Toratope Names

Discussion of shapes with curves and holes in various dimensions.

Toratope Names

Postby ndl » Wed Jan 09, 2019 1:59 am

Hi everyone,
I'm a little confused about the naming conventions of the toratopes. In 4D we have the cubinder which is short for cubic cylinder, which assumedly means a lathing of the cube (and a 3D cylinder would really be in full a "squaric cylinder"). But then what is the term spherinder? Both the duocylinder and spherinder are lathings of the 3D cylinder (in 2 different directions). A lathing of the sphere would be a glome. Also the torinder is not a lathing of the torus, that would be the spheritorus. Spheric Prism and Toric Prism would be more accurate. Same thing with the glominder in 5D and probably some others.

Thanks
ndl
Mononian
 
Posts: 12
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

Re: Toratope Names

Postby wendy » Wed Jan 09, 2019 2:25 am

Hi, ndl. Welcome.

A lot of torotope names are derived from the comb product. There are a lot of swirl toratopes too.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(Latex\) at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 1863
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Toratope Names

Postby ICN5D » Fri Jan 11, 2019 2:01 am

Hi ndl,

The suffix -inder is used to define an extrusion, or product with an interval. I don't think we have a naming convention for the bisecting rotate (lathing). It could be the prefix cyl-, come to think of it. But, we do have one for the non-bisecting rotate (embed/bundle over the circle) , which is -torus .

So, spheritorus means 2-sphere over the circle, and spherinder means prism of the sphere. Torinder can be built two ways: extrusion of the torus, and cylinder over the circle ( a 'cylindritorus' if you will). But, because of these ambiguous dual meanings, we don't really fuss over names much anymore.

Though, I do see the confusion. For the hypercylinders, we have : cylinder , cubinder , tesserinder , penterinder , etc, to define the repeated extrusion of a circle, which also happens to be equal to bisecting rotations of an n-cube. In addition, the prefix is also similar to the name of that respective n-cube. Definitely sends mixed signals, lol.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1058
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Toratope Names

Postby ndl » Fri Jan 11, 2019 6:15 am

Thank you ICN5D for your clarifications. Maybe you could also explain to me the operations that create the various types of torii. I think I understand what your calling non-bisecting rotate, I imagine it as a double bend of the square: into a cylinder and then the top edges bending towards each other inside (or outside really). But in 4D that seems to only create the torinder and the spheritorus.
The concept of spherating described in the wiki doesn't seem like a valid operation. All the other operations add one dimension to the existing toratope, but spherating adds an arbitrary amount and doesn't always represent a basic shape, for example spherating a line gets you a cylinder with two half spheres on the end.
ndl
Mononian
 
Posts: 12
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

Re: Toratope Names

Postby wendy » Fri Jan 11, 2019 7:54 am

Spheration is something i invented, to describe what happens when people use sticks and balls etc to make models. The sort of thing that Zome does.

If you draw, in a given dimension, the faintest lines representing a shape, it is hard to hold. But if you apply some arbitary thickness to it, then you can hold it.

A straight line, would give a long cylinder, but the two vertices are typically spherated at a different size, gives more like a dumbell shape. The atomium in Brussles ( http://www.atomium.be is an example of spheration.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(Latex\) at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 1863
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Toratope Names

Postby Klitzing » Fri Jan 11, 2019 12:39 pm

There is a lot being found right in this very forum, spread out on various threads of discussion.
I myself found it always inspiring to follow, even so I contributed not so much in the field of round shapes.

P. Wright, M. Četrnáct, and W. Krieger mainly invented the first type of constructive symbols,
while P. Pugeau and Quickfur invented a somewhat different type of constructive symbols.

