>} • {|>| + 13^|>> + 6^||>} • {26^|> + 3^||} • 22^| • 8:*

|>|>| - TRIANGLE PRISM PYRAMID PRISM

|>|> • {|>| + 2^|>> + 3^||>} • {11^|> + 3^||} • 15^| • 7:*
[| • 2:*]
-----------------------------------------------------------------------------------------------------------
|>|>| • {2:|>|> + |>|| + 2^|>>| + 3^||>|} • {13.|>| + 6.||> + 4.|>> + 3^|||} • {22.|> + 21.||} • 30^| • 14:*

|>||> - TRIANGLE DIPRISM PYRAMID

|>|| • {4^|>| + 3^|||} • {4:|> + 15^||} • 24^| • 12:*
+[*]
---------------------------------------------------------------------------------------------------
|>||> • {|>|| + 4^|>|> + 3^|||>} • {4^|>| + 3^||| + 4^|>> + 15^||>} • {28.|> + 15^||} • 36^| • 13:*

|>||| - TRIANGLE TRIPRISM

|>|| • {4^|>| + 3^|||} • {4:|> + 15^||} • 24^| • 12:*
[| • 2:*]
---------------------------------------------------------------------------
|>||| • {6.|>|| + 3^||||} • {12.|>| + 21.|||} • {8:|> + 54.||} • 60^| • 24:*

||>>> - SQUARE-TRIPYRAMID

||>> • {2^||> + 4^|>>} • {12^|> + ||} • 13^| • 6:*
+[*]
----------------------------------------------------------------------
||>>> • {3^||>> + 4^|>>>} • {3^||> + 16^|>>} • {25^|> + ||} • 19^| • 7:*

||>>| - SQUARE DIPYRAMID PRISM

||>> • {2^||> + 4^|>>} • {12^|> + ||} • 13^| • 6:*
[| • 2:*]
--------------------------------------------------------------------------------------------------
||>>| • {2:||>> + 2^||>| + 4^|>>|} • {4.||> + 8.|>> + 12^|>| + 2:||} • {24.|> + 15.||} • 32^| • 12:*

||>|O - CYLHEMOCTAHEDRINDER

||> • {|| + 4^|>} • 8^| • 5:*
[|O • (O)]
-------------------------------------------------------------------------------------
||>|O • {|||O + 4^|>|O + ||>(O)} • {8^||O + ||(O) + 4^|>(O)} • {5:|O + 8^|(O)} • 5:(O)

||>|> - SQUARE PYRAMID PRISM PYRAMID

||>| • {||| + 2:||> + 4^|>|} • {10^|| + 8.|>} • 21^| • 10:*
+[*]
------------------------------------------------------------------------------------------------------
||>|> • {||>| + |||> + 2^||>> + 4^|>|>} • {||| + 12.||> + 4^|>| + 8^|>>} • {10^|| + 29.|>} • 31^| • 11:*

||>|| - SQUARE PYRAMID DIPRISM

||> • {|| + 4^|>} • 8^| • 5:*
[|| • 4^| • 4:*]
-------------------------------------------------------------------------------------------
||>|| • {|||| + 4^||>| + 4^|>||} • {4:||> + 16^|>| + 12^|||} • {16.|> + 41.||} • 52^| • 20:*

|O|O| - DUOCYLDYINDER

|O|O • 2+|O(O) • [(O)(O)]
[| • 2:*]
---------------------------------------------------------------
|O|O| • {2:|O|O + 2+||O(O)} • {4.|O(O) + |[(O)(O)]} • 2:[(O)(O)]

|||O> - CUBINDRONE

|||O • {4^||O + ||(O)} • {4:|O + 4^|(O)} • 4:(O)
+[*]
-----------------------------------------------------------------------------------------------------------
|||O> • {|||O + 4^||O> + ||>(O)} • {4^||O + 4^|O> + ||(O) + 4^|>(O)} • {4:|O + 4^|(O) + 4:|^(O)} • 4:(O) • *

|||>> - CUBE DIPYRAMID

|||> • {||| + 6^||>} • {6^|| + 12^|>} • 20^| • 9:*
+[*]
----------------------------------------------------------------------------------
|||>> • {2^|||> + 6^||>>} • {||| + 12^||> + 12^|>>} • {6^|| + 32^|>} • 29^| • 10:*

|||>| - CUBE PYRAMID PRISM

|||> • {||| + 6^||>} • {6^|| + 12^|>} • 20^| • 9:*
[| • 2:*]
--------------------------------------------------------------------------------------------
|||>| • {2:|||> + |||| + 6^||>|} • {8.||| + 12.||> + 12^|>|} • {32.|| + 24.|>} • 49^| • 18:*

||||O - TESSERINDER

||| • 6^|| • 12^| • 8:*
[|O • (O)]
------------------------------------------------------------------------
||||O • {6^|||O + |||(O)} • {12^||O + 6^||(O)} • {8:|O + 12^|(O)} • 8:(O)

||||> - TESSERACT PYRAMID

|||| • 8^||| • 24^|| • 32^| • 16:*
+[*]
--------------------------------------------------------------------------
||||> • {|||| + 8^|||>} • {8^||| + 24^||>} • {24^|| + 32^|>} • 48^| • 17:*

||||| - PENTERACT

|||| • 8^||| • 24^|| • 32^| • 16:*
[| • 2:*]
----------------------------------------------
||||| • 10^|||| • 40^||| • 80^|| • 80^| • 32:*