At some time I tried to line out those constructions to myself and started to aggregate the easiest shapes for examplifying survey,
cf. https://bendwavy.org/klitzing/explain/round.htm. Perhaps this might serve to ease your entry too. ;)

--- rk
Klitzing
Pentonian
 
Posts: 1437
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Toratope Names

Postby ndl » Wed Jan 16, 2019 4:14 am

Thank you everyone for all the resources. I think I understand what spheration is I just have trouble understanding how it fits together with the other toratopic operations. I also was wondering about tapering, starting with a square isn't there a possibility of tapering of only one of the 2 dimensions to end up with the triangular prism? A square pyramid is more like a double tapering (the same way a torus is a double bending). If explained like this could tapering become commutable with the other operations?
ndl
Mononian
 
Posts: 12
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

Re: Toratope Names

Postby ICN5D » Wed Jan 16, 2019 10:49 pm

ndl wrote: I think I understand what your calling non-bisecting rotate, I imagine it as a double bend of the square: into a cylinder and then the top edges bending towards each other inside (or outside really). But in 4D that seems to only create the torinder and the spheritorus.


Not exactly. A bisecting rotate is a rotation (surface of revolution) around an n-1D plane, held at the origin. The bisecting plane behaves like a fixed axle of the rotation motion. This makes/adds rounded faces to the shape, leading to a more 'spherical thing'.

A non-bisecting rotate is when you first translate the shape away from origin (in any direction, usually coordinate), so that the origin lies outside of the interior. Then, revolve around the now outside n-1 plane (stationary axle). This motion makes donuts and adds holes to the shape.





ndl wrote:The concept of spherating described in the wiki doesn't seem like a valid operation. All the other operations add one dimension to the existing toratope, but spherating adds an arbitrary amount and doesn't always represent a basic shape, for example spherating a line gets you a cylinder with two half spheres on the end.


Spheration is tricky to explain. Think of it as more of a symbolic manipulation, that has a few different geometric and algebraic definitions, depending on what's being spherated. I trust that you've seen the open toratope notation symbols, like : (II)I , (II)(II) , (II)(II)(II) , III , etc. To spherate these shapes, simply add another pair of parentheses to the outside : ((II)I) , ((II)(II)) , ((II)(II)(II)) , (III) , etc. These are now the closed toratope forms.




ndl wrote:I also was wondering about tapering, starting with a square isn't there a possibility of tapering of only one of the 2 dimensions to end up with the triangular prism? A square pyramid is more like a double tapering (the same way a torus is a double bending). If explained like this could tapering become commutable with the other operations?


Tapering is meant to be a dimension adding operation, that's like a modified extrusion. So, instead of simply dragging a shape across n+1 dimensions, tapering means to shrink it to a point while dragging. Or, another way is to connect all points of a shape to a single point, into +1 higher dimension (convex hull with a point). But, if you assume unit edge shapes, a tesseract pyramid cannot extend into 5D, because of the whole degenerate pyramid height problem (derived through the pythagoras and long diagonal equations), and so remains in 4D, even though you've technically tapered it.

As far as I know, tapering is only commutable with the bisecting rotate. You can think of the two together as one operation that adds +2 dimensions. A cone is the only 3D example. You could either spin a line into a circle, then taper to cone, or taper the line into a triangle, then spin into a cone. Either way works. The extrude also commutes with bisecting rotate, so shapes like IO>IIO (6D cone x cylinder), have 4 forms : IO>IIO , I>OIIO , IO>IOI , I>OIOI , that are equal. But, extrude does not commute with taper.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1058
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: Toratope Names

Postby ndl » Thu Jan 17, 2019 4:11 am

ICN5D wrote:Spheration is tricky to explain. Think of it as more of a symbolic manipulation, that has a few different geometric and algebraic definitions, depending on what's being spherated. I trust that you've seen the open toratope notation symbols, like : (II)I , (II)(II) , (II)(II)(II) , III , etc. To spherate these shapes, simply add another pair of parentheses to the outside : ((II)I) , ((II)(II)) , ((II)(II)(II)) , (III) , etc. These are now the closed toratope forms.


Ok, after reading more carefully I think I understand the spheration described in the wiki as the "rss" operation, it's not something that adds dimension at all just rounds out what you already have by extrusion. It's interesting how the cylinder rounds out into a torus.

What I was describing about bending a shape in a new dimension to create a new shape, I think it's a new way way of looking at it not described before but accomplishes the same thing as the bisecting and non-bisecting rotate.

ICN5D wrote:Tapering is meant to be a dimension adding operation, that's like a modified extrusion. So, instead of simply dragging a shape across n+1 dimensions, tapering means to shrink it to a point while dragging. Or, another way is to connect all points of a shape to a single point, into +1 higher dimension (convex hull with a point). But, if you assume unit edge shapes, a tesseract pyramid cannot extend into 5D, because of the whole degenerate pyramid height problem (derived through the pythagoras and long diagonal equations), and so remains in 4D, even though you've technically tapered it.

As far as I know, tapering is only commutable with the bisecting rotate. You can think of the two together as one operation that adds +2 dimensions. A cone is the only 3D example. You could either spin a line into a circle, then taper to cone, or taper the line into a triangle, then spin into a cone. Either way works. The extrude also commutes with bisecting rotate, so shapes like IO>IIO (6D cone x cylinder), have 4 forms : IO>IIO , I>OIIO , IO>IOI , I>OIOI , that are equal. But, extrude does not commute with taper.


Yes, the way tapering is described here would not commute with extrude, but I was trying to define it differently. Instead of just the idea of tapering to a point, what about tapering to another polytope of less dimension than the starting one. The simplest is a square to a line which makes a triangular prism. Also a cube to a line (square pyramid prism) or cube to square (3,4 duoprism). These should all be valid operations. So the old tapering of a square is really a double tapering losing two dimensions at once. If these tapering operations are split up like this then it should commute with everything. As well as creating torii with double bending as I described earlier.
ndl
Mononian
 
Posts: 12
Joined: Tue Nov 27, 2018 2:13 pm
Location: Queens, NY

Re: Toratope Names

Postby wendy » Thu Jan 17, 2019 8:10 am

rss and spheration are two entirely different things.

The spheration term was devised to assist modelers describe the polytope model made. Although it involves making round things, it is different to rss.

rss or Crind Product

rss is 'root-sum-square', it is applied to the radial coordinates of several perpendicular bases. It gives a sphere from three lines, each from +1 to -1 on different axies. The centre is taken as 0, and the surface as 1. A radial solid is supposed to consist of concentric shells, where a ray through a given point of that shell is a constant r. Then the particular product of a figure gives R=f(r_x,r_y,r_z).

So if we suppose the bases are lines from -1 to +1, on each of the three axies, then 'R = max(x,y,z)' will give a cube, 'R=sum(x,y,z)' gives an octahedron, and R=rss(x,y,z) gives a sphere. The notation along the line is [max], <sum> and (rss), represent the diameters of a cube, octahedron, and sphere respectively. One can then construct various figures using these.

[iii] = cube.
[i(ii)] = cylinder
(i[ii]) = crind (intersection of two cylinders of the same diameter)
(iii) = sphere. (for different lengths, an ellipsoid)

In four dimensions, you end up with quite a few more.

[(ii)(ii)] duocylinder = bi-circular prism.
<(ii)(ii)> bi-circular tegum
[(iii)i] spherical prism.

One might note that the crind (i[ii]) has 90° edges. It is basically the same shape as a globe, but as if each line of lattitude is replaced by the square surrounding a circle. At four points, you are going to have some pretty sharp corners. Spheration, in one of its forms, softens all sharp angles.

Spheration

Spheration is a kind of surface paint added to models to enhance their appearance, or allow further mathematics. The spheration of three crossing lines, for which the rss (crind product) is a sphere, looks like a jack at https://www.wikihow.com/Play-Jacks . The essential three crossing lines are visible, as are the vertices, all of these being thickened to make them useful to hold.

Other kinds of surface-paint are things like 'surtope paint', where curved surfaces are replaced by small flat planes that follow the surface. This gives a solid to which one can apply polytope rules. A cylinder might become for example, a polygonal prism.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(Latex\) at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 1863
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Toratopes

Who is online

Users browsing this forum: Google [Bot] and 2 guests