|OO[|>] - SPHENTRIANGLINDER

|OO • (OO)
[|> • 3^| • 3:*]
-------------------------------------------------------
|OO[|>] • {3^|OO| + |>(OO)} • {3:|OO + 3^|(OO)} • 3:(OO)

|O>[|>] - CONTRIANGLINDER

|O> • {|O + |^(O)} • (O) • *
[|> • 3^| • 3:*]
-------------------------------------------------------------------------------------------------------------------
|O>[|>] • {|>|O + 3^|O>| + |>|(O)} • {3:|O> + 3^||O + |>(O) + 3^||^(O)} • {3:|O + 3^|(O) + 3:|^(O) + |>} • 3:(O) : 3^| • 3:*

||>[|>] - HEMOCTAHEDROTRIANGLINDER

||> • {|| + 4^|>} • 8^| • 5:*
[|> • 3^| • 3:*]
-----------------------------------------------------------------------------------------------
||>[|>] • {|>|| + 3^||>| + 4^|>[|>]} • {3:||> + 3^||| + 20^|>|} • {27.|| + 17.|>} • 39^| • 15:*

|>>[|>] - TETRAHEDROTRIANGLINDER

|>> • 4^|> • 6^| • 4:*
[|> • 3^| • 3:*]
--------------------------------------------------------------------------------
|>>[|>] • {3^|>>| + 4^|>[|>]} • {3:|>> + 18^|>|} • {16.|> + 18^||} • 30^| • 12:*

|>[|>]| - DUOTRIANGLINDYINDER

|>[|>] • 6^|>| • {6:|> + 9^||} • 18^| • 9:*
[| • 2:*]
--------------------------------------------------------------------------------
|>[|>]| • {2:|>[|>] + 6^|>||} • {18.|>| + 9^|||} • {12^|> + 36^||} • 45^| • 18:*

|>[|>]> - DUOTRIANGLINDRIC PYRAMID

|>[|>] • 6^|>| • {6:|> + 9^||} • 18^| • 9:*
+[*]
------------------------------------------------------------------------------------
|>[|>]> • {|>[|>] + 6^|>|>} • {6^|>| + 6^|>> + 9^||>} • {24.|> + 9^||} • 27^| • 10:*

the toratopes are next.....
Last edited by ICN5D on Tue Mar 04, 2014 4:54 am, edited 1 time in total.
It is by will alone, I set my donuts in motion
ICN5D
Pentonian

Posts: 1121
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

### Re: Yet another improved notation for shapes

Of course I couldn't help but build some awesome construction trees out of those new formulas. Here's a few I threw together:

[ * • n ] == *
+[*]
---------------
[ | • 2:* • n ] == |
[ | • 2:* • n ]
-----------------------
[ || • 4^| • 4:* • n ] == ||
+[*]
-----------------------------------------
[ ||> • {|| + 4^|>} • 8^| • 5:* • n ] == ||>
[ | • 2:* • n ]
----------------------------------------------------------------------------------
[ ||>| • {||| + 2:||> + 4^|>|} • {10^|| + 2:4^|>} • 21^| • 10:* • n ] == ||>|
[ | • 2:* • n ]
--------------------------------------------------------------------------------------------------------------------------
[ ||>|| • {|||| + 4^||>| + 4^|>||} • {4:||> + 16^|>| + 12^|||} • {16.|> + 41.||} • 52^| • 20:* • n ] == ||>||

[ * • n ] == *
+[*]
---------------
[ | • 2:* • n ] == |
+[*]
-----------------------
[ |> • 3^| • 3:* • n ] == |>
[ | • 2:* • n ]
-------------------------------------------
[ |>| • {2:|> + 3^||} • 9^| • 6:* • n ] == |>|
+[*]
------------------------------------------------------------------------------
[ |>|> • {|>| + 2^|>> + 3^||>} • {11^|> + 3^||} • 15^| • 7:* • n ] == |>|>
[| • 2:* • n]
---------------------------------------------------------------------------------------------------------------------------------------------
[ |>|>| • {2:|>|> + |>|| + 2^|>>| + 3^||>|} • {13.|>| + 6.||> + 4.|>> + 3^|||} • {22.|> + 21.||} • 30^| • 14:* • n ] == |>|>|

[ * • n ] == *
+[*]
---------------
[ | • 2:* • n ] == |
[O]-[()] • n
----------------
[ |O • (O) • n ] == |O
[ | • 2:* • n ]
--------------------------------------
[ ||O • {2:|O + |(O)} • 2:(O) • n ] == ||O
+[*]
----------------------------------------------------------------------------------------
[ ||O> • {||O + 2^|O> + |>^(O)} • {2:|O + |(O) + 2:|^(O)} • 2:(O) • * • n ] == ||O>
[ | • 2:* • n ]
---------------------------------------------------------------------------------------------------------------------------------------------------------------------
[ ||O>| • {2:||O> + 2^|O>| + |||O + |>|^(O)} • {4:||O + ||(O) + 4.|O> + 2:||^(O) + 2:|>^(O)} • {4:|O + 8^|(O)} • {4:(O) : |} • 2:* • n ] == ||O>|
It is by will alone, I set my donuts in motion
ICN5D
Pentonian

Posts: 1121
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

### Re: Yet another improved notation for shapes

For an in-depth explanation of how this math system works, see here .
It is by will alone, I set my donuts in motion
ICN5D
Pentonian

Posts: 1121
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL